Stability and Vibrations of Thin Walled Composite Structures [1st Edition] 9780081004296, 9780081004104

Table of contents :
Content:
Related titles,Front Matter,Copyright,List of contributorsEntitled to full text1 - Introduction to composite materials, Pages 1-47
2 - Sandwich Structures, Pages 49-90, Fiorenzo A. Fazzolari
3 - Classical, first order, and advanced theories, Pages 91-140, Michele D'Ottavio, Olivier Polit
4 - Stability of composite columns and plates, Pages 141-190
5 - Vibration of composite columns and plates, Pages 191-222
6 - Dynamic buckling of composite columns and plates, Pages 223-252
7 - Stability of composite shell–type structures, Pages 253-428
8 - Vibrations of composite shell-type structures, Pages 429-459, Eelco Jansen
9 - Stability of composite stringer-stiffened panels, Pages 461-507
10 - The influence of initial geometric imperfections on composite shell stability and vibrations, Pages 509-548, Eelco Jansen
11 - Test results on the stability and vibrations of composite columns and plates, Pages 549-573
12 - Test results on stability and vibrations of stringer-stiffened composite panels, Pages 575-618
13 - Test results on the stability and vibrations of composite shells, Pages 619-691, Haim Abramovich, K. Kalnins, A. Wieder
14 - Computational aspects for stability and vibrations of thin-walled composite structures, Pages 693-734, Tanvir Rahman, Eelco Jansen
Index, Pages 735-757

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Related titles Failure Mechanisms in Polymer Matrix Composites (ISBN: 978-1-84569-750-1) Creep and Fatigue in Polymer Matrix Composites (ISBN: 978-1-84569-656-6) Fatigue Life Prediction of Composites and Composite Structures (ISBN: 978-1-84569-525-5)

Woodhead Publishing Series in Composites Science and Engineering

Stability and Vibrations of Thin-Walled Composite Structures Edited by

Haim Abramovich Technion, I.I.T., Haifa, Israel

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2017 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-08-100410-4 (print) ISBN: 978-0-08-100429-6 (online) For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals

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List of contributors

Haim Abramovich

Technion, I.I.T., Haifa, Israel

Mariano Arbelo ITA, Aeronautics Institute of Technology, S~ao José dos Campos, Brazil The University of Liverpool, Liverpool, United Kingdom

Jan Błachut

Saullo G.P. Castro

Embraer, S~ao José dos Campos, Brazil

F abio Ribeiro Soares da Cunha

Embraer, S~ao José dos Campos, Brazil

Richard Degenhardt German Aerospace Center (DLR), Institute for Composite Structures and Adaptive Systems, Braunschweig, Germany; PFH, Private University of Applied Sciences G€ ottingen, Composite Engineering Campus Stade, Germany Michele D’Ottavio

UPL, Université Paris Nanterre, Ville d’Avray, France

Fiorenzo A. Fazzolari

University of Cambridge, Cambridge, United Kingdom

Christian H€ uhne German Aerospace Center (DLR), Institute for Composite Structures and Adaptive Systems, Braunschweig, Germany Leibniz Universit€at Hannover, Hannover, Germany

Eelco Jansen K. Kalnins

Riga Technical University, Riga, Latvia

Steffen Niemann German Aerospace Center (DLR), Institute for Composite Structures and Adaptive Systems, Braunschweig, Germany RMIT University, Melbourne, VIC, Australia

Adrian Orifici Olivier Polit

UPL, Université Paris Nanterre, Ville d’Avray, France

Tanvir Rahman

DIANA FEA BV, Delft, Netherlands

Khakimova Regina

INVENT GmbH, Braunschweig, Germany

Ronald Wagner German Aerospace Center (DLR), Institute for Composite Structures and Adaptive Systems, Braunschweig, Germany A. Wieder Israel

Griphus e Aerospace Engineering and Manufacturing Ltd., Tel Aviv,

1

Introduction to composite materials Haim Abramovich Technion, I.I.T., Haifa, Israel

1.1 1.1.1

Introduction General introduction

One of the definitions for a composite material, made of two constituents, one is fiber (the reinforcement) and the other is glue (the matrix), states that a combination of the two materials would result in properties better than those of the individual components when they are used alone. The main advantages of composite materials over other existing materials, such as metal or plastics, are their high strength and stiffness, combined with low density, allowing for weight reduction in the finished part. The various types of composites are usually referred in the literature, as a block diagram, as depicted in Fig. 1.1. In this chapter, when we are discussing a composite material, we restrict ourselves to continuous fibers (reinforcements) being embedded in the matrix in the form of an adequate glue. Examples of such continuous reinforcements include unidirectional, woven cloth, and helical winding (see Fig. 1.2). Continuous-fiber composites are often made into laminates by stacking single sheets of continuous fibers in different orientations to obtain the desired strength and stiffness properties with fiber volumes as high as 60%e70%. Fibers produce high-strength composites because of their small diameter;

Composites

Particle reinforced

Large particles

Dispersion strengthened

Fiber reinforced

Structural

Continuous Discontinuous (short) (aligned)

Aligned

Laminates

Sandwich

Randomly oriented

Figure 1.1 Typical composite materials. Stability and Vibrations of Thin-Walled Composite Structures. http://dx.doi.org/10.1016/B978-0-08-100410-4.00001-6 Copyright © 2017 Elsevier Ltd. All rights reserved.

2

Stability and Vibrations of Thin-Walled Composite Structures

(a)

(b)

0°

(c)

0° / 90°

±30°

Figure 1.2 Typical composite materials: (a) unidirectional fiber, (b) woven cloth (two directions), and (c) filament winding.

they contain far fewer defects (normally surface defects) compared to the material produced in bulk. In addition, because of their small diameter the fibers are flexible and suitable for complicated manufacturing processes, such as small radii or weaving. Materials such as glass, graphite, carbon, and aramid are used to produce fibers (see typical properties in Table 1.1). The present usage of composite materials is mainly driven by the aerospace sector, with a large percentage of the modern airplane structures, such as Boeing 787 or Airbus A380 (see Fig. 1.3), being manufactured from carbon, glass, and aramid fibers. The main material for the matrix is a polymer, which has low strength and stiffness. The main functions of the matrix are to keep the fibers in proper orientation and spacing and to protect the fiber from abrasion and the environment. In polymer matrix composites, the good and strong bond between the matrix and the reinforcement allows the matrix to transmit the outside loads from the matrix to the fibers through shear loading at the interface. Two types of polymer matrices are available: thermosets and thermoplastics. A thermoset starts as a low-viscosity resin that reacts and cures during processing, forming a solid. A thermoplastic is a highviscosity resin that is processed by heating it above its melting temperature. Because a thermoset resin sets up and cures during processing, it cannot be reprocessed by reheating. A thermoplastic can be reheated above its melting temperature for additional processing.

1.2

Unidirectional composites

Unidirectional composites are usually composed of two constituents, the fiber and the matrix (which is the glue holding the two components together). Based on the rule of mixtures, one can calculate the properties of the unidirectional layer based on the properties of the fibers and the matrix and their volume fracture. The assumption to be made when applying the rule of mixtures is that the two constituents are bonded together and they behave like a single body. The longitudinal modulus (or the major modulus), E11, of the layer can be written as E11 ¼ Ef Vf þ Em Vm

(1.1)

Introduction to composite materials

Table 1.1

3

Typical properties of mostly used reinforced continuous

fibers Typical fiber diameter (mm)

Young’s modulus, E (GPa)

Tensile strength (GPa)

Material

Trade name

Density, r (kg/m2)

a-Al2O3 (aluminum oxide)

FP (US)

3960

20

385

1.8

Al2O3 þ SiO2 þ B2O3 (mullite)

Nextel 480 (USA)

3050

11

224

2.3

Al2O3 þ SiO2 (alumina-silica)

Altex (Japan)

3300

10e15

210

2.0

Boron (CVDa on tungsten)

VMC (Japan)

2600

140

410

4.0

Carbon (PANb precursor)

T300 (Japan)

1800

7

230

3.5

Carbon (PANb precursor)

T800 (Japan)

1800

5.5

295

5.6

Carbon (pitchc precursor)

Thornel P755 (USA)

2060

10

517

2.1

SiC (þO) (silicon carbide)

Nicalon (Japan)

2600

15

190

2.5e3.3

SiC (low O) (silicon carbide)

Hi-Nicalon (Japan)

2740

14

270

2.8

SiC (þ O þ Ti) (silicon carbide)

Tyranno (Japan)

2400

9

200

2.8

SiC (monofilament; silicon carbide)

Sigma

3100

100

400

3.5

E-glass (silica)

2500

10

70

1.5e2.0

E-glass (silica)

2500

10

70

1.5e2.0

Quartz (silica)

2200

3e15

80

3.5

Aromatic polyamide

Kevlar 49 (USA)

1500

12

130

3.6

Polyethylene (UHMW)d

Spectra 1000 (USA)

970

38

175

3.0

High-carbon steel

E.g., Piano wire

7800

250

210

2.8 Continued

4

Stability and Vibrations of Thin-Walled Composite Structures

Table 1.1

Continued Typical fiber diameter (mm)

Young’s modulus, E (GPa)

Tensile strength (GPa)

Material

Trade name

Density, r (kg/m2)

Aluminum

Electrical wire

2680

1670

75

0.27

Titanium

Wire

4700

250

115

0.434

a

CVD, chemical vapor deposition. PAN, polyacrylonitrile. About 90% of the carbon fiber produced worldwide are made from PAN. c Pitch is a viscoelastic material that is composed of aromatic hydrocarbons. Pitch is produced by the distillation of carbon-based materials, such as plants, crude oil, and coal. d UHMW ¼ ultra-high-molecular-weight polyethylene (or polyethene, the most common plastic produced in the world) is a subset of the thermoplastic polyethylene. From B. Harris, Engineering Composite Materials, The Institute of Materials, London, UK, 1999, 193 p. and R.M. Jones, Mechanics of Composite Materials, second ed., Taylor & Francis, Philadelphia, PA 19106, USA, 1999, 519 p. b

where Ef and Em are the longitudinal moduli for the fibers and the matrix, respectively, and Vf and Vm are their respective volume fractions.1 The major Poisson’s coefficient, y12, is given by y12 ¼ yf Vf þ ym Vm

(1.2)

where yf and ym are the longitudinal moduli for the fibers and the matrix, respectively. The minor Poisson’s coefficient, y21, will be calculated to be y12 y21 E22 ¼ 0 y21 ¼ y12 E11 E22 E11

(1.3)

The transverse modulus (or the minor modulus), E22, of the layer is given as Vf Vm 1 ¼ þ 0 E22 ¼ E22 Ef Em

Em Em ¼ Em Em Vf þ Vm Vf þ ð1  Vf Þ Ef Ef

(1.4)

The shear modulus of the layer, G12, is given as Vf Vm 1 ¼ þ 0 G12 ¼ G12 Gf Gm

1

Note that Vf þ Vm ¼ 1.

Gm Gm ¼ Gm Gm Vf þ Vm Vf þ ð1  Vf Þ Gf Gf

(1.5)

Introduction to composite materials

5

Boeing 787 “dreamliner” composite structure

(a) Materials used in 787 body Carbon laminate composite Fiberglass Aluminum Carbon sandwich composite Aluminum/steel/titanium

Total materials used By weight Other Steel 5% 10%

Composites 50%

Titanium 15% Aluminum 20% By comparison, the 777 uses 12 percent composites and 50 percent aluminum.

(b) Model aeroplane A380 composite components Carbon fiber

Wing box

Outer wing

Glass fiber Hybrid (carbon + glass)

Tail cone

Vertical stabiliser

Allerons

Aramid fiber

Horizontal stabiliser outer boxes

Flap-track fairings Outer flap

Fixed leading-edge upper and lower panels

Keel beam Pressure bulkhead

Belly fairing skins

Over wing panel

Trailing-edge upper and lower panels and shroud-box Spoilers

Radome Main and centre landing-gear doors Nose landinggear doors

Central torsion box

Main landing-gear leg-fairing door

Pylon fairings Nacelle cowlings

Figure 1.3 Usage of composite materials in aerospace structures: (a) Boeing 787 and (b) Airbus A380. (a) Boeing Industry and (b) Airbus Industrie.

where Gf and Gm are the shear moduli for the fibers and the matrix, respectively. To assess and compare the differences between the properties of the fiber and the matrix, the reader is referred to Table 1.2 . As described in Ref. [1] the simple micromechanics model used in the rule of mixtures predicts well the values of the four variables, E11, E22, G12, and y12, as compared to experimental values, as can be seen in Table 1.3.

1.3

Properties of a single ply

A ply has two major dimensions and another dimension, i.e., the thickness, that is very small when compared to the two major ones. Therefore, the 3D presentation of an

6

Stability and Vibrations of Thin-Walled Composite Structures

Table 1.2

Typical properties of T300 carbon fibers and 914 epoxy

resin Property

T300 carbon fibers

914 epoxy resin matrix

Young’s modulus, E (GPa)

220

3.3

Shear modulus, G (GPa)

25

1.2

Poisson’s ratio, y

0.15

0.37

From B. Harris, Engineering Composite Materials, The Institute of Materials, London, UK, 1999, 193 p.

Predictions of unidirectional composite properties by the simple micromechanics model

Table 1.3

Predicted values [moduli in (GPa)]

Experimental values [moduli in (GPa)]

124.7

125.0

ð1Vf Þ Em

7.4

9.1

ð1Vf Þ Gm

2.6

5.0

0.25

0.34

Equations

Relationship

1.1

E11 ¼ EfVf þ Em (1  Vf)

1.4

1 E22

¼ Eff þ

V

1.5

1 G12

¼ Gff þ

V

1.2

y12 ¼ yfVf þ ym (1  Vf)

Adapted from B. Harris, Engineering Composite Materials, The Institute of Materials, London, UK, 1999, 193 p.

orthotropic material will be simplified to a 2D presentation (plane stress) by assuming that s33 ¼ 0 (see Refs. [1,2]). This leads to a reduced compliance matrix for the ply in the form 2 3 1 n21  0 7 6 8 9 6 E1 9 78 E2 6 7> s11 > ε11 > > > > > > 6 7 < = 6 n = 7< 1 12 6 7 (1.6) ε22 ¼ 6  0 7 s22 > > > E2 E1 > > > > 7> : : ; 6 ; 6 7 s g12 12 6 1 7 4 5 0 0 G12 The third equation for the strain in the thickness direction, ε33, that is seldom used has the following form ε33 ¼ 

g13 g s11  23 s22 E1 E2

(1.7)

Introduction to composite materials

7

and the remaining two equations for the shear strains are written as 3

2

1 ) 6 ( g23 6 G23 ¼6 6 g13 4 0

) 0 7( 7 s23 7 7 1 5 s13 G13

(1.8)

Calculation of the stresses as a function of strains would yield (by using Eqs. 1.6 and 1.8) 8 9 2 s11 > Q11 > > > > > > > < = 6 6 s22 ¼ 6 6 Q21 > > > > 4 > > > > : ; s12 0

Q12 Q22 0

3 2 E1 y21 E1 0 9 9 8 78 6 3 7> ε11 > 6 ð1  y12 y21 Þ ð1  y12 y21 Þ 0 > ε11 > > > > > 7 6 > > > > > > 7> 7> = 6 = 7< 7< 6 7 7 6 y E E 0 7 ε22 ¼ 6 ε22 12 2 2 7 0 7> > > > > > > 5> 6 ð1  y y Þ ð1  y y Þ > > > 12 21 12 21 > > > > 7> : ; 6 ; : 7 6 g12 g Q66 5 12 4 0 0 G12

with Q12 ¼ Q21 and y12 s y21 (1.9)

(

s23 s13

1.4

)

" ¼

Q23

0

0

Q13

#(

g23 g13

)

" ¼

G23

0

0

G13

#(

g23

)

g13

(1.10)

Transformation of stresses and strains

Consider the two coordinate systems described in Fig. 1.4. The one with indexes 1 and 2 describes the ply orthotropic coordinate system, whereas the other one (x, y) is an arbitrary axis that is rotated at a given angle q relative to the 1, 2 system. The transformation of the stresses and the strains from the 1, 2 coordinate system to the x, y coordinate system is done by multiplication of both the stresses and the strains at the ply level by the transformation matrix T as given by2: 8 9k 8 9k sx > s1 > > > > > > < = = < > s2 ¼ ½T sy > > > > > > > : ; ; : > sxy s12 2

(1.11)

See, for example, J.E. Ashton, J.C. Halpin, Primer on Composite Materials: Analysis, Technomic Publishing Co., Inc., 750 Summer St., Stamford, Conn. 06901, USA, 1969.

8

Stability and Vibrations of Thin-Walled Composite Structures

y 2

x

θ

1

Figure 1.4 Two coordinate systems: 1, 2 the ply orthotropic axis; x, y arbitrary axis.

8 9k 8 9k > > > > > > > εx > ε > > > > > 1> > > > > > < > < = = ε2 ¼ ½T εy > > > > > > > > > > > > > g12 > > gxy > > > > : 2 > : 2 ; ;

(1.12)

where k is the number of the ply for which the transformation of strains and stresses is performed.3 The transformation matrix T is given by 2

c2

6 2 ½T ¼ 6 4 s

cs

s2 c2 cs

3

2cs

7 2cs 7 5 where c2  s2

c h cos q s h sin q

(1.13)

To obtain the inverse of the matrix T, one needs simply to insert q instead of q in Eq. (1.13) to yield 2

c2

6 2 ½T1 ¼ ½TðqÞ ¼ 6 4s

cs

s2 c2

2cs

3

7 2cs 7 5

(1.14)

cs c2  s2

The ply (or lamina) strainestress relationship transformed to the laminate references axis (x, y) is written as

3

Note that s11 h s1; s22 h s2; ε11 h ε1; ε22 h ε2.

Introduction to composite materials

9

8 8 9k 9 εx >k s1 > > > > > > > > > > > < < = = 1 k s2 ¼ ½T ½Q ½T εy > > > > > > > > > > > > : : ; ; gxy s12 3k 2 Q11 Q12 0 7 6 7 6 where ½Qk ¼ 6 Q12 Q22 0 7 5 4 0

0

(1.15)

2Q66

where the expressions for Q11, Q12, Q22, and Q66 are given in Eq. (1.9). Performing the matrix multiplication in Eq. (1.15) yields 8 9k 8 9 εx > k s1 > > > > > > > < = < =  k εy s2 ¼ Q > > > > > > > > : ; : ; gxy s12 2 Q11 Q12  k 6 where Q ¼ 6 4 Q12 Q22 Q16 Q26

Q16

3k

(1.16)

7 Q26 7 5 Q66

where Q11 ¼ Q11 cos4 q þ 2ðQ12 þ 2Q66 Þsin2 qcos2 q þ Q22 sin4 q   Q12 ¼ ðQ11 þ Q22  4Q66 Þsin2 qcos2 q þ Q12 sin4 q þ cos4 q Q22 ¼ Q11 sin4 q þ 2ðQ12 þ 2Q66 Þsin2 qcos2 q þ Q22 cos4 q Q16 ¼ ðQ11  Q12  2Q66 Þsin qcos3 q þ ðQ12  Q22 þ 2Q66 Þsin3 qcos q Q26 ¼ ðQ11  Q12  2Q66 Þsin3 qcos q þ ðQ12  Q22 þ 2Q66 Þsin qcos3 q   Q66 ¼ ðQ11 þ Q22  2Q12  4Q66 Þsin2 qcos2 q þ Q12 sin4 q þ cos4 q (1.17)

10

Stability and Vibrations of Thin-Walled Composite Structures

 k Another useful way of presenting the various terms of the matrix Q is the invariant procedure presented by Tsai and Pagano [2]: Q11 ¼ U1 þ U2 cosð2qÞ þ U3 cosð4qÞ Q12 ¼ U4  U3 cosð4qÞ Q22 ¼ U1  U2 cosð2qÞ þ U3 cosð4qÞ 1 Q16 ¼  U2 sinð2qÞ  U3 sin ð4qÞ 2 1 Q26 ¼  U2 sinð2qÞ þ U3 sinð4qÞ 2 Q66 ¼ U5  U3 cosð4qÞ where

(1.18)

1 U1 ¼ ½3Q11 þ 3Q22 þ 2Q12 þ 4Q66  8 1 U2 ¼ ½Q11  Q22  2 1 U3 ¼ ½Q11 þ Q22  2Q12  4Q66  8 1 U4 ¼ ½Q11 þ Q22 þ 6Q12  4Q66  8 1 U5 ¼ ½Q11 þ Q22  2Q12 þ 4Q66  8 Note that the terms U1, U4, and U5 are invariant to a rotation relative to the 3 axis (perpendicular to the 1e2 plane). Now we shall present the addition of the properties of each lamina to form a laminate, which is the structure to be investigated when we apply loads. Referring to Fig. 1.5, one can write the displacement in the x direction of a point at a z distance from the midplane as (where w is the displacement in the z direction) u ¼ u0  z

vw vx

(1.19)

Similarly the displacement in the y direction will be v ¼ v0  z

vw vy

(1.20)

Introduction to composite materials

11

u0 B B A Zc

X A

c D

α X

Cα

Z

D Z After deformation

Before deformation

Zcα

Figure 1.5 The plate cross section before and after the deformation.

Then the strains (εx, εy, and gxy) and the curvatures (kx, ky, and kxy) can be written as εx h

vu vu0 v2 w ¼  z 2 ¼ ε0x þ zkx vx vx vx

εy h

vv vv0 v2 w ¼  z 2 ¼ ε0y þ zky vy vy vy

gxy h

(1.21)

vu vv vu0 vv0 v2 w þ ¼ ¼ g0xy þ zkxy þ  2z vy vx vxvy vy vx

where ε0x ; ε0y ; g0xy are the strains at the neutral plane. In matrix notation, Eq. (1.21) can be given as 8 8 9 8 0 9 9 εx > > kx > > εx > > > > > > > > > < < < = = > =   ε0y εy þ z ky 0fεg ¼ ε0 þ zfkg ¼ > > > > > > > > > > > > : : ; > ; : g0 > ; gxy k xy xy

(1.22)

Then the stresses at the lamina level will be given by  k  0   k ε þ z Q ½k fsgk ¼ Q

(1.23)

Now we shall deal with force (Nx, Ny, Nxy) and moment (Mx, My, Mxy) resultants. Their definitions are given as (h is the total thickness of the laminate) Z Nx h

h=2

h=2

Z Mx h

h=2

h=2

Z sx dz;

Ny h

h=2

h=2

Z sx zdz;

My h

Z sy dz;

h=2

h=2

Nxy h

h=2

h=2

Z sy zdz;

Mxy h

sxy dz; h=2

h=2

(1.24) sxy zdz.

12

Stability and Vibrations of Thin-Walled Composite Structures

By substituting the expressions of the stresses, one obtains expressions for the force and moment resultants as a function of the strain on the midplane, ε0, and the curvature, k (see also Ref. [2]). The short-written expressions are 38   9 ½B < ε0 = 5 ¼4 : fMg ; ½B ½D : fkg ; 9 8 < fNg =

2

½A

or 99 88 Nx > > > > > > > > > > > > > >> < => > > > > > > > > N > > y > > > > > > > > > > > > > > > > > : ;> > > > = < Nxy >

22

A11

A12

66 66 6 6 A12 66 64 6 6 A 16 6 ¼6 6 8 9 2 > > > 6 Mx > > > > > > 6 B11 > > > > > > > > > > > 66 > < => > 66 > > > > > 66 My > > > 6 6 B12 > > > > > > > 44 > > > > > > > >> ; :> : ; Mxy B16

A16

3

A22

7 7 A26 7 7 5

A26

A66

B12

B16

3

B22

7 7 B26 7 7 5

B26

B66

2

B11

6 6 6 B12 6 4

B12 B22

B16

B26

D11 6 6 6 D12 6 4

D12

D16

D26

2

D22

3 38 8 0 9 9 > εx > > > > > > > > >> > > > > > 7 7> >> > < = > > 7 7> > 0 > > 7> > ε B26 7 > > y 7 7> > > > > > > > 7 > > 5 >> > > > > > 7> > > > > : ; > 7 = B66 7< g0xy > 7 37 8 > k 9> > D16 7> > x > >> > 7> > > > > > > > 7 7> > > > >< > => 7 7> > > > 7 7 D26 7 7> ky > > > > > > > > 5 5> > > > > > > > > > > > > : ; : ; D66 kxy B16

(1.25) where the various constants are defined as Z Aij h

h=2 h=2

Z Bij h

h=2 h=2

Z Dij h

h=2 h=2

k

Qij dz ¼

n X

k

Qij ðhk  hk1 Þ

k¼1 k

Qij zdz ¼ k

n  1X k Qij h2k  h2k1 2 k¼1

Qij z2 dz ¼

(1.26)

n  1X k Qij h3k  h3k1 3 k¼1

where i; j ¼ 1; 1; 1; 2; 2; 2; 1; 6; 2; 6; 6; 6 The way the sum is performed in Eq. (1.26) is according to the notations in Fig. 1.6. The passage from integral over the thickness of the laminate to the sum over the thickness is dictated by the fact that the individual plies are very thin and the properties within each laminate are assumed constant in the thickness direction. Finally, using the classical lamination theory, the equations of motion for the static case, for a thin plate made of laminated composite plies, are given as (see Refs. [1,2])

Introduction to composite materials

13

z

k = n–2 k = n–3

k=n k = n–1

y

t

h3 h2 k=3

h1 h0

k=2 k=1

b

Figure 1.6 Lamina notations within a given laminate.

 vNx vNxy v2 u0 v2 vw0 þ ¼ I1 2  I2 2 vx vy vx vt vt

 vNxy vNy v2 v 0 v2 vw0 þ ¼ I1 2  I2 2 vx vy vy vt vt

v2 Mxy v2 My v v2 M x vw0 vw0 þ þ N þ 2 þ N þ xx xy vx vxvy vx vy vx2 vy2

v vw0 vw0 v2 w0 þ Nxy Nyy ¼ pz þ I1 2 vy vy vx vt

(1.27)



 v2 v2 w0 v2 w0 v2 vw0 vw0 þ I3 2 þ 2 þ I2 2 vx vy vt vx2 vy vt where pz is the load per unit area in the z direction4 and the subscript 0 represents the values at the midplane of the cross section. N represents the in-plane load,

4

Note that the coordinate z is normally used for the thickness direction, while x and y coordinates define the plate area.

14

Stability and Vibrations of Thin-Walled Composite Structures

and the various moments of inertia, I1, I2, and I3 are given by (r is the mass/unit length) Z Ij ¼

h=2

h=2

rzj1 dz;

j ¼ 1; 2; 3

(1.28)

To obtain the equations for a beam, one can use Eq. (1.27), while all the derivations with respect to y are identically zero. This yields a one-dimensional equation in the following form:

 v2 M x v vw0 v2 w v4 w v3 w þ N ¼ pz þ I1 2  I3 2 2 þ I2 2 xx 2 vx vt vt vx vt vx vx vx

(1.29)

where Nx is the axial (in-plane, in the direction of the length of the beam) load. Based on the relationship between transverse deflection, w, and the bending moment, we can rewrite Eq. (1.29) in terms of w only to yield D11



  v2 w v2 v2 w v vw0 ¼ M 0  N D þ x xx 11 vx2 vx2 vx vx2 vx ¼ pz þ I1

v2 w v4 w v3 w  I þ I 3 2 vt 2 vt 2 vx2 vt 2 vx

(1.30)

with its associated boundary conditions Geometric: specify either w or

vw vx

vM or M Natural: specify either Q h vx

(1.31)

Typical boundary conditions normally used in the literature are in the following form Simply supported: w ¼ 0 and M ¼ 0; Clamped: w ¼ 0 and Free: Q h

vw ¼ 0; vx

(1.32)

vM ¼ 0 and M ¼ 0: vx

There are thermal issues associated with the manufacturing of composite structures because of the differential thermal contraction during the postcuring phase and the temperature changes occurring during the service life of the structure. This issue is caused by

Introduction to composite materials

15

the relatively small axial thermal expansion coefficient of the modern reinforcing fibers (for carbon fibers it is even slightly negative), while the resin matrix has a large thermal coefficient. When cooling from a typical curing temperature, such as from 140 C to room temperature, the fibers of the laminate composite will be in compression, while the matrix will show tension stresses [1]. Typical residual stresses due to thermal mismatch between the two components of the laminate are presented in Table 1.4. Another important data for design is the experimental tension and compression strength measured during various laboratory tests, as presented in Table 1.5 (see Ref. [1]). Finally a table with a list of the main manufacturers of various composite materials is presented in Table 1.6. The attached list of references, aimed at presenting the reader with updated information being published in the open literature, presents typical studies and articles dealing with the behavior and calculated response of composite and sandwich structures to various external loads. The review starts at year 2000 and continues to the present time. The references are presented on a chronological order for each of the following parts: Laminated composite beams or columns; Sandwich structures: beams, columns, and bars; Sandwich plates and shells; Composite plates and shells; and Experiments on beams, columns, bars, sandwich plates and shells, and composite plates and shells.

Typical thermal stresses in some common unidirectional composites

Table 1.4

% Fiber volume, Vf

Temperature range, DT (8K)

Fiber residual stress (MPa)

Matrix residual stress (MPa)

Matrix

Fiber

Epoxy (high T cure)

T300 carbon

65

120

19

36

Epoxy (low T cure)

E-glass

65

100

15

28

Epoxy (low T cure)

Kevlar 49

65

100

16

30

Borosilicate glass

T300 carbon

50

520

93

93

CAS glasseceramic

Nicalon SiC

40

1000

186

124

CAS, CaO-Al2O3-SiO2. From B. Harris, Engineering Composite Materials, The Institute of Materials, London, UK, 1999, 193 p.

16

Stability and Vibrations of Thin-Walled Composite Structures

Typical experimental tension and compression strengths for common composite materials

Table 1.5

Tensile strength, st (GPa)

Compression strength, sc (GPa)

Ratio sc/st

Material

Layup

% Fiber volume, Vf

GRP

Unidirectional

60

1.3

1.1

0.85

CFRP

Unidirectional

60

2.0

1.1

0.55

KFRP

Unidirectional   ð45 ; 02 Þ2 S

60

1.0

0.4

0.40

65

1.27

0.97

0.77

  ð45 ; 02 Þ2 S

65

1.42

0.90

0.63

  ð45 ; 02 Þ2 S

65

1.67

0.88

0.53

SiC/CAS (CMC)

Unidirectional

37

334

1360

4.07

SiC/CAS (CMC)

[0 , 90 ]3S

37

210

463

2.20

HTA/913 (CFRP) T800/924 (CFRP) T800/ 5245 (CFRP)

CFRP, carbon fiber reinforced polymer; CMC, ceramic matrix composite; GRP, glass reinforced plastic; KFRP, Kevlar fiber reinforced plastic. From B. Harris, Engineering Composite Materials, The Institute of Materials, London, UK, 1999, 193 p.

List of the main manufacturers of various composite materials and resins

Table 1.6

Type of composite

Company

Company website

Thermoplastic composites

Milliken Tegris

tegris.milliken.com

Thermoplastic composites

Polystrand, Inc.

www.polystrand.com

Nonwoven fabrics (PolyWeb) and foam

Wm. T. Burnett & Co.

www.williamtburnett.com

Thermoplastic composites

Schappe Techniques

www.schappe.com

Thermoplastic composites

TechFiber

www.fiber-tech.net

Thermoplastic composites

TenCate

www.tencate.com

Thermoplastic composites

TherCom

www.thercom.com

Introduction to composite materials

Table 1.6

17

Continued

Type of composite

Company

Company website

Thermoplastic composites

Vectorply

www.vectorply.com

Composite materials: resins and fibers

SF Composites

www.sf-composites.com

Formulation and manufacture of epoxy-based systems

SICOMIN

www.sicomin.com

Composite materials þ resin, composite laminates

Lamiflex SPA

www.lamiflex.il

Composite materials þ polyester

AMP Composite

www.amp-composite.il

Infusion, pultrusion, wet layup, prepreg, filament winding

Applied Poleramic Inc.

www.appliedpoleramic.com

Epoxy and polyurethane

Endurance Technologies

www.epoxi.com

Composite materials

Gurit

www.gurit.com

Advanced thermoset resins

Huntsman Advanced Materials

www.huntsman.com/advanced_ materials/a/Home

Advanced thermoset resins

Lattice Composites

www.latticecomposites.com

Kevlar

DuPont Kevlar

www.dupont.com/products-andservices/fabrics-fibersnonwovens/fibers/brands/ kevlar.html

UHMWPEdultra high molecular weight, high performance polyethylene material

DuPont Tenslyon

www.dupont.com

Innegra HMPP (polypropylene), high performance fiber

Innegra Technologies

www.innegratech.com

Spread tow fabrics

TeXtreme

www.textreme.com/b2b

Adhesives and sealants

3M

solutions.3m.com

Prepreg and resins

Axiom Materials Inc.

www.axiommaterials.com

Fabrics, resins, composite materials

Barrday Advanced Materials Solutions

www.barrday.com

Continued

18

Stability and Vibrations of Thin-Walled Composite Structures

Table 1.6

Continued

Type of composite

Company

Company website

Carbon prepreg

Hankuk Carbon Co., Ltd.

www.hcarbon.com/eng/product/ overview.asp

Carbon fibers and prepregs

Hexcel

www.hexcel.com/Products/ Industries/ICarbon-Fiber

Prepregs & Compounds

Pacific Coast Composites

www.pccomposites.com

Prepregs & Compounds

Quantum Composites

www.quantumcomposites.com

References [1] B. Harris, Engineering Composite Materials, The Institute of Materials, London, UK, 1999, 193 p. [2] R.M. Jones, Mechanics of Composite Materials, second ed., Taylor & Francis, Philadelphia, PA 19106, USA, 1999, 519 p.

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[511] S. Dariushi, M. Sadighi, A new nonlinear high order theory for sandwich beams: an analytical and experimental investigation, Composite Structures 108 (February 2014) 779e788. [512] J. Romanoff, J.N. Reddy, Experimental validation of the modified couple stress Timoshenko beam theory for web-core sandwich panels, Composite Structures 111 (May 2014) 130e137. [513] D. Balkan, Z. Mecitoǧlu, Nonlinear dynamic behavior of viscoelastic sandwich composite plates under non-uniform blast load: theory and experiment, International Journal of Impact Engineering 72 (October 2014) 85e104. [514] V.P. Vavilov, A.V. Plesovskikh, A.O. Chulkov, D.A. Nesteruk, A complex approach to the development of the method and equipment for thermal nondestructive testing of CFRP cylindrical parts, Composites Part B: Engineering 68 (January 2015) 375e384. [515] S. Rohde, P. Ifju, B. Sankar, D. Jenkins, Experimental testing of bend-twist coupled composite shafts, Experimental Mechanics 55 (9) (November 2015) 1613e1625. [516] B. Yang, Z. Wang, L. Zhou, J. Zhang, W. Liang, Experimental and numerical investigation of interply hybrid composites based on woven fabrics and PCBT resin subjected to low-velocity impact, Composite Structures 132 (November 2015) 464e476. [517] K. Giasin, S. Ayvar-Soberanis, A. Hodzic, An experimental study on drilling of unidirectional GLARE fibre metal laminates, Composite Structures 133 (December 2015) 794e808. [518] X.-L. Gong, Z. Wen, Y. Su, Experimental determination of residual stresses in composite laminates [02/q2]s, Advanced Composite Materials 24 (2015) 33e47. [519] P. Wang, F. Sun, H. Fan, W. Li, Y. Han, Retrofitting scheme and experimental research of severally damaged carbon fiber reinforced lattice-core sandwich cylinder, Aerospace Science and Technology 50 (March 2016) 55e61.

Sandwich Structures Fiorenzo A. Fazzolari University of Cambridge, Cambridge, United Kingdom

2.1

2

Introduction

Sandwich structures are multilayer structural members composed of two stiff face sheets generally bonded to a honeycomb or foam core (see Fig. 2.1). The latter are the most common ones, but other typologies of core such as web core or truss core can also be employed. They are characterized by high stiffness-to-weight ratio, which makes them very versatile and then usable in several engineering applications. Sandwich panels are usually applied in civil engineering as light roof and wall panels to provide thermal insulation of buildings. They are used in marine engineering and, more specifically, in the hull and deck constructions. A significant use of sandwich structures can be found in aerospace and aeronautical engineering above all as constructions employed in aircraft and spacecraft structures. Various other applications can also be broadly found in mechanical as well as automotive engineering. Advanced sandwich-type constructions imply the presence of a thick orthotropic core with bonded anisotropic face sheets that are treated as composite laminates. In such a way, there is an opportunity to tailor both the physical and mechanical properties of the faces by proper selection of laminate materials, their stacking sequence, and orientation. Suitable selection of fiber orientation and stacking sequence can result in substantial improvements of the buckling strength and of the nonlinear response behavior to a variety of load conditions. The transverse shear elastic moduli of the core layer can also be optimized so as to enhance the overall response behavior of sandwich constructions. As expected, analytical modeling of sandwich-type panels is much more intricate than that of usual laminated composite structures. In contrast to the latter for which the axiomatic assumptions are postulated for the structure as a whole, in the case of sandwich-type constructions the assumptions involve a layer-wise kinematics description. Moreover, the analysis of sandwich panels featuring anisotropic laminated face sheets is much more complicated than that with single-layered faces. The complexity is due to the presence of three types of asymmetries resulting from the stacking sequences in the face sheets, namely (1) asymmetry with respect to the midsurface of the face sheets (referred to as face asymmetry) inducing face bendingestretching coupling; (2) asymmetry with respect to the midsurface of the core (referred to as global asymmetry), which induces global bendingestretching coupling; and (3) presence of ply angles between the principal axes of orthotropy of the face sheet materials and the geometric axes of the panel, inducing a structural coupling between stretching and shearing. As a general guideline to the proportions, an efficient sandwich is obtained when the weight of the core is roughly equal to the combined weight of the faces. Obviously the Stability and Vibrations of Thin-Walled Composite Structures. http://dx.doi.org/10.1016/B978-0-08-100410-4.00002-8 Copyright © 2017 Elsevier Ltd. All rights reserved.

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Stability and Vibrations of Thin-Walled Composite Structures

Figure 2.1 Classical sandwich structure.

bending stiffness of this arrangement is very much greater than that of a single solid plate of the same total weight made of the same material as the faces. The core has several crucial functions. It must be stiff enough in the orthogonal direction to the faces to ensure that they remain the correct distance apart. It must be stiff enough in shear to ensure that when the panel is bent, the faces do not slide over each other. If this last condition is not fulfilled, the faces merely behave as two independent beams or panels and the sandwich effect is lost. The core must also be stiff enough to keep the faces nearly flat; otherwise, it is possible for a face to buckle locally (wrinkle) under the influence of compressive stress in its own plane. The core must satisfy all these requirements, and it is also important that the adhesive should not be sufficiently flexible to permit substantial relative movements of the faces and the core. If the core is stiff enough, it may make a useful contribution to the bending stiffness of the panel as a whole. This contribution is rather small in the case of the low-density cores that are usually employed, and it is very often expedient to ignore it. This also leads to a considerable simplification of the analysis of stresses and deflections. A different typology of sandwich structures can be derived by exploiting the use of functionally graded materials (FGMs). FGMs represent a class of heterogeneous composite materials made up of a mixture of ceramics and metals that are characterized by the smooth and continuous variation in properties from the bottom to the top of the considered structural element. To this class also belong those materials in which the gradation is generated by varying the fibers volume fraction in laminated composites. In the general case, the material properties of FGMs are controlled by the variation of the volume fraction of the constituent materials. Being ultrahigh temperature-resistant materials, they are suitable for aerospace applications, such as aircraft, space vehicles, barrier coating, and propulsion systems. Moreover, they have several advantages over other types of advanced materials such as fiber-reinforced composites, indeed, problems such as delamination, fiber failure, adverse hygroscopic effects due to moisture content, etc., are effectively eliminated or nonexistent. With their potential applications, FGMs are steadfastly making headway in aerospace design. Thus, there is the need to fully analyze the free vibration and thermal stability characteristics of the FGMs. This necessity led the research community to

Sandwich Structures

51

investigate several configurations of FGM sandwich structures by using a considerable amount of new structural theories.

2.1.1

Some historical notes and literature

According to Noor et al. [1], the concept of sandwich structure can be traced back to Fairbairn in 1849 [2]. Various sandwich constructions were already employed during the World War II (see Ref. [3]). Mosquito aircraft is often quoted as being the first major application of sandwich panels, but there were numerous earlier, though less spectacular, uses of the sandwich principle. However, according to Vinson [4], the first research paper on the subject matter was written by Marguerre in 1944 [5] and was focused on the analysis of sandwich panels subjected to in-plane loadings. Since then, a significant amount of papers was written with the aim of thoroughly investigating sandwich structures in every aspect. During the decades, several comprehensive reviews have been proposed by different authors such as Librescu [6], Vinson [4,7], and among many others. It is worth also mentioning various books, which have been written on both modeling and design of sandwich-type constructions. Carlson and Kardomateas [8] focused on some important deformation and failure modes of sandwich panels such as global buckling, wrinkling and local instabilities, and face/core debonding. Allen [9] provided a simple guide to the principal aspects of the theory of sandwich constructions and to the assumptions on which it is based. Yu [10] in Chapters 4 and 8 proposed the derivation of the linear and nonlinear governing differential equations of sandwich structures, respectively. Useful design insights on lightweight sandwich constructions are given in the book edited by Davies [11], which starting from considerations of the right materials to use in the design addresses then all the major issues, such as thermal performance and water-tightness, acoustics, durability, mechanical testing, and several others. Lu and Xin [12] proposed a thorough investigation of the vibration and acoustical behavior of typical sandwich structures subject to mechanical and/or acoustical loadings. The behavior of sandwich structures in severe environments that include highly dynamic loading was studied by Abrate et al. [13]. In particular, the book proposes several analyses to better understand the dynamic response and failure of sandwich structures. Vinson [14] provided a thorough analysis in terms of elastic instability and structural optimization to obtain the minimum weight sandwich panels. As regards the theoretical and computational modeling of sandwich structures, it is worth mentioning some important contributions. Pagano [15] provided the exact solution to sandwich plates. Frostig et al. [16] derived a closed-form solution of the governing equations of curved sandwich panels with transversely flexible core. The equations were derived by using a higher-order theory. Multiskin and multilayered core constructions were addressed in Ref. [16]. Localized effects in the nonlinear behavior of sandwich panels with a transversely flexible core was investigated by Frostig and Thomsen [17]. A section with an interesting literature survey on higher-order shear deformation theories (HSDTs) was given in Ref. [18]. Pantano and Averill [19] considered the thermal stress analysis of sandwich plates in the framework of zigzag (ZZ) and layerwise (LW) theories. The response to concentrated loadings using the higher-order theory and three-dimensional (3D) elasticity analysis was addressed by Swanson [20]. The same author [21] provided an assessment of a new developed Finite

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Stability and Vibrations of Thin-Walled Composite Structures

Element (FE) model by comparison with the 3D elasticity results obtained in Ref. [20]. A local LW model for the bending of weak cores was presented by Whitney [22]. An assumed stress field in the thickness direction was assumed independently for the core and skins. The governing equations were obtained by referring to the stress method already used in the paper by Pagano [23]. The thick plate first order shear deformation theory (FSDT) analysis was used by Liu and Zhao [24] for a comparison with a thin plate model analysis. The appropriate choice of shear correction factor was discussed by Birman and Bert [25]. Various HSDTs were discussed by Matsunaga [26] for both sandwich and laminated plates. A finite element analysis of bended sandwich plates was considered by Topdar et al. [27], who used an equivalent single-layer (ESL) Ambartusumian-type theory. The already mentioned Frostig HSDT was employed by Lyckegaard and Thomsen [28] to analyze the junction between flat and curved sandwich panels. An HSDT finite element analysis for a free vibration of an anisotropic sandwich plate was conducted by Garg et al. [29]. A higher-order approach for the vibration analysis of a sandwich plate with a viscoelastic core was proposed by Malekzadeh et al. [30] for the damping of local and global vibrations. Several advanced and refined plate theories were employed by Fazzolari [31,32] to cope with the free vibration and thermomechanical stability of sandwich structures in thermal environment. Roque et al. [33] applied an LW theory based on radial basis functions to the vibration response of sandwich plates. Finally, various kinematics have recently been compared by Hu et al. in Ref. [34] for the analysis of sandwich beams made of viscoelastic materials. As regards the FGM concept, it is worth highlighting the fact that it was originated in Japan in 1984 during the space plane project, in the form of a proposed thermal barrier material capable of withstanding a surface temperature of 2000 K and a temperature gradient of 1000 K across a cross section of h/2); furthermore, employing CLPT instead of FSDT in the faces introduces some minor discrepancies only for l < h/3; finally, the ESL model T32ZZ differs from the reference solution already for l < 5h and it predicts a nonconservative wrinkling load that is more than 30% higher than the reference value. As a closing remark, note that the wrinkling results fully substantiate Koiter’s recommendation, for they confirm that the transverse normal deformation is necessary for capturing the short wavelength response, i.e., for enhancing the accuracy of twodimensional shell/plate models.

Acknowledgments The authors gratefully acknowledge A. Loredo and P. Vidal for their support in the numerical evaluation of exact solutions and refined Sinus-based theories.

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[23] K. Niu, R. Talreja, Modeling of wrinkling in sandwich panels under compression, Journal of Engineering Mechanics 125 (1999) 875e883. [24] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, second ed., CRC Press, 2004. [25] K.-J. Bathe, Finite Element Procedures, Prentice-Hall, New Jersey, 1996. [26] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics 12 (1945) A69eA77. [27] R.D. Mindlin, Influence of rotatory inertia and shear on flexural vibrations of isotropic, elastic plates, Journal of Applied Mechanics 18 (1951) 31e38. [28] J.M. Whitney, Shear correction factors for orthotropic laminates under static load, Journal of Applied Mechanics 40 (1973) 302e304. [29] W.H. Wittrick, Analytical three-dimensional elasticity solutions to some plate problems and some observations on Mindlin’s plate theory, International Journal of Solids and Structures 23 (1987) 441e464. [30] V. Birman, C.W. Bert, On the choice of shear correction factor in sandwich structures, Journal of Sandwich Structures and Materials 4 (2002) 83e95. [31] W.T. Koiter, A consistent first approximation in the general theory of thin elastic shells, in: W.T. Koiter (Ed.), The Theory of Thin Elastic Shells, IUTAM, North-Holland, Delft, 1959, pp. 12e33. [32] E. Carrera, Cz0 -requirements e models for the two dimensional analysis of multilayered structures, Composite Structures 37 (1997) 373e383. [33] K.P. Soldatos, T. Timarci, A unified formulation of laminated composite, shear deformable, five-degrees-of-freedom cylindrical shell theories, Composite Structures 25 (1993) 165e171. [34] O. Polit, M. Touratier, High-order triangular sandwich plate finite element for linear and non-linear analyses, Computer Methods in Applied Mechanics and Engineering 185 (2000) 305e324. [35] M. Touratier, An efficient standard plate theory, International Journal of Engineering Science 29 (1991) 901e916. [36] S. Cheng, Elasticity theory of plates and a refined theory, Journal of Applied Mechanics 46 (1979) 644e650. [37] J.N. Reddy, A simple higher-order theory for laminated composite plates, Journal of Applied Mechanics 51 (1984) 745e752. [38] A. Bhimaraddi, L.K. Stevens, A higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates, Journal of Applied Mechanics 51 (1984) 195e198. [39] H. Murakami, Laminated composite plate theory with improved in-plane response, Journal of Applied Mechanics 53 (1986) 661e666. [40] M. D’Ottavio, B. Kr€oplin, An extension of Reissner mixed variational theorem to piezoelectric laminates, Mechanics of Advanced Materials and Structures 13 (2006) 139e150. [41] A. Robaldo, E. Carrera, Mixed finite elements for thermoelastic analysis of multilayered anisotropic plates, Journal of Thermal Stresses 30 (2007) 165e194. [42] E. Carrera, On the use of Murakami’s zig-zag function in the modeling of layered plates and shells, Computers and Structures 82 (2004) 541e554. [43] E. Carrera, S. Brischetto, Analysis of thickness locking in classical, refined and mixed multilayered plate theories, Composite Structures 82 (2008) 549e562. [44] J.-C. Paumier, A. Raoult, Asymptotic consistency of the polynomial approximation in the linearized plate theory. Application to the Reissner-Mindlin model, in: ESAIM: Proceedings, vol. 2, SMAI, 1997, pp. 203e213.

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[67] V.Z. Vlasov, Thin-Walled Elastic Beams, second ed., Israel Program for Scientific Translations Ltd., 1963. [68] P. Ladeveze, J. Simmonds, New concepts for linear beam theory with arbitrary geometry and loading, European Journal of Mechanics A Solids 17 (1998) 377e402. [69] R. El Fatmi, Non-uniform warping including the effects of torsion and shear forces. Part I: a general beam theory, International Journal of Solids and Structures 44 (2007) 5912e5929. [70] V. Giavotto, M. Borri, P. Mantegazza, G. Ghiringhelli, V. Carmaschi, G.C. Maffioli, F. Mussi, Anisotropic beam theory and applications, Computers and Structures 16 (1983) 403e413. [71] D.H. Hodges, Nonlinear Composite Beam Theory, AIAA, Inc., 2006. [72] V.L. Berdichevsky, On the energy of an elastic rod, Journal of Applied Mathematics and Mechanics 45 (1981) 518e529. [73] A.R. Atilgan, D.H. Hodges, Unified nonlinear analysis for nonhomogeneous anisotropic beams with closed cross sections, AIAA Journal 29 (1991) 1990e1999. [74] W. Yu, D.H. Hodges, Generalized Timoshenko theory of the variational asymptotic beam sectional analysis, Journal of the American Helicopter Society 50 (2005) 46e55. [75] W. Yu, D.H. Hodges, J.C. Ho, Variational asymptotic beam sectional analysis e an updated version, International Journal of Engineering Science 59 (2012) 40e64. [76] P. Vidal, O. Polit, A sine finite element using a zig-zag function for the analysis of laminated composite beams, Composites Part B Engineering 42 (2011) 1671e1682. [77] P. Vidal, O. Polit, A family of sinus finite elements for the analysis of rectangular laminated beams, Composite Structures 84 (2008) 56e72. [78] P. Vidal, O. Polit, Assessment of the refined sinus model for the non-linear analysis of composite beams, Composite Structures 87 (2009) 370e381. [79] P. Vidal, O. Polit, A refined sine-based finite element with transverse normal deformation for the analysis of laminated beams under thermomechanical loads, Journals of Mechanics of Materials and Structures 4 (2009) 1127e1155. [80] E. Carrera, G. Giunta, M. Petrolo, Beam Structures e Classical and Advanced Theories, John Wiley & Sons, Ltd., 2011.

Appendices A.

Beam models

Beam structures1 are slender bodies that are loaded in the most general, threedimensional, case by axial and shear forces as well as bending and torsion moments. As outlined in Section 3.1, a beam model is formulated by taking advantage of the slenderness and it consists of (1) a closed set of PDE or ODE that depends on only one independent variable (the coordinate along the beam axis); (2) the definition of the stiffnesses of the cross-sectional surface, which in turn depend on the geometric and material properties; (3) some relations for recovering the stress state in any point 1

Following Antman [7], such structural members are also called “rods.” “Column” or “truss” most often indicates a beam that is subjected to only axial loads, the latter term being preferred for geometrically linear formulations. So, “beam-columns” indicate one-dimensional structures subjected to axial and lateral loads simultaneously [66].

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of the body from the solution of the one-dimensional problem. The key point is thus to define the cross-sectional deformation, generally referred to as warping, and various approaches have been followed for accomplishing this task. The classical axiomatic approach is referred to as the Saint-Venant’s theory: based on static assumptions (stresses perpendicular to the beam axis are nil), it considers a cross section that remains plane when the beam is loaded by axial force and bending moments, but that is free to warp uniformly out of its plane under the action of a torsion moment and shearing forces. In-plane warping is included in Saint-Venant’s theory through the Poisson effect. Classical beam theories such as Euler-Bernoulli’s or Timoshenko’s are formulated within the framework of Saint-Venant’s approach on introducing further restrictive assumptions that limit the out-of-plane warping to torsional deformation: a rigid cross section is thus postulated for the extension, bending, andd within Timoshenko’s theorydshearing, while uniform Saint-Venant’s warping is retained for torsion. Refined models that go beyond Saint-Venant’s theory are required in presence of nonuniform (restrained) warping [67], which typically occurs in thin-walled beams, or whenever a short wavelength response is to be evaluated [68,69]. Furthermore, composite cross sections require high-order warping functions for including the elastic couplings between extension, bending, shearing, and torsion, which are of utmost importance in a number of specific and complex applications such as beam models for helicopter rotor blades. In this context, it is worth mentioning the early work by Giavotto et al. [70], in which Saint-Venant’s theory was enriched through warping functions obtained from a cross-sectional finite element analysis, as well as the unified, nonlinear approach of Hodges summarized in Ref. [71]. The framework of this latter approach is Berdichevsky’s variational-asymptotic method [72], which allows to formulate one-dimensional beam models with high-order warping functions and whose accuracy is defined in an asymptotic sense (see also Refs. [73e75]). The remainder of this Appendix will limit the attention to composite beam models without considering torsion, i.e., for which out-of-plane warping may be induced under bending load due to the transverse shear flexibility of the composite cross section. For the sake of conciseness, complicating effects such as taper ratio, initial twist and curvature, and thin-walled open cross sections will be left out of the scope. Straight prismatic beams are considered that occupy the volume V ¼ S  ½0; L, where L is the length of the beam measured along the x1 axis and S is the cross section in the (x2, x3)-plane. Let further the cross section S be rectangular of dimension b  h PNp ðpÞ and composed of Np plies, where h ¼ p¼1 h is the thickness measured along the x3 ¼ z coordinate and b is the width measured along the x2 ¼ y coordinate. The procedure for constructing the reduced one-dimensional model starting from the elasticity relations can be still formally introduced as in Section 3.2, but the two-dimensional domain U has to be replaced by the one-dimensional domain x1 ˛ {0, L} and the thickness direction z by the two-dimensional cross section ðy; zÞ ˛ S. The kinematic approximations are thus introduced into Hamilton’s principle as ui ðx1 ; x2 ; x3 ; tÞ ¼ Fsi ðy; zÞe usi ðx1 ; tÞ withsi ¼ 0; 1; 2; .Ni

(A.1)

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133

and the differentiations and integrations of Eq. (3.9) can be explicitly carried out over S.

A.1. Classical theories Classical theories rely on the hypothesis of an infinitely rigid cross section, thus decoupling bending and torsion. In analogy to the CLPT and FSDT theories for plates, one discerns the EulereBernoulli beam theory (EBBT) and the Timoshenko beam theory (TBT) that are described below. EulereBernoulli beam theory. The EBBT can be formulated according to the following kinematic assumptions: 1. The cross section is infinitely rigid in its plane. 2. The cross section remains plane and perpendicular to the beam axis during deformation.

The first assumption states that neither stress nor deformation exists in the crosssectional (y, z) plane, that is, a one-dimensional constitutive relation is used for relating the axial strain and the axial stress. The planarity of the cross section enforces a linear distribution of the displacement field across S, i.e., the displacement field is constructed as a superposition of an axial displacement and a rigid rotation of S. The perpendicularity condition states the equivalence between the rotations of the cross section and the slope of the deformed beam axis, which means that the transverse shear strains gxy and gxz are nil. One notices the correspondence with the CLPT discussed in Section 3.1 and illustrated in Fig. 3.2(b). The displacement field for the EBBT thus reads u1 ðx1 ; y; z; tÞ ¼ e u01 ðx1 ; tÞ  y

ve u02 ðx1 ; tÞ ve u03 ðx1 ; tÞ z vx1 vx1

u2 ðx1 ; y; z; tÞ ¼ e u02 ðx1 ; tÞ; u3 ðx1 ; y; z; tÞ ¼ e u03 ðx1 ; tÞ

(A.2a) (A.2b)

Note that only three unknown functions suffice to express the EBBT kinematics, which correspond to the displacement vector ½e u01 ; e u02 ; e u03  of a generic point of the beam axis x1. Timoshenko beam theory. TBT extends EBBT on introducing a transverse shear strain field that is constant across S, and it relies on the following kinematic assumptions: 1. The cross section is infinitely rigid in its plane. 2. The cross section remains plane during deformation.

Based on these assumptions, the displacement field for the TBT in the threedimensional space can be written as u1 ðx1 ; y; z; tÞ ¼ e u01 ðx1 ; tÞ þ y e u11 ðx1 ; tÞ þ z e u21 ðx1 ; tÞ

(A.3a)

u02 ðx1 ; tÞ; u3 ðx1 ; y; z; tÞ ¼ e u03 ðx1 ; tÞ u2 ðx1 ; y; z; tÞ ¼ e

(A.3b)

134

Stability and Vibrations of Thin-Walled Composite Structures

The five unknown functions that define the TBT kinematics are thus the displacement vector e u0i of a point on the beam axis and the rotations e u11 and e u21 about the z- and y-axes, respectively, of the rigid cross section. Note the correspondence with the FSDT discussed in Section 3.2 and illustrated in Fig. 3.2(c). Shear correction coefficients can be included for reducing the transverse shear deformation energy predicted by the uniform transverse shear stresses of TBT.

A.2.

Refined theories

Refinements to the previously outlined classical theories are formulated upon relaxing the assumption of a rigid cross section, i.e., including a warping deformation thatdin the present context that neglects torsiondis due to transverse shear strains. In the following and in view of the numerical assessment proposed in Eq. (A.3), the discussion shall be limited to refined theories formulated within the framework of the Sinus model. The HSDT kinematics reads  ve u02 ðx1 ; tÞ ve u02 ðx1 ; tÞ þ f ðyÞ e u11 ðx1 ; tÞ þ vx1 vx1  ve u03 ðx1 ; tÞ ve u03 ðx1 ; tÞ þ f ðzÞ e u21 ðx1 ; tÞ þ z vx1 vx1

u1 ðx1 ; y; z; tÞ ¼ e u01 ðx1 ; tÞ  y

(A.4a) u2 ðx1 ; y; z; tÞ ¼ e u02 ðx1 ; tÞ;

u3 ðx1 ; y; z; tÞ ¼ e u03 ðx1 ; tÞ

(A.4b)

where one retrieves EBBT and TBT by setting f (y) ¼ f(z) ¼ 0 and f (y) ¼ y, f (z) ¼ z, respectively, while for a Sinus theory one sets f ðyÞ ¼

h py sin ; p b

f ðzÞ ¼

h pz sin p h

(A.5)

In case of beam problems defined in the (x1, z)-plane, the kinematics Eq. (A.4) reduces simply to  ve u03 ðx1 ; tÞ ve u03 ðx1 ; tÞ u01 ðx1 ; tÞ  z þ f ðzÞ e u21 ðx1 ; tÞ þ u1 ðx1 ; y; z; tÞ ¼ e vx1 vx1 (A.6a) u3 ðx1 ; y; z; tÞ ¼ e u03 ðx1 ; tÞ

(A.6b)

ZZTs can be formulated, which account for the slope discontinuity of the axial displacement u1 along the stacking direction x3 ¼ z. For this, one may directly superimpose Murakami’s zigzag function (MZZF) defined in Eq. (3.33) to the HSDT kinematics (see Ref. [76]); alternatively, the slope discontinuity is introduced by fulfilling the interlaminar continuity of the transverse shear stress (see the AMT

Classical, first order, and advanced theories

135

approach outlined in Section 3.4.2 and the formulation presented in Refs. [77,78]). One may note that the dependence on the x2 ¼ y coordinate remains unaffected in either case because the plies are assumed to be stacked along x3 ¼ z only. All theories mentioned above retain the static assumption of Saint-Venant’s theory, which calls for the use of the one-dimensional constitutive law for plane stress. The full three-dimensional constitutive law can be employed on retaining the direct stresses in the cross-sectional plane and following the same arguments outlined in Section 3.4.3. In the case of a plane problem, the HSDT of Eq. (A.6) can be enhanced with a quadratic expansion for the transverse displacement as follows [79].  ve u03 ðx1 ; tÞ ve u03 ðx1 ; tÞ u1 ðx1 ; y; z; tÞ ¼ e u01 ðx1 ; tÞ  z þ f ðzÞ e u21 ðx1 ; tÞ þ vx1 vx1 (A.7a) u03 ðx1 ; tÞ þ z e u13 ðx1 ; tÞ þ z2 e u23 ðx1 ; tÞ u3 ðx1 ; y; z; tÞ ¼ e

(A.7b)

All beam theories discussed above have a number of unknown functions that is independent of the number of plies constituting the composite cross section. Refined Layerwise beam models are obtained in the very same manner as mentioned in Section 3.5 and lead to models with a number of unknown functions and, hence, DOF that depends on the number of the considered layers. CUF permits to build classical and refined beam models, based on ESL or LW descriptions, in the same way as outlined in Section 3.6 by directly employing the generic expansion given in Eq. (A.1): the cross section S is discretized in a number s ¼ 1, 2,. N of points that are interpolated by a given polynomial basis Fs(y, z)

First natural frequency of simply-supported cross-ply laminated beams

Table A.1

2 Plies (0 degree, 90 degrees) S

4

10

100

Ref

4.5105

5.7700

6.1672

S2ZC

4.5525

5.7827

S2

4.6390

S0ZC

3 Plies (0 degree, 90 degrees, 0 degree) 4

10

100

5.8545

10.334

13.930

6.1673

5.8569

10.335

13.931

5.8104

6.1712

6.0488

10.599

13.940

4.8284

5.8580

6.1682

5.9758

10.345

13.930

S0Z

4.6169

5.8029

6.1676

5.9660

10.417

13.932

S0

4.7040

5.8254

6.1678

6.0502

10.629

13.939

FSDT

4.6784

5.8254

6.1678

6.8997

11.418

13.955

CLPT

5.9253

6.1305

6.1712

13.933

13.989

13.644

CLPT, classical laminated plate theory; FSDT, first-order shear deformation theory.

136

Stability and Vibrations of Thin-Walled Composite Structures

Bending of the simply-supported (0 degree, 90 degrees) beam under sinusoidal load

Table A.2

z

U 1 ðh=2Þ

U3 ð0Þ

s11 ð  h=2Þ

s13 ðmaxÞ

Ref

0.0714

4.7081

1.8762

0.6764

S2ZC

0.0720

4.6231

1.9367

0.6371

S2

0.0546

4.2833

1.8087

0.5838

S0ZC

0.0544

4.1878

2.0958

0.4193

S0Z

0.0722

4.5438

1.9903

0.6918

S0

0.0613

4.4027

2.1200

0.6297

FSDT

0.0603

4.4347

1.7496

0.4547

Ref

0.0623

2.9611

1.7652

0.7230

S2ZC

0.0623

2.9481

1.7750

0.6687

S2

0.0582

2.8445

1.7450

0.6021

S0ZC

0.0593

2.8836

1.8068

0.4331

S0Z

0.0623

2.9374

1.7872

0.7039

S0

0.0604

2.9156

1.8101

0.6426

FSDT

0.0603

2.6254

1.7496

0.4547

Ref

0.0603

2.6288

1.7443

0.7337

S2ZC

0.0602

2.6160

1.7373

0.6763

S2

0.0601

2.6173

1.7413

1.5774

S0ZC

0.0603

2.6280

1.7502

0.4358

S0Z

0.0603

2.6285

1.7500

0.7064

S0

0.0603

2.6283

1.7502

0.6451

FSDT

0.0603

2.6283

1.7496

0.4547

CLPT

0.0603

2.6255

1.7496

e

S¼4

S ¼ 10

S ¼ 100

CLPT, classical laminated plate theory; FSDT, first-order shear deformation theory.

Classical, first order, and advanced theories

137

Bending of the simply-supported (0 degree, 90 degrees, 0 degree) beam under sinusoidal load

Table A.3

Model z

U1 ðh=2Þ

U 3 ð0Þ

s11 ð  h=2Þ

s13 ð0Þ

Ref

0.0148

2.8901

1.1304

0.3580

S2ZC

0.0146

2.8913

1.1871

0.3545

S2

0.0135

2.6853

1.0974

0.2950

S0ZC

0.0158

2.7916

1.2469

0.3855

S0Z

0.0155

2.8027

1.2224

0.3363

S0

0.0139

2.7258

1.0974

0.2904

FSDT

0.0080

2.0941

0.6324

0.1592

Ref

0.0094

0.9332

0.7361

0.4239

S2ZC

0.0093

0.9331

0.7374

0.4336

S2

0.0090

0.8719

0.7250

0.3254

S0ZC

0.0095

0.9321

0.7459

0.4450

S0Z

0.0095

0.9193

0.7519

0.4043

S0

0.0090

0.8828

0.7105

0.3048

FSDT

0.0080

0.7642

0.6324

0.1592

Ref

0.0080

0.5153

0.6315

0.4421

S2ZC

0.0080

0.5128

0.6289

0.4556

S2

0.0080

0.5131

0.6317

0.4497

S0ZC

0.0080

0.5153

0.6335

0.4583

S0Z

0.0080

0.5151

0.6336

0.4225

S0

0.0080

0.5147

0.6331

0.3076

FSDT

0.0078

0.5135

0.6324

0.1561

CLPT

0.0080

0.5109

0.6324

e

S¼4

S ¼ 10

S ¼ 100

CLPT, classical laminated plate theory; FSDT, first-order shear deformation theory.

138

Stability and Vibrations of Thin-Walled Composite Structures

(Taylor, Lagrange, Legendre,.), and the solution of each point us is only dependent on the axial coordinate, i.e., us ¼ us(x1). The interested reader may consult [80], a book entirely dedicated to the description of CUF for beam structures. It is emphasized that CUF can be effectively employed for determining, in a combined axiomaticeasymptotic sense, the “best” model for a given application, i.e., the model of lowest order that provides with a predefined accuracy a specific result for a specific problem. The specific result may be related to global or local response, and the specific problem is defined by geometry, material, boundary, and loading conditions.

A.3.

Assessment on some benchmark problems

The same benchmark configurations addressed for the assessment of plate models in Section 3.7 are here employed to assess several classical and refined ESL theories based on the Sinus model. The different models are identified following the same naming convention summarized in Table 3.1. The proposed assessment concerns the effect of the length-to-thickness ratio S ¼ a/h for straight beam problems in the (x1, z) plane. Nondimensional fundamental frequencies for the simply-supported nonsymmetric 2-ply and symmetric 3-ply laminated beams are displayed in Table A.1. The nondimensional static   response of simply-supported beams subjected to the sinusoidal load p3 x1 ; z ¼ h2 ¼ p0 sin pxa 1 are given in Table A.2 for the 2-ply laminate and in Table A.3 for the 3-ply laminate. The reported nondimensional quantities are defined according to Eq. (3.37). Results for simply-supported sandwich beams are displayed in Table A.4 (nondimensional fundamental frequencies defined by Eq. (3.38)) and Table A.5 (static response defined according to Eq. (3.37a)). Table A.4

First natural frequency of the simply-supported sandwich

beam S

4

10

100

Ref

7.6880

15.677

25.335

S2ZC

7.9322

16.022

25.358

S2

8.0566

16.201

25.368

S0ZC

7.7196

15.702

25.336

S0Z

7.9807

16.074

25.351

S0

8.0750

16.240

25.359

FSDT

11.133

19.656

25.458

CLPT

24.292

25.333

25.545

CLPT, classical laminated plate theory; FSDT, first-order shear deformation theory.

Classical, first order, and advanced theories

139

Bending of the simply-supported sandwich beam under sinusoidal load

Table A.5

z

U1 ð  h=2Þ

U3 ð0Þ

s11 ðh=2Þ

s13 ð0Þ

Ref

0.0299

11.061

2.3841

0.3392

S2ZC

0.0288

10.408

2.2999

0.4153

S2

0.0276

10.067

2.2263

0.4064

S0ZC

0.0305

11.028

2.3995

0.3655

S0Z

0.0297

10.315

2.3384

0.4094

S0

0.0276

10.076

2.1730

0.3942

FSDT

0.0158

5.2869

1.2476

0.1290

Ref

0.0182

2.6688

1.4317

0.3504

S2ZC

0.0180

2.5563

1.4186

0.4286

S2

0.0176

2.4785

1.4075

0.4192

S0ZC

0.0182

2.6621

1.4378

0.3771

S0Z

0.0182

2.5397

1.4282

0.4218

S0

0.0178

2.4878

1.3986

0.4020

FSDT

0.0158

1.6927

1.2476

0.1291

Ref

0.0159

1.0248

1.2456

0.3526

S2ZC

0.0158

1.0229

1.2465

0.4312

S2

0.0158

1.0201

1.2478

0.5062

S0ZC

0.0159

1.0247

1.2496

0.3794

S0Z

0.0159

1.0235

1.2465

0.4243

S0

0.0159

1.0229

1.2492

0.4035

FSDT

0.0158

1.0149

1.2476

0.1291

CLPT

0.0158

1.0081

1.2476

e

S¼4

S ¼ 10

S ¼ 100

CLPT, classical laminated plate theory; FSDT, first-order shear deformation theory.

It is apparent that the comments made in Section 3.7 for the plate problem assessments are valid for the considered plane beam problems as well. Classical theories (EBBT and TBT) can be employed for thin beams (S ¼ 100) for all models and yield very similar results in this case. When dealing with moderately thick (S ¼ 10) and

140

Stability and Vibrations of Thin-Walled Composite Structures

thick (S ¼ 4) beams, the results show that the most accurate model is S2ZC: on the one hand, the zigzag effect is required for capturing the effects introduced by the layers’ interfaces within the composite stack, and on the other hand, the three-dimensional constitutive law should be retained for including the transverse normal deformation energy, which is no longer negligible when the beam is thick (S ¼ 4). These conclusions demonstrate once again the validity of Koiter’s recommendation [31] (see Section 3.4) in the case of composite structures.

Stability of composite columns and plates

4

Haim Abramovich Technion, I.I.T., Haifa, Israel

4.1

Introduction

This chapter aims at presenting the stability analysis of composite columns and plates using the classical lamination plate theory (CLPT) and the application of the first-order shear deformation plate theory (FSDPT) to obtain buckling loads.

4.1.1

The classical lamination plate theory approach

The CLPT [1e6] is an extension of the well-known KirchhoffeLove classical plate theory that is applied to laminated composite plates. The displacement field assumed has the following form:

uðx; y; z; tÞ ¼ u0 ðx; y; tÞ  z

vw0 vx

vðx; y; z; tÞ ¼ v0 ðx; y; tÞ  z

vw0 vy

(4.1)

wðx; y; z; tÞ ¼ w0 ðx; y; tÞ where the assumed variables u0, v0, and w0 are displacements in the x, y, and z directions, respectively, of a point on the midplane of the plate (namely, at z ¼ 0). The theory neglects transverse shear and transverse normal effect, leaving the bending and in-plane stretching deformations the only effective deformation influencing the plate behavior. Then the equations of motion can be written as [2,5,6]

Stability and Vibrations of Thin-Walled Composite Structures. http://dx.doi.org/10.1016/B978-0-08-100410-4.00004-1 Copyright © 2017 Elsevier Ltd. All rights reserved.

142

Stability and Vibrations of Thin-Walled Composite Structures


 vNxx vNxy v 2 u0 v2 vw0 þ ¼ I0 2  I1 2 vx vy vx vt vt
 vNxy vNyy v 2 v0 v2 vw0 þ ¼ I0 2  I1 2 vx vy vy vt vt
 v2 Mxy v2 Myy v v2 Mxx vw0 vw0 þ þ N þ 2 þ N xx xy vx vxvy vx vy vx2 vy2

  v vw0 vw0 v2 w0 v2 vu0 vv0 þ Nyy þ þ Nxy ¼ q þ I0 2 þ I1 2 vy vx vy vx vy vt vt

(4.2)


 v2 v2 w 0 v2 w 0  I2 2 þ 2 vt vx2 vy where q is the distributed pressure on the surface of the plate and the mass moments of inertia I0, I1, and I2 are defined as 8 9 8 9 I0 > > >1> > < > = Z þh=2 > < > = z $r0 $dz I1 ¼ > > > h=2 > > > : > ; : 2> ; z I2

(4.3)

where h is the total thickness of the plate and r0 is the relevant density. In Eq. (4.2), Nxx, Nxy, and Nyy are the force resultants per unit length and Mxx, Mxy, and Myy are the moment resultants per unit length defined as 9 8 9 8 sxx > Nxx > > > > > > > Z = = < þh=2 < Nyy ¼ syy dz; > > > h=2 > > > > > ; : ; : Nxy sxy

8 9 8 9 Mxx > sxx > > > > > > > Z < = = þh=2 < Myy ¼  syy $z$dz. > > > h=2 > > > > > : ; : ; Mxy sxy

(4.4)

In Eq. (4.4), z is a coordinate normal to the surface of the plate and sxx and syy are the normal stresses in the x and y directions, respectively, while sxy is the shear stress. The stress resultants Nxx, Nxy, and Nyy and Mxx, Mxy, and Myy (see their definition in Figs. 4.1 and 4.2) can be defined using the assumed displacements (Eq. 4.1) to yield

Stability of composite columns and plates

8 9 2 Nxx > A11 > > > > > < = 6 6 Nyy ¼ 6 A12 > > 4 > > > > : ; Nxy A16

A12 A22 A26

2

B11 6 6  6 B12 4

B12

B16

B26

B22

143

9 8
2 > > > > vu 1 vw > > 0 0 > > þ > > > > > > 2 vx vx > 3> > > > A16 > > > > > >
2 > = 7< 7 vv0 1 vw0 A26 7 þ > 5> vy 2 vy > > > > > > > > > A66 > > > > > > > 2 > vu vv v w > > 0 0 0> > > þ þ > > : vy vx vxvy ;

z

Nxx Nxy

Nyx

Nyy

(4.5)

8 9 2 > > v w > > 0 > > > > > 2 > > > vx > 3> > > > > B16 > > > > > = 7< v2 w > 7 0 B26 7 5> vy2 > > > > > > > > > > B66 > > > > > 2 > > > v w 0> > > > > :2 ; vxvy

Nyy Nxy

Nyx

y

Nxx x

Figure 4.1 Definition of the force resultants.

Mxy

Myx

z Myy

Mxy

x

Figure 4.2 Definition of the moment resultants.

Myy

Mxy

Mxx

y

144

Stability and Vibrations of Thin-Walled Composite Structures

8 9 2 Mxx > B11 > > > > > < = 6 6 Myy ¼ 6 B12 > > 4 > > > > : ; Mxy B16

2

B12 B22 B26

9 8
2 > > > > vu 1 vw > > 0 0 > > þ > > > > > > 2 vx vx > > 3> > > B16 > > > > > >
2 > = 7< 7 vv0 1 vw0 B26 7 þ > 5> vy 2 vy > > > > > > > > > B66 > > > > > > > 2 > vu vv v w > > 0 0 0> > > þ þ > > : vy vx vxvy ;

D11 6 6  6 D12 4

D12

D16

D26

D22

8 9 2 > > v w > > 0 > > > > > 2 > > > vx > 3> > > > > D16 > > > > > = 7 < v2 w > 7 0 D26 7 5> vy2 > > > > > > > > > > D66 > > > > > 2 > > > v w 0> > > > > :2 ; vxvy

(4.6)

where Aij ¼

N X

ðkÞ

Qij ðzkþ1  zk Þ;

k¼1

Bij ¼

N  1X ðkÞ  Q z2  z2k ; 2 k¼1 ij kþ1

N  1X ðkÞ  Dij ¼ Qij z3kþ1  z3k 3 k¼1

(4.7)

ðkÞ

with Qij being the lamina stiffness after transformation. Substituting Eq. (4.6) into Eq. (4.2) leads to the equations of motion for a laminated composite plate expressed by the three assumed displacements (Eq. 4.1):



2 
2  v2 u0 v3 w 0 v v0 v3 w0 v u0 v3 w 0 v2 v 0 A11 þ þ3 2 þ 2 þ 3 þ A12 þ A16 2 vxvy vxvy2 vxvy vx2 vx vx vy vx
2 
 v v 0 v3 w 0 v2 u0 v2 v 0 v3 w0 v3 w0 v3 w0 þ 2 þ A26 þ þ  B þ A  B 11 12 66 vxvy vy2 vy3 vy2 vxvy2 vx3 vxvy2  3B16

v3 w 0 v3 w 0 v3 w 0 v2 u0 v3 w 0  B  2B ¼ I  I 0 1 26 66 vx2 vy vy3 vxvy2 vt 2 vxvt 2 (4.8)

Stability of composite columns and plates

145





2
2   v2 v0 v3 w0 v u0 v3 w0 v u0 v3 w0 þ þ þ þ A þ A 12 16 vxvy vx2 vy vy2 vy3 vx2 vx3
2 
2  v u0 v2 v 0 v3 w0 v u0 v2 v 0 v3 w0 v3 w 0 þ A26 þ 3 þ þ 2 þ 2 þ A  B12 2 66 2 2 2 2 vxvy vxvy vx vy vxvy vx vy vx vy

A22

 B22

v3 w0 v3 w 0 v3 w 0 v3 w 0 v2 v 0 v3 w0 ¼ I  B  3B  2B  I 0 1 16 26 66 vy3 vx3 vxvy2 vx2 vy vt 2 vyvt 2 (4.9)





3  v3 u0 v4 w0 v v0 v3 u0 v3 w0 þ þ 4 þ B12 þ4 2 2 B11 vx3 vx vx2 vy vxvy2 vx vy
3
3   3 4 v u0 v v 0 v w0 v v0 v4 w0 þ B16 3 2 þ 3 þ 8 3 þ 2 þ B22 vx vy vx vx vy vy3 vy4
3 
3  v u0 v3 v 0 v4 w 0 v u0 v3 v 0 v4 w 0 þ B26 þ 4 þ 3 þ 8 þ þ 2B 66 vy3 vxvy2 vxvy3 vxvy2 vx2 vy vx2 vy2  D11

v4 w0 v4 w0 v4 w 0 v4 w0  2D  D  4D 12 22 16 vx4 vx2 vy2 vy4 vx3 vy

v4 w0 v4 w0  4D þ Pðw0 Þ 66 vxvy3 vx2 vy2

 v2 w0 v2 v2 w0 v2 w0 v2 vu0 vv0 þ þ ¼ q þ I0 2  I2 2 þ I 1 vt vt vx2 vy2 vt 2 vx vy

 4D26

(4.10) where

  v vw0 vw0 v vw0 vw0 þ Nxy þ Nyy Nxx Nxy Pðw0 Þ ¼ þ vx vy vx vy vx vy

4.1.2

(4.11)

The first-order shear deformation plate theory approach

The FSDPT approach (see, for example, Refs. [7e13]) is an extension of the well-known Timoshenko beam theory [14,15] and/or the MindlineReissner [7,8] plate theory that is applied to laminated composite plates. The assumed displacement field has the following form: uðx; y; z; tÞ ¼ u0 ðx; y; tÞ þ zfx ðx; y; tÞ vðx; y; z; tÞ ¼ v0 ðx; y; tÞ þ zfy ðx; y; tÞ wðx; y; z; tÞ ¼ w0 ðx; y; tÞ

(4.12)

146

Stability and Vibrations of Thin-Walled Composite Structures

where the assumed variables u0, v0, and w0 are displacements in the x, y, and z directions, respectively, of a point on the midplane of the plate (namely, at z ¼ 0) and fx and fy are rotations about the x and y axes, respectively. The theory takes into account the transverse shear assuming it is constant over the thickness coordinate. The theory requires the calculation of shear deformation correction factors [11,12]. Then the equations of motion can be written as [6,8e10] vNxx vNxy v 2 u0 v2 f þ ¼ I0 2 þ I1 2 x vx vy vt vt v2 f y vNxy vNyy v 2 v0 þ ¼ I0 2 þ I1 2 vx vy vt vt vQx vQy v2 w0 þ þ Pðw0 Þ ¼ q þ I0 2 vx vy vt

(4.13)

vMxx vMxy v2 f v2 u0 þ  Q x ¼ I2 2 x þ I1 2 vx vy vt vt v2 f y vMxy vMyy v2 v 0 þ  Q y ¼ I2 2 þ I1 2 vx vy vt vt In Eq. (4.13), P(w0) was defined by Eq. (4.11), while Qx and Qy are the shear forces or the transverse force resultants defined as (

Qx Qy

)

Z ¼K

þh=2 h=2

(

sxy syz

) dz

(4.14)

where K is the shear correction coefficient computed by equating the strain energy due to transverse shear stresses to the strain energy due to true transverse shear as calculated by a 3D elasticity theory, while sxy and syz are the transverse shear stresses. For a homogenous beam having a rectangular cross section, K ¼ 5/6.

Stability of composite columns and plates

147

The relations among the stress resultants and the assumed displacements are given by

8 9 2 Nxx > A11 > > > > > < = 6 6 Nyy ¼ 6 A12 > > 4 > > > > : ; Nxy A16

A12 A22 A26

2

B11 6 6 þ 6 B12 4

B12

B16

B26

8 9 2 Mxx > B11 > > > > > < = 6 6 Myy ¼ 6 B12 > > 4 > > > > : ; Mxy B16

2

8 9
2 > > > > vu 1 vw 0 0 > > > > þ > > > > > > 2 vx vx > > 3> > > A16 > > > > > >
2 > = 7< 7 vv0 1 vw0 A26 7 þ > 5> vy 2 vy > > > > > > > > > A66 > > > > > > > 2 > vu vv v w > > 0 0 0> > > þ þ > > : vy vx vxvy ; 8 9 > > vfx > > > > > > > vx > > 3> > > > > B16 > > > > > > = 7< vf y 7 B26 7 vy > 5> > > > > > > > > > B66 > > > > > vf vf > > y x > > > : vy þ vx > ;

B22

B12 B22 B26

8 9
2 > > > > vu0 1 vw0 > > > > þ > > > > > > 2 vx vx > > 3> > > B16 > > > > > >
2 > = 7< 7 vv0 1 vw0 B26 7 þ > 5> vy 2 vy > > > > > > > > > B66 > > > > > > > 2 > vu vv v w > > 0 0 0> > > þ þ > > : vy vx vxvy ;

D11 6 6 þ 6 D12 4

D12

D16

D26

D22

8 9 > > vfx > > > > > > > vx > > 3> > > > D16 > > > > > > > = 7< vfy 7 D26 7 vy > 5> > > > > > > > > > D66 > > > > > vf vf > > y x > > > : vy þ vx > ;

(4.15)

(4.16)

148

Stability and Vibrations of Thin-Walled Composite Structures

(

Qy

)

" ¼K

Qx

9 8 > > vw0 > > #> þ fy > > > = < A45 vy

A44

(4.17)

> > > A55 > vw0 > > > þ fx > ; : vx

A45

With the constants Aij, Bij, and Dij being defined in Eq. (4.7), A44, A45, and A55 are given by A44 ¼

N X

ðkÞ

Q44 ðzkþ1  zk Þ;

A45 ¼

k¼1

A55 ¼

N X

N X

ðkÞ

Q45 ðzkþ1  zk Þ;

k¼1

(4.18)

ðkÞ

Q55 ðzkþ1  zk Þ

k¼1 ðkÞ

ðkÞ

ðkÞ

where Q44 , Q45 , and Q55 are the lamina stiffness after transformation. The equations of motion in terms of displacements for the FSDPT approach can be written as (see also Ref. [6])
A11

v2 u0 v3 w 0 þ 3 vx2 vx




þ A12

v2 v0 v3 w0 þ vxvy vxvy2




þ A16

v2 u0 v3 w 0 v2 v 0 þ3 2 þ 2 2 vxvy vx vy vx




2 
2  v2 fy v v0 v3 w0 v u0 v2 v 0 v3 w 0 v2 fx þ 2 þA26 þ þ þ B þ A þ B 11 12 66 vxvy vxvy vy2 vy3 vy2 vxvy2 vx2 ! ! 2 2 v2 fy v2 fx v fy v2 fx v fy v2 u0 v2 f x þ 2 þ B26 2 þ B66 þ þ I þB16 2 ¼ I 0 1 vxvy vxvy vx vy vy2 vt 2 vt 2 (4.19) A22


2 
2
2   v v 0 v3 w 0 v u0 v3 w 0 v u0 v3 w 0 þ þ þ þ A þ A 12 16 vxvy vx2 vy vy2 vy3 vx2 vx3


þ A26

þ B22

¼ I0

v2 u0 v2 v 0 v3 w0 þ3 þ2 2 vxvy vy vxvy2

v 2 fy




þ A66

 v2 u0 v2 v 0 v3 w 0 v2 fx þ 2 þ2 2 þ B12 vxvy vx vxvy vx vy

v2 fy v2 fx v2 f x þ 2 þ B þ B 2 16 26 vxvy vy2 vx2 vy

! þ B66

2 v2 fx v fy þ 2 vxvy vx

!

v2 f y v 2 v0 þ I1 2 2 vt vt (4.20)

Stability of composite columns and plates

149





2
2    v2 w0 vfy v w0 vfy v w0 vfx þ þ þ þ KA þ KA 45 45 vy vxvy vx vxvy vy vy2
2  v w0 vfx v2 w0 þ KA55 þ þ Pðw0 Þ ¼ q þ I0 2 2 vx vx vt

KA44


B11

v2 u0 v3 w0 þ 3 vx2 vx


þ B26

þ D16


þ B12

v2 v0 v3 w0 þ vxvy vxvy2




2  v u0 v2 v 0 v4 w0 þ 2 þ3 3 þ B16 2 vxvy vx vx vy


2  v2 fy v u0 v2 v 0 v3 w0 v2 fx þ 2 þ þ D þ B66 þ D 11 12 vxvy vxvy vy2 vxvy2 vx2 ! !
 v2 f y v2 fx v2 fy v2 fx v2 fy vw0 þ 2 þ D26 2 þ D66 þ f þ 2  KA 55 x vxvy vxvy vx vx vy vy2

v2 v0 v3 w0 þ 3 vy2 vy


 KA45



(4.21)

vw0 þ fy vy



 ¼ I2

v2 fx v2 u0 þ I 1 vt 2 vt 2 (4.22)





2 
2  v u0 v3 w 0 v v0 v3 w0 þ þ þ B 22 vx2 vx3 vy2 vy3
2 
2  v u0 v2 v 0 v3 w0 v v 0 v2 u0 v3 w0 þ B26 þ 3 þ 2 þ 2 þ þ B 66 vxvy vy2 vxvy2 vx2 vxvy vxvy2 ! ! v2 fy v2 f y v2 fy v2 fx v2 fx v2 fx v2 fx þ D22 2 þ D16 2 þ D26 þ2 þ þ D12 þ D66 vxvy vxvy vxvy vy vx vy2 vx2 B12

v2 u 0 v3 w 0 þ vxvy vxvy2


 KA45



vw0 þ fx vx

þ B16




 v2 fy vw0 v2 v 0 þ f y ¼ I2 2 þ I1 2  KA44 vy vt vt (4.23)

4.2

Buckling of columns: the classical lamination theory approach

Buckling of columns, which is an one-dimensional element (see Fig. 4.3), can be obtained directly from Eqs. (4.8)e(4.10) by setting all the terms involving differentiation to zero with respect to y. Note that another one-dimensional element is achieved by running the cylindrical bending approach. In this approach, an infinite element in the y direction is used, assuming no change in this direction. Therefore the displacements of

150

Stability and Vibrations of Thin-Walled Composite Structures

z y z

b q x

h L

Figure 4.3 The laminated composite column model.

the model, u0, v0, and w0, are a function of the x coordinate only. Note that this case can be considered as plane strain, whereas the column (beam) is a plane stress case. The equations of motion for a general laminate using the classical lamination theory (CLT) can therefore be written as (assuming v0 ¼ 0)
2  v u0 v3 w0 v3 w 0 v 2 u0 v3 w 0 A11 þ ¼ I  I  B 11 0 1 vx2 vx3 vx3 vt 2 vxvt 2
B11

v3 u0 v4 w 0 þ 4 vx3 vx

  D11

v4 w0 v2 w0 ¼ q þ Pðw Þ þ I 0 0 vx4 vt 2

 v 2 v2 w 0 v2 vu0  I2 2 þ I 1 vt vx2 vt 2 vx

(4.24)

(4.25)

where Pðw0 Þ ¼


 v vw0 Nxx vx vx

(4.26)

Eqs. (4.24) and (4.25) constitute the general form for a column under axial and shear loads (Nxx and Nxy) with high order terms and dependence of time. To analyze the stability problem of the columns, all time-dependent terms will be set to zero, the lateral load q will also be assumed to be zero. This leads to the following coupled equations: A11


2  d u0 d 3 w 0 d 3 w0 þ ¼0  B 11 dx2 dx3 dx3


3  d u0 d 4 w 0 d 4 w0 d2 w0 þ þ N ¼0  B11 þ D xx 11 dx3 dx4 dx4 dx2

(4.27)

(4.28)

Stability of composite columns and plates

151

N xx stands for the axial compression per unit width (b) and it was assumed that it is not a function of x. The two coupled equations of motion (Eqs. 4.27 and 4.28) can be decoupled to yield one equation having the following form:
 B2 d 4 w0 d2 w0 þ N ¼0 D11  11 xx A11 dx4 dx2

(4.29)

d4 w0 N xx d2 w0    þ ¼0 dx4 D11  B211 A11 dx2

(4.30)

or

4.2.1

Symmetric laminate (B11 ¼ 0)

For the case of symmetric laminate B11 ¼ 0, multiplying the equation by the width of the beam, b, while w0 ¼ w0p þ wb [where w0p is the prebuckling deflection satisfying Eq. (4.30) and wb is the buckling deflection], yields d4 wb b$N xx d2 wb d 4 wb P d 2 wb þ ¼ 0 0 þ ¼0 Exx $Iyy dx2 dx4 b$D11 dx2 dx4

(4.31)

where b$D11 ¼ Exx $Iyy and b$N xx ¼ 0

(4.32)

Eq. (4.32) is exactly the BernoullieEuler equation of buckling for columns. Now we shall deal with the general equation (Eq. 4.30) for the buckling of symmetric laminated composite columns (B11 ¼ 0). By substituting w0 ¼ w0p þ wb , we get 2 d4 wb N xx d2 wb d 4 wb 2 d wb þ ¼ 0 0 þ l ¼0 dx4 D11 dx2 dx4 dx2

(4.33)

where l2 ¼

N xx D11

(4.33a)

The solution of Eq. (4.33) has the following general form: wb ðxÞ ¼ C1 sinðlxÞ þ C2 cosðlxÞ þ C3 x þ C4

(4.34)

152

Stability and Vibrations of Thin-Walled Composite Structures

where C1, C2, C3, and C4 are determined using the four boundary conditions of the problem. For example, assuming a simply supported column having a length L, one can write its boundary conditions as d2 wb ð0Þ ¼0 dx2

wb ð0Þ ¼ 0;

Mxx ð0Þ ¼ 0 0

wb ðLÞ ¼ 0;

d 2 wb ðLÞ Mxx ðLÞ ¼ 0 0 ¼0 dx2

(4.35)

Substituting the boundary conditions in Eq. (4.34) yields a set of four equations with four unknowns, which have the following matrix form: 2 6 6 6 6 6 4

0

1

0

0

l2

0

sinðlLÞ

cosðlLÞ

L

l2 sinðlLÞ l2 cosðlLÞ

0

38 9 8 9 1 > C1 > > 0 > > > > > > > > > > > > 7> > =

< C2 > = > 07 7 ¼ 7 > > C3 > > > > 0> 17 > > > > 5> > > > > > > > ; : ; : > 0 0 C4

(4.36)

To obtain a unique solution, the determinant of the matrix appearing in Eq. (4.36) must vanish. This leads to the following characteristic equation: sinðlLÞ ¼ 0 0 lL ¼ np;

n ¼ 1; 2; 3; 4.

(4.37)

The critical buckling load of the column (the lowest one, n ¼ 1) will then be 

N xx

 cr

¼

p2 D11 L2

(4.38)

The term in Eq. (4.38) should be multiplied by the width of the beam, b, to obtain Pcr [N]. The beam buckling shape (the eigenfunction) has the following form (this is obtained by back-substituting the eigenvalue from Eq. (4.37) into Eq. (4.36)): wb ðxÞ ¼ C1 sinðlxÞ ¼ C1 sin

px L

(4.39)

Tables 4.1 and 4.2 present some of the most encountered column cases, while the schematic drawing of the various out-of-plane boundary conditions are presented in Fig. 4.4. Only the out-of-plane boundary conditions can influence the critical buckling load and the shape during buckling (see Appendix A). To conclude this subchapter, a beam with general boundary conditions is presented in Fig. 4.5.

Stability of composite columns and plates

153

Buckling of laminated composite columns: out-of-plane boundary conditions using the CLT approach

Table 4.1

Out-of-plane boundary conditions No.

x[0

Name

x[L d 2 wb dx2

¼0

d 2 wb ¼0 dx2 dwb wb ¼ 0; ¼0 dx

1

SS-SS

wb ¼ 0;

2

C-C

wb ¼ 0;

3

C-F

4

F-F

d 2 wb d 3 wb dwb ¼0 ¼ 0; þ l2 2 dx dx3 dx

5

SS-C

wb ¼ 0;

d 2 wb ¼0 dx2

d 2 wb d 3 wb dwb ¼0 ¼ 0; þ l2 2 dx dx3 dx dwb wb ¼ 0; ¼0 dx

6

SS-F

wb ¼ 0;

d 2 wb ¼0 dx2

d 2 wb d 3 wb dwb ¼0 ¼ 0; þ l2 2 3 dx dx dx

7

G-F

dwb d 3 wb dwb ¼ 0; ¼0 þ l2 dx dx3 dx

d 2 wb d 3 wb dwb ¼0 ¼ 0; þ l2 dx2 dx3 dx

8

G-SS

dwb d 3 wb dwb ¼ 0; ¼0 þ l2 dx dx3 dx

wb ¼ 0;

9

G-G

dwb d 3 wb dwb ¼ 0; ¼0 þ l2 dx dx3 dx

10

G-C

dwb d 3 wb dwb ¼ 0; ¼0 þ l2 dx dx3 dx

dwb ¼0 dx dwb ¼0 wb ¼ 0; dx

wb ¼ 0;

d 2 wb d 3 wb dwb ¼0 ¼ 0; þ l2 dx2 dx3 dx

d 2 wb ¼0 dx2

dwb d 3 wb dwb ¼ 0; ¼0 þ l2 dx dx3 dx dwb wb ¼ 0; ¼0 dx

C, clamped; F, free; G, guided; SS, simply supported.

Defining the following variables as k1 ¼

k1 k2 kq kq ; k2 ¼ ; k q1 ¼ 1 ; kq2 ¼ 2 D11 D11 D11 D11

(4.40)

B2

where D11 h D11  A1111 and k1 and k2 are linear springs, while k q1 and kq2 are torsion springs. The boundary conditions of the problem depicted in Fig. 4.5 are given by d3 wð0Þ d2 wð0Þ dwð0Þ 2 dwð0Þ ¼ k þ l $wð0Þ ¼ k q1 1 3 2 dx dx dx dx d3 wðLÞ dwðLÞ d 2 wðLÞ dwðLÞ ¼ k2 $wðLÞ þ l2 ¼ kq2 3 dx dx dx dx2

(4.41)

Table 4.2

Buckling loads and relevant buckling modes of laminated composite columns using the CLT approach Name

Characteristic equation

Critical buckling load

Mode shape

1

SS-SS

sin(lL) ¼ 0

  p2 N xx cr ¼ 2 D11 L

sin

lL ¼ np 2

C-C

lLsin(lL) ¼ 2[1  cos(lL)] lL ¼ 2p,8.987,4p,.

3

C-F

cos(lL) ¼ 0 lL ¼ ð2n  1Þ

4

F-F

p 2

sin(lL) ¼ 0 tan(lL) ¼ lL lL ¼ 1.430p,2.459p,.

6

SS-F

sin(lL) ¼ 0 lL ¼ np

7

G-F

cos(lL) ¼ 0 lL ¼ ð2n  1Þ

8

G-SS

cos(lL) ¼ 0 lL ¼ ð2n  1Þ

9

G-G

sin(lL) ¼ 0 lL ¼ np

10

G-C

p 2

sin(lL) ¼ 0 lL ¼ np

p 2

px L


 2px 1  cos L px 1  cos 2L

sin

px

  p2 N xx cr ¼ 2 D11 L

h L xi sinðaxÞ þ aL 1  cosðaxÞ  L p while a ¼ 1:4318 L px sin L

  p2 N xx cr ¼ 2 D11 4L

px cos 2L

  p2 N xx cr ¼ 2 D11 4L

px cos 2L

  p2 N xx cr ¼ 2 D11 L

px cos L

  p2 N xx cr ¼ 2 D11 L

px  1  cos L

  p2 N xx cr ¼ 2:045 2 D11 L

Stability and Vibrations of Thin-Walled Composite Structures

SS-C

  p2 N xx cr ¼ 2 D11 4L   p2 N xx cr ¼ 2 D11 L

lL ¼ np 5

  p2 N xx cr ¼ 4 2 D11 L

154

No.

Stability of composite columns and plates

Simply supported

155

Guided

k

Linear spring

Clamped

Free

kθ

Torsion spring

Figure 4.4 Typical out-of-plane boundary conditions for laminated composite columns under axial compression.

Nx

kθ

1

k1

kθ

2

Nx

k2

Figure 4.5 A typical laminated composite column under axial compression having spring-based boundary conditions.

Applying the boundary conditions to the general solution presented in Eq. (4.34) yields the following characteristic equation (see details in Ref. [16]): h i n      k 1 þ k 2 l6 þ kq1 $kq2 k1 þ k2 þ k 1 $k 2 $L l4  o  þ kq1 $kq2 kq1 þ k q2  kq1 $kq2 $L l2 sinðlLÞ  n    þ k1 þ k2 kq1 þ k q2 l3  k 1 $k 2 $L kq1 þ k q2 l3 o  2$k1 $k2 $kq1 $kq2 $l cosðlLÞ þ k 1 $k 2 $k q1 $kq2 $l ¼ 0

(4.42)

Eq. (4.42) can be used to solve problems involving not only boundary conditions with springs but also classical boundary conditions. Letting the linear springs k1 and k2 tend to infinity (N) while the torsional ones, kq1 and kq2 , are set to zero leads to the classical simply supported boundary conditions at both ends of the compressed beam. Similarly, setting all the springs to zero would yield freeefree boundary conditions. Clamped boundary conditions are obtained when both types of springs (linear and torsion) tend to infinity.

156

Stability and Vibrations of Thin-Walled Composite Structures

4.2.2

Nonsymmetric laminate (B11 s 0)

To solve the case of a nonsymmetric laminate (B11 s 0), the coupled Eqs. (4.27) and (4.28) have to be uncoupled. The result presented by Eq. (4.29) enables to find the solution for the lateral displacement w0 (a complete derivation is presented in Appendix A). This is given by _  _  w0 ðxÞ ¼ C1 sin l x þ C2 cos l x þ C3 x þ C4

(4.43)

where _2

l ¼

N xx   D11  B211 A11

(4.44)

The in-plane displacement will then be presented as (see Appendix A) _  _  u0 ðxÞ ¼ C5 sin l x þ C6 cos l x þ C7 x þ C8

(4.45)

where B11 _ C5 ¼  $ l $C2 A11

(4.46)

B11 _ C6 ¼ þ $ l $C1 A11

Solving for a pinnedepinned column (simply supported at both ends), with the following boundary conditions (Appendix A) du0 ð0Þ d2 w0 ð0Þ þ D11 ¼0 dx dx2

(4.47)

du0 ðLÞ d 2 w0 ðLÞ þ D11 ¼0 dx dx2

(4.48)

w0 ð0Þ ¼ 0;

Mxx ð0Þ ¼ 0 0  B11

w0 ðLÞ ¼ 0;

Mxx ðLÞ ¼ 0 0  B11

u0 ð0Þ ¼ 0 A11

(4.49)

du0 ðLÞ d2 w0 ðLÞ  B11 ¼ N xx dx dx2

(4.50)

yields the following eigenvalue _  _ sin l L ¼ 0 0 l L ¼ np;

n ¼ 1; 2; 3; 4.

(4.51)

Stability of composite columns and plates

157

which gives the critical buckling load per unit width having the following form: 

N xx



¼ cr


 B211 p2  D 11 L2 A11

(4.52)

Thebuckling shape will be similar to the symmetric case (see Table 4.2), i.e.,  sin px L . Note that as expected, the buckling load for a symmetric laminate will be higher than that for a nonsymmetric one with the same number of layers. Tables 4.3(a) and (b) present the boundary conditions for the nonsymmetric case, whereas Table 4.4 gives the buckling loads and the associated buckling mode for the most used cases. Table 4.3(a) Buckling of nonsymmetric laminated composite columns: outof-plane boundary conditions using the CLT approach Out-of-plane boundary conditions No.

Name

x[0

x[L

1

SS-SS

w0 ¼ 0

w0 ¼ 0

B11 2

3

C-C

C-F

du0 d 2 w0 þ D11 2 ¼ 0 dx dx

5

F-F

SS-C

w0 ¼ 0

dw0 ¼0 dx

dw0 ¼0 dx

w0 ¼ 0

G-F

B11

du0 d 2 w0 þ D11 2 ¼ 0 dx dx

B11

d 2 u0 d 3 w0 dw0 ¼0 þ D11 3 þ N xx 2 dx dx dx

B11

du0 d 2 w0 þ D11 2 ¼ 0 dx dx

B11

du0 d 2 w0 þ D11 2 ¼ 0 dx dx

B11

d2 u0 d 3 w0 dw0 ¼0 þ D11 3 þ N xx 2 dx dx dx

B11

d 2 u0 d 3 w0 dw0 ¼0 þ D11 3 þ N xx 2 dx dx dx

w0 ¼ 0 B11

6

du0 d 2 w0 þ D11 2 ¼ 0 dx dx

w0 ¼ 0

dw0 ¼0 dx 4

B11

du0 d 2 w0 þ D11 2 ¼ 0 dx dx

dw0 ¼0 dx B11

d2 u0 d 3 w0 dw0 ¼0 þ D11 3 þ N xx 2 dx dx dx

C, clamped; F, free; G, guided; SS, simply supported.

w0 ¼ 0 dw0 ¼0 dx B11

du0 d 2 w0 þ D11 2 ¼ 0 dx dx

B11

d 2 u0 d3 w0 dw0 ¼0 þ D11 3 þ N xx 2 dx dx dx

158

Stability and Vibrations of Thin-Walled Composite Structures

Buckling of nonsymmetric laminated composite columns: in-plane boundary conditions using the CLT approach

Table 4.3(b)

In-plane boundary conditions No.

Name

x[0

x[L

1

SS-SS

u0 ¼ 0

A11

du0 d 2 w0  B11 2 ¼ N xx dx dx

2

C-C

u0 ¼ 0

A11

du0 d 2 w0  B11 2 ¼ N xx dx dx

3

C-F

u0 ¼ 0

A11

du0 d 2 w0  B11 2 ¼ N xx dx dx

4

F-F

u0 ¼ 0

A11

du0 d 2 w0  B11 2 ¼ N xx dx dx

5

SS-C

u0 ¼ 0

A11

du0 d 2 w0  B11 2 ¼ N xx dx dx

6

G-F

u0 ¼ 0

A11

du0 d 2 w0  B11 ¼ N xx dx dx2

4.3

Buckling of columns: the first-order shear deformation theory approach

Buckling of columns using the first-order shear deformation theory (FSDT) can be obtained directly from Eqs. (4.19)e(4.23) by setting all the terms involving differentiation to zero with respect to y. As described in Section 4.2, another approach would be to run the cylindrical bending model leading to the displacements, u0, v0, and w0, being a function of the x coordinate only. The equations of motion for a general laminate using the FSDT approach (see also Refs. [17e21]) can therefore be written as (assuming v0 ¼ fy ¼ 0)
2  v u0 v3 w0 v2 f x v2 u0 v2 f x A11 þ ¼ I þ I þ B 11 0 1 vx2 vx3 vx2 vt 2 vt 2
KA55

v2 w0 vfx þ vx2 vx

  N xx

v2 w 0 v2 w0 ¼ q þ I 0 vx2 vt 2

(4.53)

(4.54)


2 
 v u0 v3 w0 v2 f x vw0 v2 f x v2 u0 þ f þ  KA þ I B11 þ D ¼ I 11 2 1 55 x vx2 vx3 vx2 vx vt 2 vt 2 (4.55)

Table 4.4

Buckling loads and relevant buckling modes of laminated composite columns using the CLT approach Name

Characteristic equation

Critical buckling load per unit width

1

SS-SS

_  sin l L ¼ 0



N xx



p2 D11 a L2

cr

¼

cr

¼4

Mode shape sin

px L

_

l L ¼ np

2

C-C

_  h _ i l L sin l L ¼ 2 1  cos l L

_



N xx



p2 D11 L2

1  cos

cr

¼

p2 D11 4L2

1  cos

cr

¼

p2 D11 L2

sin

cr

¼ 2:045

cr

¼


2px L

_

l L ¼ 2p; 8:987; 4p; .

3

C-F

_  cos l L ¼ 0 _

l L ¼ ð2n  1Þ

4

F-F



N xx



px

Stability of composite columns and plates

No.

2L

p 2

_  sin l L ¼ 0



N xx



px L

_

5

SS-C

l L ¼ np _  _ tan l L ¼ l L



N xx



p2 D11 L2

_

6

G-F

l L ¼ 1:430p; 2:459p; . _  cos l L ¼ 0 _

l L ¼ ð2n  1Þ

 B2 D11  A1111 .

N xx



p2 D11 4L2

p 2 159


D11 ¼

a



h xi sinðaxÞ þ aL 1  cosðaxÞ  L p while a ¼ 1:4318 L px cos 2L

160

Stability and Vibrations of Thin-Walled Composite Structures

One should note that Eqs. (4.53)e(4.55) are for the case of uniform properties along the beam (Fig. 4.3). For the particular case of properties varying along the x coordinate of the beam, the reader is referred to [19e21]. To solve the buckling problem, the lateral load, q, and time derivatives are set to zero, leading to the following three coupled equations of motion1:

A11


2  d u0 d 3 w b d2 f þ þ B11 2x ¼ 0 2 3 dx dx dx


KA55

d2 wb dfx þ dx2 dx

  N xx

(4.56)

d2 wb ¼0 dx2

(4.57)


2 
 d u0 v3 w b d2 fx dwb þ fx ¼ 0 þ 3 þ D11 2  KA55 B11 dx2 dx dx dx

(4.58)

Decoupling the equations provides the following uncoupled ones2 (see also Ref. [22]): d4 wb b2 d2 wb þl ¼0 dx4 dx2

(4.59)

d3 fx b2 d2 fx ¼0 þl dx3 dx

(4.60)

d 4 u0 b 2 d 2 u0 þl ¼0 dx4 dx2

(4.61)

where

2 N xx b l ¼

 0 N xx B211 N xx D11  1 A11 KA55

1 2


 2 B2 b l D11  11 A11 ¼
 2 B2 b l D11  11 A11 1þ KA55

w0 ¼ w0p þ wb (where w0p is the prebuckling deflection and wb is the buckling deflection). 3 To uncouple the equations, one needs to neglect high-order terms such as vdxw3b .

(4.62)

Stability of composite columns and plates

161

Accordingly, the solutions for Eqs. (4.59)e(4.61) have a form similar to that presented in Eq. (4.34), i.e.,     lx þ C2 cos b lx þ C3 x þ C4 wb ðxÞ ¼ C1 sin b

(4.63)

    fx ðxÞ ¼ C5 sin b lx þ C6 cos b lx þ C7

(4.64)

    u0 ðxÞ ¼ C8 sin b lx þ C9 cos b lx þ C10 x þ C11

(4.65)

One should note that the 11 constants (C1eC11) are not independent. Their dependency among them can be obtained by back-substituting the solutions (Eqs. 4.63e4.65) in the coupled equations (Eqs. 4.56 and 4.58) to yield the following five relations: A11 $C8 þ B11 $C5 ¼ 0

(4.66)

A11 $C9 þ B11 $C6 ¼ 0

(4.67)

  2 2 l  D11 $C5 $b l  KA55  C2 $b l þ C5 ¼ 0  B11 $C8 $b

(4.68)

  2 2 l  D11 $C6 $b l  KA55 C1 $b l þ C6 ¼ 0  B11 $C9 $b

(4.69)

 KA55 $ðC3 þ C7 Þ ¼ 0 0 C3 þ C7 ¼ 0

(4.70)

These 5 relationships together with the 6 boundary conditions of the problem will provide the needed 11 equations with 11 unknowns, C1eC11. The six boundary conditions of the problem are (see also footnote 1) du0 df þ B11 x ¼ N xx or u0 ¼ 0 dx dx
 dwb dwb KA55 þ fx  N xx ¼ 0 or wb ¼ 0 dx dx

A11

B11

du0 df þ D11 x ¼ 0 or fx ¼ 0 dx dx

(4.71)

(4.72)

(4.73)

162

Stability and Vibrations of Thin-Walled Composite Structures

Let us present the general case of a simply supported laminated composite beam having the following boundary conditions: du0 ðLÞ df ðLÞ þ B11 x ¼ N xx dx dx

u0 ð0Þ ¼ 0;

A11

wb ð0Þ ¼ 0;

wb ðLÞ ¼ 0

B11

(4.74) (4.75)

du0 ð0Þ df ð0Þ þ D11 x ¼ 0; dx dx

B11

du0 ðLÞ df ðLÞ þ D11 x ¼0 dx dx

(4.76)

Solving for the 11 unknowns, C1eC11, we get the following results for the first 4 constants: C1 ¼

B11 A11

 

1  cos b lL 

sin b lL



;

C2 ¼

B11 ; A11

C3 ¼ 0;

B11 C4 ¼  . A11

(4.77)

To obtain the buckling load, one should demand that w0(x) /N, leading to the same procedure applied for a general beam using the CLT approach. Observing the constant C1 leads to the following eigenvalue solution:   sin b lL ¼ 0 0 b lL ¼ np;

n ¼ 1; 2; 3; 4.

(4.78)

which leads to the critical buckling load per unit width having the following form:  p 2


 h  i B211 D11  N xx cr   L A11 h  CLT i N xx cr ¼ ¼ p2
2 B11 N xx cr CLT D11  1þ L A11 1þ KA55 KA55

(4.79)

If one defines the following term as h Gh

N xx

 i cr CLT

KA55

(4.80)

Stability of composite columns and plates

163

Buckling loads and relevant buckling modes of laminated composite columns using the FSDT approach

Table 4.5

a

No.

Name

1

SS-SS

2

C-C

3

C-F

4

F-F

5

G-F

D11 ¼ Gh

b

h  







N xx

N xx

N xx

N xx

N xx

N xx

 i cr CLT

 cr

 cr



¼

p2 D11 a L2

p2 ¼ 4 2 D11 L

cr

¼

p2 D11 4L2

cr

¼

p2 D11 L2

cr

¼

p2 D11 4L2





Critical buckling load per unit width h  i N xx cr b   CLT N xx cr ¼ 1þG h  i N xx cr   CLT N xx cr ¼ 1þG h  i N xx cr   CLT N xx cr ¼ 1þG h  i N xx cr   CLT N xx cr ¼ 1þG h  i N xx cr   CLT N xx cr ¼ 1þG

Mode shape

sin

px L


2px 1  cos L px 1  cos 2L sin

px

cos

L px 2L


B2 D11  A1111 .

½ðN xx Þcr CLT KA55

.

then Eq. (4.79) can be written as 

N xx

h

 cr

¼

N xx

 i cr CLT

1þG

(4.81)

The shape of the buckling will be sinðpx L Þ like for the symmetric case, which is calculated using the CLT approach. The buckling load calculated using the FSDT approach is lower than that calculated according to the CLT. Like before, a nonsymmetric laminate would yield a lower buckling load as than a symmetric one having the same layers. Some typical buckling loads are presented in Table 4.5.

4.4 4.4.1

Buckling of plates: the classical lamination plate theory approach Simply supported special orthotropic plates

Presenting the buckling problem of laminated composite plates using the CLPT approach is done based on Eqs. (4.8)e(4.10), in which the time-dependent terms

164

Stability and Vibrations of Thin-Walled Composite Structures

and the lateral surface load, q, are set to zero. The plate buckling issue is complicated and only a few closed form solutions are available for certain boundary conditions and layups. Based on the extensive work done by Leissa [23], we shall present those available solutions. Other rigorous plate solutions can be solved based on the Navier method that treats rectangular plates on simply supported boundary conditions all around it or the Lévy method suitable for plates with two opposite simply supported edges, whereas the other two can have any combination of boundary conditions. Approximate solutions can be obtained using the RayleigheRitz, GalerkineBubnov, or the extended Kantorovich method, all based on energy approaches. The first case to be solved is sometimes called special orthotropic plates for which the bendingestretching coupling terms Bij and the bendingetwisting coefficients D16 and D26 are set to zero. Then from Eq. (4.10) one obtains v4 w 0 v4 w 0 v4 w0 þ 2ðD þ 2D Þ þ D 12 22 66 vx4 vx2 vy2 vy4

  v vw0 vw0 v vw0 vw0 þ Nxy þ Nyy ¼ Nxx Nxy  vx vy vx vy vx vy

D11

(4.82)

Assuming that the loads per unit width are constant and taking account that they are compressive loads, one can rewrite Eq. (4.82) as D11

v4 w0 v4 w0 v4 w0 v2 w0 v2 w 0 v2 w0 þ þ 2ðD þ 2D Þ þ D ¼ N þ 2N N xx xy yy 12 22 66 vx4 vx2 vy2 vy4 vx2 vxvy vy2 (4.83)

Eq. (4.83) presents a plate under compression loads in the x and y directions, N xx and N yy , respectively, and shear loads N xy . All those loads are forces per unit width of the plate (see Fig. 4.6). For a plate, the following boundary conditions are used: 1. Simply supported on all four sides of the plate w0 ðx; 0Þ ¼ 0

w0 ðx; bÞ ¼ 0

w0 ð0; yÞ ¼ 0

w0 ða; yÞ ¼ 0

(4.84)

Mxx ð0; yÞ ¼ 0 Mxx ða; yÞ ¼ 0 Myy ðx; 0Þ ¼ 0 Myy ðx; bÞ ¼ 0 y

Nxx

b a

Figure 4.6 A schematic plate under unidirectional compression.

Nxx x

Stability of composite columns and plates

165

where

v2 w0 v2 w0 Mxx ¼  D11 2 þ D12 2 ; vx vy

v2 w0 v2 w0 Myy ¼  D12 2 þ D22 2 ; vx vy

Mxy ¼ 2D66

v2 w0 vxvy (4.85)

2. Clamped on all four sides of the plate w0 ðx; 0Þ ¼ 0

w0 ðx; bÞ ¼ 0

w0 ð0; yÞ ¼ 0

vw0 ðx; 0Þ vw0 ðx; bÞ ¼0 ¼0 vx vx

w0 ða; yÞ ¼ 0

vw0 ð0; yÞ vw0 ða; yÞ ¼0 ¼0 vy vy

(4.86)

3. Free on all four sides of the plate Mxx ð0; yÞ ¼ 0 Mxx ða; yÞ ¼ 0 Myy ðx; 0Þ ¼ 0 Myy ðx; bÞ ¼ 0 Vy ð0; yÞ ¼ 0

Vy ða; yÞ ¼ 0

Vx ðx; 0Þ ¼ 0

Vy ðx; bÞ ¼ 0

(4.87)

where Vx ¼ Qx þ

vMxy vy

Vy ¼ Qy þ

vMxy vx

(4.88)

and 
vMxx vMxy v vw0 vw0 þ þ  N xx Nxy Qx ¼ vx vy vy vx vx
 vMyy vMxy v vw0 vw0 Nxy þ þ  N yy Qy ¼ vy vy vx vx vy

(4.89)

Note that Eq. (4.85) presents the expressions for the moments per unit width, Mxx, Myy, and Mxy. Solving Eq. (4.83) for a rectangular plate loaded in the x direction only (Nxy ¼ Nyy ¼ 0) for all around simply supported boundary conditions (Eq. 4.84) can be shown to reduce to w0 ðx; 0Þ ¼ 0

w0 ðx; bÞ ¼ 0

w0 ð0; yÞ ¼ 0

w0 ða; yÞ ¼ 0

v2 w0 ðx; 0Þ ¼0 vx2

v2 w0 ðx; bÞ ¼0 vx2

v2 w0 ð0; yÞ ¼0 vy2

v2 w0 ða; yÞ ¼0 vy2

(4.90)

and substituting the following out-of-plane deflection w0, which satisfies the boundary conditions (Eq. 4.90), into Eq. (4.83)

166

Stability and Vibrations of Thin-Walled Composite Structures

mpx npy sin a b

w0 h w0 ¼ Cmn sin

m; n ¼ 1; 2; 3.

(4.91)

where Cmn is a small arbitrary amplitude coefficient, yields the following critical load per unit width (b in the current case): 

N xx





¼ p D11 2

cr

m2 a

n2 n4  a 2

þ 2ðD12 þ 2D66 Þ þ D22 b b m

(4.92)

Eq. (4.92) is a function of both m and n, which are the number of half-waves in the x and y directions, respectively. It is clear that the minimum is obtained for n ¼ 1. Defining the following nondimensional term (like in Ref. [23]) Kx h

N xx $b2 D22

(4.93)

enables the rewriting of Eq. (4.92) in the following form: " #

    Kx D11 b 2 2 D12 D66 a 21 ¼ m þ2 þ2 þ b m2 p2 D22 a D22 D22

(4.94)

Eq. (4.94) enables to find the lowest values for Kx as a function of the aspect ratio a/ b and m for given values of D11, D22, D12, and D66. Fig. 4.7 presents Eq. (4.94) for 32 28 24 20 Kx

D11 D22

16

= 10

2

π 12 D11

8

D22

=1

m=1

4 m=1

m=2

m=1

2

4

3 m=4

m=3

m=2

m=3

D22

0 0

0.5

1

1.5

2

2.5

3

D11

3.5

4

= 0.1 4.5

5

a/b

Figure 4.7 Uniaxial buckling of a rectangular plate simply supported on all sides for three values of D11/D22, while (D12 þ 2D66)/D22 ¼ 1.

Stability of composite columns and plates

167

three values of D11/D22, while (D12 þ 2D66)/D22 ¼ 1. For other values of those parameters, one would get similar curves. The minimum of each curve generated by Eq. (4.94) is given at rffiffiffiffiffiffiffiffi a 4 D11 ¼ m$ b D22

(4.95)

yielding
rffiffiffiffiffiffiffiffi
  Kx D11 D12 D66 þ ¼2 þ2 p2 min D22 D22 D22

(4.96)

Another case is when the forces are applied in both the directions of the plate, as depicted in Fig. 4.8. In the case of simply supported boundary conditions all around the plate, we can use Eq. (4.91) and insert it in Eq. (4.83) for the case N xy ¼ 0. This yields the following equation (taking the absolute value only): m2 n2 m2 n2 m4 n4

þ N yy ¼ p2 D11 þ 2ðD12 þ 2D66 Þ þ D22 a b a a b b (4.97)  2 N Dividing Eq. (4.97) by ma and taking a h N yy yields N xx

xx

 n 2 h m2 n4  a 2 i þ 2ðD12 þ 2D66 Þ þ D22 D11   a b b m N xx cr ¼ p2 a2  n 2 1þa b m

y Nyy

Nxx

b a

Nxx x

Nyy

Figure 4.8 A schematic plate under bidirectional compression.

(4.98)

168

Stability and Vibrations of Thin-Walled Composite Structures

y

Nxy

Nxy

b

Nxy

a

x

Nxy

Figure 4.9 A schematic plate under shear.

or h Kx ¼ p2

D11

n2 n4  a 2 i þ 2ðD12 þ 2D66 Þ þ D22 b b m a2  n 2 1þa b m

m2 a

(4.99)

For the biaxial loads the n ¼ 1 value will not necessarily lead to the lowest value of the compression load, and therefore, its lowest value will be decided after calculating the critical buckling load for a few combinations of m and n. The case of a rectangular plate on simply supported boundary conditions under pure shear is presented schematically in Fig. 4.9. This case is more complicated than the previous two cases. The equation to be solved is given by D11

v4 w 0 v4 w0 v4 w0 v2 w 0 þ 2ðD þ 2D Þ þ D ¼ 2N xy 12 22 66 vx4 vx2 vy2 vy4 vxvy

(4.100)

There are no analytical solutions for this case, even for an all around simply supported rectangular plate. The reader is referred to the results presented in Ref. [24] for an infinitely long plate in the x direction (a/b /N), with the D11 bending rigidity being neglected. The exact solution for this case, in which simply supported boundary conditions were assumed along the long edges, has the following form: N xy $b2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 11:71 4 ½D22 ðD12 þ 2D66 Þ

(4.101)

Another interesting approach is presented by Leissa [23]. According to his way of solving Eq. (4.100), he assumes a deflection having the following form: x

wðx; yÞ ¼ f ðyÞeikb

(4.102)

Stability of composite columns and plates

169

pffiffiffiffiffiffiffi where i ¼ 1, b is the plate’s width, and k is a wavelength constant to be determined. Substituting Eq. (4.102) into Eq. (4.100) leads to D11

k4 b

f ðyÞ  2ðD12 þ 2D66 Þ

k2 d 2 f ðyÞ b

dy2

þ D22

k df ðyÞ d 4 f ðyÞ ¼0  2iN xy dy4 b dy (4.103)

The solution of Eq. (4.103) has the following form: y

y

y

y

f ðyÞ ¼ A1 eib1 b þ A2 eib2 b þ A3 eib3 b þ A4 eib4 b

(4.104)

where b1, b2, b3, and b4 are roots of the fourth-degree polynomial equation coming from Eq. (4.103). Substituting the simply supported boundary conditions at y ¼ 0, b into Eq. (4.104), leads to a fourth-order characteristic determinant, with k and N xy being free parameters. For each value of k, there is at least one value of N xy . The critical value for the shear load per unit width would be the lower one. It is shown in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ref. [23] that the parameter D11 =D22 ½ðD12 þ 2D66 Þ=D22  determines the solution. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For 1  D11 =D22 ½ðD12 þ 2D66 Þ=D22   N the critical value of N xy is determined from Kshear

N xy b2 ¼ ¼ k1 D22

rffiffiffiffiffiffiffiffi 4 D11 D22

(4.105)

rffiffiffiffiffiffiffiffi 4 D11 k ¼ k2 $b$ (4.106) D22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For 0  D11 =D22 ½ðD12 þ 2D66 Þ=D22   1 the critical value of N xy is calculated from Kshear

N xy b2 ¼ ¼ k3 D22

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D12 þ 2D66 D22

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D12 þ 2D66 k ¼ k4 $b$ D22

(4.107)

(4.108)

The various coefficients, k1ek4, are presented in Table 4.6. For other conditions, the approximate solutions using energy methods are advised (see Appendix B).

170

Stability and Vibrations of Thin-Walled Composite Structures

Coefficients for the buckling parameters of a simply supported infinite strip loaded in shear

Table 4.6

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11 =D22 ðD12 þ 2D66 Þ=D22

k1

k2

k3

k4

0.0

46.84



0.05



1.92

0.20

47.20

1.94

0.50

48.80

2.07

52.68

2.49

1.00

52.68

2.49

2.00

43.20

2.28

3.00

39.80

2.16

5.00

37.00

2.13

10.00

35.00

2.08

20.00

34.00



30.00



2.05

40.00

33.0



N

32.5



From A.W. Leissa, Buckling of Laminated Composite Plates and Shell Panels, AFWAL-TR-85e3069, June 1985.

4.4.2

Simply supported on two opposite edges of special orthotropic plates

Using the Lévy method, one can solve the buckling of rectangular plates, in which two opposite edges are on simply supported boundary conditions, whereas the other two sides can be clamped, free, or any other combination of boundaries. The solution is presented for constant in-plane loads, without shear type loads. Therefore the equilibrium equation, Eq. (4.83), can be written as D11

v4 w 0 v4 w0 v4 w0 v2 w 0 v2 w0 þ 2ðD12 þ 2D66 Þ 2 2 þ D22 4 ¼ N xx 2 þ N yy 2 4 vx vx vy vy vx vy

(4.109)

Assuming the following solution for Eq. (4.109) w0 ðx; yÞ ¼ Fn ðxÞsin

npy b

and substituting in it yields

n ¼ 1; 2; 3

(4.110)

Stability of composite columns and plates

D11

171



2 np2 d 4 Fn ðxÞ d Fn ðxÞ  2ðD þ 2D Þ þ N xx 12 66 4 dx b dx2

np4 np2 þ D22 þ N yy Fn ðxÞ ¼ 0 b b

(4.111)

A general solution for Eq. (4.111) can be assumed to have the following form: Fn ðxÞ ¼ C1 sinhðl1 xÞ þ C2 coshðl1 xÞ þ C3 sinðl2 xÞ þ C4 cosðl2 xÞ

(4.112)

where l1 and l2 are the roots of the following characteristic equation

np2 np4 np2

þ N xx l2 þ D22 þ N yy D11 l4  2ðD12 þ 2D66 Þ ¼0 b b b (4.113) The roots are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 B þ B2  D11 C D11 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2 ðl2 Þ ¼  B þ B2  D11 C D11 ðl1 Þ2 ¼

(4.114)

where B h ðD12 þ 2D66 Þ

np2 b

N xx þ 2



h

C h D22

np4 b

þ N yy

np2

b

(4.115)

The constants C1eC4 are to be determined using the appropriate boundary conditions at x ¼ 0, a. For example, for clamped boundary conditions at x ¼ 0 that are written as Fn ð0Þ ¼ Fn ðaÞ ¼ 0

dFn ð0Þ dFn ðaÞ ¼ dx dx

(4.116)

the relevant eigenvalue problem would have the following form 2

0

6 6 l1 6 6 6 sinhðl1 aÞ 4

1 0 coshðl1 aÞ

l1 coshðl1 aÞ l1 sinhðl1 aÞ

38 9 8 9 0 C > > > > > 1> > > > > > > > > > 7> > > > > = < < = 7 0 l2 0 C 2 7 ¼ 7 > > C3 > > >0> > sinðl2 aÞ cosðl2 aÞ 7 > > 5> > > > > > > > ; : > : > ; > 0 l2 cosðl2 aÞ l2 sinðl2 aÞ C4 (4.117) 0

1

172

Stability and Vibrations of Thin-Walled Composite Structures

To obtain a nontrivial solution the determinant of the coefficients should vanish, yielding the following characteristic equation:   2l1 l2 ½1  coshðl1 aÞcosðl2 aÞ þ l21  l22 sinhðl1 aÞsinðl2 aÞ ¼ 0

(4.118)

Numerically solving Eq. (4.118) would yield the buckling loads of the plate as a function of its aspect ratio, a/b. Similarly, other boundary conditions can be evaluated, while two opposite sides of the rectangular plate are on simply supported boundary conditions.

4.4.3

Unsymmetric orthotropic plates

In general, laminated composite plates will be fabricated with ply arrangements that are not necessarily symmetric with respect to the midplane of the plate. This case is named the unsymmetric laminated plate. Typical examples of symmetric, antisymmetric, and unsymmetric cross-ply layups are shown in Fig. 4.10, where regular means plies of equal thickness. First we shall deal with the case of simply supported boundary condition all around the rectangular plate. Because in-plane boundary conditions are added to the outof-plane boundary conditions, there are four types of simply supported boundary conditions in the unsymmetric case. These can be written as (see Ref. [23]) SS1: w ¼ Mn ¼ un ¼ ut ¼ 0 SS2: w ¼ Mn ¼ Nn ¼ ut ¼ 0

(4.119)

SS3: w ¼ Mn ¼ un ¼ Nnt ¼ 0 SS4: w ¼ Mn ¼ Nn ¼ Nnt ¼ 0

0 degree

0 degree

0 degree

90 degree

90 degree 0 degree

0 degree 0 degree

0 degree

90 degree

90 degree

Regular antisymmetric

Regular unsymmetric

Regular symmetric 0 degree

0 degree

90 degree

90 degree 0 degree

0 degree

0 degree

90 degree

90 degree

Antisymmetric

Unsymmetric

Symmetric

Figure 4.10 Symmetric, antisymmetric, and unsymmetric regular and general cross-ply laminates.

Stability of composite columns and plates

173

where n and t represent normal and tangent directions to a given boundary, respectively. Accordingly, Mn, Nn, and Nnt are the bending moment, the normal force, and the shear force per unit width, respectively, at plate’s edges x ¼ const. or y ¼ const. and can be written as a function of the displacements, u, v, and w0. According to Ref. [23], only two closed form solutions are available for unsymmetric laminated rectangular plates under uniform biaxial loading (N xx ¼ const., N yy ¼ const., and N xy ¼ 0). One solution is for cross-ply having SS2 boundary conditions all around the plate and the second one for angle-ply plates on SS3-type edges. For unsymmetric cross-ply laminates, we have A16 ¼ A26 ¼ B16 ¼ B26 ¼ D16 ¼ D26 ¼ 0. For an antisymmetric lamination, the previous problem is further simplified by adding A11 ¼ A22, D11 ¼ D22, B11 ¼  B22, B12 ¼ B66 ¼ 0. SS2 conditions at x ¼ 0, a and y ¼ 0, b for unsymmetric cross-ply laminates have the following exact solutions for the three displacements: uðx; yÞ ¼ Amn cos vðx; yÞ ¼ Bmn sin

mpx npy sin a b

mpx npy cos a b

w0 ðx; yÞ ¼ Cmn sin

(4.120)

mpx npy sin a b

Substituting Eq. (4.120) into Eq. (C.1) (for the matrix form of the equilibrium equations, see Appendix C) yields N xx

mp2 a

þ N yy

np2 b

2

b 33 þ ¼a

b 11 a b 13 a b 23  a b 13 2b a 12 a 2

b 12 b 11 a b 22  a a

(4.121)

Note that the in-plane resultants, N xx and N yy , are compressive and the various   b ij appearing in Eq. (4.121) are the results of the application of the various terms a operators given in Appendix C (Eq. C.2) on the expressions of the displacements (Eq. 4.120) for the unsymmetric cross-ply laminate case. They are given as mp2 np2 np2 mp2 b 22 ¼ A22 þ A66 þ A66 a a b b a mp2 np2 mp4 np4 ¼ D11 þ 2ðD12 þ 2D66 Þ þ D22 a a b b mpnp b 21 ¼ ðA12 þ A66 Þ ¼a a b mpnp2 mp3 b 31 ¼ B11 ¼a þ ðB12 þ 2B66 Þ a a b mp2 np np3 b 32 ¼ ðB12 þ 2B66 Þ ¼a þ B22 a b b

b 11 ¼ A11 a b 33 a b 12 a b 13 a b 23 a

(4.122)

174

Stability and Vibrations of Thin-Walled Composite Structures

It is interesting to note that for symmetric laminates (Bij ¼ 0), the right-hand side of b 33 , which is exactly similar to Eq. (4.83) when the shear resulEq. (4.121) reduces to a tant N xy is zero. The results calculated by Jones [25] for antisymmetric cross-ply graphite/epoxy plates having various number of layers is presented in Fig. 4.11. As the number of layers tends to infinity, the term B11 will go to zero, bringing the results for an antisymmetric laminate to those of an orthotropic plate. Fig. 4.12 presents the variation of the uniaxial buckling load as a function of the Young’s moduli ratio E1/E2 for a square antisymmetric cross-ply laminated plate. The loading parameter is normalized by the critical load for orthotropic plate, N xx0 ðB11 ¼ 0Þ. As the modulus ratio decreases from the graphite/epoxy value of 40 (see Fig. 4.12), the influence of bendingeextension coupling decreases slowly. The reduction in the buckling load of a squared two-layered graphite/epoxy plate from the orthotropic value is about 65%; however, the influence of this coupling dies out rapidly as the number of layers increases. For example, for six layers, the reduction in the buckling load is only about 7%.

3.5

Nxx

b

Nxx

a

3.0

Orthotropic solution (B11=0) 2.5

Nxx⋅b2

∞ 6 4

2.0

π 2D22 1.5

Number of layers

1.0 2 E1 G12 = 40 = 0.5 ν 12 = 0.25 E2 E2

0.5 0 0

0.5

1.0

1.5 2.0 2.5 Plate aspect ratio, a/b

3.0

Figure 4.11 Uniaxial nondimensional buckling loads of rectangular antisymmetric cross-ply laminated plates. From R.M. Jones, Buckling and vibration of unsymmetrically laminated cross-ply rectangular plates, AIAA Journal 11 (12) (1973) 1626e1632.

Stability of composite columns and plates

175

∞ 6

1.0

4

0.8

Number of layers

0.6 Nxx Nxx0 0.4

2 Nxx

0.2

Nxx

a

ν12 =0.25

a 0

10

0

G12 =0.5 E2

20

30

40

E Modulus ratio, 1 E2

50

Figure 4.12 Relative uniaxial buckling loads of a square antisymmetric cross-ply laminated plates. From R.M. Jones, Buckling and vibration of unsymmetrically laminated cross-ply rectangular plates, AIAA Journal 11 (12) (1973) 1626e1632.

Another closed form solution was found and described in Refs. [5,26] for antisymmetric angle-ply laminates, for which we have A16 ¼ A26 ¼ D16 ¼ D26 ¼ B11 ¼ B22 ¼ B12 ¼ 0. For the SS3 boundary conditions at x ¼ 0, a and y ¼ 0, b, the expressions for the three displacements are uðx; yÞ ¼ Amn sin

mpx npy cos a b

vðx; yÞ ¼ Bmn cos

mpx npy sin a b

w0 ðx; yÞ ¼ Cmn sin

(4.123)

mpx npy sin a b

As done before for the SS2 boundary conditions, we substitute Eq. (4.123) into Eq. (C.1) (for the matrix form of the equilibrium equations see Appendix C), yielding

N xx

mp2 a

þ N yy

np2 b

_

¼ a 33 þ

_

_

_

_

_2

2a 12 a 13 a 23  a 11 a 13 _

_

_2

a 11 a 22  a 12

(4.124)

176

Stability and Vibrations of Thin-Walled Composite Structures

As before,  the in-plane resultants, N xx and N yy , are compressive and the various _

terms a ij appearing in Eq. (4.121) are given as _

a 11 ¼ A11

_

a 33 ¼ D11

_

mp2

a mp4 a

 A66

np2

_

_

a 13 ¼ a 31 ¼ 3B16

_

_

a 23 ¼ a 32 ¼ B16

a 22 ¼ A22

np2 b

 A66

mp2 a

mp2 np2 np4 þ D22 a b b mpnp

þ 2ðD12 þ 2D66 Þ

a 12 ¼ a 21 ¼ ðA12 þ A66 Þ

_

b

_

a mp2 np

a mp3 a

b

b

(4.125)

np3 b mpnp2

þ B26

þ 3B26

a

b

The critical buckling load per unit width can then be written as N xx ¼

"

p2

a2 m 2 þ n2

! # D11 m4 þ 2ðD12 þ 2D66 Þm2 n2 r 2 þ D22 n4 r 4

N yy r N xx

    1 b m2 B16 m2 þ 3B26 n2 r 2 þ b3 n2 r 2 3B16 m2 þ B26 n2 r 2  b1 2

 (4.126)

where r ¼ a/b and    b1 ¼ A11 m2 þ A66 n2 r 2 A66 m2 þ A22 n2 r 2  ðA12 þ A66 Þm2 n2 r 2      b2 ¼ A11 m2 þ A66 n2 r 2 B16 m2 þ 3B26 n2 r 2  ðA12 þ A66 Þ 3B16 m2 þ B26 n2 r 2 n2 r 2      b2 ¼ A66 m2 þ A22 n2 r 2 3B16 m2 þ B26 n2 r 2  ðA12 þ A66 Þ B16 m2 þ 3B26 n2 r 2 m2 (4.127) Figs. 4.13 and 4.14 present typical buckling loads for square antisymmetric angleply laminated plates for uniaxial and biaxial loadings, respectively, as calculated in Ref. [27]. Typical values can be seen in Table 4.7 from the same Ref. [27].

80 Orthotropic solution (B16=B26=0)

Nxx

∞ 60

Nxx

b

θ

6

b

4

Nxx⋅b2 E2t 3

40

Number of layers

20

2

E1 = 40 E2 0

0

15

G12 = 0.5 E2

ν 12 = 0.25

30 45 60 Lamination angle, θ

75

90

Figure 4.13 Uniaxial buckling loads for square antisymmetric angle-ply laminated plates. From R.M. Jones, H.S. Morgan, J.M. Whitney, Buckling and vibration of antisymmetrically laminated angle-ply rectangular plates, Transactions of the ASME, Journal of Applied Mechanics 12 (1973) 1143e1144. N 40

b

θ

Orthotropic solution (B16=B26=0)

b

N 30

∞ 6 4

N⋅b2 E2t 3

20 Number of layers 10

2 E1 = 40 E2

0 0

G12 = 0.5 E2

ν 12 = 0.25

15 30 Lamination angle, θ

45

Figure 4.14 Biaxial compression buckling loads for square antisymmetric angle-ply laminated plates. From R.M. Jones, H.S. Morgan, J.M. Whitney, Buckling and vibration of antisymmetrically laminated angle-ply rectangular plates, Transactions of the ASME, Journal of Applied Mechanics 12 (1973) 1143e1144.

178

Stability and Vibrations of Thin-Walled Composite Structures

Uniaxial and biaxial buckling loads for antisymmetrically laminated graphite/epoxy plates (E1/E2 [ 40, G12/E2 [ 0.5, n12 [ 0.25) Table 4.7

Number of layers 2

4 

q (degrees)

6   N$b2 E2 t3

N(Orthotropic)

Uniaxial loading 0

35.831

35.831

35.831

35.831

15

21.734

38.253

41.313

43.760

30

20.441

49.824

55.265

59.619

45

21.709

56.088

62.455

67.548

60

19.392 (m ¼ 2)

45.434 (m ¼ 2)

50.257 (m ¼ 2)

54.115 (m ¼ 2)

75

12.915 (m ¼ 2)

22.075 (m ¼ 2)

23.772 (m ¼ 2)

25.129 (m ¼ 2)

90

13.132 (m ¼ 3)

13.132 (m ¼ 3)

13.132 (m ¼ 3)

13.132 (m ¼ 3)

Biaxial loading 0

10.871

10.871

10.871

10.871

15

10.332

17.660

19.017

20.103

30

10.220

24.912

27.633

29.809

45

10.854

28.044

31.227

33.774

From R.M. Jones, H.S. Morgan, J.M. Whitney, Buckling and vibration of antisymmetrically laminated angle-ply rectangular plates, Transactions of the ASME, Journal of Applied Mechanics 12 (1973) 1143e1144.

For plates having all around clamped boundary conditions, the in-plane boundary conditions (like in the case of simply supported plates) enable four types of clamping given by C1: w ¼

vw ¼ un ¼ ut ¼ 0 vn

C2: w ¼

vw ¼ Nn ¼ ut ¼ 0 vn

vw ¼ un ¼ Nnt ¼ 0 C3: w ¼ vn C4: w ¼

vw ¼ Nn ¼ Nnt ¼ 0 vn

(4.128)

Stability of composite columns and plates

179

For these boundary conditions (and also for other combinations of boundary conditions), no closed-form solutions are available. Whitney [28,29] calculated buckling loads for C1, C2, and C3 boundary conditions applicable to 45-degree angle-ply square plates using a series method [5,26].

Buckling of plates: the first-order shear deformation plate theory approach

4.5 4.5.1

Simply supported symmetric plates

Let us try to present a solution for a rectangular laminated plate, for the SS1 boundary conditions, using the Navier approach. These boundary conditions are u0 ðx; 0Þ ¼ 0; u0 ðx; bÞ ¼ 0; v0 ð0; yÞ ¼ 0; v0 ð0; bÞ ¼ 0 w0 ðx; 0Þ ¼ 0; w0 ðx; bÞ ¼ 0; w0 ð0; yÞ ¼ 0; w0 ð0; bÞ ¼ 0 fx ðx; 0Þ ¼ 0; fx ðx; bÞ ¼ 0; fy ð0; yÞ ¼ 0; fy ð0; bÞ ¼ 0

(4.129)

Nxx ð0; yÞ ¼ 0; Nxx ða; yÞ ¼ 0; Nyy ðx; 0Þ ¼ 0; Nyy ðx; bÞ ¼ 0 Mxx ð0; yÞ ¼ 0; Mxx ða; yÞ ¼ 0; Myy ðx; 0Þ ¼ 0; Myy ðx; bÞ ¼ 0 The following expressions satisfy the boundary conditions presented in Eq. (4.129) uðx; yÞ ¼ Umn cos vðx; yÞ ¼ Vmn sin

mpx npy sin a b

mpx npy cos a b

w0 ðx; yÞ ¼ Wmn sin

mpx npy sin a b

fy ðx; yÞ ¼ Emn cos

mpx npy sin a b

fx ðx; yÞ ¼ Emn sin

mpx npy cos a b

(4.130)

Substituting Eq. (4.130) into Eqs. (4.19)e(4.23) shows that the terms A45, A16, A26, B16, B26, D16, and D26 must vanish to enable a solution for the Navier problem, which means that the assumed trigonometric terms cancel out leaving only constants. Therefore, the equations should be written as (after neglecting higher order terms and the external load per unit area, q, and dropping the time derivatives)

180

Stability and Vibrations of Thin-Walled Composite Structures


2 
2 
2  v2 f y v u0 v v0 v u0 v2 v 0 v2 f x þ þ B þ A þ A þ B 12 11 12 66 vx2 vxvy vy2 vxvy vx2 vxvy ! v2 fx v2 fy þ B66 þ ¼0 vy2 vxvy

(4.131)


2 
2 
2  v2 f y v v0 v u0 v u0 v2 v 0 v2 f x þ þ B A22 þ A þ B þ A 12 12 22 66 vy2 vxvy vxvy vx2 vxvy vy2 ! v2 fx v2 fy þ 2 ¼0 þ B66 vxvy vx

(4.132)

A11


KA44

v2 w0 vfy þ vy2 vy


B11

þ KA55



v2 w0 vfx þ vx2 vx

 þ



  v vw0 v vw0 Nxx Nyy þ ¼0 vx vy vx vy (4.133)


2   v2 f y v2 v 0 v u0 v2 v 0 v2 f x þ þ D þ B66 þ D 11 12 vxvy vxvy vxvy vy2 vx2 !
 2 v2 fx v fy vw0 þ fx ¼ 0 þ  KA55 vxvy vx vy2

v2 u0 vx2

þ D66









þ B12

(4.134)



2 
2   v2 f y v2 u0 v v0 v v 0 v2 u0 v2 f x þ D þ þ B þ B22 þ D 12 22 66 vxvy vxvy vy2 vx2 vxvy vy2 !
 v2 fy v2 fx vw0 þ fy ¼ 0 þ D66 þ  KA44 vxvy vy vx2 B12

(4.135) Following Ref. [6], the results of the substitution for constant in-plane loads can be shown in the following matrix form: 2 6a 6 b 11 6 6a 6 b 21 6 6 0 6 6a 6 b 41 6 4a b 51

b 12 a

0

b 14 a

b 22 a

0

b 24 a

0

b 33 a

b 34 a

b 42 a

b 43 a

b 44 a

b 52 a

b 53 a

b 54 a

3 8 9 8 9 0 > > U > > > > > > b 15 7 a 7> > > > > mn > > > > > > 7> > > > > 0 > > > > V b 25 7 a > mn = > > = < > 7< 7 b 34 7 Wmn ¼ 0 a > > > 7> > > > > >E > > > > > b 45 7 a mn > > > > > 7> 0 > > > > > > > 7> > > > > b 55 5: Emn ; : ; a 0

(4.136)

Stability of composite columns and plates

181

b ij , are given in Appendix D. To obtain a unique where the terms of the matrix, a solution the determinant in Eq. (4.136) should vanish, yielding the buckling loads per unit width, according to the case selected. Of course, for a symmetric laminate the term Bij would be identically zero, leading to the uncoupling of the in-plane and out-ofplane displacements and the determinant would be 3  3, reducing drastically the calculation of the buckling loads. Closed-form solutions for other boundary conditions are based on the Lévy approach, i.e., two opposite edges are on simply supported boundary conditions, whereas the other two can have any boundary conditions. According to Refs. [30,31], for such cases the assumed expressions for the various displacements for the symmetric case are written as w0 ðx; yÞ ¼ Wmn ðyÞsin

mpx a

fy ðx; yÞ ¼ Emn ðyÞcos

mpx a

fx ðx; yÞ ¼ Emn ðyÞsin

mpx a

(4.137)

The expressions in Eq. (4.137) satisfy the following boundary conditions (rectangular plate): At x ¼ 0; a:

w0 ¼ Mxx ¼ fy ¼ 0

(4.138)

Substituting Eq. (4.137) into the equations of motion for the FSDPT (Eqs. 4.19e4.23) yields the required buckling solution. In Ref. [30], a solution for antisymmetric angle-ply laminated rectangular plate is presented for a Lévy type beam. The assumed displacements are u0 ðx; yÞ ¼ Umn ðyÞsin

mpx a

V0 ðx; yÞ ¼ Vmn ðyÞcos

mpx a

w0 ðx; yÞ ¼ Wmn ðyÞsin

mpx a

fy ðx; yÞ ¼ Emn ðyÞcos

mpx a

fx ðx; yÞ ¼ Emn ðyÞsin

mpx a

(4.139)

182

Stability and Vibrations of Thin-Walled Composite Structures

and the boundary conditions investigated, using a coordinate system located at midwidth of the plate, were At x ¼ 0; a Simply supported: u0 ¼ w0 ¼ fy ¼ 0
 vfy vfx vu0 vv0 Mxx h D11 þ D12 þ B16 þ ¼0 vx vy vy vx 
vfy vfx vu vv þ þ B26 þ A66 ¼0 Nxy h B16 vy vx vx vy At y ¼ b=2 Simply supported: v0 ¼ w0 ¼ fx
 vfy vfx vu0 vv0 þ D22 þ B26 þ Myy h D12 ¼ Nxy ¼ 0 vx vy vy vx

(4.140)

or Clamped: u0 ¼ v0 ¼ w0 ¼ fx ¼ fy ¼ 0 or




 vw0 Free : Myy ¼ Nxy ¼ Qyy h KA44 þ fy ¼ 0 vx
 vu0 vv0 vfx vfy þ B26 þ D66 þ Mxy h B16 ¼0 vx vy vy vx
 vu0 vv0 vfx vfy þ A22 þ B26 þ Nyy h A12 ¼0 vx vy vy vx Substitution of the expressions of the displacements in the equations of motion yields the relevant buckling solutions.

References [1] S. Timoshenko, S. Woinowsky-Krieger, Theory of plates and shells, McGraw-Hill New York (1959) 580. [2] E. Reissner, Y. Stavsky, Bending and stretching of certain types of heterogeneous aelotropic elastic plates, Journal of Applied Mechanics 28 (3) (1961) 402e408. [3] S.B. Dong, K.S. Pister, R.L. Taylor, On the theory of laminated anisotropic shells and plates, Journal of Aerospace Sciences 29 (8) (1962) 969e975.

Stability of composite columns and plates

183

[4] J.M. Whitney, Cylindrical bending of unsymmetrically laminated plates, Journal of Composite Materials 3 (4) (1969) 715e719. [5] J.M. Whitney, A.W. Leissa, Analysis of heterogeneous anisotropic plates, Transactions of the ASME, Journal of Applied Mechanics 36 (2) (1969) 261e266. [6] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells e Theory and Analysis, second ed., CRC Press, 2004, 831 pp. [7] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics 12 (2) (1945) A69eA77. [8] R.D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, ASME Journal of Applied Mechanics 18 (1) (1951) 31e38. [9] J.M. Whitney, The effect of transverse shear deformation in the bending of laminated plates, Journal of Composite Materials 3 (4) (1969) 534e547. [10] J.M. Whitney, N.J. Pagano, Shear deformation in heterogeneous anisotropic plates, Journal of Applied Mechanics 37 (4) (1970) 1031e1036. [11] J.M. Whitney, Shear correction factors for orthotropic laminates under static load, Journal of Applied Mechanics 40 (1) (1973) 302e304. [12] E. Reissner, Note on the effect of transverse shear deformation in laminated anisotropic plates, Computer Methods in Applied Mechanics and Engineering 20 (1979) 203e209. [13] J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, second ed., John Wiley, New York, 2002, 608 pp. [14] S.P. Timoshenko, On the correction factor for shear of the differential equation for transverse vibrations of bars of uniform cross-section, Philosophical Magazine 41 (245) (1921) 744e746. [15] S.P. Timoshenko, On the transverse vibrations of bars of uniform cross-section, Philosophical Magazine 43 (253) (1922) 125e131. [16] G.J. Simitses, An Introduction to the Elastic Stability of Structures, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976, 253 pp. [17] H. Abramovich, O. Hamburger, Vibration of a uniform cantilever Timoshenko beam with translational and rotational springs and with a tip mass, Journal of Sound and Vibration 154 (1) (1992) 67e80. [18] H. Abramovich, A note on experimental investigation on a vibrating Timoshenko cantilever beam, Journal of Sound and Vibration 160 (1) (1993) 167e171. [19] H. Abramovich, A. Livshits, Dynamic behavior of cross-ply laminated beams with piezoelectric layers, Composite Structures 25 (1e4) (1993) 371e379. [20] H. Abramovich, A. Livshits, Free vibrations of non-symmetric cross-ply laminated composite beams, Journal of Sound and Vibration 176 (5) (1994) 597e612. [21] H. Abramovich, Thermal buckling of cross-ply composite laminates using a first-order shear deformation theory, Composites Structures 28 (1994) 201e213. [22] H. Abramovich, Deflection control of laminated composite beams with piezoceramic layers- closed form solutions, Composite Structures 43 (1998) 217e231. [23] A.W. Leissa, Buckling of Laminated Composite Plates and Shell Panels, AFWALTR-85e3069, June 1985. [24] S. Bergmann, H. Reissner, Neuere probleme aus der flugzeugstatik uber die knickung von wellblechstreifen bei schubbeanspruchung, Zeitschrift fur Flugtechnik und Motorluftschiffahrt (Z.F.M.) 20 (18) (September 1929) 475e481. [25] R.M. Jones, Buckling and vibration of unsymmetrically laminated cross-ply rectangular plates, AIAA Journal 11 (12) (1973) 1626e1632.

184

Stability and Vibrations of Thin-Walled Composite Structures

[26] J.M. Whitney, A Study of the Effects of Coupling Between Bending and Stretching on the Mechanical Behavior of Layered Anisotropic Composite Materials (Ph.D. dissertation), Ohio State University, 1968. Also Tech. Rept. AFML-TR-68-330, 80 pp, April 1969. [27] R.M. Jones, H.S. Morgan, J.M. Whitney, Buckling and vibration of antisymmetrically laminated angle-ply rectangular plates, Transactions of the ASME, Journal of Applied Mechanics 12 (1973) 1143e1144. [28] J.M. Whitney, Bending, Vibrations, and Buckling of Laminated Anisotropic Rectangular Plates, Wright Patterson AFBML, Ohio, Technical Rept. AFML-TR-70-75, 35 pp, August 1970. [29] J.M. Whitney, The effect of boundary conditions of the response of laminated composites, Journal of Composite Materials 4 (1970) 192e203. [30] A.A. Khdeir, Stability of antisymmetric angle-ply laminated plates, Journal of Engineering mechanics 115 (5) (May 1989) 952e962. [31] J.N. Reddy, A.A. Khdeir, L. Librescu, Lévy type solutions for symmetrically laminated rectangular plates using first order shear deformation theory, Journal of Applied Mechanics 54 (September 1987) 740e742. [32] H. Abramovich, M. Eisenberger, O. Shulepov, Vibrations and buckling of cross-ply nonsymmetric laminated composite beams, AIAA Journal 34 (5) (May 1996) 1064e1069.

Further reading [1] H. Abramovich, Shear deformation and rotary inertia effects of vibrating composite beams, Composite Structures 20 (1992) 165e173.

Appendices A.

Nonsymmetric laminated composite beam: the classical lamination theory approach

Neglecting high-order terms in Eqs. (4.27) and (4.28) leads to the following two coupled equations A11

d 2 u0 d 3 w0  B ¼0 11 dx2 dx3

 B11

d 3 u0 d 4 w0 d2 w0 þ D þ N ¼0 xx 11 dx3 dx4 dx2

(A.1)

(A.2)

Derivation with respect to x of Eq. (A.1) and substituting in Eq. (A.2) leads to the following decoupled equation: 

B11 d 4 w0 d 4 w0 d2 w0 þ D þ N ¼0 xx 11 A11 dx4 dx4 dx2

(A.3)

Stability of composite columns and plates

185

The solution of Eq. (A.3) has the following form: _  _  w0 ðxÞ ¼ C1 sin l x þ C2 cos l x þ C3 x þ C4

(A.4)

where _2

N xx l ¼   D11  B211 A

(A.5)

11

Assuming that the axial displacement u0(x) has the same form as the lateral displacement (see, for example, Refs. [19e22,32]), namely, _  _  u0 ðxÞ ¼ C5 sin l x þ C6 cos l x þ C7 x þ C8

(A.6)

The four constants in Eq. (A.6) are connected to the four constants in Eq. (A.4). By substituting Eqs. (A.4) and (A.6) in Eq. (A.1), we get the following relationships: B11 _ C5 ¼  $ l $C2 A11

(A.7)

B11 _ C6 ¼ þ $ l $C1 A11

The rest of the constants, C1, C2, C3, C4, C7, and C8 will be determined using the six boundary conditions of the problem. For the case of simply supported at both ends of the column, the out-of-plane boundary conditions have the following expressions du0 ð0Þ d2 w0 ð0Þ þ D11 ¼0 dx dx2

(A.8)

du0 ðLÞ d2 w0 ðLÞ þ D11 ¼0 dx dx2

(A.9)

w0 ð0Þ ¼ 0;

Mxx ð0Þ ¼ 0 0  B11

w0 ðLÞ ¼ 0;

Mxx ðLÞ ¼ 0 0  B11

and the in-plane boundary conditions can be written as (assuming that at x ¼ 0 there is no axial displacement, whereas at the other end of the beam, at x ¼ L, there is a compression load) u0 ð0Þ ¼ 0 A11

du0 ðLÞ d2 w0 ðLÞ  B11 ¼ N xx dx dx2

(A.10) (A.11)

186

Stability and Vibrations of Thin-Walled Composite Structures

Applying the four out-of-plane boundary conditions (Eqs. A.8 and A.9) leads to the following four equations: C2 þ C4 ¼ 0 _2 B211 _2 $ l $C2  B11 $C7  D11 $ l $C2 ¼ 0 A11 _  _  C1 sin l L þ C2 cos l L þ C3 x þ C4 ¼ 0



(A.12)

_  B2 _2 _  B211 _2 $ l $C2 $cos l L  11 $ l $C1 $sin l L þ B11 $C7 A11 A11

_  _  _2 _2 þD11 $ l $C1 $sin l L þ D11 $ l $C2 $cos l L ¼ 0 Applying the two in-plane boundary conditions (Eqs. A.10 and A.11) yields B11 _ $ l $C1 þ C8 ¼ 0 A11 _  _  _2 _2 B11 $ l $C2 $cos l L  B11 $ l $C1 $sin l L þ A11 $C7 _  _  _2 _2 N xx þB11 $ l $C1 $sin l L þ B11 $ l $C2 $cos l L ¼ N xx 0 C7 ¼  A11 (A.13) Solving for the six constants we get the following results: C1 ¼

B11 A11

_  cos l L  1 _  ; sin l L

C2 ¼

B11 ; A11

C3 ¼ 0

B11 C4 ¼  ; A11

C7 ¼ 

N xx ; A11

_ B2 11 A211

C8 ¼  l

_  cos l L  1 . _  sin l L

(A.14)

To obtain the buckling load, one should demand that w0(x) /N, i.e., finding when one of the constants would tend to infinity. Observing the constant C1 leads to the following eigenvalue solution _  _ sin l L ¼ 0 0 l L ¼ np;

n ¼ 1; 2; 3; 4.

(A.15)

Stability of composite columns and plates

187

which leads to the critical buckling load per unit width, b, having the following form: 

N xx



¼ cr


 B211 p2  D 11 L2 A11

(A.16)

  case. The shape of the buckling will be sin px L like for the symmetric   One should note that the critical buckling load per unit width N xx cr is influenced by the coupling coefficient B11, showing that for nonsymmetric cases the buckling load will be lower than that for symmetric cases having the same number of laminates. For the clampedeclamped case the involved boundary conditions are w0 ð0Þ ¼ 0;

dw0 ð0Þ ¼0 dx

(A.17)

w0 ðLÞ ¼ 0;

dw0 ðLÞ ¼0 dx

(A.18)

u0 ð0Þ ¼ 0;

A11

du0 ðLÞ d 2 w0 ðLÞ  B11 ¼ N xx dx dx2

(A.19)

Applying the boundary conditions yields the following characteristic equation: _  _  h _ i _ l L $sin l L ¼ 2 1  cos l L 0 l L ¼ 2p; 8:987; 4p.

(A.20)

with C1 ¼ C3 ¼ C4 ¼ C8 ¼ 0, C2 s 0, and C7 ¼ NA11xx . The critical buckling load per unit width can then be written as 

N xx




 B211 4p2 ¼ 2 D11  cr L A11

The buckling shape can be written as 


2px 1 w0 ðxÞ ¼ C2 cos L

(A.21)

(A.22)

One should note that for the clampedeclamped case, which involves only outof-plane geometric boundary conditions (see Eqs. A.19 and A.20), both displacements contain an undetermined term (C2), like in the case of symmetric laminate.

B. Energy terms for plates using the classical lamination theory approach According to Ref. [23], the total energy of a general orthotropic plate in buckling can be written as U ¼ Vbend þ Vstretch þ Vbendestretch þ Vload

(B.1)

188

Stability and Vibrations of Thin-Walled Composite Structures

where Vbend is the plate’s internal strain energy due to bending presented by Vbend

1 ¼ 2

ZZ " S


2 2
2 2 v w0 v2 w0 v2 w0 v w0 D11 þ 2D12 2 þ D22 vx2 vx vy2 vy2
2 2 # ZZ v w0 1 v2 w0 v2 w0 þ 4D66 4D16 2 dxdy þ 2 S vxvy vx vxvy þ 4D26

(B.2)

v2 w0 v2 w0 dxdy vy2 vxvy

Vstretch is the plate’s internal strain energy due to in-plane stretching given by Vstretch

 #
2
2
vu vu vv vv vu vv 2 þ D22 þ þ 2A12 þ A66 A11 dxdy vx vx vy vy vy vx S



ZZ 1 vu vu vv vv vu vv þ þ þ þ 2A26 dxdy 2A16 2 S vx vy vx vy vy vx (B.3)

1 ¼ 2

ZZ "

Vbendstretch is the plate’s internal strain energy due to bendingestretching coupling presented by

Vbendstretch

ZZ "


 vu v2 w0 vv v2 w0 vu v2 w0 þ 2B12 þ B11 vx vx2 vy vx2 vx vy2 S # 
2 vv v2 w0 vu vv v2 w0 þ y þ 4B66 þ B22 dxdy vy vy2 vx vx vxvy
 ZZ 1 vu v2 w0 vv v2 w0 vu v2 w0 þ þ2 2B16  2 S vy vx2 vx vx2 vx vxvy


2 2 2 vu v w0 vv v w0 vv v w0 þ 2B26 þ þ 2 dxdy vy vy2 vx vy2 vy vxvy

1 ¼ 2

(B.4)

and Vload is the negative work of the in-plane forces during buckling given by the following expression

Vload

1 ¼ 2


  # ZZ "
vw0 2 vw0 vw0 vw0 2 þ N yy N xx þ 2N xy dxdy vx vx vy vy S

(B.5)

For a specially orthotropic plate, Eqs. (B.1)e(B.5) transform into U ¼ Vbend þ Vload

(B.6)

Stability of composite columns and plates

Vbend

1 ¼ 2

ZZ "


D11

S

v2 w0 vx2


þ D22

Vload

1 ¼ 2

ZZ " S

189

2

v w0 vy2 2

þ 2D12

v2 w0 v2 w0 vx2 vy2

2


þ 4D66

v w0 vxvy 2

2 #

(B.7) dxdy



  # vw0 2 vw0 vw0 vw0 2 þ N yy N xx þ 2N xy dxdy vx vx vy vy

(B.8)

The term S is the area of the plate (a  b).

C. Matrix notation for the equilibrium equations using the classical lamination theory approach A convenient way of presenting the equilibrium equations at buckling is the following matrix form: 2

a11

6 6 a21 4 a31

a12 a22 a32

9 8 9 38 uðx; yÞ > >0> > > > = < > = > 7< 7 ¼ 0 a23 5 vðx; yÞ > > > > > > ; : > : ; > 0 w0 ðx; yÞ ½a33  N a13

(C.1)

where the various operators are given by a11 h A11

v2 v2 v2 þ A þ 2A 16 66 vxvy vx2 vy2

a22 h A22

v2 v2 v2 þ A þ 2A 26 66 2 vxvy vy2 vx

a33 h D11

v4 v4 v4 v4 v4 þ 2ðD þ 4D þ 2D Þ þ 4D þ D 12 22 16 66 26 vx4 vx3 vy vx2 vy2 vxvy3 vy4

a12 ¼ a21 h A16

v2 v2 v2 þ A þ ðA þ A Þ 12 66 26 2 vxvy vx2 vy

a13 ¼ a31 h  B11

v3 v3 v3 v3  ðB  3B þ 2B Þ  B 12 16 66 26 vx3 vx2 vy vxvy2 vy3

a23 ¼ a32 h  B16

v3 v3 v3 v3  3B  ðB þ 2B Þ  B 12 22 3 66 26 vx3 vx2 vy vxvy2 vy

N h N xx

v2 v2 v2 þ þ 2N N xy yy vxvy vx2 vy2 (C.2)

190

D.

Stability and Vibrations of Thin-Walled Composite Structures

The terms of the matrix notation for the equilibrium equations using the first-order shear deformation plate theory approach 2 6a 6 b 11 6 6a 6 b 21 6 6 0 6 6a 6 b 41 6 4a b 51

b 12 a

0

b 14 a

b 22 a

0

b 24 a

0

b 33 a

b 34 a

b 42 a

b 43 a

b 44 a

b 52 a

b 53 a

b 54 a

3 8 9 8 9 0> > > > U > > > > b 15 7 a mn 7> > > > > > > > > > > 7> > > > > 0 > > > > V b 25 7 a mn > > > = = < > 7< 7 b 34 7 Wmn ¼ 0 a > 7> > > > > > > > > >0> > Emn > > > b 45 7 a > > > 7> > > > > > > > 7> > > > > ; : : ; b 55 5 Emn a 0

(D.1)

where b 11 h A11 a

mp2 a

b 14 ¼ a b 41 h B11 a b 22 h A66 a

np2 b

b 21 h ðA12 þ A66 Þ b 12 ¼ a a

mpnp a

b

mpnp mp2 np2 b 51 h ðB12 þ B66 Þ b 15 ¼ a þ B66 a a b a b

mp2 a

þ A66

þ A22

np2 b

b 42 h ðB12 þ B66 Þ b 24 ¼ a a

mpnp a

b

mp2 np2 þ B22 a b   mp2

mp2 np2 np 2 b 33 h K A44 þ A55  N yy a  N xx b a a b b 25 ¼ a b 52 h B66 a

b 43 h KA55 b 34 ¼ a a b 44 h D11 a

mp a

b 53 h KA44 b 35 ¼ a a

np b

mp2 np2 þ D66 þ KA55 a b

b 54 h ðD12 þ D66 Þ b 45 ¼ a a

mpnp a

b

b 55 h D22 a

np2 b

þ D66

mp2 a

þ KA44 (D.2)

Vibration of composite columns and plates

5

Haim Abramovich Technion, I.I.T., Haifa, Israel

5.1

Introduction

This chapter is complementary to Chapter 4, which dealt with stability issues for composite columns and plates. It is aimed at presenting the vibration analysis of composite columns and plates based on the classical lamination theory (CLT) and the first-order shear deformation theory (FSDT) developed in Chapter 3. Some typical examples dealing with one-dimensional behavior characteristic to beams are given in Refs. [1e16].

5.1.1

The classical lamination plate theory approach

As described in the previous chapter, the classical lamination plate theory (CLPT) is an extension of the well-known KirchhoffeLove classical plate theory that is applied to laminated composite plates. The equations of motion are
 vNxx vNxy v2 u0 v2 vw0 þ ¼ I0 2  I1 2 vx vy vx vt vt
 vNxy vNyy v2 v 0 v2 vw0 þ ¼ I0 2  I1 2 vx vy vy vt vt
 v2 Mxy v2 Myy v v2 Mxx vw0 vw0 þ þ N þ 2 þ N xx xy vx vxvy vx vy vx2 vy2 þ



  v vw0 vw0 v2 w 0 v2 vu0 vv0 þ Nyy þ þ I Nxy ¼ q þ I0 1 vy vx vy vy vt 2 vt 2 vx


 v2 v2 w0 v2 w0  I2 2 þ 2 vt vx2 vy

Stability and Vibrations of Thin-Walled Composite Structures. http://dx.doi.org/10.1016/B978-0-08-100410-4.00005-3 Copyright © 2017 Elsevier Ltd. All rights reserved.

(5.1)

192

Stability and Vibrations of Thin-Walled Composite Structures

where q is the distributed pressure on the surface of the plate and the mass moments of inertia I0, I1, and I2 are defined as 8 9 8 9 I0 > > >1> > > Z < = < > = þh=2 > z $r0 $dz I1 ¼ (5.2) > > > h=2 > > > : > ; : 2> ; z I2 where h is the total thickness of the plate and r0 is the relevant density. In Eq. (5.1), Nxx, Nxy, and Nyy are the force resultants per unit length and Mxx, Mxy, and Myy are the moment resultants per unit length defined as 9 8 9 8 sxx > Nxx > > > > > > > = Z þh=2 < = < Nyy ¼ syy dz; > > > h=2 > > > > > ; : ; : Nxy sxy

8 9 8 9 Mxx > sxx > > > > > > > Z þh=2 < < = = Myy ¼  syy $z$dz. > > > h=2 > > > > > : ; : ; Mxy sxy

(5.3)

In Eq. (5.3), z is a coordinate normal to the surface of the plate and sxx and syy are the normal stresses in the x and y directions, respectively, while sxy is the shear stress. The stress resultants Nxx, Nxy, and Nyy and Mxx, Mxy, and Myy, can be defined using the assumed displacements (see Eq. 4.1 in Chapter 4) to yield 8 9
2 > > > > vu 1 vw 0 0 > > > > þ > > > > > > vx 2 vx 8 9 2 > 3> > > > > N A A A xx 11 12 16 > > > > > > > > > > > >
 < = 6 7< 2 = 6 7 vv0 1 vw0 Nyy ¼6 A12 A22 A26 7 þ > > > 4 5> > vy 2 vy > > > > > > > > > : ; > > > Nxy A16 A26 A66 > > > > > > > 2 > > vu vv v w > > 0 0 0 > > þ þ > > : vy vx vxvy ; 8 9 2 > > v w > > 0 > > > > > > 2 > vx > > 2 3> > > > B11 B12 B16 > > > > > > = 6 7< v2 w > 6 7 0  6 B12 B22 B26 7 4 5> vy2 > > > > > > > > > > B16 B26 B66 > > > > > 2 > > > > v w 0 > > > > :2 ; vxvy

(5.4)

Vibration of composite columns and plates

8 9 2 Mxx > B11 > > > > > < = 6 6 Myy ¼6 B12 > > 4 > > > > : ; Mxy B16

B12 B22 B26

2

193

9 8
2 > > > > vu 1 vw > > 0 0 > > þ > > > > > > 2 vx vx > > 3> > > B16 > > > > > >
2 > = 7< 7 vv0 1 vw0 B26 7 þ > 5> vy 2 vy > > > > > > > > > B66 > > > > > > > 2 > vu vv v w > > 0 0 0> > > þ þ > > : vy vx vxvy ;

D11 6 6  6 D12 4

D12

D16

D26

D22

8 9 2 > > v w > > 0 > > > > > 2 > > > vx > 3> > > > > D16 > > > > > = 7< v2 w > 7 0 D26 7 5> vy2 > > > > > > > > > > D66 > > > > > 2 > > > v w 0> > > > > :2 ; vxvy

(5.5)

where Aij ¼

N X

ðkÞ

Qij ðzkþ1  zk Þ;

k¼1

Bij ¼

N  1X ðkÞ  Q z2  z2k ; 2 k¼1 ij kþ1

N  1X ðkÞ  Dij ¼ Qij z3kþ1  z3k 3 k¼1

(5.6)

ðkÞ

with Qij being the lamina stiffness after transformation. Substituting Eq. (5.5) into Eq. (5.1) provides the equations of motion for a laminated composite plate expressed by the three assumed displacements (u0, v0, and w0):



2 
2  v2 u0 v3 w0 v v0 v3 w0 v u0 v3 w0 v2 v0 þ þ 3 þ þ þ A þ A 2 12 16 vxvy vxvy2 vxvy vx2 vx3 vx2 vy vx2
2 
2  v v 0 v3 w 0 v u0 v2 v 0 v3 w0 v3 w0 v3 w0 þ A26 þ 2 þ þ  B þ A  B 11 12 66 vxvy vy2 vy3 vy2 vxvy2 vx3 vxvy2 A11

 3B16

v3 w0 v3 w 0 v3 w 0 v2 u0 v3 w 0  B  2B ¼ I  I 0 1 26 66 vx2 vy vy3 vxvy2 vt 2 vxvt 2 (5.7)

194

Stability and Vibrations of Thin-Walled Composite Structures





2
2   v2 v 0 v3 w 0 v u0 v3 w0 v u0 v3 w 0 þ þ þ þ A þ A 12 16 vxvy vx2 vy vy2 vy3 vx2 vx3
2 
2  v u0 v2 v 0 v3 w0 v u0 v2 v 0 v3 w 0 v3 w 0 þ A26 þ 3 þ þ 2 þ 2 þ A  B12 2 66 2 2 2 2 vxvy vxvy vx vy vxvy vx vy vx vy

A22

 B22

v3 w0 v3 w 0 v3 w 0 v3 w 0 v2 v 0 v3 w0 ¼ I  B  3B  2B  I 0 1 16 26 66 vy3 vx3 vxvy2 vx2 vy vt 2 vyvt 2 (5.8)


3 
3  v u0 v4 w 0 v v0 v3 u0 v3 w 0 þ þ 4 þ B12 þ4 2 2 B11 vx3 vx vx2 vy vxvy2 vx vy
3  v u0 v 3 v 0 v4 w0 þ B16 3 2 þ 3 þ 8 3 vx vy vx vx vy
3 
3  v v0 v4 w0 v u0 v3 v 0 v4 w0 þ B22 þ 2 4 þ B26 þ3 þ8 vy3 vy vy3 vxvy2 vxvy3
þ 2B66

 D11

v3 u0 v3 v 0 v4 w 0 þ 4 þ vxvy2 vx2 vy vx2 vy2



v4 w0 v4 w 0 v4 w0 v4 w0  2D  D  4D 12 22 16 vx4 vx2 vy2 vy4 vx3 vy

 4D26

v4 w0 v4 w 0  4D þ Pðw0 Þ 66 vxvy3 vx2 vy2



 v2 w 0 v2 v2 w0 v2 w0 v2 vu0 vv0 þ  I2 2 þ 2 þ I1 2 ¼ q þ I0 vx vy vt 2 vt vx2 vy vt (5.9) where

  v vw0 vw0 v vw0 vw0 þ Nxy þ Nyy Nxx Nxy Pðw0 Þ ¼ þ vx vy vx vy vx vy

(5.10)

Vibration of composite columns and plates

195

The first-order shear deformation plate theory approach

5.1.2

The first-order shear deformation plate theory (FSDPT), described in detail in Chapter 4, is an extension of the well-known Timoshenko beam theory and/or the Mindline Reissner plate theory that is applied to laminated composite plates. The equations of motion in terms of displacements for the FSDPT approach can be written as
A11

v2 u0 v3 w0 þ 3 vx2 vx






v2 v0 v3 w0 þ vxvy vxvy2

þ A12




þ A16

v2 u0 v3 w0 v2 v0 þ3 2 þ 2 2 vxvy vx vy vx




2 
2  v2 fy v v 0 v3 w 0 v u0 v2 v 0 v3 w0 v2 fx þ 2 þ þ þ B þ A þ B 11 12 66 vxvy vxvy vy2 vy3 vy2 vxvy2 vx2 ! ! 2 2 v2 fy v2 fx v fy v2 fx v fy v2 u0 v2 f x þ 2 þ B26 þ B þ þ I þ B16 2 ¼ I 0 1 66 vxvy vxvy vx vy2 vy2 vt 2 vt 2

þ A26

(5.11)
A22

v2 v0 v3 w0 þ 3 vy2 vy




þ A12


2   v2 u0 v3 w0 v u0 v3 w0 þ þ 3 þ A16 vxvy vx2 vy vx2 vx


2 
2  v u0 v2 v 0 v3 w0 v u0 v2 v 0 v3 w 0 v 2 fx þ A26 þ 3 þ þ 2 þ 2 þ A þ B 12 66 vxvy vxvy vx2 vxvy vy2 vxvy2 vx2 vy þ B22

v2 fy

¼ I0

vy2

þ B16

v2 fy v2 fx v2 fx þ 2 þ B 2 26 vxvy vx2 vy

! þ B66

v2 fx v2 fy þ 2 vxvy vx

!

v2 fy v2 v 0 þ I 1 vt 2 vt 2 (5.12)


2
2
2    v w0 vfy v w0 vfy v w0 vfx þ þ þ KA44 þ KA45 þ KA45 vy2 vy vxvy vx vxvy vy
2  v w0 vfx v2 w 0 þ Þ ¼ q þ I þ KA55 þ Pðw 0 0 vx2 vx vt 2

(5.13)

196

Stability and Vibrations of Thin-Walled Composite Structures

! !
2  v u0 v3 w0 v2 v 0 v3 w 0 v2 u0 v2 v 0 v4 w0 þ þ B11 þ 3 þ B12 þ3 3 þ B16 2 vxvy vxvy2 vxvy vx2 vx2 vx vx vy !

þ B26

v2 v 0 v3 w 0 þ 3 vy2 vy

þ D16

2 v2 fx v fy þ 2 2 vxvy vx


 KA45

vw0 þ fy vy

þ B66

v2 u0 v2 v 0 v3 w0 þ 2 þ vxvy vy2 vxvy2

! þ D26

 ¼ I2

v2 f y vy2

þ D66

! þ D11

2 v2 fx v fy þ vxvy vy2

!

v2 fy v2 fx þ D 12 vxvy vx2


 vw0 þ fx  KA55 vx

v2 fx v2 u0 þ I 1 vt 2 vt 2 (5.14)





2  v2 u0 v3 w 0 v v0 v3 w0 þ 3 þ B22 þ 3 B12 þ B16 vx2 vx vy2 vy
2 
2  v u0 v2 v 0 v3 w 0 v v 0 v2 u0 v3 w0 þ 3 þ 2 þ B26 þ 2 þ þ B 66 vxvy vy2 vxvy2 vx2 vxvy vxvy2 ! ! v2 fy v2 f y v2 fy v2 fx v2 fx v2 fx v2 f x þ D22 þ D16 þ D26 þ2 þ þ D12 þ D66 vxvy vxvy vxvy vy2 vx2 vy2 vx2 v2 u0 v3 w0 þ vxvy vxvy2


 KA45



vw0 þ fx vx







 KA44

vw0 þ fy vy

 ¼ I2

v2 fy vt 2

þ I1

v2 v 0 vt 2 (5.15)

where P(w0) is given by Eq. (5.10); u0, v0, and w0 are the displacements in the directions x, y, and z, respectively; fx, fy are rotations about the x and y axes, respectively; and the various terms Aij, Bij, and Dij are defined in detail in Chapter 4.

5.2

Vibrations of columns: the classical lamination theory approach

The equations of motion for calculating a general laminate column can be derived from Eq. (5.1) by assuming that the displacement of the model is only a function of the x coordinate, yielding the following equations (v0 is assumed to be zero):
2  v u0 v3 w0 v3 w 0 v2 u0 v3 w 0 A11 þ ¼ I  I  B 11 0 1 vx2 vx3 vx3 vt 2 vxvt 2

(5.16)

Vibration of composite columns and plates


B11

v3 u0 v4 w0 þ 4 vx3 vx

  D11

where Pðw0 Þ ¼

197


 v4 w0 v2 w 0 v2 v2 w 0 ¼ q þ Pðw Þ þ I  I 0 0 2 vx4 vt 2 vt 2 vx2
 v2 vu0 þ I1 2 vt vx (5.17)


 v vw0 Nxx vx vx

8 9 8 9 I0 > >1> > > > Z < > = < = þh=2 > z $r0 $dz I1 ¼ > > > h=2 > > > ; : 2> ; : > z I2

(5.18)

(5.19)

and h is the total thickness of the plate and r0 is the relevant density. To analyze the vibration problem of the columns, the lateral load q will be assumed to be zero, whereas the axial compressive load, Pðw0 Þ, will be assumed to be nonzero. This leads to the following coupled equations:
2 
 v u0 v3 w0 v3 w 0 v2 u0 v2 vw0 A11 þ 3  B11 3 ¼ I0 2  I1 2 (5.20) vx2 vx vx vt vt vx




v4 w0 vx4

 v2 w0 v2 v2 w0 v2 vu0 ¼ Pðw0 Þ þ I0 2  I2 2 þ I1 2 vt vt vx2 vt vx

B11

5.2.1

v3 u0 v4 w0 þ 4 vx3 vx

 D11

(5.21)

Symmetric laminate (B11 ¼ 0, I1 ¼ 0)

In the case of symmetric laminate B11 ¼ 0 and I1 ¼ 0 the following single equation is v3 w0 obtained, whereas Eq. (5.20) (assuming the term is neglected) is solved vx3 independently:
 v4 w0 v2 w0 v2 w0 v2 v2 w0 ¼ I0  I2 2  D11 4  N xx (5.22) vx vx2 vt 2 vt vx2 Assuming that w0(x, t) ¼ W(x)eiut, where u is the natural frequency, and substituting it in Eq. (5.22) yields the following differential equation:

D11

 d2 W d4 W  2 þ N þ u I  u2 I0 $W ¼ 0 xx 2 dx4 dx2

(5.23)

198

Stability and Vibrations of Thin-Walled Composite Structures

The solution of Eq. (5.23) has the following general form: WðxÞ ¼ A1 sinh ða1 xÞ þ A2 cosh ða1 xÞ þ A3 sin ða2 xÞ þ A4 cos ða2 xÞ

(5.24)

where A1, A2, A3, and A4 are constants to be determined using boundary conditions and the terms a1 and a2 are defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u    2 u 2 N xx þ u2 I2 þ 4D11 u2 I0 t Nxx þ u I2 þ a1 ¼ 2D11 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u   2 u 2 N xx þ u2 I2 þ 4D11 u2 I0 t Nxx þ u I2 þ a2 ¼ 2D11

(5.25)

To investigate the influence of the compression load, N xx , and the rotary inertia, I2, on the natural frequency, the natural frequency squared (u2) is rewritten using Eq. (5.25) to yield

u2 ¼

u2 ¼

D11 a41 þ N xx a21 D11 ¼ a41 2 I0 I0  I2 a1

D11 a42  N xx a22 D11 ¼ a42 2 I0 I0 þ I2 a2

! N xx 1þ D11 a21 
I2 1  a21 I0 ! N xx 1 D11 a22 
I2 2 1 þ a2 I0

(5.26)

The expressions in Eq. (5.26) are the same and one of them can be used once, for instance, a2 is known. Also, it is clear from Eq. (5.26) (the second expression) that the compressive load N xx reduces the natural frequency, and the same tendency is obtained by including the rotary inertia, I2. Assuming that N xx ¼ I2 ¼ 0, one obtains the known frequency equation used by the BernoullieEuler theory, namely, u2 ¼ a42

D11 I0

(5.27)

Vibration of composite columns and plates

199

Let us show the application of boundary conditions in Eq. (5.24) for the simply supported case. The boundary conditions for this case are d2 Wð0Þ ¼0 dx2

Wð0Þ ¼ 0;

Mxx ð0Þ ¼ 00

WðLÞ ¼ 0;

d 2 WðLÞ ¼0 Mxx ðLÞ ¼ 00 dx2

(5.28)

Substituting the boundary conditions in Eq. (5.24) yields a set of four equations with four unknowns, having the following matrix form 2

0

6 6 0 6 6 6 6 sinhða1 LÞ 4 a21 sinhðlLÞ

38 9 8 9 0> C > > > > > 1> > > > > 7> > > > > > > > 7> a21 0 a22 7< C2 = < 0 = 7 ¼ 7 > > > > C3 > > coshða2 LÞ sinða2 LÞ cosða2 LÞ 7> 0> > > > > > > > 5> > > > ; : ; : > 2 2 2 a1 coshðlLÞ a2 sinða2 LÞ a2 cosða2 LÞ 0 C4 1

0

1

(5.29) To obtain a unique solution, the determinant of the matrix appearing in Eq. (5.29) must vanish. This leads to the following characteristic equation:

sinða2 LÞ ¼ 00a2 L ¼ np;

n ¼ 1; 2; 3; 4.0a22 ¼

n 2 p2 L2

(5.30)

Substituting the result of Eq. (5.30) into the second expression in Eq. (5.26) leads to the natural frequencies of a symmetric layered column, having a length L, under compressive load, including rotary inertia on simply supported boundary conditions v0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u u uB uB N xx C u@1   2 C A u np
2 rffiffiffiffiffiffiffiffiu D 11 L np D11 u u
u¼
2  ; L I0 u I2 np t 1þ I0 L

n ¼ 1; 2; 3; 4.

(5.31)

200

Stability and Vibrations of Thin-Walled Composite Structures

Neglecting the rotary inertia leads to the following expression vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1ffi u0 u
2 rffiffiffiffiffiffiffiffiu C B np D11 u N xx C uB u¼ B1 
2 C; n ¼ 1; 2; 3; 4. u L I 0 t@ np A D11 L

(5.32)

showing that the axial compression load reduces the natural frequencies of a column, whereas for the case of no axial compression, but including the rotary inertia, the expression has the following form
u¼

np L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rffiffiffiffiffiffiffiffiv D11 u 1 u u
2 !; I0 u I np 2 t 1þ I0 L

n ¼ 1; 2; 3; 4.

(5.33)

displaying the same tendency as before, namely, the rotary inertia tends to reduce the natural frequencies of the column. Using the classical beam theory, for the simple case of a column without axial compression and rotary inertia, the expression for the natural frequencies is similar to that for an isotropic case
un ¼

np L

2 rffiffiffiffiffiffiffiffi D11 ; I0

n ¼ 1; 2; 3; 4.

(5.34)

Tables 5.1 and 5.2 present some of the most encountered column cases having various boundary conditions (see Fig. 4.4 and Table 4.1). The various expressions given in Table 5.1 are for the case without axial compression load and neglecting the rotary inertia. For this case, Eq. (5.24) simplifies into the following equation: WðxÞ ¼ A1 sinh ða2 xÞ þ A2 cosh ða2 xÞ þ A3 sin ða2 xÞ þ A4 cos ða2 xÞ

5.2.2

(5.35)

Nonsymmetric laminate (B11 s 0, I1 s 0)

To solve the case of a nonsymmetric laminate, it is easier to present Eqs. (5.20) and (5.21) in a matrix form, in which higher order terms are neglected 2

3 v2 v3 B11 3 6 A11 vx2 7( u0 ðx; tÞ ) vx 6 7 6 7 6 7 3 4 2 5 w0 ðx; tÞ 4 v v v B11 3 D11 4 þ N xx 2 vx vx vx 3 2 v I1 7 2 ( u0 ðx; tÞ ) ( 0 ) 6 I0 vx 7 v 6 7 þ6 ¼ 7 vt 2 6 4 v w0 ðx; tÞ 0 v2 5 I0  I2 2 I1 vx vx

(5.36)

No.

Name

Characteristic equation

Eigenvalues

1

SS-SS

sinða2 LÞ ¼ 0 ða2 LÞn ¼ np;


2 rffiffiffiffiffiffiffiffi np D11 L I0 2 rffiffiffiffiffiffiffiffi
4:73004 D11 u1 ¼ L I0

2

C-C

un ¼

n ¼ 1; 2; 3.

cosða2 LÞcoshða2 LÞ ¼ 1 ða2 LÞn ¼ 4:73004; 7:85321; 10:9956; .

3

C-F

cosða2 LÞcoshða2 LÞ ¼ 1 ða2 LÞn ¼ 1:87510; 4:69409; 7:85340; .

4

F-F

SS-C

SS-F

G-F

ð4n þ 1Þp 4

tanða2 LÞ ¼ tanhða2 LÞ ða2 LÞn ¼ 3:9266; 7:0686; 10:2102; :::

7

ð2n þ 1Þp 2

tanða2 LÞ ¼ tanhða2 LÞ ða2 LÞn ¼ 3:9266; 7:0686; 10:2102; :::

6

ð2n  1Þp 2

cosða2 LÞcoshða2 LÞ ¼ 1 ða2 LÞn ¼ 4:73004; 7:85321; 10:9956; .

5

ð2n þ 1Þp 2

ð4n þ 1Þp 4

tanða2 LÞ ¼ tanhða2 LÞ ða2 LÞn ¼ 2:3650; 5:4978; 8:6394; .

ð4n  1Þp 4

u1 ¼

 rffiffiffiffiffiffiffiffi
1:87351 2 D11 L I0

Vibration of composite columns and plates

Characteristic equations and eigenvalues for natural vibrations of laminated composite columns using the CLT approach

Table 5.1

 rffiffiffiffiffiffiffiffi
4:73004 2 D11 u1 ¼ L I0

u1 ¼

 rffiffiffiffiffiffiffiffi
3:9266 2 D11 L I0

u1 ¼


 rffiffiffiffiffiffiffiffi 3:9266 2 D11 I0 L

u1 ¼

 rffiffiffiffiffiffiffiffi
2:3650 2 D11 L I0 201

Continued

Continued

202

Table 5.1 No.

Name

Characteristic equation

8

G-SS

cosða2 LÞ ¼ 0 ða2 LÞn ¼ ð2n  1Þ

9 10

G-G G-C

sinða2 LÞ ¼ 0 ða2 LÞn ¼ np;

p 2

n ¼ 1; 2; 3; .

n ¼ 1; 2; 3.

tanða2 LÞ ¼ tanhða2 LÞ ða2 LÞn ¼ 2:3650; 5:4978; 8:6394; .

ð4n  1Þp 4

un ¼

rffiffiffiffiffiffiffiffi ð2n  1Þp D11 2L I0

ffi np2 rffiffiffiffiffiffiffi D11 un ¼ L I0 2 rffiffiffiffiffiffiffiffi
2:3650 D11 u1 ¼ L I0 Stability and Vibrations of Thin-Walled Composite Structures

C, clamped; F, free; G, guided; SS, simply supported.

Eigenvalues

Mode shapes and their relevant eigenvalues for natural vibrations of laminated composite columns using the CLT approach

Table 5.2

1

Name

Mode shape

Eigenvalues

SS-SS

npx Wn ðxÞ ¼ sin L

un ¼

 







) cosh ða2 LÞn  cos ða2 LÞn ða2 LÞnx ða2 LÞnx ða2 LÞnx ða2 LÞnx  sinh  Wn ðxÞ ¼ cosh  cos   sin L L L L sinh ða2 LÞn  sin ða2 LÞn

a

C-C

3a

C-F

2

 







) cosh ða2 LÞn þ cos ða2 LÞn ða2 LÞnx ða2 LÞnx ða LÞ x ða LÞ x  sinh 2 n  sin 2 n   cos  L L L L sinh ða2 LÞn þ sin ða2 LÞn

un ¼

 







) cosh ða2 LÞn  cos ða2 LÞn ða2 LÞnx ða2 LÞnx ða2 LÞnx ða2 LÞnx    cos   sin sinh Wn ðxÞ ¼ cosh L L L L sinh ða2 LÞn  sin ða2 LÞn

un ¼

 







) cosh ða2 LÞn  cos ða2 LÞn ða2 LÞnx ða2 LÞnx ða2 LÞnx ða2 LÞnx  sinh   cos   sin Wn ðxÞ ¼ cosh L L L L sinh ða2 LÞn  sin ða2 LÞn

un ¼

 







) cosh ða2 LÞn þ cos ða2 LÞn ða2 LÞnx ða2 LÞnx ða LÞ x ða LÞ x  sinh 2 n þ sin 2 n  þ cos  L L L L sinh ða2 LÞn þ sin ða2 LÞn

un ¼

 







) cosh ða2 LÞn  cos ða2 LÞn ða2 LÞnx ða2 LÞnx ða2 LÞnx ða2 LÞnx    cos   sin Wn ðxÞ ¼ cosh sinh L L L L sinh ða2 LÞn  sin ða2 LÞn

un ¼

Wn ðxÞ ¼ cosh

4

a

5

a

SS-C

6a

SS-F

F-F



Wn ðxÞ ¼ cosh

7

a

G-F



8

G-SS

9

G-G

10a

G-C

ð2n  1Þpx 2L 
npx Wn ðxÞ ¼ cos L



Wn ðxÞ ¼ sin

Wn ðxÞ ¼ cosh











np L

ð2n  1Þp 2L

2 rffiffiffiffiffiffiffiffi D11 I0

ð2n þ 1Þp 2L

2 rffiffiffiffiffiffiffiffi D11 I0

ð4n þ 1Þp 4L

2 rffiffiffiffiffiffiffiffi D11 I0

ð4n þ 1Þp 4L

2 rffiffiffiffiffiffiffiffi D11 I0

ð4n  1Þp 4L

2 rffiffiffiffiffiffiffiffi D11 I0

rffiffiffiffiffiffiffiffi ð2n  1Þp D11 I0 2L ffi np2 rffiffiffiffiffiffiffi D11 un ¼ L I0

rffiffiffiffiffiffiffiffi ð4n  1Þp 2 D11 un ¼ 4L I0

un ¼

203

 







) sinh ða2 LÞn þ sin ða2 LÞn ða2 LÞnx ða2 LÞnx ða LÞ x ða LÞ x  sinh 2 n  sin 2 n   cos  L L L L cosh ða2 LÞn  cos ða2 LÞn

The expression for the eigenvalues is given for n >> 1. For the first modes please refer to Table 5.1.

a

2 rffiffiffiffiffiffiffiffi D11 I0

rffiffiffiffiffiffiffiffi ð2n þ 1Þp 2 D11 un ¼ I0 2L


Vibration of composite columns and plates

No.

204

Stability and Vibrations of Thin-Walled Composite Structures

The matrix presentation in Eq. (5.36) includes axial compression N xx , rotary inertia I2, and the coupling mass moment of inertia I1, which couples both structurally and dynamically the two assumed deflections, u0 and w0. There is no general solution for Eq. (5.36). For simply supported boundary conditions we can present a relatively simple way to determine the natural frequencies of a nonsymmetric laminated beam. For this case, we assume that the deflections u0 and w0 have the following form: (

u0 ðx; tÞ w0 ðx; tÞ

) ¼

9 8
 M > > P mpx > iut > > Um cos e > > > > > = < m¼1 L
 > > M > P mpx iut > > > > > > Wm sin e > ; : L

h

8 9 M P > > iut > > > U cosðlxÞe > > > < m¼1 m = > > M > > P > > > Wm sinðlxÞeiut > : ;

m¼1

m¼1

(5.37) pffiffiffiffiffiffiffi 2 where i ¼ 1, u is the natural frequency squared, and l ¼ mp/L. Substituting Eq. (5.37) into Eq. (5.36) yields 2 4

A11 l2

B11 l3

B11 l3

D11 l4  N xx l2

3( 5

Um Wm

)

3 u2 I 1 l 5 þ4   u2 I1 l u2 I0  I2 l2 ( ) ( ) Um 0 ¼ 0 Wm 2

u2 I0

(5.38)

Then Eq. (5.38) can be casted in the following form: ) ( ) #( " 0 Um B11 l3 þ u2 I1 l A11 l2  u2 I0 ¼   B11 l3 þ u2 I1 l D11 l4  N xx l2  u2 I0  I2 l2 0 Wm (5.39) To obtain a unique solution, the determinant of the matrix in Eq. (5.39) must vanish, leading to the following characteristic equation:  2   A u2 þ B u 2 þ C ¼ 0

(5.40)

where   A h I12 l2  I0 I0  I2 l2     B h A11 l2 I0  I2 l2 þ I0 D11 l2  N xx l2  2I1 B11 l4 h  i  C h A11 N xx  A11 D11  B211 l2 l4

(5.41)

Vibration of composite columns and plates

205

Solution of Eq. (5.40) would provide the natural frequency for the simply supported nonsymmetric case. For a symmetric case (B11 ¼ I1 ¼ 0), Eq. (5.39) will not be coupled and the result will be the expression presented in Eq. (5.31). Appendix A presents a procedure to solve the case of a nonsymmetric beam having other boundary conditions.

5.3

Vibrations of columns: the first-order shear deformation theory approach

As derived in Chapter 4, the equations of motion for a general laminate using the FSDT approach (see also Refs. [9e13]) can therefore be written as (assuming v0 ¼ fy ¼ 0)
2  v u0 v3 w0 v2 f v2 u0 v2 f A11 þ 3 þ B11 2x ¼ I0 2 þ I1 2x 2 vx vx vx vt vt
KA55

B11

v2 w0 vfx þ vx2 vx

  N xx

v2 w0 v2 w0 ¼ q þ I0 2 2 vx vt

(5.42)

(5.43)


2 
 v u0 v3 w0 v2 f x vw0 v2 f x v2 u 0 þ f þ  KA þ I þ D ¼ I 11 2 1 55 x vx2 vx3 vx2 vx vt 2 vt 2 (5.44)

Eqs. (5.42)e(5.44) are for the case of uniform properties along the beam. For the particular case of properties varying along the x coordinate of the beam, the reader is referred to Refs. [12,13]. To solve the vibration problem, the lateral load, q, is set to zero resulting in the following three coupled equations of motion:

A11


2  v u0 v3 w0 v2 fx v2 u0 v2 fx þ ¼ I þ I þ B 11 0 1 vx2 vx3 vx2 vt 2 vt 2


KA55

v2 w0 vfx þ vx2 vx

  N xx

v2 w0 v2 w0 ¼ I 0 vx2 vt 2

(5.45)

(5.46)


2 
 v u0 v3 w0 v2 f x vw0 v2 f x v2 u 0 þ f þ  KA þ I B11 þ D ¼ I 11 2 1 55 x vx2 vx3 vx2 vx vt 2 vt 2 (5.47)

206

Stability and Vibrations of Thin-Walled Composite Structures

Assuming harmonic vibrations with a squared frequency, u2, and neglecting highorder terms, we can rewrite Eqs. (5.45)e(5.47) as

A11

d2 U d2 F þ B þ u2 I 0 U þ u2 I 1 F ¼ 0 11 dx2 dx2


KA55

 d2 W dF d2 W þ N þ u2 I 0 W ¼ 0  xx dx2 dx dx2


d2 U d2 F dW þ F þ u2 I 2 F þ u 2 I 1 U ¼ 0 B11 2 þ D11 2  KA55 dx dx dx

(5.48)

(5.49)

(5.50)

For symmetric laminates, we can write the following expressions: A11

d2 U þ u2 I0 U ¼ 0 dx2


KA55

 d2 W dF d2 W þ N þ u2 I 0 W ¼ 0  xx dx2 dx dx2


d2 F dW þ F þ u2 I2 F ¼ 0 D11 2  KA55 dx dx

(5.51)

(5.52)

(5.53)

For the symmetric case, Eq. (5.51) is not coupled with Eqs. (5.52) and (5.53), which are coupled and have to be solved together.

5.3.1

Symmetric laminate (B11 ¼ 0, I1 ¼ 0)

First we shall present a general solution for the symmetric case presented by the two coupled equations Eqs. (5.52) and (5.53). The two equations are decoupled to yield uncoupled equations






 2 N xx d4 W N xx d W 2 D11 I0 þ u þ 1  þ N I xx 2 dx2 KA55 dx4 KA55 KA55

I 2 u2 2 þ u I0 1 W ¼0 KA55

(5.54)






 2 N xx d4 F N xx d F 2 D11 I0 þ u þ 1  þ N I xx 2 dx2 KA55 dx4 KA55 KA55

I 2 u2 þ u2 I 0 1 F¼0 KA55

(5.55)

D11 



D11 

Vibration of composite columns and plates

207

One should note that without axial compression, N xx ¼ 0, Eqs. (5.54) and (5.55) will have the following form, which is similar to that in Ref. [10]. D11



2 d4 W d W I 2 u2 2 D11 I0 2 þ u þ I þ u I  1 W ¼0 2 0 dx4 dx2 KA55 KA55



2 d4 F d F I2 u 2 2 D11 I0 2 D11 4 þ u þ I2 þ u I0 1 F¼0 dx dx2 KA55 KA55

(5.56)

(5.57)

The general solutions for Eqs. (5.54) and (5.55) (as well as for Eqs. 5.56 and 5.57) have the following form: WðxÞ ¼ A1 coshðs1 xÞ þ A2 sinhðs1 xÞ þ A3 cosðs2 xÞ þ A4 sinðs2 xÞ

(5.58)

FðxÞ ¼ B1 coshðs1 xÞ þ B2 sinhðs1 xÞ þ B3 cosðs2 xÞ þ B4 sinðs2 xÞ

(5.59)

whereas the constants A1, A2, A3, and A4 and B1, B2, B3, and B4 are interconnected (by back-substituting Eqs. 5.58 and 5.59 into the coupled equations, Eqs. 5.52 and 5.53) and have the following form B1 ¼ B2 ¼ B3 ¼

KA55 $s1 A2 h aA2  KA55 þ u2 I2

(5.60)

KA55 $s1 A1 h aA1 D11 $s21  KA55 þ u2 I2

(5.61)

KA55 $s2 A4 h bA4 2 2  D11 $s1  KA55

(5.62)

D11 $s21

u2 I

B4 ¼ 

u2 I

KA55 $s2 A3 h  bA3 2 2  D11 $s1  KA55

(5.63)

and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ðb2  4acÞ þ s1 ¼  2a 2a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ðb2  4acÞ s2 ¼ þ þ 2a 2a

(5.64)

(5.65)

208

Stability and Vibrations of Thin-Walled Composite Structures

Out-of-plane boundary conditions for a nonsymmetric laminate column

Table 5.3

Name

Boundary conditions

Simply supported (or hinged) end

W ¼ 0 and B11

Clamped end

W ¼ 0 and F ¼ 0

dU dF þ D11 ¼0 dx dx


dU dF dW dW þ D11 ¼ 0 and KA55 þ F  N xx ¼0 dx dx dx dx 
dW dW F ¼ 0 and KA55 þ F  N xx ¼0 dx dx

Free end

B11

Guided end

where a h D11 

N xx ; KA55

I2 u 2 c h u I0 1 KA55

b h u2

D11 I0 þ KA55


1



N xx I2 þ N xx ; KA55

(5.66)

2

The remaining constants, A1, A2, A3, and A4, will be found after imposing the relevant boundary conditions. The general boundary conditions for a nonsymmetric laminate are given in Table 5.3 and Eq. (5.67). The in-plane boundary condition for the nonsymmetric laminate is given in Eq. (5.67): A11

dU dF þ B11 ¼ N xx or U ¼ 0 dx dx

(5.67)

For a symmetric case without axial compression load the expressions for the boundary condition simplify and are given in Table 5.4 and Eq. (5.68). The in-plane boundary condition for the symmetric laminate axial noncompressed column is given in Eq. (5.68): dU ¼ 0 or U ¼ 0 dx

(5.68)

Table 5.5 presents the characteristic equations for various boundary conditions for symmetric laminate axial noncompressed columns.

Vibration of composite columns and plates

209

Out-of-plane boundary conditions for a symmetric laminate axial noncompressed column

Table 5.4

5.3.2

Name

Boundary conditions

Simply supported (or hinged) end

W ¼ 0 and

Clamped end

W ¼ 0 and F ¼ 0

Free end

dF dW ¼ 0 and þF¼0 dx dx

Guided end

F ¼ 0 and

dF ¼0 dx

dW þF¼0 dx

Nonsymmetric laminate (B11 s 0, I1 s 0)

To solve a nonsymmetric laminate for beams or columns, we shall follow the derivation proposed by Abramovich and Livshits [15]. The equations of motion, Eqs. (5.45)e(5.47), will be presented in a matrix formulation (for N xx ¼ 0), namely, 2

3 v2 v2 6 A11 2 7 0 B11 2 6 78 vx vx 6 7 u ðx; tÞ 9 0 > 6 7> > > > 6 7> < = 2 6 7 v v 6 0 7 w ðx; tÞ KA KA 55 2 55 6 7> 0 > vx vx > 6 7> > : ; 6 7> 6 7 fx ðx; tÞ
 6 7 2 2 v v v 4 5 B11 2 KA55 D11 2  KA55 vx vx vx 8 9 8 9 2 3 I0 u0 ðx; tÞ > 0 I1 > > > >0> > > > = < > < = > 6 7 v2 > 6 7 þ6 0 I0 0 7 2 w0 ðx; tÞ ¼ 0 > 4 5 vt > > > > > > > > > ; : > : ; > 0 I1 fx ðx; tÞ 0 I2

(5.69)

The associated boundary conditions are given by Eq. (5.63) and Table 5.3. Following Ref. [15], the nondimensional displacements and the nondimensional beam’s length are defined as fqg ¼ fu0 =L; w0 =L; fx gT ;

x h x=L

(5.70)

210

Characteristic equations for natural vibrations of laminated composite columns using the FSDT approach

Table 5.5

No.

Name

Boundary conditions

Characteristic equation

1

SSSS

Wð0Þ ¼ 0 dFð0Þ=dx ¼ 0 WðLÞ ¼ 0 dFðLÞ=dx ¼ 0

sinðs2 LÞ ¼ 0

2

C-C

Wð0Þ ¼ 0 Fð0Þ ¼ 0 WðLÞ ¼ 0 FðLÞ ¼ 0

2a  2a cosðs2 LÞcoshðs1 LÞþ 
2 a  b2 sinðs2 LÞsinhðs1 LÞ ¼ 0 b

3

C-F

Wð0Þ ¼ 0 Fð0Þ ¼ 0 dFðLÞ=dx ¼ 0

4

F-F

dFð0Þ=dx ¼ 0 dWð0Þ=dx þ Fð0Þ ¼ 0 dFðLÞ=dx ¼ 0 dWðLÞ=dx þ FðLÞ ¼ 0

n ¼ 1; 2; 3.

 2 h i þ aðbs2  as1 Þ coshðs1 LÞcosðs2 LÞ a½s1 ðs1  aÞ  s2 ðs2 þ bÞ þ s1 s2 ba b að2s1 s2 þ s1 b þ s2 aÞsinhðs1 LÞsinðs2 LÞ ¼ 0

2as1 ðs1 þ aÞ  2 cosðs2 LÞcoshðs1 LÞþ b2 s22 ðs1 þ aÞ2  a2 s21 ðs2 þ bÞ2 sinðs2 LÞsinhðs1 LÞ ¼ 0 bs2 ðs2 þ bÞ

5

SS-C

Wð0Þ ¼ 0 dFð0Þ=dx ¼ 0 WðLÞ ¼ 0 FðLÞ ¼ 0

b tanh (s1L) ¼ a tan (s2L)

6

SS-F

Wð0Þ ¼ 0 dFð0Þ=dx ¼ 0 dFðLÞ=dx ¼ 0 dWðLÞ=dx þ FðLÞ ¼ 0

s1a(s2 þ b) tanh (s1L) ¼ s2b(s1 þ a) tan (s2L)

Stability and Vibrations of Thin-Walled Composite Structures

dWðLÞ=dx þ FðLÞ ¼ 0

ðs2 LÞn ¼ np;

G-F

Fð0Þ ¼ 0 dWð0Þdx þ Fð0Þ ¼ 0 dFðLÞ=dx ¼ 0 dWðLÞ=dx þ FðLÞ ¼ 0

8

G-SS

Fð0Þ ¼ 0 dWð0Þdx þ Fð0Þ ¼ 0 WðLÞ ¼ 0 dFðLÞ=dx ¼ 0

9

G-G

Fð0Þ ¼ 0 dWð0Þdx þ Fð0Þ ¼ 0 FðLÞ ¼ 0 dWðLÞdx þ FðLÞ ¼ 0

10

G-C

Fð0Þ ¼ 0

s1a(s2 þ b) tan (s2L) ¼ s2b(s1 þ a) tanh (s1L)

cosðs2 LÞ ¼ 0 ðs2 LÞn ¼ ð2n  1Þ sinðs2 LÞ ¼ 0 ðs2 LÞn ¼ np;

p 2

n ¼ 1; 2; 3; .

n ¼ 1; 2; 3.

Vibration of composite columns and plates

7

A tanh (s1L) ¼ b tan (s2L)

dWð0Þdx þ Fð0Þ ¼ 0 WðLÞ ¼ 0 FðLÞ ¼ 0 C, clamped; F, free; G, guided; SS, simply supported.

211

212

Stability and Vibrations of Thin-Walled Composite Structures

yielding fqg ¼ diagonalfL; L; 1gfqg

(5.71)

To obtain expressions for free vibrations, we assume that the solution for Eq. (5.69) has the following form: fqg ¼ fU; W; FgT eiut h fQgeiut

(5.72)

Substituting Eq. (5.72) into Eq. (5.69) provides the following matrix equation 2

2

3

3

6 v2 66 66 A11 2 66 vx 66 66 66 66 66 0 66 66 66 66 66 v2 66 64 B11 2 4 vx

0 2

KA55

v vx2

v KA55 vx

v2 vx2

7 7 7 2 7 I0 7 7 6 7 v 7 þ u2 6 60 KA55 7 4 vx 7 7 I1
7 7 v2 7 D11 2  KA55 5 vx B11

0 I0 0

7 7 7 37 I1 7 7 77 77 0 77 57 7 7 I2 7 7 7 7 5

(5.73)

9 8 9 8 L$U > > > > >0> > > > > = < > = > < L$W ¼ 0 > > > > > > > > > > > ; : > ; : 0 F By introducing nondimensional parameters and after some algebraic transformation, Eq. (5.73) can be written as 22

z21

0

66 66 66 66 0 64 4 z2 b2 2 6 6 þ6 4

1 0 p

2

z2

3

2

0

7 2 6 7v 6 0 07 7 vx2 þ 6 4 5 0 b2 0

0

b2 p2

h 2 b2 p2

0

0 0 1

0

3

7v 7 17 5 vx 0

3 3 8 9 U> h p > 7> > > > = 77< 77 0 77 W ¼ 0 57> > > >  2 2 2  5> ; : > F r b p 1 2 2

(5.74)

Vibration of composite columns and plates

213

where the various parameters are defined as p2 h

u2 I 0 L 4 ; D11

h2 h

I1 . I0 L

b2 h

D11 ; KA55 L2

z2 h

B11 L ; D11

z21 h

A11 L2 ; D11

r2 h

I2 ; I0 L2

(5.75) Assuming that the general solution of Eq. (5.74) has the following form   fQg ¼ Q eimx

(5.76)

  with m being the eigenvalue and Q the eigenvector and substituting it into Eq. (5.74) leads to the following cubic algebraic equation As3 þ Bs2 þ Cs þ D ¼ 0

sh

m2 p2

(5.77)

with     A h z21  z4 ; B h 1 þ r 2 þ b2 z21  z2 2h2 þ z2 b2 ;
     1 C h z21 r 2 b2  2 þ r 2 þ b2  h2 h2 þ 2z2 b2 ; p D h r 2 b2 

(5.78)

1  b2 h 4 p2

By solving Eq. (5.77), we can write the general solution for Eq. (5.74) in the following form (see a detailed discussion regarding the format of the solution in Ref. [15]): U ¼ A1 gm sinhðm1 xÞ þ A2 gm coshðm1 xÞ þ A3 ld sinðm2 xÞ  A4 ld cosðm2 xÞ þ A5 ab sinðm3 xÞ  A6 ab cosðm3 xÞ W ¼ A1 coshðm1 xÞ þ A2 sinhðm1 xÞ þ A3 cosðm2 xÞ þ A4 sinðm2 xÞ þ A5 cosðm3 xÞ þ A6 sinðm3 xÞ F ¼  A1 m sinhðm1 xÞ  A2 m coshðm1 xÞ  A3 l sinðm2 xÞ þ A4 l cosðm2 xÞ  A5 a sinðm3 xÞ þ A6 a cosðm3 xÞ

(5.79)

214

Stability and Vibrations of Thin-Walled Composite Structures

where the various terms in Eq. (5.79) are defined as follows: pffiffiffiffi m1 h s1 p; gh

pffiffiffiffiffiffiffiffi m2 h s2 p;

z2 s1 þ h2 ; z21 s1 þ 1

s1 þ b2 m h pffiffiffiffi p; s1

dh

pffiffiffiffiffiffiffiffi m3 h s3 p;

z2 s2 þ h2 ; z21 s2 þ 1

s 2 þ b2 l h pffiffiffiffiffiffiffiffi p; s2

bh

z 2 s 3 þ h2 ; z21 s3 þ 1

(5.80)

s3 þ b2 a h pffiffiffiffiffiffiffiffi p. s3

Imposing the adequate boundary conditions would result in finding both the eigenvalues (the natural frequencies) and the eigenvectors (the modes of vibration). For this problem, six boundary conditions should be imposed, as presented in Table 5.6. One should note (see also Ref. [15]) that only for a column or a beam with two simply supported (hinged) movable ends an analytical solution exists. The characteristic equation obtained after demanding the determinant of the coefficients in Eq. (5.79) is sinðm3 Þsinðm2 Þ ¼ 0

(5.81)

The solution of Eq. (5.81) has two series: the first one has bending-dominated vibrations m3 ¼ kp

k ¼ 1; 2; 3; .; n

(5.82)

whereas the second series has longitudinal-dominated vibrations m2 ¼ kp

k ¼ 1; 2; 3; .; n.

(5.83)

Boundary conditions for the nonsymmetric laminated columns using the FSDT approach

Table 5.6

Name

Boundary conditions

Clamped immovable end

U¼W¼F¼0

Clamped movable end

z21

dU dF þ z2 ¼W ¼F¼0 dx dx

Simply supported immovable end

z2

dU dF þ ¼W ¼U¼0 dx dx

Simply supported movable end

dU dF dU dF þ ¼ W ¼ z21 þ z2 ¼0 dx dx dx dx  dU dF 1  dU dF ¼ z21 þ ¼ b2 F þ dW þ z2 ¼0 z2 dx dx dx dx dx

Free end

z2

Vibration of composite columns and plates

5.4

215

Vibrations of plates: the classical lamination plate theory approach

5.4.1

Simply supported special orthotropic plates

The first case to be solved is sometimes called special orthotropic plates for which the bending-stretching coupling terms Bij and the bending-twisting coefficients D16 and D26 are set to zero. Then from Eq. (4.10) in Chapter 4, by assuming symmetry (Bij ¼ D16 ¼ D26 ¼ I1 ¼ 0) and zeroing the in-plane and out-of-plane loads, one obtains D11

v4 w 0 v4 w0 v4 w0 v2 w0 þ 2ðD þ 2D Þ þ D þ I 12 22 0 66 vx4 vx2 vy2 vy4 vt 2
 v2 v2 w0 v2 w0  I2 2 þ ¼0 vt vx2 vy2

(5.84)

where I0 and I2 are defined in Eq. (5.2). Let us assume the following solution for the outof-plane displacement w0 (tacitly assuming harmonic vibrations with a frequency u)


w0 ðx; y; tÞ ¼ Wmn

mpx sin a




 npy iut e b

(5.85)

where a is the length and b is the width of the plate having a total thickness of t, and by substituting in Eq. (5.84), we get
D11

mp a

4


2
2
4 mp np np þ 2ðD12 þ 2D66 Þ þ D22 a b b


2
2  mp np  u2 I0 þ I2 þ ¼0 a b

(5.86)

The solution of Eq. (5.86) has the following form, which presents the natural frequencies for a special orthotropic laminated plate

u2mn ¼


4 D11 p a

m4


2
4 a a 4 þ 2ðD12 þ 2D66 þ D22 n b b
2 "
2 # p a I0 þ I2 m2 þ n2 a b

while the mode of vibration is given by mpx npy w0 ð x; yÞ ¼ Wmn sin sin a b

Þm2 n2

(5.87)

(5.88)

216

Stability and Vibrations of Thin-Walled Composite Structures

One can see from Eq. (5.87) that inclusion of the rotary inertia I2 tends to reduce the natural frequencies. For a square plate, a ¼ b, when neglecting the rotary inertia, the general frequency would show as u2mn ¼

p4  D11 m4 þ 2ðD12 þ 2D66 Þm2 n2 þ D22 n4 4 I0 a

(5.89)

The lowest frequency, sometimes called also the fundamental frequency, will occur at m ¼ n ¼ 1; that is, for a rectangular plate, we will have
2
4

p4 a a ¼ þ D22 D11 þ 2ðD12 þ 2D66 Þ b b I 0 a4

u211

5.4.2

(5.90)

Simply supported on two opposite edges of special orthotropic plates

Using the Lévy method1, one can solve the vibrations of rectangular plates, in which two opposite edges are on simply supported boundary conditions, whereas the other two sides can be clamped, free, or any other combination of boundaries. Therefore the equilibrium equation, without in-plane loads, is given by Eq. (5.84). The out-ofplane deflections can be written as w0 ðx; y; tÞ ¼ Wm ðxÞ

npy eiut b

(5.91)

Substituting Eq. (5.91) into Eq. (5.84) leads to the following differential equation:

D11


2 2 d 4 Wm np d Wm 2 þ u I  2ðD þ 2D Þ 2 12 66 4 b dx dx2


2


4  np np 2  u I0 þ I2  D22 Wm ¼ 0 b b

(5.92)

or

b a

1

d 4 Wm b d 2 Wm þb  bc Wm ¼ 0 dx4 dx2

(5.93)

M. Lévy, Memoire sur la theorie des plaques elastiques planes, Journal de Mathematiques Pures et Appliquees 3 (1899) 219.

Vibration of composite columns and plates

217

where
2 np 2 b b ; a h D11 ; b h u I2  2ðD12 þ 2D66 Þ b
2


4 np np bc h u2 I0 þ I2  D22 b b

(5.94)

The general solution of Eq. (5.92) has the following form: WðxÞ ¼ A1 sinh ða1 xÞ þ A2 cosh ða1 xÞ þ A3 sin ða2 xÞ þ A4 cos ða2 xÞ

(5.95)

where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u u u 2 2 t b tb bþ b b þ 4b a bc a bc bþ b b þ 4b s1 ¼ ; s2 ¼ 2b a 2b a

(5.96)

Application of the boundary conditions in the x direction will lead to the natural frequencies and their associated mode shapes to be in the x direction.

References [1] A.V.K. Murty, R.P. Shimpi, Vibrations of laminated beams, Journal of Sound and Vibration 36 (2) (1974) 273e284. [2] L.S. Teoh, C.C. Huang, The vibration of beams of fiber-reinforced material, Journal of Sound and Vibration 51 (4) (1977) 467e473. [3] K. Chandrashekhara, K. Krishnamurthy, S. Roy, Free vibration of composite beams including rotary inertia and shear deformation, Composite Structures 14 (1990) 269e279. [4] G. Singh, G.V. Rao, N.G.R. Iyengar, Analysis of the nonlinear vibrations of unsymmetrically laminated composite beams, AIAA Journal 29 (10) (1991) 1727e1735. [5] S. Krishnaswamy, K. Chandrashekhara, W.Z.B. Wu, Analytical solutions to vibration of generally layered composite beams, Journal of Sound and Vibration 159 (1) (1992) 85e99. [6] K. Chandrashekhara, K.M. Bangera, Free vibration of composite beams using a refined shear flexible beam element, Computers and Structures 43 (4) (1992) 719e727. [7] T. Kant, K. Swaminathan, Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher-order refined theory, Composite Structures 53 (2001) 73e85. [8] M.R. Aagaah, M. Mahinfalah, G.N. Jazar, Natural frequencies of laminated composite plates using third order shear deformation theory, Composite Structures 72 (2006) 273e279.

218

Stability and Vibrations of Thin-Walled Composite Structures

[9] H. Abramovich, O. Hamburger, Vibration of a uniform cantilever Timoshenko beam with translational and rotational springs and with a tip mass, Journal of Sound and Vibration 154 (1) (1992) 67e80. [10] H. Abramovich, Shear deformation and rotary inertia effects of vibrating composite beams, Composite Structures 20 (1992) 165e173. [11] H. Abramovich, A note on experimental investigation on a vibrating Timoshenko cantilever beam, Journal of Sound and Vibration 160 (1) (1993) 167e171. [12] H. Abramovich, Thermal buckling of cross-ply composite laminates using a first-order shear deformation theory, Composites Structures 28 (1994) 201e213. [13] H. Abramovich, Deflection control of laminated composite beams with piezoceramic layers e closed form solutions, Composite Structures 43 (1998) 217e231. [14] H. Abramovich, M. Eisenberger, O. Shulepov, Vibrations and buckling of cross-ply non-symmetric laminated composite beams, AIAA Journal 34 (5) (May 1996) 1064e1069. [15] H. Abramovich, A. Livshits, Free vibrations of non-symmetric cross-ply laminated composite beams, Journal of Sound and Vibration 176 (5) (1994) 597e612. [16] H. Abramovich, A. Livshits, Dynamic behavior of cross-ply laminated beams with piezoelectric layers, Composite Structures 25 (1e4) (1993) 371e379.

Appendices A.

General solution for a nonsymmetric beam resting on any boundary conditions

  Using Eq. (5.36) and by assuming that no axial compression is applied N xx ¼ 0 , while I1 ¼ 0 (by placing the beam’s coordinate system in the middle plane of the beam) and the rotary moment of inertia is assumed to be negligible (I2 ¼ 0), we have the following matrix notation: 2

v2 6 A11 2 6 vx 6 6 4 v3 B11 3 vx

3 v3 ) " B11 3 7( I0 vx 7 u0 ðx; tÞ 7 þ 7 0 v4 5 w0 ðx; tÞ D11 4 vx

0 I0

#

v2 vt 2

(

u0 ðx; tÞ w0 ðx; tÞ

) ¼

( ) 0 0 (A.1)

Assuming that the nondimensional displacements have the following form 8 9 u0 ðx; tÞ > > > > ( ) > < L > = UðxÞeiut ¼ > > > > WðxÞeiut w ðx; tÞ 0 > > : ; L

(A.2)

Vibration of composite columns and plates

219

while the axial nondimensional axis being x ¼ x/L (L is the length of the beam) and by substituting the nondimensional expressions of the two beams’ displacements into Eq. (A.1), we get 2 6 z21 6 6 6 6 4

d2 þ p2 2 dx

d3 z2 3 dx

3 d3 z 3 7( ) ( ) dx 7 0 7 UðxÞ ¼ 7 7 WðxÞ 0 d4 25  p dx4 2

(A.3)

where

p2 h

u2 I0 L4 ; D11

b2 h

D11 ; KA55 L2

z2 h

B11 L ; D11

z21 h

A11 L2 . D11

(A.4)

The characteristic equation of Eq. (A.3) has the following form 

 z2 1 m2 z21  z4 s3 þ s2  12 s  2 ¼ 0 s h 2 p p p

(A.5)

By solving Eq. (A.5), we can write the general solution of Eq. (A.3) in the following form (see a detailed discussion regarding the format of the solution in Ref. [15]): U ¼ A1 gm sinhðm1 xÞ þ A2 gm coshðm1 xÞ þ A3 ld sinðm2 xÞ  A4 ld cosðm2 xÞ þ A5 ab sinðm3 xÞ  A6 ab cosðm3 xÞ W ¼ A1 coshðm1 xÞ þ A2 sinhðm1 xÞ þ A3 cosðm2 xÞ

(A.6)

þ A4 sinðm2 xÞ þ A5 cosðm3 xÞ þ A6 sinðm3 xÞ where the various terms in Eq. (A.6) are defined as follows: pffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi m1 h s1 p; m2 h s2 p; m3 h s3 p; z2 s1 z2 s2 z2 s3 ; d h ; b h ; z21 s1 þ 1 z21 s2 þ 1 z21 s3 þ 1 pffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi m h s1 p; l h s2 p; a h s3 p. gh

(A.7)

220

Stability and Vibrations of Thin-Walled Composite Structures

Boundary conditions for a nonsymmetric laminate beam (the CLT approach)

Table A.1

Name

Boundary conditions

Clamped immovable end

U¼W¼

Clamped movable end

z21

Simply supported immovable end Simply supported movable end Free end

dW ¼0 dx

dU d2 W dW  z2 2 ¼ W ¼ ¼0 dx dx dx

U ¼ W ¼ z2

dU d2 W  2 ¼0 dx dx

z2

dU d 2 W dU d2 W  2 ¼ W ¼ z21  z2 2 ¼ 0 dx dx dx dx

z2

dU d 2 W d2 U d3 W dU d2 W  2 ¼ z21 2  z2 3 ¼ z21  z2 2 ¼ 0 dx dx dx dx dx dx

Imposing the adequate boundary conditions would result in finding both the eigenvalues (the natural frequencies) and the eigenvectors (the modes of vibration). For this problem, six boundary conditions should be imposed at each end, as presented in Table A.1.

B.

Matrix notation for the equilibrium equations using the classical lamination theory approach

A convenient way of presenting the equilibrium equations at buckling and/or vibration is the following matrix form: 2

a11

6 6 6 a21 4 a31

a12 a22 a32

9 38 2 uðx; y; tÞ > m11 > > > > > = v2 6 7< 7 6 þ 26 0 a23 7 vðx; y; tÞ > 5> vt 4 > > > > : ; w0 ðx; y; tÞ ½a33  N m13 8 9 8 9 uðx; y; tÞ > > > > >0> > > > > = < > < = > vðx; y; tÞ ¼ 0  > > > > > > > > > > > > ; : : ; q w0 ðx; y; tÞ a13

0

m13

3

m22

7 7 m23 7 5

m23

m33

(B.1)

Vibration of composite columns and plates

221

where the various operators are given by a11 h A11

v2 v2 v2 þ A þ 2A 16 66 vxvy vx2 vy2

a22 h A22

v2 v2 v2 þ A66 2 þ 2A26 2 vxvy vy vx

a33 h D11

v4 v4 v4 v4 v4 þ 2ðD þ 4D þ 2D Þ þ 4D þ D 12 22 16 66 26 vx4 vx3 vy vx2 vy2 vxvy3 vy4

a12 ¼ a21 h A16

v2 v2 v2 þ A þ ðA þ A Þ 12 66 26 vxvy vx2 vy2

a13 ¼ a31 h  B11

v3 v3 v3 v3  ðB  3B þ 2B Þ  B 12 16 66 26 vx3 vx2 vy vxvy2 vy3

a23 ¼ a32 h  B16

v3 v3 v3 v3  3B  ðB þ 2B Þ  B 12 22 66 26 vx3 vx2 vy vxvy2 vy3

N h N xx

v2 v2 v2 þ N yy 2 þ 2N xy 2 vxvy vx vy (B.2)

and m11 h  I0 ;

m22 h  I0 ;

v m13 h I1 ; vx

v m23 h I1 ; vy

m33 h I0  I2

v2 v2 þ vx2 vy2

! (B.3)

C. The terms of the matrix notation for the equilibrium equations using the first-order shear deformation plate theory approach 2

b 11 a

6 6a 6 b 21 6 6 0 6 6a 4 b 41 b 51 a

b 12 a

0

b 14 a

b 22 a

0

b 24 a

0

b 33 a

b 34 a

b 42 a

b 43 a

b 44 a

b 52 a

b 53 a

b 54 a

9 8 9 38 0> > > Umn > > > > b 15 > a > > > > > > > > > > > 7> > > > > 0 > > > > V b 25 7 a mn > > > = = < > 7< 7 b 34 7 Wmn ¼ 0 a > > 7> > > > > > > > > > > > Emn > b 45 7 a > > > > 5> 0 > > > > > > > > > > > > b 55 : Emn ; : ; a 0

(C.1)

222

Stability and Vibrations of Thin-Walled Composite Structures

where
b 11 h A11 a

mp a

2 þ A66

b 14 ¼ a b 41 h B11 a
b 22 h A66 a

mp a

" b 33 h K A44 a

b 34 a




b 21 h ðA12 þ A66 Þ b 12 ¼ a a


2
2 mp np þ B66 a b

2

b 25 ¼ a b 52 h B66 a


2 np b

þ A22


2 np b



 mp np a b

b 51 h ðB12 þ B66 Þ b 15 ¼ a a

b 42 h ðB12 þ B66 Þ b 24 ¼ a a



 mp np a b



 mp np a b


2
2 mp np þ B22 a b

np b

2


þ A55


 mp b 43 h KA55 ¼a a

mp a

b 35 a

2 #  N xx


2
2 mp np  N yy a b


 np b 53 h KA44 ¼a b


2
2 mp np b 44 h D11 a þ D66 þ KA55 a b b 54 h ðD12 þ D66 Þ b 45 ¼ a a



 mp np a b


2
2 np mp b 55 h D22 a þ D66 þ KA44 b a (C.2)

6

Dynamic buckling of composite columns and plates Haim Abramovich Technion, I.I.T., Haifa, Israel

6.1

Introduction

This chapter will deal with what is called in the literature “dynamic” buckling of columns and plates (metal and composite materials). First the term “dynamic” buckling will be explained and defined followed by examples from the literature. Then the equations of motions for columns and plates will be presented, and numerical and experimental results of tests performed on columns, plates, and shells will be highlighted. The topic of applying an axially time-dependent load onto a column, thus inducing lateral vibrations and eventually causing the buckling of the column, was studied for many years. Sometimes this is called vibration buckling, as proposed by Lindberg [1]. As it is described in his fundamental report [1], the axial oscillating load might lead to unacceptable large vibration amplitudes at a critical combination of the frequency and amplitude of the axial load and the inherent damping of the column. This behavior is presented in Fig. 6.1(a), in which an oscillating axial load induces bending moments

(a)

F(t)

F(t)

F(t)

t(time)

(b)

F(t)

F(t)

F(t)

t(time)

Figure 6.1 (a) Buckling under parametric resonance and (b) pulse-type buckling. Stability and Vibrations of Thin-Walled Composite Structures. http://dx.doi.org/10.1016/B978-0-08-100410-4.00006-5 Copyright © 2017 Elsevier Ltd. All rights reserved.

224

Stability and Vibrations of Thin-Walled Composite Structures

that cause lateral vibrations of the column. As described in Ref. [1] the column will laterally vibrate at large amplitudes when the loading frequency will be twice the natural lateral bending frequency of the column. The term used by Lindberg, vibration buckling, presents some kind of similarity to vibration resonance. However, in the case of vibration resonance the applied load is in the same direction as the motion, namely, in our case lateral to the column, and the resonance will occur when the loading frequency equals the natural frequency of the column. This type of vibration buckling is called by Lindberg as the dynamic stability of vibrations induced by oscillating parametric loading. This type of resonance is also called in the literature as parametric resonance (see an application of this type of dynamic stability in Refs. [2,3]). Another type of vibration type is sometimes also called pulse buckling, where the structure will be deformed to unacceptably large amplitudes as a result of a transient response of the structure to the dynamic axially applied load [1]. The sudden applied load might cause a permanent deformation because of the plastic response of the column, a snap to a larger postbuckling deformation, or simply a return to its undeformed state. This is pictured in Fig. 6.1(b), in which the response of the column to a sudden short-time axial load is shown. Buckling will occur when an unacceptably large deformation or stress is encountered by the column. The column can withstand a large axial load before reaching the buckling condition, provided the load duration is short enough. Under an intense, short-duration axial load, the column would buckle into a very-high-order mode, as shown in Fig. 6.1(b). Lindberg [1] claims that pulse buckling falls under the following mathematical definition: dynamic response of structural systems induced by timevarying parametric loading. Throughout this chapter, the pulse buckling will be considered equivalent to dynamic buckling. The dynamic buckling of structures has been widely addressed in the literature. It started with the famous paper by Budiansky and Roth [4], through Hegglin’s report on dynamic buckling of columns [5], and continued with Budiansky and Hutchinson [6] and Hutchinson and Budiansky [7] in the mid-sixties. Then more structures have been addressed as presented in Refs. [1,2,8e43]. It is most intriguing and challenging to define a criterion for the critical load causing the structure to buckle under the subjected pulse loading. As presented by Kubiak [32] and Ari-Gur [13,19,22] as well as by other authors [18,20,24e26], a new quantity called dynamic load factor (DLF) is introduced to enable the use of the dynamic buckling criteria. It is defined as DLF h

Pulse buckling amplitude ðPcr Þdyn: h Static buckling amplitude ðPcr Þstatic

(6.1)

According to Kubiak [32] the most popular criterion had been proposed by Volmir [10] for plates subjected to in-plane pulse loading. As quoted in Ref. [32], Volmir proposed the following criterion: Dynamic critical load corresponds to the amplitude of pulse load (of constant duration) at which the maximum plate deflection is equal to some constant value k (k - half or one plate thickness).

Dynamic buckling of composite columns and plates

(b)

λ

L

Quadratic

⎛x x ⎛2 ⎛ F = kL ⎜ – – α ⎛ – ⎝ ⎝ ⎝⎜ L L ⎝

(x + x)

(c)

R (λ ,t)

(a)

225

Δmax 1.5 1.0

L

λ=5

Or cubic

⎛x x ⎛3 ⎛ F = kL ⎜ – – β ⎛ – ⎝ ⎝ ⎝⎜ L L ⎝

Pcr = 0.52

0.5

(λ D)cr

λ

0 0 0.2 0.4 0.6 0.8 P

Figure 6.2 (a) The nonlinear model, (b) the Budiansky and Hutchinson (B&H) schematic criterion, and (c) the application of the B&H criterion to axisymmetric dynamic buckling of clamped shallow spherical shells. (a, b) Adapted from B. Budiansky, J.W. Hutchinson, Dynamic buckling of imperfection sensitive structures, in: H. G€otler (Ed.), Proceedings of the 11th International Congress of Applied Mechanics, 1964, Springer-Verlag, Berlin, 1966, pp. 636e651. (c) Adapted from B. Budiansky, R.S. Roth, Axisymmetric Dynamic Buckling of Clamped Shallow Spherical Shells, Collected Papers on Instability of Shell Structures, NASA TN-D-1510, 761 p., 1962, pp. 597e606.

Another, very widely used, criterion has been formulated and proposed by Budiansky and Hutchinson [6] based on an earlier work [4] and was later extended [7]. Originally the criterion was formulated for shell-type structures but was also used for columns and plates. The criterion claims that “Dynamic stability loss occurs when the maximum deflection grows rapidly with the small variation of the load amplitude.” This criterion is schematically presented in both Fig. 6.2(b), where R(l, t) is the response of the simply nonlinear model assumed in Ref. [6], and Fig. 6.2(a), whereas l is the nondimensionally applied dynamic pulse-type compressive load. Fig. 6.3(c) presents the application of the criterion for the axisymmetric dynamic buckling of clamped shallow spherical shells as presented in Ref. [4]. Other dynamic buckling criteria were suggested and applied by Ari-Gur and Simonetta [22]. They formulated four criteria that are presented schematically in Fig. 6.3(a)e(d). The first criterion (see Fig. 6.3(a)) correlates the maximum lateral deflection, Wm, to the pulse intensity defined as Lm. The first criterion is stated as “Buckling occurs when, for a given pulse shape and duration, a small increase in the pulse intensity causes a sharp increase in the rate of growth of the peak lateral deflection” [22]. The authors claim that this criterion can be used to both displacement and force loading types for a wide range of pulse frequencies; however, for very short pulse durations the results might be misleading. The reason for this is connected to the characteristics of the out-of-plane deflections that, for short pulse duration in the vicinity of buckling loads, turn out to have short-wavelength patterns that are associated with smaller peak deflections. Therefore the authors present the second criterion

226

Stability and Vibrations of Thin-Walled Composite Structures

Lm

Lm

Wm

Wm Wm

Lm

Wm Lm

(a)

(b)

Um

Fm

Fm

Um Um

Fm (Compression) Um (Tension) Fm

(c)

(d)

Figure 6.3 Buckling criteria: (a) the first criterion, (b) the second criterion, (c) the third criterion, and (d) the fourth criterion. Adapted from B. Budiansky, J.W. Hutchinson, Dynamic buckling of imperfection sensitive structures, in: H. G€otler (Ed.), Proceedings of the 11th International Congress of Applied Mechanics, 1964, Springer-Verlag, Berlin, 1966, pp. 636e651.

(Fig. 6.3(b)), which answers the deficiency of the first criterion, that is suitable to patterns of short-wavelength deflection shapes. It claims that “dynamic buckling occurs when a small increase in the pulse intensity causes a decrease in the peak lateral deflection” and is relevant to only impulsive loads and may be used in addition to the first criterion The last two criteria presented in Ref. [22] are related to the intensity of the applied load versus the maximum response of the loaded edge, say at x ¼ 0. According to the third buckling criterion (Fig. 6.3(c)), “buckling occurs when a small increase in the force intensity Fm causes a sharp increase in the peak longitudinal displacement Um at x ¼ 0.” The buckling phenomenon under pulse loading is due to the diminishing of the structural resistance to an in-plane compressive load when the dynamic lateral deflections grow rapidly. The fourth buckling criterion, associated to a displacement pulse (Fig. 6.3(d)), states that “buckling occurs when a small increase in the pulse displacement intensity Um causes a transition of the peak reaction force Fm at x ¼ 0 from compression to tension.” This transition is possible when the tensile force needed to keep the deforming structure at the prescribed Um is greater than the maximum compression at its loaded edge.

Dynamic buckling of composite columns and plates

227

Another interesting criterion is suggested by Petry and Fahlbusch [24], claiming that the BudianskyeHutchinson criterion is very conservative for structures with stable postbuckling equilibrium path because it does not take into account the loadcarrying capacity of the structure. The criterion proposed is based on stress analysis and claims that “a stress failure occurs if the effective stress sE exceeds the limit stress of the material; a dynamic response caused by an impact is defined to be dynamically stable if sE  sL is fulfilled at every time everywhere in the structure.” Using this criterion, which is claimed to be practical also for ductile materials (by using the yield stress sY instead of sL), the DLF (see Eq. 6.11) is modified to DLF h

ðNF Þdyn.

(6.2)

ðNF Þstatic

where ðNF Þdyn. and ðNF Þstatic are defined as the dynamic failure load and static failure load, respectively.

6.2

Dynamic buckling of columns

6.2.1

Dynamic buckling of columns using the CLT

The column differential equations of motion under time-dependent axial compression, as presented in Fig. 6.4, are given by Nx;x ¼ I0 u€  I1 w€;x

(6.3)


Mx;x þ Nx $w;x þ I2 $w€;xx  ¼ I0 w€ þ I1 u€;x

(6.4)

where Z ðI0 ; I1 ; I2 Þ ¼ r

h=2 

h=2

 1; z; z2 dz

(6.5)

and ( ),x is the first partial differentiation with respect to x; u€; w€ are the second partial differentiation of the axial and lateral displacements, respectively, with respect to time, t; h is the total thickness of the column; and r is the relevant density for each layer in the laminate. Assuming that the force and moment resultants can be written as a function of the strains and curvatures, according to the CLT approach, we obtain (

Nx Mx

)

" ¼

A11

B11

B11

D11

#(

εx kx

) (6.6)

228

Stability and Vibrations of Thin-Walled Composite Structures

N0 (t) z, w w0

L

x,u

Figure 6.4 A column with geometric initial imperfection, w0, subjected to time-dependent axial compression [N0(t) is the axial compression at x ¼ 0].

where Z ðA11 ; B11 ; D11 Þ ¼

h=2 h=2

  Q11 1; z; z2 dz

(6.7)

and Q11 is the plane stress-reduced stiffness for each layer after transformation (for an exact expression see Chapters 4 and/or 5). The strain displacement (u, longitudinal displacement; w, lateral displacement) relationships are (

εx kx

)

8 9 1 > = < u;x þ w2;x  w20;x > 2 ¼ > > ; : ðw  w0 Þ;xx

(6.8)

Substitution of Eqs. (6.6)e(6.8) into Eqs. (6.3) and (6.4) yields   A11 u;xx þ w;x $w;xx  w0;x $w0;xx  B11 ðw  w0 Þ;xxx ¼ I0 u€  I1 w€;x

(6.9)

Dynamic buckling of composite columns and plates

229

"

 # 3 2 1 A11 ðux $wx Þx þ wx $wxx  w0;x w0;xx $wx  w0;x $wxx 2 2
þ B11 uxx þ w0;xx ðw  w0 Þx x  D11 ½w  w0 xxxx ¼ I0 w€ þ I1 u€x

(6.10)

Assuming a symmetric laminate, B11 ¼ I1 ¼ 0, then Eqs. (6.9) and (6.10) are simplified to yield   A11 u;xx þ w;x $w;xx  w0;x $w0;xx ¼ I0 u€

A11



u;x $w;x



(6.11)

  3 2 1 þ w;x $w;xx  w0;x w0;xx $w;x  w0;x $w;xx ;x 2 2

 D11 ½w  w0 ;xxxx ¼ I0 w€

(6.12)

One should note that all the terms in Eqs. (6.11) and (6.12) are a function of both x and t. The associated boundary conditions (pinned-pinned or simply supported) for the symmetric laminate are Lateral deflection :

wð0; tÞ ¼ wðL; tÞ ¼ 0;

(6.13)

Bending moment:

D11 $w;xx ð0; t Þ ¼ D11 $w;xx ðL; t Þ ¼ 0;

(6.14)

Axial force:

A11 $u;x ð0; t Þ ¼ Nx ð0; t Þ ¼ N0 ðt Þ ¼ N0 sin

Axial displacement:

u;x ðL; t Þ ¼ 0:

pt ; T

(6.15) (6.16)

while the initial conditions assume that wð x; 0Þ ¼ w0 w;x ð x; 0Þ ¼ uð0; 0Þ ¼ u_ð0; 0Þ ¼ 0. The solution for the equations of motion, Eqs. (6.11) and (6.12), can be obtained using energy methods, such as the Galerkin method, using the boundary and initial conditions presented in Eqs. (6.13)e(6.16). To apply the Galerkin approach, one should assume displacements that satisfy the boundary conditions of the problem, such as for the present case the following functions will be assumed: wðx; tÞ ¼ AðtÞsin

px ; L

px uðx; 0Þ ¼ BðtÞcos ; L

w0 ðxÞ ¼ A0 sin

px L

pt N0 ðtÞ ¼ N0 sin . T

(6.17)

230

Stability and Vibrations of Thin-Walled Composite Structures

Column A0/h = 0.1

50

2T/Tb = 1

Wc max (mm)

40

30

2T/Tb = 4

2T/Tb = 2

2T/Tb = 6

20 2T/Tb = 20

10 β

Static

– a = 1/tg β "Dynamic buckling” load

0 0

1000

2000 N0 (N)

3000

4000

Figure 6.5 The influence of the loading duration (T) on the column response; Tb, the first natural period of the column; and A0, the amplitude of the initial geometric imperfection. Adapted from T. Weller, H. Abramovich, R. Yaffe, Dynamic buckling of beams and plates subjected to axial impact, Computers and Structures 32 (3e4) (1989) 835e851.

Multiplying Eq. (6.11) by the expression for u(x, 0) while Eq. (6.12) is multiplied by w(x, t) and integrating both from x ¼ 0 till x ¼ L will exclude the x dependence, yielding two nonlinear time-dependent equations that have to be solved numerically. For each value of N0, the response of the beam, w and u, can be calculated for an assumed initial geometric imperfection, w0 ðx; tÞ ¼ A0 sin px L , where A0 is a known amplitude (usually percentage of the column thickness) for a constant time period T. A typical drawing is presented in Fig. 6.5. Calculating the dynamic buckling for each value of 2T/Tb and dividing it by the static buckling load will yield the DLF for the tested or calculated case (see Eq. 6.1).

6.2.2

Dynamic buckling of columns using the FSDT

To solve the dynamic buckling of columns (see Fig. 6.4) using the FSDT, we shall assume the following displacement field: ux ðx; z; tÞ ¼ uðx; tÞ þ zfx ðx; tÞ uz ðx; z; tÞ ¼ wðx; tÞ  w0 ðxÞ

(6.18)

where the assumed variables u and w are displacements in the x and z directions, respectively, of a point on the midplane of the plate (namely, at z ¼ 0); fx is a rotation about the x axis; and w0 is the initial geometric imperfection of the column. The equations of motion can be written as (see Chapters 4 and 5)

Dynamic buckling of composite columns and plates

231

€x Nx;x ¼ I0 u€ þ I1 f
  Qx;x þ Nx $ wx  w0;x x ¼ I0 w€ € x þ I1 u€ Mx;x  Qx ¼ I2 f

(6.19)

In Eq. (6.19), Qx is the shear force or the transverse force resultant defined as Z þh=2 Qx ¼ K sxy $dz (6.20) h=2

where K is the shear correction coefficient computed by equating the strain energy due to transverse shear stresses to the strain energy due to true transverse shear as calculated by a 3D elasticity theory and sxy is the transverse shear stress. For a homogenous beam having a rectangular cross section, K ¼ 5/6 (see Chapter 4). The force and moment resultants as a function of the displacements are given by 8 9 9 2 8 1 2 3> 2 > > > A11 B11 Nx > 0 > > > > > u;x þ 2 w;x  w0;x > > = = 6 < 7< 7 (6.21) B Mx ¼ 6 D 0 11 11 4 4 5 x;x > > > > > > > > > ; > : > ; : 0 0 KA55 > Qx 4x þ w;x  w0;x As was assumed in section Dynamic buckling of columns using the CLT, the strains have the following form 9 8  9 8   ( ) > = > = < ux;x þ 1 w2  w2 > < u;x þ 1 w2  w2 þ z4x;x > εx 0;x 0;x 2 ;x 2 ;x ¼ ¼ > > > ; > ; : : gxz ux;z þ uz;x 4x þ w;x  w0;x (6.22) Substituting Eq. (6.21) into Eq. (6.19) yields the following equations of motion expressed by the three assumed displacements u(x, t), w(x, t), and fx(x, t) and the known initial geometric imperfection, w0(x) (see also a similar derivation in Ref. [25])   €x A11 u;xx þ w;x $w;xx  w0;x $w0;xx þ B11 4x;xx ¼ I0 u€ þ I1 4
    A11 u;xx þ w;x $w;xx  w0;x $w0;xx þ B11 4x;xx w;x  w0;x

     1 1 þ A11 u;x þ w2;x  w20;x þ B11 4x;x w;xx  w0;xx 2 2   þ KA55 4x;x þ w;xx  w0;xx ¼ I0 w€   B11 u;xx þ w;x $w;xx  w0;x $w0;xx   € x þ I1 u€ þ D11 4x;xx  KA55 4x þ w;x  w0;x ¼ I2 4

(6.23)

232

Stability and Vibrations of Thin-Walled Composite Structures

For a symmetric layup, B11 ¼ I1 ¼ 0, Eq. (6.23) is simplified to yield   A11 u;xx þ w;x $w;xx  w0;x $w0;xx ¼ I0 u€
    A11 u;xx þ w;x $w;xx  w0;x $w0;xx w;x  w0;x þ

      1 1 A11 u;x þ w2;x  w20;x w;xx  w0;xx þ KA55 4x;x þ w;xx  w0;xx ¼ I0 w€ 2 2   €x D11 4x;xx  KA55 4x þ w;x  w0;x ¼ I2 4 (6.24) As in the previous section, all the terms in Eq. (6.24) are a function of both x and t. The associated boundary conditions (pinned-pinned or simply supported) for the symmetric laminate are Lateral deflection:

wð0; tÞ ¼ wðL; tÞ ¼ 0;

(6.25)

Bending moment:

D11 $fx;x ð0; tÞ ¼ D11 $fx;x ðL; tÞ ¼ 0;

(6.26)

Axial force:

A11 $u;x ð0; t Þ ¼ Nx ð0; t Þ ¼ N0 ðt Þ ¼ N0 sin

pt ; T

(6.27)

u;x ðL; t Þ ¼ 0:

Axial displacement:

(6.28)

while the initial conditions assume that wð x; 0Þ ¼ w0 w;x ð x; 0Þ ¼ uð0; 0Þ ¼ u_ð0; 0Þ ¼ 0. As discussed earlier, the analytic solution of the equations of motion cannot be obtained; therefore, energy methods, such as the Galerkin method, are suggested. Suitable assumed solutions for the case of simply supported boundary conditions are uðx; 0Þ ¼ AðtÞcos

px ; L

px w0 ðxÞ ¼ A0 sin L

wðx; tÞ ¼ BðtÞsin

pt N0 ðtÞ ¼ N0 sin . T

px ; L

fx ðx; tÞ ¼ CðtÞcos

px ; L

(6.29)

The procedure described in section Dynamic buckling of columns is again used to obtain the response of the column to a pulse-type loading. As can be seen in Fig. 6.6, based on the results presented in Ref. [25], the response of the column has a similar behavior to what had been presented in Fig. 6.5. Note that in Fig. 6.6(b), the compressive strain is a linear function of the applied axial compression load; therefore, it can be compared with what it is presented in Fig. 6.5.

Dynamic buckling of composite columns and plates

233

Initail geometric imperfections

2.0

2.0

1.5

1.5

Deflection (mm)

Deflection (mm)

W0 = 0.001 h W0 = 0.01 h W0 = 0.1 h

1.0 0.5 0.0 0.0

0.5

1.0 1.5 2.0 Axial displacement (mm)

2.5

3.0

1.0 0.5 0.0 0.0

(a)

5000

10,000 15,000 Strain (μstrain)

20,000

25,000

(b)

Figure 6.6 The dynamic pulse buckling response of a column (L ¼ 150 mm, width ¼ 20 mm) for various initial geometric imperfections: (a) mid-span deflection versus maximal axial displacement at the impacted end and (b) mid-span deflection versus compressive strain at the neutral axis of the column mid-span. Adapted from Z. Zheng, T. Farid, Numerical studies on dynamic pulse buckling composite laminated beams subjected to an axial impact pulse, Composite Structure 56 (3) (2002) 269e277.

6.3

Dynamic buckling of plates

The present derivation for the dynamic buckling of orthotropic rectangular plates under uniaxial loading is based on Ekstrom’s fundamental study [11] presented already in 1973 using the models developed by Lekhnitskii for anisotropic plates [44]. Let us assume a simply supported rectangular orthotropic plate uniaxial loaded as presented in Fig. 6.7. We choose the natural axes of the material to coincide with the coordinate axes leading to the following in-plane stressestrain relations:

y

Nx

Nx

b a

Figure 6.7 A thin orthotropic plate uniaxially loaded.

x

234

Stability and Vibrations of Thin-Walled Composite Structures

8 9 2 3 sx > Ex yxy Ey 0 > > > < = 6 7 1 6 yyx Ex 7 sy ¼ Ey 0 4 5 > > 1  yxy $yyx > > : ; 0 0 ð1  yxy $yyx ÞGxy sxy 9 2 9 8 38 εx > ε Q11 Q12 0 > x > > > > > > = 6 = < < 7 7 ε h6 Q  εy Q 0 22 4 21 5> y > > > > > > > ; ; : : gxy gxy 0 0 Q66 2

yxy Ex

1 6 E 8 9 6 x εx > 6 > > > < = 6 6 yyx εy ¼6 6  Ey > > > > : ; 6 6 gxy 6 4 0



Q12 ¼ Q21 0 yyx ¼

Ex yxy Ey

1 Ey 0

(6.30a)

3 0 7 78 9 7> sx > > 7> 7< = 7 0 7 sy > > > 7> 7: sxy ; 7 1 5 Gxy

(6.30b)

and (6.31)

The dynamic buckling problem can be solved by assuming out-of-plane initial geometric imperfections (w0), and therefore, the strains will be assumed to have the following expressions "  8  2 # 9 2 > > vu 1 vw vw 0 > > > >  > þ > > > > > vx 2 vx vx > > 9 > 8 > > > > > ε x > > > > " # > >  2  2 > < = = > < vv 1 vw vw 0 εy ¼  > vy þ 2 > > > > > > > vy vy > ; > : > > > > gxy > > > > > > > > 2 2 > > vu vv v w v w > > 0 > > : ; þ þ  vx vy vxvy vxvy

(6.32)

where w(x, y, t) is the total out-of-plane geometric displacement, w0(x, y) is the initial out-of-plane geometric displacement, and u(x, y, t) and v(x, y, t) are the in-plane displacements in the x and y directions, respectively.

Dynamic buckling of composite columns and plates

235

The compatibility equation for the stress function F(x, y, t) and the plate equations of motion (see Refs. [11,44]) can be written as    2 2  2 2 yxy 1 v4 F 1 v4 F 1 v w v w0 v4 w0 $ 4 þ $ 4 þ2   þ 2 2 ¼ Ey vx Ex vy 2Gxy Ex vxvy vxvy vx vy 

v4 w vx2 vy2 (6.33a)

v4 ðw  w0 Þ v4 ðw  w0 Þ v4 ðw  w0 Þ þ 2ðD1 þ 2Dxy Þ þ Dy 4 2 2 vx vx vy vy4

2  v w v2 F v2 w v2 F v2 w v2 F v2 w $ þ ¼h $ 2 $ r 2 vx2 vy2 vxvy vxvy vy2 vx2 vt

Dx

(6.33b)

where h is the plate thickness and r is the plate density. The flexural densities of the plate are defined as follows Dx h

Ex h3 ; 12ð1  yxy yyx Þ

Dy h

Ey h3 ; 12ð1  yxy yyx Þ

Dxy ¼

Gxy h3 ; 12 (6.34)

Ex yyx h3 Ey yxy h3 ¼ . D1 h 12ð1  yxy yyx Þ 12ð1  yxy yyx Þ The out-of-plane boundary conditions of the present problem, assuming that the initially straight edges of the plate remain straight after buckling, can be written as v2 w v2 w0 ¼ ¼ 0; vx2 vx2

at x ¼ 0; a w ¼ w0 ¼ 0 and

v2 w v2 w0 ¼ ¼ 0: at y ¼ 0; b w ¼ w0 ¼ 0 and vy2 vy2

(6.35)

The in-plane boundary conditions have no restrain at y ¼ 0, b and in the x direction at x ¼ 0; a

1 b

1 at y ¼ 0; b a

Z

b

0

Z 0

a

sx dy ¼

1 b

1 sy dx ¼ a

Z 0

Z 0

b 2

v F dy ¼ P; vy2 (6.36)

a 2

v F dx ¼ 0: vx2

where P is the compression load acting on the plate.

236

Stability and Vibrations of Thin-Walled Composite Structures

The assumed solution for w and the initial geometric imperfection, w0, has the following forms wðx; y; tÞ ¼ AðtÞsin

mpx npy sin a b

(6.37)

w0 ðx; y; tÞ ¼ Ao sin

mpx npy sin a b

(6.38)

The expressions in Eqs. (6.37) and (6.38) satisfy the boundary conditions assumed in Eq. (6.35). Substituting those equations in the compatibility equation leads to the following relationship between the stress function F and the displacements   

yxy 1 v4 F 1 v4 F 1 m 2 n2 p 4 2mpx 2npy þ cos $ 4 þ $ 4 þ2  ¼ cos Ey vx Ex vy 2Gxy Ex a b 2a2 b2  2  2  A  A0 (6.39) The solution of Eq. (6.39), which also satisfies the in-plane boundary conditions in Eq. (6.36), can be written as 

Fðx; y; tÞ ¼ A

2

 A20



 a2 n2 Ey 2mpx b2 m2 Ex 2npy P þ cos cos  y2 2 2 2 2 a b 2 32b m 32a n

(6.40)

Eq. (6.40) is then inserted into Eq. (6.32) yielding 

p4 m4 m2 n2 n4 m 2 p2 Dx 4 þ 2ðD1 þ 2Dxy Þ 2 2 þ Dy 4 ½AðtÞ  A0  ¼ 2 PðtÞ$AðtÞ h a a b b a 

d 2 AðtÞ p4 m4 2npy n4 2mpx
2 þ 4 Ey cos A ðtÞ  A20 AðtÞ r þ E cos x dt 2 b a 8 a4 b (6.41) Note that Eq. (6.41) is a nonlinear equation containing functions of the variables x, y, and t. To eliminate the x, y-dependency, the Galerkin method is applied, which mpx npy sin dxdy demands the multiplication of both sides of the equation by sin a b

Dynamic buckling of composite columns and plates

237

and integrating over the middle plane of the plate yielding the following nonlinear time-dependent expression: 

d2 AðtÞ p4 m4 m 2 n2 n4 þ þ 2ðD þ 2D Þ þ D ½AðtÞ  A0  D x xy y 1 dt 2 rh a4 a2 b 2 b4 


2 m2 p2 p 4 m4 n4 2  2 PðtÞ$AðtÞ þ E þ E x y A ðtÞ  A0 AðtÞ ¼ 0 4 4 a r 24r a b

(6.42)

where m and n are both odd integers. The time dependence of the term P(t) is chosen according to its time dependence, namely, for a step function P(t) ¼ P0 (constant), for an impulse P(t) ¼ P0d(t) (where (t) is the Kronecker delta1), or any other time function. Eq. (6.42) can be nondimensionalized (assuming n ¼ 1) to yield the following expression (see discussion in Ref. [11]):

4 ð1  yxy $yyx Þ d2 x m þ 2R12 m2 b2 þ R22 b4 ðx  x0 Þ  m2 sx þ þ S 2 ds 8 4   4   m þ R22 b4 x2  x20 x ¼ 0

(6.43)

where P1 h

4Dx p2 ; a2 h

A xh ; h

D1 þ 2Dxy R12 h ; Dx

x0 h

A0 ; h

Dy Ey R22 h ¼ ; Dx Ex

a bh ; b

sh

P3 p2 S h 21 2 : c a r

P c$t ¼ ; P1 P1 (6.44)

Eq. (6.43) is solved using numerical methods for integration of nonlinear equations, such as the famous RungeeKutta methods.2 Solutions for the response of the plate for various S values are presented in [11]. It is interesting to note that Eq. (6.42) can be used to obtain the static plate buckling load and its natural frequencies, as well as the expression for large plate deflections.

1

2

The function is 1 if the variables are equal, and 0 otherwise, that is, dij ¼ 0

if i s j;

dij ¼ 1

if i ¼ j.

The RungeeKutta methods are a family of implicit and explicit iterative methods (including the Euler method’s routine) used in temporal discretization for the approximate solutions of ordinary differential equations developed approximately in 1900 by C. Runge and M. W. Kutta.

238

Stability and Vibrations of Thin-Walled Composite Structures

By assuming the static case for a perfect plate d2 AðtÞ ¼ 0 and A0 ¼ 0 dt 2

(6.45)

and neglecting high-order terms such as A3, Eq. (6.42) yields 

p2 m4 m 2 n2 n4 m2 Dx 4 þ 2ðD1 þ 2Dxy Þ 2 2 þ Dy 4  2 P ¼ 0 h a a b b a

(6.46)

The critical buckling load of the plate is obtained from Eq. (6.46) by assuming n ¼ 1, i.e., p 2 Dx Pcr ¼ 2 b h

# "  mb 2 2ðD1 þ 2Dxy Þ Dy  a 2 þ þ a Dx Dx mb

a 0ðPcr Þmin. occurs when b

rffiffiffiffiffiffi Dy ¼ integer Dx

(6.47)

By neglecting high-order terms such as A3 and assuming a perfect plate (A0 ¼ 0), Eq. (6.42) is written as 

d2 AðtÞ p4 m4 m 2 n2 n4 m2 p 2 þ þ 2ðD þ 2D Þ þ D D x 4 xy y 4 AðtÞ  2 PðtÞ$AðtÞ ¼ 0 1 2 2 2 dt rh a a b b a r (6.48) The second term in Eq. (6.48) is the square of the natural frequency of the perfect plate, namely, u2mn ¼



p4 m4 m2 n2 n4 Dx 4 þ 2ðD1 þ 2Dxy Þ 2 2 þ Dy 4 rh a a b b

(6.49)

Finally, assuming the static case, the expression for the postbuckling behavior for a perfect plate with large deflections is obtained from Eq. (6.42)

 

p4 m4 m2 n2 n4 m 2 p2 Dx 4 þ 2ðD1 þ 2Dxy Þ 2 2 þ Dy 4  2 PAðtÞ h a a b b a 

p 4 m4 n4 þ Ex þ 4 Ey A3 ¼ 0 4 24 a b

(6.50)

Dynamic buckling of composite columns and plates

239

Eq. (6.50) shows that the lateral deflection would be zero until the critical buckling load given by Eq. (6.47) is reached. After that point, nonzero deflections are possible, while the loadedeflection relation is cubic. For further information on the behavior of plates under dynamic buckling, the reader is referred to Refs. [16e20,22,24,32,36]. The definition of the buckling loads for plates, both for a static and dynamic case, is one of the issues to be solved consistently. To assist the definition of the buckling load for a plate, a method is described in Appendix A, which can be easily applied for experimental and numerical data consisting of deflections versus applied load.

6.4

Dynamic buckling of thin-walled structures: numerical and experimental results

The dynamic buckling of thin-walled structures has been dealt in depth also from the experimental point of view, in addition to various calculations using different approaches. Fig. 6.8(a) presents a shallow clamped spherical cap loaded by a sudden rectangular pressure q applied at time t ¼ 0 and held constant for a given period and then suddenly removed (as described in Ref. [4]). Typical numerical results are then shown in Fig. 6.8(b), showing the reduction of the critical nondimensional pressure with increasing the duration of the applied pressure (see other results in Ref. [4]). In the static critical load, from a certain nondimensional time duration (in this case from approximately s ¼ 3) the static critical pressure is above the dynamic critical pressure, whereas for short duration the dynamic critical load is higher than the static

(a)

(b) 2.0

2a h

λ=5

1.5 Z0

(pcr)dynamic

H pcr

r

1.0 (pcr)static

R

0.5

0

0

2

4

6

–τ

8

10

12

Figure 6.8 (a) A clamped shallow spherical cap and (b) variation of the nondimensional critical pressure with nondimensional load duration. Adapted from B. Budiansky, R.S. Roth, Axisymmetric Dynamic Buckling of Clamped Shallow Spherical Shells, Collected Papers on Instability of Shell Structures, NASA TN-D-1510, 761 p., 1962, pp. 597e606.

240

Stability and Vibrations of Thin-Walled Composite Structures

(b) 4

(a) a

0

3

x DLF

b z y

a/b=1 h / b = 0.005 W11 / h = 0.2 σL = 100 MPa 0

2

h 1 0 0

1

2

3

4

TS /TP

5

Figure 6.9 (a) An isotropic flat plate and (b) variation of DLF with the nondimensional load duration. From D. Petry, G. Fahlbusch, Dynamic buckling of thin isotropic plates subjected to in-plane impact, Thin-Walled Structures 38 (3) (2000) 267e283.

one, as is normally known for short-duration loads. The various nondimensional terms used in Fig. 6.8 were defined in Ref. [4] as rffiffiffiffi ct E q s¼ ; c¼ ; p¼ ; R r q0 (6.51)  1=2  1=2
  2E h 2 1=4 H l¼2 3 1y ; q0 ¼
  1=2 R h 3 1  y2 where E is the Young’s modulus, y is the Poisson’s ratio, and r is the density. Similar results were reported in Ref. [24], in which an isotropic plate (Fig. 6.9(a)) is sudden loaded by a rectangular impulse and the response of the plate is presented using the term DLF as defined in Eq. (6.2) as a function of the nondimensional time duration, where Ts is the impact period and Tp is the natural period of the plate. As shown by other researchers, in the vicinity of Ts/Tp ¼ 1 the term DLF is less than unity, implying that the static load is higher than the dynamic one. Shifting this period ratio toward 0.5 shows a rapid increase in the value of DLF, reaching a maximal value of 3.6 in the vicinity of Ts/Tp ¼ 0.1 (approximately); that is, as expected, the plate can withstand high dynamic compressive stresses (3.6 is the static buckling load of the plate), provided the time duration is very short. Impacting the plate with sinusoidal or triangular impulses displays the same behavior as the rectangular one (see Fig. 6.10). Ref. [20] describes the experimental results for laminated composite plates under impulse-type uniaxial loading obtained by dropping masses and measuring the response using a back-to-back strain gages bonded in the center of the plate. Fig. 6.11 presents the experimental variation of the DLF term defined as DLF h

ðεcr Þdyn: ðεcr Þstatic

(6.52)

Dynamic buckling of composite columns and plates

241

4 a/b=1 h / b = 0.005 σ = 100 MPa L

Rectangular impact

3

DLF

Sinusoidial impact Triangular impact

2

1

0 0.0

0.5

1.0

1.5 TS /TP

2.0

2.5

3.0

Figure 6.10 Variation of DLF with the nondimensional load duration for three types of impulse. Taken from D. Petry, G. Fahlbusch, Dynamic buckling of thin isotropic plates subjected to in-plane impact, Thin-Walled Structures 38 (3) (2000) 267e283. 2 CS21 CS22 CM32 KM32

DLF

1.5

1

0.5

0

0

2

4

6 Timp/Tb

8

10

12

Figure 6.11 Variation of DLF with the nondimensional load durationdexperimental results. Adapted from H. Abramovich, A. Grunwald, Stability of axially impacted composite plates, Composite Structures 32 (1e4) (1995) 151e158.

as a function of the nondimensional period ratio Timp/Tb, where Timp is the period of the impact as measured by the bonded strain gages and Tb is the lowest natural frequency of the plate. As shown earlier, DLF less than 1 was measured for some values of nondimensional period ratio, whereas for short periods, the DLF is higher than unity, reaching experimental values of up to 2. The properties of the specimens presented in Fig. 6.11 are summarized in Table 6.1, while the material properties are presented in Table 6.2 (using the data published in Ref. [20]). Table 6.3 presents additional

242

Stability and Vibrations of Thin-Walled Composite Structures

Table 6.1

The properties of the tested plates Total thickness measured (mm)

Specimen

Material

Layup

Total number of layers

CS21

Graphite-epoxy HT-T300 (Torey)

(45 , 45 , 45 )sym

12

1.63

CS22

Graphite-epoxy HT-T300 (Torey)

(45 , 45 , 45 )sym

12

1.63

CM32

Graphite-epoxy HT-T300 (Torey)

(0 , 45 , 90 , 45 , 0 )

9

1.125

KM32

Kevlar (DuPont)

(0 , 45 , 90 , 45 , 0 )

9

1.125

From H. Abramovich, A. Grunwald, Stability of axially impacted composite plates, Composite Structures 32 (1e4) (1995) 151e158.

Table 6.2

Material properties of the tested plates

Material

E11 (MPa)

E22 (MPa)

G12 (MPa)

y12

y21

Graphite-epoxy HT-T300 (Torey)

122

8.55

3.88

0.32

0.022

Kevlar (DuPont)

70.8

5.5

2.05

0.34

0.026

From H. Abramovich, A. Grunwald, Stability of axially impacted composite plates, Composite Structures 32 (1e4) (1995) 151e158.

Table 6.3

Specimen

AR[a 3 b] (mm2)

fexp (Hz)

(εcr )static (ms)

CS21

2[150  300]

350

2600

2[150  300]

250

2413

CM32

1[225  225]

393

90

KM32

1[225  225]

186

400

CS22 a

a

Experimental results of the tested plates

Nominal clamped boundary conditions. From H. Abramovich, A. Grunwald, Stability of axially impacted composite plates, Composite Structures 32 (1e4) (1995) 151e158.

Dynamic buckling of composite columns and plates

243

data in the form of natural frequencies and static buckling strains as obtained during the tests described in Ref. [20]. One should note that except plate CM32, which was on clamped boundary conditions all around the perimeter of the plate, all the other plates had simply supported boundary conditions along the four edges of the plate. As the measured response of the plates was in the form of strains, measured by two back-to-back bonded strain gages, the compression strain εc and the bending strain εb were calculated3 and the axial compression load and lateral out-of-plane deflection were replaced by the compression and bending strains, respectively. The definition of the static load was performed using the modified Donnell’s approach and was found to be most appropriate for plates with initial geometric imperfections, w0, (see a discussion in Ref. [45]), having the following expression εb ε þ w0 b  ¼ εc ðεcr Þ 2 2 static þ a εb þ 3εb w0 þ 2w0

(6.53)

Using experimental data in the form of εb versus εc one can curve-fit it using Eq. (6.53) to yield the static buckling strain ðεcr Þstatic together with the initial geometric imperfection w0 and the constant a. Based on the work performed in Ref. [18], the dynamic buckling strain ðεcr Þdyn: is determined by curve-fitting the following equation to the experimental data (εb vs. εc) εb εb þ w0 ¼ εc ðεcr Þdyn. þ aεb

(6.54)

As for the static case, the two terms, the constant a and the initial geometric imperfection w0, are also determined from the curve-fitting process, yielding consistent results as presented in Ref. [20]. A dynamic buckling investigation was numerically performed in Ref. [39] on a curved laminated composite stringer stiffened panel. Details of the actual model, the dimensions of the stringer, the finite element model, and the mode shape at f ¼ 424 Hz (the lowest frequency of the panel) are presented in Fig. 6.12. The variation of the DLF (defined by Eq. 6.1) versus the nondimensional load duration is shown in Fig. 6.13, where T is the period of the applied load and Tb ¼ 1/fb ¼ 2358.5 ms is the lowest natural period of the stringer stiffened panel (fb ¼ 424 Hz). As shown earlier for other structural cases, for this stringer stiffened panel also the DLF is lower than unity in the vicinity of the lowest natural frequency of the specimen, returning to values above unity for very short periods, whereas for long periods the DLF tends to unity. Investigations of shells under dynamic type loading have been performed by many investigators, such as in Refs. [21,23,26,27,30,31,38,46,47]. A comprehensive experimental and numerical investigation was performed in the PhD thesis written by Eglitis

3

2 εc h ε1 þε 2 ;

2 εb h ε1 ε 2 , where ε1 and ε2 are the measured strains.

244

Stability and Vibrations of Thin-Walled Composite Structures

(a)

(b)

Stringer geometry (mm)

18

19.5

20.5

60

3.0

(c)

(d)

SS

SS

CL

Figure 6.12 Dynamic buckling investigation of a curved laminated composite stringer stiffened panel: (a) the stringer stiffened panel model, (b) geometric dimensions of the stringer, (c) the finite element (FE) model, and (d) the lowest mode shape of the stringer stiffened panel at f ¼ 424 Hz. CL, clamped; SS, simply supported. Adapted from H. Abramovich, H. Less, Dynamic buckling of a laminated composite stringerstiffened cylindrical panel, Composites Part B 43 (5) (July 2012) 2348e2358.

1.2 1 DLF

0.8 0.6 0.4 0.2 0

0

1

2

3

4

5

6

7

8

2T/Tb

Figure 6.13 Variation of DLF with the nondimensional load duration: experimental results. Adapted from H. Abramovich, H. Less, Dynamic buckling of a laminated composite stringerstiffened cylindrical panel, Composites Part B 43 (5) (July 2012) 2348e2358

Dynamic buckling of composite columns and plates

245

(a) 6

Model RTU #16 Model RTU #4 Model RTU #1–4

5

DLF

4 3 2 1 0 0

1

2

3

4

5

Load duration (ms)

(b) 3.0 Model RTU #16 Model RTU #4 Model RTU #1–4 Experimental RTU #16 Experimental RTU #4 Experimental RTU #1–4

2.5

DLF

2.0

1.5

1.0

0.5 0

10

20

30

40

50

Load duration (ms)

Figure 6.14 Typical results from Ref. [38]. Variation of DLF with load duration. Experimental and numerical results of specimens, type 1, at (a) 0 < t s/2 leads to axisymmetric buckling mode together with a slight drop in DLF at a load duration of T ¼ 2a/c (where a is the shell length, c is the speed of sound in the shell). The natural periods for RTU #16, RTU #4, and RTU #1-4 tested shells (see Fig. 6.14) were s ¼ 2.92, 4.04, and 6.06 ms, respectively (see Ref. [38]). Similar results are presented in Ref. [46], in which numerical and experimental investigations of a composite laminated cylindrical shell under impulsive loading are presented. The tested shell (see Fig. 6.16(a)) had a laminate of [0 , 45 , þ45 ]; was nominally clamped at its edges, with a length of 230 mm (between the clamped edges) and diameter of 250 mm; and was manufactured from graphite-epoxy with the following properties: E11 ¼ 137.0 GPa, E22 ¼ 9.81 GPa, G12 ¼ 5.886 GPa, and n12 ¼ 0.34, and 0.125 mm layer thickness. The dynamic loading was calculated using a dropping mass on the end plate with a mass of 32 kg. The calculated shell static mode shape is presented in Fig 6.16(b), while the dynamic responses for 746 and 2170 ms impact duration are shown in Fig. 6.16(c) and (d), respectively. As expected, the static and the dynamic modes have different patterns. The calculated DLF for numerical and experimental results is presented in Fig. 6.17 for various initial geometric imperfections. Although the experimental results do not go below DLF ¼ 1, for zero initial geometric imperfections, the numerical results show a DLF lower than unity in the vicinity of T/Tb ¼ 1, whereas for higher values of geometric imperfections, the DLF curve is higher than unity.

Dynamic buckling of composite columns and plates

247

Figure 6.16 Typical results from Ref. [46]: (a) the tested cylindrical shell, (b) the static mode shape, (c) the dynamic mode shape at 746 ms, and (d) the dynamic mode shape at 2170 ms. Adapted from H. Abramovich, T. Weller, P. Pevzner, Dynamic buckling behavior of thin walled composite circular cylindrical shells under axial impulsive loading, in: Proc. of AIAC-12, Twelfth Australian International Aerospace Congress, Melbourne, March 19e22, 2007.

A study [47] presents results similar to those described in Ref. [46] for the behavior of composite cylindrical shells having laminates of [0 /90 /0 /90 /90 /0 /90 /0 ] and [0 /0 /60 /60 /60 /60 /0 /0 ] under various durations of load impulse. A step pulse was used for the compressive loading, while calculations using the ABAQUS/ Standard code yielded a lowest natural frequency of 427 Hz and a static linear buckling load of 97.19 kN for the [0 /90 /0 /90 /90 /0 /90 /0 ] laminate and 120.39 kN for the [0 /0 /60 /60 /60 /60 /0 /0 ] laminate. ABAQUS/Explicit was used for the dynamic buckling analysis of the shells under the impulsive loading. Fig. 6.18 presents the DLF, calculated using Eq. (6.1), as a function of the period of the applied step impulse loading. For short periods, the DLF is higher than unity, whereas from T ¼ 5 ms, it drops below unity for both laminates. As the largest natural period can be calculated to 2.342 ms, it again shows that in the vicinity of the lowest natural frequency the DLF might fall below unity for laminated composite cylindrical shells also.

248

Stability and Vibrations of Thin-Walled Composite Structures

2.0

W0/h = 0.0 W0/h = 0.6 W0/h = 1.2 Experiment

DLF

1.6 1.2 0.8 0.4 0.0 0.0

1.0

2T/Tb

3.0

2.0

Figure 6.17 DLF versus nondimensional impulse period (Tb ¼ 1/fb, where fb is the lowest natural frequency of the shell). Adapted from H. Abramovich, T. Weller, P. Pevzner, Dynamic buckling behavior of thin walled composite circular cylindrical shells under axial impulsive loading, in: Proc. of AIAC-12, Twelfth Australian International Aerospace Congress, Melbourne, March 19e22 2007. 2.5 (0°/90°/0°/90°/90°/0°/90°/0°) (0°/0°/60°/–60°/–60°/60°/0°/0°)

2.0

DLF

1.5 1.0

0.5

0.0 0.0

2.5

5.0

7.5

10.0

12.5

15.0

Impulse time duration (ms)

Figure 6.18 DLF versus impulse time duration. Adapted from H. Abramovich, T. Weller, P. Pevzner, Dynamic buckling behavior of thin walled composite circular cylindrical shells under axial impulsive loading, in: Proc. of AIAC-12, Twelfth Australian International Aerospace Congress, Melbourne, March 19e22, 2007.

References [1] H.E. Lindberg, Dynamic Pulse Buckling e Theory and Experiment, DNA 6503H, Handbook, SRI International, 333 Ravenswood Avenue, Menlo Park, California 94025, USA, 1983. [2] G.J. Simitses, Dynamic Stability of Suddenly Loaded Structures, Springer Verlag, New York, USA, 1990, 290 p.

Dynamic buckling of composite columns and plates

249

[3] M. Chung, H.J. Lee, Y.C. Kang, W.-B. Lim, J.H. Kim, J.Y. Cho, W. Byun, S.J. Kim, S.-H. Park, Experimental study on dynamic buckling phenomena for supercavitating underwater vehicle, International Journal of Naval Architecture and Ocean Engineering 4 (2012) 183e198, http://dx.doi.org/10.2478/IJNAOE-2013-0089. [4] B. Budiansky, R.S. Roth, Axisymmetric Dynamic Buckling of Clamped Shallow Spherical Shells, Collected Papers on Instability of Shell Structures, NASA TN-D-1510, 761 p., 1962, pp. 597e606. [5] B. Hegglin, Dynamic Buckling of Columns, SUDAER No. 129, Department of Aeronautics & Astronautics, Stanford University, Stanford, California, USA, June 1962, 55 p. [6] B. Budiansky, J.W. Hutchinson, Dynamic buckling of imperfection sensitive structures, in: H. G€otler (Ed.), Proceedings of the 11th International Congress of Applied Mechanics, 1964, Springer-Verlag, Berlin, 1966, pp. 636e651. [7] W.J. Hutchinson, B. Budiansky, Dynamic buckling estimates, AIAA Journal 4 (3) (1966) 525e530. [8] M.H. Lock, A Study of Buckling and Snapping Under Dynamic Load, Air Force Report No. SAMSO-TR-68-100, Aerospace Report No. TR-0158(3240-30)-3, December 1967, 55 p. [9] J.A. Burt, Dynamic Buckling of Shallow Spherical Caps Subjected to a Nearly Axisymmetric Step Pressure Load (Master thesis), Naval Postgraduate School, Monterey, California 93940, USA, September 1971, 75 p. [10] S.A. Volmir, Nonlinear Dynamics of Plates and Shells, Science, USSR, Moscow, 1972. [11] R.E. Ekstrom, Dynamic buckling of a rectangular orthotropic plate, AIAA Journal 11 (12) (1973) 1655e1659. [12] L.H.N. Lee, Dynamic buckling of an inelastic column, International Journal of Solids and Structures 17 (3) (1981) 271e279. [13] J. Ari-Gur, T. Weller, J. Singer, Experimental and theoretical studies of columns under axial impact, International Journal of Solids and Structures 18 (7) (1982) 619e641. [14] L.H.N. Lee, K.L. Ettestad, Dynamic buckling of an ice strip by axial impact, International Journal of Impact Engineering 1 (4) (1983) 343e356. [15] G. Gary, Dynamic buckling of an elastoplastic column, International Journal of Impact Engineering 1 (4) (1983) 357e375. [16] M. Dannawi, M. Adly, Constitutive equation, quasi-static and dynamic buckling of 2024T3 plates e experimental result and analytical modelling, Journal de Physique Colloques 49 (C3) (1988). C3-575eC3-588. [17] V. Birman, Problems of dynamic buckling of antisymmetric rectangular laminates, Composite Structures 12 (1) (1989) 1e15. [18] T. Weller, H. Abramovich, R. Yaffe, Dynamic buckling of beams and plates subjected to axial impact, Computers and Structures 32 (3e4) (1989) 835e851. [19] J. Ari-Gur, D.H. Hunt, Effects of anisotropy on the pulse response of composite panels, Composite Engineering 1 (5) (1991) 309e317. [20] H. Abramovich, A. Grunwald, Stability of axially impacted composite plates, Composite Structures 32 (1e4) (1995) 151e158. [21] A. Schokker, S. Sridharan, A. Kasagi, Dynamic buckling of composite shells, Computers and Structures 59 (1) (1996) 43e53. [22] J. Ari-Gur, R. Simonetta, Dynamic pulse buckling of rectangular composite plates, Composites Part B 28 (1997) 301e308. [23] M.R. Eslami, M. Shariyat, M. Shakeri, Layerwise theory for dynamic buckling and postbuckling of laminated composite cylindrical shells, AIAA Journal 36 (10) (1998) 1874e1882.

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Stability and Vibrations of Thin-Walled Composite Structures

[24] D. Petry, G. Fahlbusch, Dynamic buckling of thin isotropic plates subjected to in-plane impact, Thin-Walled Structures 38 (3) (2000) 267e283. [25] Z. Zheng, T. Farid, Numerical studies on dynamic pulse buckling composite laminated beams subjected to an axial impact pulse, Composite Structure 56 (3) (2002) 269e277. [26] R. Yaffe, H. Abramovich, Dynamic buckling of cylindrical stringer stiffened shells, Computers and Structures 81 (9e11) (2003) 1031e1039. [27] C.C. Chamis, G.H. Abumeri, Probabilistic Dynamic Buckling of Smart Composite Shells, NASA/TM-2003-212710, 2003. [28] Z. Zhang, Investigation on Dynamic Pulse Buckling and Damage Behavior of Composite Laminated Beams Subject to Axial Pulse (Ph.D. thesis), Faculty of Engineering, Civil Engineering, Dalhousie University, Halifax, Nova Scotia, Canada, 2003, 228 p. [29] H.E. Lindberg, Little Book of Dynamic Buckling, LCE Science/Software, September 2003. [30] C. Bisagni, Dynamic buckling tests of cylindrical shells in composite materials, in: 24th International Congress of the Aeronautical Sciences (ICAS 2004), 29 Auguste3 September, Yokohama, Japan, 2004. [31] C. Bisagni, Dynamic buckling of fiber composite shells under impulsive axial compression, Thin-Walled Structures 43 (3) (2005) 499e514. [32] T. Kubiak, Dynamic buckling of thin-walled composite plates with varying widthwise material properties, International Journal of Solids and Structures 42 (2005) 5555e5567. [33] W. Ji, A.M. Waas, Dynamic bifurcation buckling of an impacted column, International Journal of Engineering Science 46 (10) (2008) 958e967. [34] T. Kubiak, Dynamic buckling estimation for beam-columns with open cross-sections, Paper No. 13, in: B.H.V. Topping, L.F. Costa Neves, R.C. Barros (Eds.), Proceedings of the Twelfth International Conference on Civil, Structural and Environmental Engineering Computing, Civil-Comp Press, Stirlingshire, Scotland, 2009. [35] M. Jabareen, I. Sheinman, Dynamic buckling of a beam on a nonlinear elastic foundation under step loading, Journal of Mechanics of Materials and Structures 4 (7e8) (2009) 1365e1373. [36] K.K. Michalska, About some important parameters in dynamic buckling analysis of plated structures subjected to pulse loading, Mechanics and Mechanical Engineering 14 (2) (2010) 269e279. [37] R.J. Mania, Membrane-flexural coupling effect in dynamic buckling of laminated columns, Mechanics and Mechanical Engineering 14 (1) (2010) 137e150. [38] E. Eglitis, Dynamic Buckling of Composite Shells (Ph.D. thesis), Riga Technical University, Faculty of Civil Engineering, Institute of Materials and Structures, Riga, Latvia, 2011, 172 p. [39] H. Abramovich, H. Less, Dynamic buckling of a laminated composite stringer-stiffened cylindrical panel, Composites Part B 43 (5) (July 2012) 2348e2358. [40] A. Landa, The Buckling Resistance of Structures Subjected to Impulsive Type Actions (Master thesis), Norwegian University of Science and Technology, Department of Marine Technology NTNU Trondheim, Norway, February 2014, 104 p. [41] J.G. Straume, Dynamic Buckling of Marine Structures (Master thesis), Norwegian University of Science and Technology, Department of Marine Technology NTNU Trondheim, Norway, June 2014, 148 p. [42] V.A. Kuzkin, Structural Model for the Dynamic Buckling of a Column Under Constant Rate Compression, arXiv:1506.00427 [physics.class-ph], 2015.

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[43] O. Mouhata, K. Abdellatif, Dynamic buckling of stiffened panelsThe 5th International Conference of Euro Asia Civil Engineering Forum (EACEF-5), Procedia Engineering 125 (2015) 1001e1007. [44] S.G. Lekhnitskii, Anisotropic Plates, second ed., Gordon and Breach, New York, 1968 translation from Russian. [45] H. Abramovich, T. Weller, R. Yaffe, Application of a modified Donnell technique for the determination of critical loads of imperfect plates, Computers and Structures 37 (1990) 463e469. [46] H. Abramovich, T. Weller, P. Pevzner, Dynamic buckling behavior of thin walled composite circular cylindrical shells under axial impulsive loading, in: Proc. of AIAC-12, Twelfth Australian International Aerospace Congress, Melbourne, March 19e22, 2007. [47] V. Citra, R.S. Priyadarsini, Dynamic buckling of composite cylindrical shells subjected to axial impulse, International Journal of Scientific and Engineering Research 4 (5) (May 2013) 162e165.

Appendix A.

Calculation of the critical buckling load of a uniaxial loaded plate from test results

One of the issues facing a researcher is to correctly define the buckling load of a plate for the static and dynamic cases, based on experimental points obtained either by tests or numerically. The following method was already published and applied in Ref. [39] and is based on the work performed by Brown.4 It was shown that for practical structural geometries and loading conditions the following equation holds:

d2i



d20

Pi d0 ¼a h q¼a h 1þ Pcr di 2 2



2 2

(A.1)

where di is the lateral plate deflection due to corresponding in-plane loading Pi, d0 is the initial lateral plate deflection; h is the thickness of the plate, and a is a constant accounting for the load configuration and boundary conditions. After some mathematical manipulations, Eq. (A.1) can be rewritten as Pi di ¼ A1 þ A2 di þ A3 d3i

(A.2)

where

A1 ¼ Pcr d0 ; A2 ¼ Pcr

4

 d20 Pcr 1  2 2 ; A3 ¼ 2 2 : a h a h

(A.3)

V.L. Brown, Linearized least-squared technique for evaluating plate-buckling loads, Journal Engineering Mechanics 116 (5) (1990) 1050e1057.

252

Stability and Vibrations of Thin-Walled Composite Structures

Eq. (A.3) is presented in a suitable form for the application of the least squares method to fit a curve to a series of given data points, Pi, di. To do so, let us define the sum of the squares of the residuals for m data points as SUM ¼

m  X

Pi di  A1  A2 di  A3 d3i

2

(A.4)

i¼1

Taking partial derivatives of Eq. (A.4) with respect to the coefficients A1, A2, and A3 (see Eq. A.3) and equating them to zero (thus minimizing the error) yields the following system of equations written in a matrix form: 2 6 m 6 6 6P 6 m d 6 i 6 i¼1 6 6 m 4P 3 di i¼1

m P

di

i¼1 m P i¼1 m P i¼1

d2i d4i

8 m 9 3 P > > > d3i 7 P d > > i i > > > 8 9 > > i¼1 i¼1 > > 7 A > > 1 > > > 7> > > > > < < = = 7 m m P P 47 2 di 7 A2 ¼ Pi di > > > 7> i¼1 i¼1 > > : > > ; > 7> > > > > 7 A 3 > > m m P P > > 5 > 6 4 > Pi di > > di : ; m P

i¼1

(A.5)

i¼1

The three linear equations, presented in a matrix form in Eq. (A.5), are then solved for the three unknowns A1, A2, and A3. Substituting them into Eq. (A.3) and the results in Eq. (A.2) leads to the following cubic equation for the single unknown Pcr: P3cr  A2 P2cr  A21 A3 ¼ 0

(A.6)

Solving Eq. (A.6) will yield the buckling load of the plate based on the displacementeforce curve that is generated either by experiments or by numerical calculations.

Stability of composite shelletype structures

7

7.1 Introduction Richard Degenhardt 1, 2 1

German Aerospace Center (DLR), Institute for Composite Structures and Adaptive Systems, Braunschweig, Germany; 2PFH, Private University of Applied Sciences G€ottingen, Composite Engineering Campus Stade, Germany

This chapter deals with the structural stability behavior of unstiffened composite shelletype structures that are cylindrical, conical, or spheroidal. This type of structures is mainly used in space applications. Currently, buckling, for example, of launcher structures is designed by the rather conservative NASA SP-8007 guideline, which was developed in 1968 for metallic structures using conservative knockdown factors depending only on the slenderness (radius/over thickness) value. The buckling load of such lightweight structures is very imperfection sensitive, and its reliable determination requires knowledge of the worst realistic imperfections. Recent projects demonstrated that by using structures from composite materials the load-carrying capacity can be improved significantly. However, the imperfection sensitivity is still not fully understood and improved design guidelines for such materials still do not exist. Owing to this reason, many research projects were performed to study the buckling behavior of such imperfection-sensitive structures. This chapter deals with the following topics: Chapter 7.2, Geometric imperfections and lower-bound methods used to calculate knockdown factors for composite cylindrical shells (Saullo Castro) There are many approaches to consider geometric imperfections that have the most influence on buckling. This section gives a comprehensive overview of the existing lower-bound methods to calculate knockdown factors for composite cylindrical shells. Chapter 7.3, Semianalytical approaches for linear and nonlinear buckling analyses of imperfect composite cylinders under axial, torsional, and pressurization loads (Saullo Castro) The numerical analysis of composite structures using nonlinear finite element methods is very time-consuming. This section presents a new approach that allows a fast calculation of such structures for different loads and boundary conditions. In addition, even the realistic imperfections can be taken into account. Chapter 7.4, Composite spheroidal shells under external pressure (Jan Blachut) This section deals with the stability of externally pressurized composite spheroidal shells, including domed closures. It presents the manufacturing routes, selective numerical results, and references to the past experimental investigations. Chapter 7.5, Vibration correlation technique (VCT) for the estimation of real boundary conditions and buckling load of unstiffened plates and cylindrical shells (Mariano Arbelo)

Stability and Vibrations of Thin-Walled Composite Structures. http://dx.doi.org/10.1016/B978-0-08-100410-4.00007-7 Copyright © 2017 Elsevier Ltd. All rights reserved.

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Stability and Vibrations of Thin-Walled Composite Structures

The VCT is a promising method for the nondestructive estimation of buckling load in experiments, as the structure does not need to be loaded until buckling. This section describes a new approach for the empirical estimation of the buckling load based on existing test results. Chapter 7.6, New robust knockdown factors for the design of axially loaded composite shells (Ronald Wagner) According to the past design guidelines (e.g., NASA SP-8007 for cylindrical shells), the knockdown factors depend strongly on the slenderness of the structure. This section presents a new study that demonstrates that the knockdown factors for axially loaded composite shells are much less dependent on the slenderness than expected. Chapter 7.7, Design and manufacture of composite cones (Regina Khakimova) Currently cones are designed by the constant knockdown factor 0.33. The value is independent of the material or the kind of cone. This section describes a new design concept for composite cones, taking into account the effects of the manufacturing process.

Disclaimer This chapter presents a summary of the work published by the authors during EFRE and DESICOS projects. For a more detailed insight, please refer to the original papers.

Stability of composite shelletype structures

255

7.2 Geometric imperfections and lower-bound methods used to calculate knockdown factors for composite cylindrical shells Saullo G.P. Castro Embraer, S~ao José dos Campos, Brazil The effects of geometric imperfections on the buckling behavior of thin-walled cylinders have been observed already since the beginning of 1900s, e.g., Southwell [1]. Initially this effect was noticed by the discrepancy between theoretical and experimental results. Southwell found that his theory could not be applied for real cases where there are geometric imperfections and load asymmetries. Fl€ugge [2] and Donnell [3] were the first authors to develop formulations by taking into account the effects of initial geometric imperfections, but the nonlinear analyses failed to predict the experimental buckling loads. Their analyses required the use of largemagnitude geometric imperfections that “could scarcely have escaped the notice of the investigators” [4]. Fl€ ugge’s and Donnell’s theories produce a gradual appearance of buckles with increase in the compression load, whereas in the experiments, buckling is typically characterized by a sudden dynamic buckling event and a corresponding reduction in load. Koiter’ theory (1945, which was translated from Dutch to English in the 1960s by Riks [4]) was the first to accurately predict the imperfection sensitivity trends that were observed experimentally [4]. In 1950, Donnell and Wan [5] independently, from the study of Koiter, modified the procedure adopted by Donnell [3] 16 years earlier and proposed a new method, which was followed by several investigators with some modifications [6]. Arbocz [7] states that Koiter’s “General Theory of Elastic Stability” is widely accepted, and it is important to emphasize that Koiter’s theory is valid in the elastic regime and limited to small-magnitude imperfections [8]. In the meanwhile, the design of imperfection-sensitive structures required guidelines explaining how to take imperfection sensitivity into account, for instance, in the calculation of rocket and launcher structures [9]. In 1960, Seide, Weingarten, and Morgan (see Refs. [10,11]) published a collection of experimental results that was one of the main precursors for the well-known NASA SP-8007 guideline, published in 1965 and revised in 1968 to its most popular version [12]. The reduced stiffness method (RSM) developed by Croll [13], Batista and Croll [14], and collaborators is another method for calculating lower bounds. Croll and Batista [15] used this concept to find lower bounds for axially compressed linear-elastic isotropic cylinders. The idea of a single buckle as the worst imperfection was first pointed out by Esslinger [16] using high-speed cameras. Deml and Wunderlich [17] came to the same conclusion using a modified finite element formulation in which the nodal positions are treated as extra degrees of freedom (DOF), which will vary along the solution within a preset amplitude. In this optimization problem the new nodal

256

Stability and Vibrations of Thin-Walled Composite Structures

positions that were found seemed always as a single buckle, meaning that this is the geometric pattern that leads to the minimum buckling loads. H€uhne suggested the single perturbation load approach (SPLA) as a robust method for creating single buckle imperfections [18], and this method will also be further discussed in the following sections. Section 7.2.1 explains in detail how NASA SP-8007 and the reduced energy method (REM) can be used for calculating lower-bound knockdown factors (KDFs). In Section 7.2.2, five types of geometric imperfection commonly used in the literature are presented and it is explained about how each imperfection pattern can be applied to a finite element framework. Section 7.2.3 summarizes the study of Castro et al. [19] in which the nonlinear buckling load obtained with different imperfection patterns is compared with the lower-bound KDFs. Section 7.2.4 presents new studies related to the SPLA, dealing with its applicability and possible enhancements. Finally, Section 7.2.5 contains important considerations regarding the loading scheme and boundary conditions of simulations and tests aiming to create a database that can be used for less conservative design guidelines.

7.2.1 7.2.1.1

Lower-bound methods to calculate knockdown factors NASA SP-8007

Fig. 7.2.1 shows a collection of experimental results and the lower-bound curve that gives the shell buckling KDF, denoted by g in Eq. (7.2.1). This figure consists one of the main experimental results that supports the KDFs recommended by the NASA SP-8007 guideline. Calculating g for isotropic unstiffened cylinders requires only the cylinder radius and the wall thickness, as shown in Eq. (7.2.1). This equation also shows the equivalent thickness teq that is often used when calculating the KDF for Fdesign = Ftheoretical γ

ρ=P/PCL

1,0

0,5

0

P / PCL = 1 – 0.902 (1 – e1/16√R / t ) 0

500

1000 R/t

1500

2000

Figure 7.2.1 Test data for isotropic cylinders subjected to axial compression. Modified from J. Arbocz, J.H. Starnes Jr., Future directions and challenges in shell stability analysis, Thin-Walled Structures 40 (2002) 729e754.

Stability of composite shelletype structures

257

orthotropic materials. However, the approach for calculating KDFs for orthotropic materials does not consider all the orthotropic stiffness terms, for example, membrane-bending coupling, the two laminate directions, and tensioneshear are not included. These stiffness terms can have a significant influence on the buckling behavior and consequently on the resulting KDFs, as demonstrated by Geier et al. [20]. g ¼ 1  0:902ð1  e4 Þ rffiffiffi 1 R ðisotropicÞ 4¼ 16 t sffiffiffiffiffi 1 R 4¼ 16 teq

(7.2.1)

with heq

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 D11 D22 ðorthotropicÞ ¼ 3:4689 A11 A22

where A11 , A22 , D11 , and D22 are the extensional and bending stiffnesses extracted from the composite ABD matrix; R is the cylinder radius; t is the thickness; and teq is the equivalent thickness corrected for orthotropic cylinders. In the NASA SP-8007 guideline the KDF denoted by g in Eq. (7.2.1) is called the correlation factor, accounting for the disparity between experiments and theory. Theoretical equations for the buckling load for both isotropic and orthotropic cylinders are also provided in NASA SP-8007, where the correlation factor is used, but in modern applications of g, the theoretical buckling load is usually calculated using linear buckling analysis and the design load obtained by multiplying this theoretical buckling load by g, as shown in Eq. (7.2.2). Fdesign ¼ Ftheoretical $g

(7.2.2)

Comparative studies performed by Hilburger et al. [21], H€uhne et al. [18,22], and Degenhardt et al. [23] have shown that the lower bound given by the NASA SP-8007 guideline can lead to conservative designs. Moreover, the space industry experience has shown that structures designed for buckling using the NASA SP-8007 guideline can be so conservative that when tested after manufactured, they fail for strength.

7.2.1.2

Reduced stiffness method

RSM has been mostly applied in the civil engineering field [24] and it is based on three postulates [25]. First, significant geometric nonlinearities appear because of changes in membrane resistance. For instance, in the buckling of an in-plane loaded plate, there is no nonlinearity up to the point where some disturbance causes a normal deformation.

258

Stability and Vibrations of Thin-Walled Composite Structures

The normal deformation causes a load eccentricity that creates bending, interacting nonlinearly with the reduction of the membrane stiffness, which decreases in the postbuckled range of loading. In any case where a thin-walled structural member is initially under high compressive stress levels, with a high membrane component of the strain energy, the displacements are predicted linearly up to the point where the membrane stiffness starts to decrease. Second, for thin-walled structures, the postbuckling loss of stiffness can only occur when there is membrane resistance at the prebuckled state, meaning that if the shell does not have membrane energy before buckling, there will be no loss of stiffness after buckling. Third, the lower-bound buckling load for a particular load case will be given by an analysis in which the membrane stiffness is removed. Sosa et al. [26] showed the equivalence between the RSM and the REM. Along this study the REM will be implemented in a general finite element solver following the procedure explained by Sosa et al., in which a reduction factor a is applied to the membrane stiffness components instead of completely eliminating it, as originally proposed by Croll [25]. This approach assumes that the shell with degraded membrane stiffness will have a postbuckled shape similar to an eigenvector obtained by linear buckling analysis. Sosa and Godoy [27] compared the REM using this assumption with nonlinear postbuckling analysis and showed that this assumption may not be valid in some cases, leading to nonconservative estimates. In such cases the computation of correction coefficients is required, making the REM less straightforward. As described in detail by Sosa et al. [26], three steps can be followed to implement the REM in a general purpose finite element code (ABAQUS). In the first step, the first 50 eigenmodes are extracted from a linear buckling analysis. The strain energy of the perfect shell is then computed in the second step by applying the eigenmode obtained in the first step as an initial prescribed displacement, which will result in the energy terms used in the denominator of Eq. (7.2.3). The third step consists of reducing the membrane stiffness as presented in Eq. (7.2.4) [26] and then computing the total strain energy using the eigenmodes from the perfect shell as initial prescribed displacements, resulting in the energy terms used in the numerator of Eq. (7.2.3). The authors showed that the element type STRI3 (in ABAQUS) is the only one in which the discrete Kirchhoff constraint is imposed analytically, whereas in other element types, it is imposed numerically, making them diverge from the correct solution for high values of reduction factor [26].

Fdesign

  1 UB þ UM a ¼ Ftheoretical UB þ UM

(7.2.3)

with Ftheoretical from a linear buckling analysis and a is the reduction factor. Eq. (7.2.4) shows how the laminate stiffness matrix [ABD]degraded can be used to represent the laminate stiffness matrix with the membrane terms “A” degraded using the approach presented by Sosa et al. [26]. Note that the coupling matrix “B,” which couples the

Stability of composite shelletype structures

259

curvature “k” with the normal loads “N,” and extensional “ε” strains with moments “M” are also removed in the suggested approach. 8 9 8 9

> > > > > > > > > > > > > ε > > > 2 > > > > > < = > < ¼ ε¼ ε6 > > > > > > > > > > > > > >ε > > > > > > > 3 > > : ; > : ε4

εxx εqq gxq gqz gxz

8 ð0Þ > εxx > > > > > ð0Þ > > εqq > > > > < ð0Þ ¼ gxq > > > > > > > > ð0Þ > > > > >g > > > qz ; > > > > : ð0Þ gxz 9 > > > > > > > > =

291

8 ð1Þ > εxx > > > > > ð1Þ > > εqq > > > > < þ z gð1Þ xq > > > > > > > > > ð1Þ > > > > > > > > gqz > > > > > > > : ð1Þ ; gxz 9 > > > > > > > > > > > > =

9 > > > > > > > > > > > > =

8 ð2Þ > εxx > > > > > ð2Þ > > εqq > > > > < þ z2 gð2Þ xq > > > > > > > > > > ð2Þ > > > > > > > > gqz > > > > > > ; : ð2Þ gxz

9 > > > > > > > > > > > > = > > > > > > > > > > > > ;

(7.3.20)

Further simplification of the strain terms of Eq. (7.3.19) is possible by keeping only the nonlinear terms that mostly affect typical shell structures. In Eq. (7.3.19) the nonlinear terms are separated in those corresponding to the equations of Donnell [20], Sanders [21], and Timoshenko and Gere and other nonlinear terms not used in any of these three nonlinear theories. The Donnell equations are obtained with d1 ¼ d2 ¼ d3 ¼ 0. Sanders’ equations are obtained with d1 ¼ 1 and d2 ¼ d3 ¼ 0. Timoshenko and Gere’s equations are obtained with d1 ¼ d2 ¼ 1 and d3 ¼ 0. The case with d1 ¼ d2 ¼ d3 ¼ 1 results in the full strainedisplacement relations. Eq. (7.3.19) is in agreement with Simitses et al. [22,23], Tong and Wang [16], Goldfeld et al. [24], Goldfeld [25], and Shadmehri et al. [13]. Eq. (7.3.19) can readily be applied to both the FSDT and the CLPT making 4x ¼ wx and 4q ¼ wq/r. Focusing on the CLPT and keeping Donnell’s and Sanders’ nonlinear terms, Eq. (7.3.21) is obtained [1], where the presence of an initial imperfection field represented by w0 has been added as presented by Simitses et al. [23], Arbocz [26], Yeh et al. [27], Almroth [28], and Yamada et al. [29]. The partial derivatives related to the imperfection field w0,x and w0,q added to the terms corresponding to Sanders’ equations were taken from the work by Simitses [23]. Note that in Eq. (7.3.21) the radius r is a function of x, varying linearly between the bottom and top edges. 1 2 εð0Þ xx ¼ ux þ wx þ w0;x wx 2 1 1 1 ð0Þ εqq ¼ ðsin au þ vq þ cos awÞ þ 2 w2q þ 2 w0;q wq r 2r r  

cos a cos a 1 v v wq þ w0;q þ d1 r 2r r

1 1 1 1 cos a ð0Þ v wx þ w0;x gxq ¼ vx þ ðuq  sin avÞ þ wx wq þ w0;x wq þ w0;q wx  d1 r r r r r εð1Þ xx ¼ wxx   1 1 cos a ð1Þ εqq ¼  sin awx þ wqq þ d1 2 vq r r r     1 sin a cos a sin a ð1Þ wq þ d1 v gxq ¼  2 wxq  vx  r r r r (7.3.21)

292

Stability and Vibrations of Thin-Walled Composite Structures

Eq. (7.3.21) is valid for both conical and cylindrical shells. Keeping a ¼ 0 and defining the coordinate y ¼ rq will result in Eq. (7.3.22), which is valid for curved cylindrical panels, where the radius r is constant. 1 2 εð0Þ xx ¼ ux þ wx þ w0;x wx 2 εð0Þ yy ¼ vy þ

 

w 1 2 1 1 þ wy þ w0;y wy þ d1 v v  wy þ w0;y r 2 r 2r

1 gð0Þ xy ¼ vx þ uy þ wx wy þ w0;x wy þ wx w0;y  d1 v wx þ w0;x r εð1Þ xx

(7.3.22)

¼ wxx

1 εð1Þ yy ¼ wyy þ d1 vy r 1 gð1Þ xy ¼ 2 wxy þ d1 vx r Keeping a ¼ p/2 will result in Eq. (7.3.23), which is valid for circular flat panels, where Sander’s nonlinear terms vanish. Note that in Eq. (7.3.23) the radius varies linearly over the x coordinate. 1 2 εð0Þ xx ¼ ux þ wx þ w0;x wx 2 1 1 1 ð0Þ εqq ¼ ðu þ vq Þ þ 2 w2q þ 2 w0;q wq r 2r r 1 1 1 1 ð0Þ gxq ¼ vx þ ðuq  vÞ þ wx wq þ w0;x wq þ w0;q wx r r r r εð1Þ xx ¼ wxx   1 1 ð1Þ εqq ¼  wx þ wqq r r   1 1 ð1Þ gxq ¼  2wxq  wq r r

7.3.2

(7.3.23)

General nonlinear formulation

The total potential energy of a solid system can be expressed as P¼UþV

(7.3.24)

Stability of composite shelletype structures

293

where U is the internal energy and V is the external energy. The stationary condition of the total potential energy P can be expressed as [3,8] dP ¼ dU þ dV ¼ 0

(7.3.25)

In a nonlinear incremental analysis, Eq. (7.3.25) is usually not fulfilled at any arbitrary iteration because of the imbalance between the internal and external forces, and the resulting inequality can be represented by a residual force vector R: dP ¼ dU þ dV ¼ dcT R

(7.3.26)

where dc is the variation of a vector containing the unknown set of Ritz constants, used in the displacement field approximation. The definition of the residual force vector for a given iteration (k  1) is ðk1Þ

Rðk1Þ ¼ Fint

   Fext0 þ lðk1Þ Fextl

(7.3.27)

with l being a scalar load multiplier for the variable part of the external force vector Fextl ; the vector Fext0 contains the external forces not subjected to the load multiplier and Fint is the internal force vector. For load-controlled iterative methods, such as those belonging to the NewtoneRaphson family, the multiplier l is set constant (l(k1) ¼ l) for a given load increment, whereas when using the Riks (arc-length) method the multiplier l is variable within the load increment [30,31]. The equilibrium is approached when R / {0} such that there will be a given iteration k where the equilibrium is finally achieved and R(k) ¼ {0}. Using the Taylor expansion at R(k1) to find R(k) in terms of R(k1), the following expression is obtained: ðkÞ

R

ðk1Þ

¼R

vR

þ dcðkÞ þ Oð2Þ ¼ f0g vc ðk1Þ

(7.3.28)

Ignoring the second- and higher-order terms, Eq. (7.3.28) becomes RðkÞ ¼ Rðk1Þ þ dRðkÞ ¼ f0g

In Eq. (7.3.28) the term vR vc

ðk1Þ

(7.3.29) corresponds to the structural tangent stiffness ma-

trix KT at iteration k  1, such that ¼

vR

vc ðk1Þ

rdRðkÞ ¼

ðk1Þ K T dc

ðk1Þ

KT

(7.3.30)

294

Stability and Vibrations of Thin-Walled Composite Structures

and by applying the definition of Eq. (7.3.30) in Eq. (7.3.28) leads to ðk1Þ

Rðk1Þ þ K T

dcðkÞ ¼ f0g

(7.3.31)

which can be solved for dc ðk1Þ1

dcðkÞ ¼ K T

Rðk1Þ

(7.3.32)

The set of Ritz constants is then updated with cðkÞ ¼ cðk1Þ þ dcðkÞ

(7.3.33)

To solve the nonlinear problem, the tangent stiffness matrix KT and the residual force vector R must be calculated for every iteration in the full NewtoneRaphson method and the vector of Ritz constants c(k) is updated for each iteration until the convergence is achieved, following a given criterion [1]. In other nonlinear algorithms discussed in the following sections the calculation of KT at each iteration is avoided because it is usually computationally expensive. Based on the definition of R from Eq. (7.3.26), it can be seen that the energy functionals must be known. The internal energy can be represented for a general 3D system by the following integral: Z 1 U¼ sT ε dV (7.3.34) 2 V and its variation renders Z

1 T s dε þ dsT ε dV dU ¼ 2 V

(7.3.35)

The stress vector s can be expressed in terms of the strain vector using a symmetric constitutive matrix C [3] as s ¼ Cε, such that Z

1 T ε Cdε þ dεT Cε dV dU ¼ 2 V Z ¼ dεT CεdV (7.3.36) V

Z ¼

V

dεT sdV

Using the Ritz method, the unknown displacement field u is approximated, and the general expression can be used to represent this approximation: u ¼ Sc

(7.3.37)

where S is a matrix containing the shape of each term in the approximation functions and c is a vector containing the respective amplitude of each term.

Stability of composite shelletype structures

295

With the approximation of Eq. (7.3.37) the strain vector can be written in the following general form [32]: ε ¼ ε0 þ εL þ εL0 ε0 ¼ B0 c (7.3.38)

1 εL ¼ BL c 2 εL0 ¼ BL0 c

where ε0 and εL contain the linear and the nonlinear strain terms, respectively, due to large deflections and εL0 contains the nonlinear strain terms due to the presence of an initial imperfection field w0. The matrices B0 and BL contain the linear and nonlinear kinematic terms, respectively, and BL0 contains the nonlinear kinematic terms due to the initial imperfection field w0. The matrices B0, BL, and BL0 can be directly obtained from the strainedisplacement relations of Eq. (7.3.21), and the following relations define each of these matrices for the CLPT using Donnell’s equations: 2

vSu 6 vx 6 6 6 sin a u 6 6 r S 6 6 6 1 vSu 6 6 r vq 6 6 B0 ¼ 6 6 6 0 6 6 6 6 6 0 6 6 6 6 4 0

3 0 1 vSv r vq vSv sin a v S  r vx 0 0

0

0

7 7 7 7 cos a w 7 S 7 r 7 7 7 7 0 7 7 7 7 v2 Sw 7 7  2 7 vx 7 7   w 2 w 7 1 vS 1v S 7 þ  sin a r r vq2 7 vx 7 7  2 w w 7 1 v S sin a vS 5  2  r r vq vxvq

(7.3.39)

where Su, Sv, and Sw are the first, second, and third rows, respectively, of the matrix of shape functions S for a displacement field vector defined by uT ¼ {u, v, w}. The matrices BL and BL0 are conveniently written as BL ¼ AG BL ¼ A0 G

(7.3.40)

296

Stability and Vibrations of Thin-Walled Composite Structures

where A, A0, and G are matrices defined as 2 6 A¼4

wx

0

1w r q

0

0 0

0

1w r q

wx

0

0 0

2 6 A0 ¼ 4

3T 7 5

w0;x

0

1w r 0;q

0

0

0

0

1w r 0;q

w0;x

0

0

0

3T

(7.3.41)

7 5

and 2 60 0 6 G¼6 4 0 0

3 vSw vx 7 7 7 1 vSw 5

(7.3.42)

r vq

where wx and wq are nonlinear field quantities that should be calculated from a previous iteration. The field quantities w0,x and w0,q represent an imperfection field over the 2D domain. The previous definitions of B0, BL, and BL0 apply for the conical/cylindrical shells under discussion. The definition of the strain vector using Eq. (7.3.38) applies to any general 3D system and its variation is given by dε ¼ dε0 þ dεL þ dεL0 dε ¼ Bdc

(7.3.43)

with B ¼ B0 þ BL þ BL0

(7.3.44)

The variation of the external energy dV when only conservative forces are present (i.e., forces not depending on the displacement field) can be represented by the general expression: dV ¼ dcT ðFext0 þ lFextl Þ

(7.3.45)

Substituting the strain definitions of Eq. (7.3.43) and the definition of dV given in Eq. (7.3.45) into Eq. (7.3.26) will result in the following expression for the residual force vector R: Z ¼ dcT

V

dcT R ¼ dU þ dV T

B sdV  dcT ðFext0 þ lFextl Þ

(7.3.46)

Stability of composite shelletype structures

which can be rearranged to obtain the expression for R: Z T R ¼ B sdV  ðFext0 þ lFextl Þ V

297

(7.3.47)

Comparing Eq. (7.3.47) with the definition of the external force vector given in Eq. (7.3.27), we obtain Z T Fint ¼ B sdV (7.3.48) V

which is in accordance with the definition found in the literature [31]. The tangent stiffness matrix KT can be calculated using the definition of Eq. (7.3.30), in which the variation dR can be calculated from Eq. (7.3.47), giving Z  T dR ¼ K T dc ¼ d B sdV  dðFext0 þ lFextl Þ (7.3.49) V

Considering only conservative external forces, i.e., no contacts or forces depending on the displacement field, such as follower forces, Eq. (7.3.49) becomes Z  T K T dc ¼ d B sdV V (7.3.50) Z Z T T ¼ dB sdV þ B dsdV V

V

The left-hand-side integral, as shown in the following sections, can be written as Z V

T

dB sdV ¼ K G dc

(7.3.51)

where KG is the geometric stiffness matrix, calculated as detailed in the following discussion. The right-hand-side integral of Eq. (7.3.50) can be developed as Z Z T T B dsdV ¼ B CdεdV ¼ K L dc V V Z (7.3.52) T K L ¼ B CBdV V

where KL is the constitutive matrix including large displacements.

7.3.2.1

Geometric stiffness matrix KG

Eq. (7.3.51) defines the geometric stiffness matrix KG for any general 3D solid structure. For conical and cylindrical shells, the integration over the 2D domain of conical shells renders Z q¼2p Z x¼L T dB Nrdxdq (7.3.53) K G dc ¼ q¼0

x¼0

298

Stability and Vibrations of Thin-Walled Composite Structures

From the definition of B in Eq. (7.3.44), dB in Eq. (7.3.53) can be written as dB ¼ dB0 þ dBL þ dBL0

(7.3.54)

Inspecting the elements of B0, BL, and BL0 in Eqs. (7.3.39) and (7.3.40), it comes that for any type of nonlinear shell theory dB ¼ dBL ¼ dAG

(7.3.55)

Using Eq. (7.3.55) in Eq. (7.3.53) and expanding using the Donnell’s equations given in Eq. (7.3.41), we get 2 3 1 Z q¼2p Z x¼L 6 dwx Nxx þ dwq Nxq 7 r 6 7 K G dc ¼ (7.3.56) GT 6 7rdxdq 4 5 q¼0 x¼0 1 dwq Nqq þ dwx Nxq r Recalling that w ¼ Swc when using an approximated displacement field, Eq. (7.3.56) can be rearranged as 2 3 vSw =vx Z q¼2p Z x¼L 6 7 K G dc ¼ GT N K G 6 . 7 41 5rdxdq dc q¼0 x¼0 vSw vq r (7.3.57) 3 2 Nxx Nxq 5 NKG ¼ 4 Nxq Nqq By using the definition of matrix G given in Eq. (7.3.42), 0 1 Z q¼2p Z x¼L T G N KG G rdxdqAdc K G dc ¼ @ q¼0

Z rK G ¼

x¼0

q¼2p Z x¼L

q¼0

(7.3.58)

GT N KG G rdxdq

x¼0

This format for KG is general for any nonlinear theory, and Castro et al. [19] give the definition of G and NK G for Sanders’ nonlinear theory [1].

7.3.3

Nonlinear algorithms

This section gives more details about the iterative algorithms used to solve the nonlinear systems of equations discussed in Section 7.3.2. All the algorithms presented here fall in the category of the so-called incremental algorithms in which the load is not applied at once but divided in load increments, with the first load increment determined

Stability of composite shelletype structures

299

by a parameter called initialInc and the next load increment adjusted using many empirically determined formulas, as detailed in the following subsections. For a given load increment, many iterations are executed until the convergence criteria are fulfilled, or until a divergent behavior is identified. In case of convergence the algorithm moves to the next load increment, whereas in case of divergence the load increment is reduced and the current incremental step is restarted. Following this general scheme, the following subsections will detail different ways to solve each load increment and present the convergence and divergence criteria adopted in the later sections.

7.3.3.1

Full NewtoneRaphson method

Eq. (7.3.33) shows how the Ritz constant vector c is updated at each iteration up to convergence. Note that at each iteration the increment dc is calculated using the tangent stiffness matrix KT, using Eq. (7.3.32). In the full NewtoneRaphson method, KT is updated at every iteration, presenting a quadratic convergence rate and a high cost for each iteration because the computation of the tangent stiffness matrix is usually time consuming [31].

7.3.3.2

Modified NewtoneRaphson method

In the modified NewtoneRaphson method the tangent stiffness matrix KT is calculated at the beginning of each load increment and updated at every n iteration [31]. Although KT is updated only at some iterations, the internal force vector Fint is updated at every iteration, which only slightly increases the computational cost, as the numerical integration of Fint is considerably cheaper than that of KT. Although the convergence is slower than that in the full NewtoneRaphson method, the computational cost of each iteration is drastically reduced, usually compensating the need for more iterations. The tests performed for this thesis showed that using the modified NewtoneRaphson method results in a faster nonlinear algorithm, especially when line search algorithms, as the one described in Section 7.3.3.3, are applied in conjunction. The author verified a high dependency of the computational cost with the maximum number of iterations n between each KT update, where n ¼ 1 will result in the full NewtoneRaphson method and a high value for n will result in a poor convergence behavior, requiring smaller load increments. The value n ¼ 6 was adopted and for all the cases a good convergence behavior was obtained for increment sizes up to 0.2.

7.3.3.3

Line search algorithms

The use of line search algorithms is mainly intended to prevent divergence during the iterations, increasing the robustness of the nonlinear algorithm [30]. As mentioned in Section 7.3.3.2 a high number of iterations for an update in KT may result in poor convergence, but the author verified that this effect is attenuated when using the line search. The line search technique is an important feature of most numerical techniques applied in unconstrained optimization problems and can be used with a wide range of iterative solution procedures [31]. For application in the nonlinear problem under

300

Stability and Vibrations of Thin-Walled Composite Structures

discussion, let us consider the update of the Ritz constant vector based on Eq. (7.3.33), rewriting it as cðkÞ ¼ cðk1Þ þ hdcðkÞ

(7.3.59)

where h is a scalar set to unity in case no line search is desired. When using line searches, h becomes another parameter to be determined to minimize the total potential energy functional P. Assuming that inside the iteration P becomes only a function of h Pðh þ dhÞ ¼ PðhÞ þ ¼ PðhÞ þ

vP dh þ / vh vP vc dh þ / vc vh

(7.3.60)

¼ PðhÞ þ RðhÞT dcdh þ / For the solution at h to be stationary, f ðhÞ ¼

vP ¼ dcT RðhÞ ¼ 0 vh

(7.3.61)

Note that in Eqs. (7.3.60) and (7.3.61) the residual force vector R is written as a function of h, which holds true, as the internal force vector Fint is calculated with the updated c according to Eq. (7.3.59). The system of Eq. (7.3.61) is nonlinear and can be solved using the iterative scheme shown in Eq. (7.3.62), taken from Crisfield [31], which consists of subsequent linear interpolations that calculate the value for h(i) that renders f(h) to be zero. ! ði2Þ

  f h



hðiÞ ¼ hði1Þ  hði2Þ (7.3.62) f hði1Þ  f hði2Þ In the current implementation [33] the starting values for the iterative scheme are h(0) ¼ 0 and h(1) ¼ 1 and the stopping criterion is   abs hðiÞ  hði1Þ < 0:01

(7.3.63)

giving satisfactory results.

7.3.3.4

Convergence criteria and other nonlinear parameters

All the convergence and divergence criteria are applied to the third or higher iterations, i.e., for k > 2, meaning that at least three iterations are allowed. The convergence criterion adopted is

 

ðkÞ Rmax ¼ max RðkÞ  0:001 N (7.3.64)

Stability of composite shelletype structures

301

where R(k) is the residual force vector calculated at iteration k. The divergence criterion is ðkÞ

ðk1Þ

Rmax > Rmax

(7.3.65)

meaning that the current load step is restarted with a smaller increment size when the maximum residue increases from the previous to the current iteration. The increment size is reduced to a minInc parameter, below which the analysis is terminated. The maximum number of iterations maxNumIter used is 30, and if k > maxNumIter the current load step is restarted with a smaller increment size. Another criterion that considerably accelerated the nonlinear analyses in cases in which the convergence was slow is the slow convergence criterion:

ðk1Þ ðkÞ

Rmax  Rmax < 0:05 (7.3.66) ðk1Þ Rmax such that the load step is restarted with a smaller increment size when the maximum residual force is less than 5%.

7.3.4

Linear buckling formulation

The linear buckling behavior can be calculated by applying the neutral equilibrium criterion [14,34]: d2 P ¼ dðdU þ dVÞ ¼ 0

(7.3.67)

Using the definition of dU from Eq. (7.3.36) and the definition of dV from Eq. (7.3.45),  Z 

T T (7.3.68) d dc B sdV  d dcT ðFext0 þ lFextl Þ ¼ 0 V

When only conservative forces are used,  Z Z T T dcT dB sdV þ B dsdV ¼ 0 V

V

(7.3.69)

which has the integrals analogous to Eq. (7.3.50), whose solution is known, leading to dcT ðK G þ K L Þdc ¼ 0

(7.3.70)

As Eq. (7.3.70) must hold for any variation dc, it comes that at the buckling point detðK G þ K L Þ ¼ 0

(7.3.71)

302

Stability and Vibrations of Thin-Walled Composite Structures

By evaluating the elements of the constitutive stiffness matrix KL, it can be seen that only positive terms are possible, and therefore, KL is positive definite. On the other hand, KG may contain negative terms when compressive stresses are present. Assuming that all the stresses are linearly adjusted by an unknown load multiplier l, Eq. (7.3.71) can be rewritten as a generalized eigenvalue problem [35], giving K L F ¼ lK G F

(7.3.72)

where F is an eigenvector that corresponds to the eigenvalue l. As KL is not a singular matrix the eigenvalue problem can also be written as a standard eigenvalue problem [35]: K 1 L KG F ¼

1 IF l

(7.3.73)

where I is an identity matrix. The use of Eq. (7.3.73) with the sparse matrix solvers of SciPy [36,37] enables a high-performance calculation of a desired number of eigenvalue/eigenvector pairs. The author observed that some numerical adjusts may considerably speed up the solver, and in the implementation adopted for this thesis [33], a much higher convergence was achieved by changing the sign of the right-hand side of Eq. (7.3.73), using an initial guess of 1 for the eigenvalue and then correcting the sign of the obtained eigenvalues. Castro et al. [1,19] present how Eq. (7.3.73) should be decomposed and rearranged to calculate the linear buckling with a diversity of combined load cases.

7.3.5

Semianalytical approach using a single domain for the approximation functions

7.3.5.1

Achieving diverse boundary conditions

A diversity of boundary conditions is added to the formulation using one approach, which is an extended version of the approach published by Som and Deb [38] for isotropic cylinders. The distributed penalty stiffnesses for the bottom and top edges are schematically shown in Fig. 7.3.6. The strain energy associated with the elastic constraints can be written as Ue ¼

1 2

I 

KuBot u2 x¼L þ KvBot v2 x¼L þ KwBot w2 x¼L

 þ K4xBot 42x x¼L þ K4qBot 42q x¼L R1 dq I

1  KuTop u2 x¼0 þ KvTop v2 x¼0 þ KwTop w2 x¼0 þ 2

 þ K4xTop 42x x¼0 þ K4qTop 42q x¼0 R2 dq

(7.3.74)

Stability of composite shelletype structures

303 ν

KTop u

z, w

KTop

w

KTop

R2

θ, ν

φ

θ KTop

φ

x KTop

x, u

R1 w

KBot φ

θ KBot

u KBot

φ

x KBot

ν KBot

Figure 7.3.6 Penalty stiffnesses used to achieve diverse boundary conditions.

Writing in matrix form and calculating the first variation dUe ¼ cT K e dc I   Ke ¼ R1 SjTx¼L K Bot Ajx¼L þ R2 SjTx¼0 K Top Sjx¼0 dq

(7.3.75)

with 2

K Bot

6 6 6 6 6 ¼6 6 6 6 6 4

3

K uBot K vBot

7 7 7 7 7 7 7 7 7 7 5

0 K wBot 4

x K Bot

0

4

q K Bot

2

K Top

6 6 6 6 6 6 ¼6 6 6 6 6 4

3

K uTop K vTop

0 K wTop

0

4

x K Top

4

q K Top

7 7 7 7 7 7 7 7 7 7 7 5

(7.3.76)

304

Stability and Vibrations of Thin-Walled Composite Structures

In the following sections the elastic stiffnesses of Eq. (7.3.76) will be referred to as K u ; K v ; .; K 4q , omitting the subscripts “Bot” and “Top” when applicable to both edges. The penalty constraint contributions of KBot and KTop are added to the constitutive stiffness matrix KL, and the new linear stiffness matrix considering the elastic boundary conditions becomes K Le ¼ K L þ K e

(7.3.77)

By definition the CLPT matrix S in Eq. (7.3.75) has only three lines: ST ¼ ½ Su

Sv

Sw 

(7.3.78)

such that the rotations 4x and 4q have to be calculated and included as additional rows S4x ¼ 

vSw vx

S4q ¼ 

1 vSw r vq

STextended ¼ ½ Su

(7.3.79) Sv

Sw

S4x

S4q 

Eqs. (7.3.74)e(7.3.77) are general for any distribution of elastic stiffness, but in the current discussion, all the stiffnesses will be assumed constant to impose the penalty effect. By using the penalty boundary condition formulation presented here and the proper set of approximation functions for u, v, w, 4x, 4q, it is possible to obtain many boundary conditions by setting the right values for each elastic constant. Table 7.3.1 shows the values that should be adopted for the constants to obtain the Table 7.3.1

Elastic constants for each boundary condition Simply supported

SS1: u¼v¼ w¼0

Ku ¼ Kv ¼ Kw ¼N K 4x ¼ K 4q ¼ 0

SS2: v ¼ w¼0

Kv ¼ Kw ¼N

SS3: u ¼ w¼0 SS4: w ¼ 0

Clamped CC1: u ¼ v ¼ w ¼ wx ¼ 0

Ku ¼ Kv ¼ Kw ¼ K 4x ¼ N K 4q ¼ 0

K u ¼ K 4x ¼ K 4q ¼ 0 Ku ¼ Kw ¼N K v ¼ K 4x ¼ K 4q ¼ 0 Kw ¼N K u ¼ K v ¼ K 4x ¼ K 4q ¼ 0

CC2: v ¼ w ¼ wx ¼ 0

K v ¼ K w ¼ K 4x ¼ N

CC3: u ¼ w ¼ wx ¼ 0

K u ¼ K w ¼ K 4x ¼ N

CC4: w ¼ wx ¼ 0

K w ¼ K 4x ¼ N

K u ¼ K 4q ¼ 0 K v ¼ K 4q ¼ 0 K u ¼ K v ¼ K 4q ¼ 0

Stability of composite shelletype structures

305

four most common types of boundary conditions usually found in the literature and that will be investigated in more detail along this thesis. Note that an infinite value (N) is actually implemented using a very high number (108 when not otherwise specified) and that any behavior between simply supported and clamped can be achieved by changing the K 4x constants. In the current implementation, it is possible to use different boundary conditions between the bottom and top edges [33].

7.3.5.2

Approximation functions

One of the limitations of the Ritz method is the difficulty to find a correct set of approximation functions for a given problem [3,8]. Some considerations are important when seeking for the right of approximation functions and Reddy [8] presents a detailed description of two properties that these functions should have: 1. Convergence: the error should decrease up to a required tolerance when the number of terms is increased 2. Completeness: the increment in the number of terms must be so that it will pass through the required order of the real solution. In a polynomial approximation, for example, if the real solution is u(x) ¼ ax2 þ bx3 þ cx5, one sequence of approximations of the form uapproxm ðxÞ ¼ cm x2mþ1 ; m ¼ 0; 1; 2., is not complete and will never achieve the true solution when m is increased

The fact that the rotations are connected to the normal displacements in the CLPT makes it harder to find approximation functions that satisfy the clamped boundary conditions because the first derivate must also fulfill the boundary conditions where w ¼ 0 and 4x ¼ 0. For the FSDT, since w and 4x have independent approximation functions, it is usually simpler to cope with the boundary conditions and therefore it is simpler to find the right set of approximation functions. The inclusion of elastic restraints as explained in Ref. [19] allowed approximation functions of the same complexity for both the CLPT and the FSDT, revealing an additional benefit of this approach. In the Ritz method the approximation function is composed of a base function Si and the Ritz constant ci that gives the amplitude of the function. It is crucial for the base function Si to 1. satisfy the essential boundary conditions of the problem, 2. be contiguous along the domain, in order to be used in the variational statement, 3. be independent from any other base function Sj.

A complete Fourier Series [39] can be used to approximate a given field variable in the 2D domain of the structural model shown in Fig. 7.3.1, giving f ðx; q; cÞ ¼

 x  x cija sin ip sinðjqÞ þ cijb sin ip cosð jqÞ L L i¼0 j¼0  x  x þ cijc cos ip sinð jqÞ þ cijd cos ip cosðjqÞ L L

m X n X

(7.3.80)

306

Stability and Vibrations of Thin-Walled Composite Structures

where cij represents the Ritz constants contained in c. Note that for a complete series the indices must start at zero. When j ¼ 0 the expressions reduce to f ðx; cÞ ¼

m X i¼0

 x  x ci0b sin ip þ ci0d cos ip L L

(7.3.81)

which is a function of x only. Considering this observation, Eq. (7.3.80) can be rewritten as f ðx; q; cÞ ¼

m1  X i1 ¼0

þ

ci1a Si1a þ ci1b Si1b

m2 X n2  X i2 ¼0 j2 ¼1



ci2 j2a Si2 j2a þ ci2 j2b Si2 j2b þ ci2 j2c Si2 j2c þ ci2 j2d Si2 j2d



(7.3.82)

with  x Si1a ¼ sin i1 p L

 x Si2 j2a ¼ sin i2 p sinð j2 qÞ L

 x Si1b ¼ cos i1 p L

 x Si2 j2b ¼ sin i2 p cosð j2 qÞ L 

S i 2 j 2c

x ¼ cos i2 p sinð j2 qÞ L

(7.3.83)

 x Si2 j2d ¼ cos i2 p cosð j2 qÞ L which can be written in matrix form as f ðx; q; cÞ ¼ Sf1 cf1 þ Sf2 cf2

(7.3.84)

where cf1 and cf2 contain the Ritz constants for the field variable f : ci1a ; ci1b ; . and

ci2 j2a ; ci2 j2b ; ., respectively, and the matrices Sf1 and Sf2 are the corresponding base functions. The format of Eqs. (7.3.82) and (7.3.84) is preferred to the format of Eq. (7.3.80) because it allows the use of a different number of terms for the functions depending only on x (axisymmetric) by setting m1 s m2. B€urmann et al. [40] used a similar separation for approximation functions applied to stiffened panels. Additional functions will be required and added to Eq. (7.3.82) to account for the two nonhomogeneous boundary conditions of the proposed model (cf. Fig. 7.3.1), i.e., the axial shortening caused by the axial compression and the torsion, both discussed in this section. The nonhomogeneous boundary conditions will be included in the third set of

Stability of composite shelletype structures

307

base functions Sf0 and the Ritz constants c f0 , such that the field variable f can be approximated as f ðx; q; cÞ ¼ Sf0 c f0 þ Sf1 c f1 þ Sf2 c f2

(7.3.85)

From Fig. 7.3.1, it can be seen that the field variables being approximated for the conical and cylindrical shells are those contained in the displacement vector uT ¼ {u v w} for the CLPT. A proper choice of which terms of Eq. (7.3.82) will be selected for each field variable must be done considering the boundary conditions. In this discussion the approximation functions that cover all the boundary condition shown in Table 7.3.1 will be developed and they will result in four different models as detailed in the proceeding discussion. The base functions can be arranged to include all the field variables in a single matrix, such that, for the CLPT: ST0 ¼ ½ Su0

Sv0

Sw0 

ST1 ¼ ½ Su1

Sv1

Sw1 

½ Su2

Sv2

Sw2 

ST2

¼

(7.3.86)

Using the base functions presented in the forms of Eq. (7.3.86), the displacement vector u can be written as 8 9 8 9 c0 > u> > > > > = = < > < > (7.3.87) u ¼ v ¼ S0 c0 þ S1 c1 þ S2 c2 ¼ ½ S0 S1 S2  c1 ¼ gc > > > > > > ; ; : > : > w c2 The nonhomogeneous boundary condition terms that form matrix S0 are shown in Eq. (7.3.88), which for the model of Fig. 7.3.1 is the top axial displacement of the “testing machine” uTM. Note how the functions contained in S0 will allow a linear shortening and a linear torsion. An important observation is that uTM is not measured along the x-axis, but along the axial direction (which coincide for cylindrical shells), allowing a convenient correlation between uTM and the applied axial load FC, as detailed in Ref. [1]. 3 ðL  xÞ 6 L cos a 7 7 6 7 6 T c0 ¼ f uTM g S0 ¼ 6 0 7 7 6 5 4 0 2

(7.3.88)

Eq. (7.3.89) shows a submatrix of the base functions contained in S1. It starts at column q and the calculation of the column index is also given for both the CLPT and the FSDT.

308

Stability and Vibrations of Thin-Walled Composite Structures

2 6 6 S1 ¼ 6 6/ 4

S1 uq

0 S1 vq S1 wq

0

3 7 7 /7 7 5

(7.3.89)

p ¼ 3i1 þ 1 q ¼ 3k1 þ 1 The base functions for S1 must be evaluated according to the boundary conditions of Table 7.3.1 and using the proposed field variable function of Eq. (7.3.82). In the proposed model. all the edges are restrained to expand and there is no rigid body motion along x and q, the rotation 4x is zero for clamped boundary conditions and nonzero for simply supported boundary conditions. As the penalty constraints presented in Section 7.3.5.1 will be applied, a model that allows the development of the rotation 4x can be used, and this will also serve to simulate clamped boundary conditions when the corresponding elastic stiffness is properly set according to Table 7.3.1. Based on this discussion, the final proposal for the base functions of S1 can be defined as shown in Eq. (7.3.90).  x S1 uq ¼ S1 vq ¼ S1 wq ¼ S1 q4q ¼ sin k1 p L   x S1 q4x ¼ cos k1 p L

(7.3.90)

Note that more terms of Eq. (7.3.82) could have been kept without negatively affecting the predicted displacements because the elastic constraints will enforce the desired boundary conditions, but using a minimum number of terms to produce the right response is preferred because of the reduced computational cost and to avoid numerical errors that often arise when using high values for the elastic constraints (using double precision the author found such numerical instabilities when K u;v;w;4x ;4q > 108 ). Eq. (7.3.91) shows a submatrix of the base functions contained in S2 with the corresponding functions for both the CLPT. The formula for the index q where this submatrix starts is also given. 2 6 6 S2 ¼ 6 6/ 4

S2 uqa

S2 uqb

3

0 S2 vqa

0 p ¼ 6½m2 ð j2  1Þ þ i2  þ 1 q ¼ 6½m2 ð‘2  1Þ þ k2  þ 1

S2 vqb S2 wqa

S2 wqb

7 7 /7 7 5

(7.3.91)

Stability of composite shelletype structures

309

Recalling Eq. (7.3.82), the author verified that for all the boundary conditions only two terms, Si2 j2a and Si2 j2b or Si2 j2c and Si2 j2d , are required to achieve a proper representation of the displacement fields. Writing all the base functions of Eq. (7.3.91) in the form x ;4q ¼ f ðxÞu;v;w;4x ;4q sinð‘ qÞ S2q u;v;w;4 2 a

u;v;w;4x ;4q

S2q b

(7.3.92)

¼ f ðxÞu;v;w;4x ;4q cosð‘2 qÞ

where functions f(x)w, f ðxÞ4x , and f ðxÞ4q can use a single expression for all the boundary conditions based on the same arguments given for S1 f ðxÞw ¼ sinðbx Þ f ðxÞ4x ¼ cosðbx Þ f ðxÞ4q ¼ sinðbx Þ

(7.3.93)

with: bx ¼ k 2 p

x L

and the expressions for f(x)u and f(x)v depend on the boundary condition type, as shown in Table 7.3.2. In the subsequent sections the suffixes BC1, BC2, BC3, and BC4 will be used to identify the models of Table 7.3.2. It will be shown how model BC4 can be used with the proper set of penalty stiffness constants to achieve BC3, BC2, or BC1 boundary conditions. Similarly, models BC2 and BC3 can be used to achieve BC1 [19].

7.3.5.3

Fitting measured imperfection data into a continuous function

The nonlinear equations discussed in Section 7.3.2 can take into account an initial imperfection field w0 that will directly influence the nonlinear strains grouped in the

Base functions for S2 for many boundary conditions

Table 7.3.2

Boundary condition

f(x)u

f(x)v

BC1

sin(bx)

sin(bx)

BC2

cos(bx)

sin(bx)

BC3

sin(bx)

cos(bx)

BC4

cos(bx)

cos(bx)

310

Stability and Vibrations of Thin-Walled Composite Structures

vector component εL0 . Arbocz [41] proposed in 1969 the half-cosine function for the imperfection field, which can be written as w0 ¼

n0 X m0 X

 cos

j¼0 i¼0

 ipx ðAij cosð jqÞ þ Bij sinðjqÞÞ L

(7.3.94)

where Aij and Bij are the amplitudes of each corresponding base function. The derivatives w0,x and w0,q used in the nonlinear kinematic equations are w0;x ¼

w0;q ¼

  n0 X m0 X ip ipx  sin ðAij cosðjqÞ þ Bij sinð jqÞÞ L L j¼0 i¼0 n0 X m0 X j¼0 i¼0

  ipx cos j ð Aij sinð jqÞ þ Bij cosðjqÞÞ L

(7.3.95)

The coefficients Aij and Bij are calculated in the implemented routines [33] using the least squares method of NumPy [36], and one implementation using pure Python/ NumPy is given by Castro et al. [1,42]. The 3D points corresponding to the geometric imperfections used in this thesis are obtained using the advanced topometric sensor (ATOS) imperfection measurement system [43], when not otherwise specified. In the current implementation [33], Eq. (7.3.94) is represented in matrix form as w0 ¼ aT c0  a ¼ fx0 sinð0qÞ fx0 cosð0qÞ fx1 sinð0qÞ fx1 cosð0qÞ / fxm0 sinð0qÞ fxm0 cosð0qÞ  / fxi sinð jqÞ fxi cosðjqÞ / fxm0 sinðn0 qÞ fxm0 cosðn0 qÞ   ipx fxi ¼ cos L (7.3.96) where c0 contains the coefficients Aij and Bij arranged in the vector form. Analogously, the functions w0,x(x, q) and w0,q(x, q) are also implemented in the matrix form of Eq. (7.3.96), as shown in Eq. (7.3.97).  w0;x ¼

va vx

T

c0

w0;q

 T va ¼ c0 vq

(7.3.97)

Figs. 7.3.7e7.3.9 show the measured imperfection field for cylinders Z23, Z25, and Z26, respectively, [44] (cf. Table 7.3.4) and the approximated imperfection field using

Stability of composite shelletype structures

311

m0 = 20, n0 = 30

m0 = 10, n0 = 15

x, mm

500

x, mm

500

0

0 –π

θ , rad

π

–π

θ , rad

π

m0 = 40, n0 = 60

m0 = 30, n0 = 45

x, mm

500

x, mm

500

0

0 –π

θ , rad

–π

π

θ , rad

π

m0 = 60, n0 = 90

m0 = 50, n0 = 75

x, mm

500

x, mm

500

0

0 –π

θ , rad

π

–π

θ , rad

Measured imperfection

x, mm

500

0 –π

θ , rad

π

Figure 7.3.7 Number of terms to approximate measured imperfection data: cylinder Z23.

π

312

Stability and Vibrations of Thin-Walled Composite Structures

m0 = 20, n0 = 30

m0 = 10, n0 = 15

x, mm

500

x, mm

500

0

0 –π

θ , rad

θ , rad

π

m0 = 40, n0 = 60

m0 = 30, n0 = 45

x, mm

500

x, mm

500

–π

π

0

0 –π

θ , rad

–π

π

θ , rad

π

m0 = 60, n0 = 90

m0 = 50, n0 = 75 500

x, mm

x, mm

500

0

0 –π

θ , rad

π

–π

θ , rad

Measured imperfection

x, mm

500

0 –π

θ , rad

π

Figure 7.3.8 Number of terms to approximate measured imperfection data: cylinder Z25.

π

Stability of composite shelletype structures

313

m0 = 20, n0 = 30

m0 = 10, n0 = 15 500

x, mm

x, mm

500

0

0 –π

θ , rad m0 = 30, n0 = 45

π

–π

θ , rad

π

m0 = 40, n0 = 60 500

x, mm

x, mm

500

0

0 –π

θ , rad

π

–π

m0 = 50, n0 = 75

θ , rad

π

m0 = 60, n0 = 90 500

x, mm

x, mm

500

0

0 –π

θ , rad

π

–π

θ , rad

Measured imperfection

x, mm

500

0 –π

θ , rad

π

Figure 7.3.9 Number of terms to approximate measured imperfection data: cylinder Z26.

π

314

Stability and Vibrations of Thin-Walled Composite Structures

Table 7.3.3

Material properties (modulus in GPa)

Material name

References

E11

E22

n12

G12

G13

G23

Geier 1997

[46,47]

123.550

8.708

0.319

5.695

5.695

5.695

Deg Cocomat

[44]

142.500

8.700

0.28

5.100

5.100

5.100

Shadmehri

[13]

210.290

5.257

0.25

3.154

3.154

2.764

Eq. (7.3.94) with different values for m0 and n0. All the contours are plotted with the same color scale of the corresponding measured imperfections. From these figures, one clearly sees how the approximated pattern approaches the measured pattern by increasing m0 and n0, and in theory, one could choose m0 and n0 as high as necessary to obtain a given accuracy, but in practice the least squares algorithms applied to obtain the Aij and Bij coefficients of Eq. (7.3.94) will usually require a high amount of RAM that limits the maximum m0 and n0 that can be chosen. Two strategies are suggested to increase the maximum values of m0 and n0 aiming to achieve a better accuracy for the approximated imperfection field, given the amount of RAM of the computer used in the calculations. The first strategy is based on the geometry of the structures under evaluation. For cylinders Z23, Z25, and Z26, described in Table 7.3.4, 2pR1/H z 3 and the first approximation functions over x has half the frequency of the approximation function over q, such that at the convergence, it is expected that n0 z 1.5  m0, assuming a similar imperfection resolution along x and q. This relation allows one to use less terms over x and more terms over q without losing the accuracy. The second strategy is to avoid using the full sample of measured points when building the coefficient matrix used for the least squares fit. For example, in the case of cylinder Z23 the geometric imperfection is represented in a text file with 341,099 lines, having one measured point per line; for cylinder Z25, there are 340,357 points; and for cylinder Z26, there are 331,307 points. A coefficient matrix built with these points using double precision (64 bits for each entry) will have approximately m0  n0  5.4 MB, which would limit the maximum number of terms to m0 ¼ 30 and n0 ¼ 45 (6.86 GB), considering that in the applied least squares routine [36] this amount will be doubled and that the computer used for the calculations has 16 GB of RAM available. The author suggests a number of randomly chosen measured points according to the formula: npoints ¼ nsample ð2 m0 n0 Þ nsample ¼ 10

(7.3.98)

which allowed to achieve successful approximations up to m0 ¼ 60 and n0 ¼ 90 terms. If the reader identifies that the approximated field does not correspond to the measured imperfection pattern when the reduced sample is used, it is likely that a higher value other than nsample ¼ 10 in the formula of Eq. (7.3.98) has to be adopted. Highly discrepant results were verified for nsample ¼ 1, i.e., when the number of measured

Geometric and laminate data from different sources in the literature Stacking sequence

Cone/cylinder name

References

Material

R1 (mm)

H (mm)

Z07

[48]

Deg Cocomat

250

510

0

0.125

[24/41]

Z11

[46,49]

Geier 1997

250

510

0

0.125

[60/02/68/52/37]

Z12

[46,49]

Geier 1997

250

510

0

0.125

[51/45/37/19/02]

Z28

[49]

Geier 1997

250

510

0

0.125

[38/68/902/8/53]

Z23

[44]

Deg Cocomat

250

500

0

0.1195

[24/41]

Z25

[44]

Deg Cocomat

250

500

0

0.117

[24/41]

Z26

[44]

Deg Cocomat

250

500

0

0.1195

[24/41]

Z32

[47,50]

Geier 1997

250

510

0

0.125

[H51/H45/H37/H19/02]

Z33

[47,50]

Geier 1997

250

510

0

0.125

[02/19/37/45/51]

Zsym

[51]

Geier 1997

250

510

0

0.125

[45/0]sym

C01

None

Geier 1997

400

200

30

0.125

[þ60/60]

C02

[2]

Deg Cocomat

400

200

45

0.125

[30/30/60/60/0]sym

C14

None

Deg Cocomat

400

300

35

0.125

[0/0/60/60/45/45]

C26

None

Deg Cocomat

400

300

35

0.125

[45/0/45/45/0/45]

ShadC02

[13]

Shadmehri

254

H

30

0.635

[þg/g]

ShadC04

[13]

Shadmehri

254

H

30

0.635

[þg/g/g/þg]

a (degrees)

Ply thickness (mm)

Inwardeoutward

Stability of composite shelletype structures

Table 7.3.4

315

316

(a)

Stability and Vibrations of Thin-Walled Composite Structures

(b)

(c)

Figure 7.3.10 Measured imperfections mapped to cone C02: (a) C02 with Z23 imperfection, (b) C02 with Z25 imperfection, and (c) C02 with Z26 imperfection.

points is equal to the number of rows of the coefficient matrix used in the least squares routine, and therefore, it is recommended to keep nsample  2. As the database for conical shells is limited and a higher amount of data is available for cylinders, it becomes convenient to define a methodology to map imperfection from cylindrical shells to cones. The approach suggested here consists of simply mapping the imperfection amplitudes in a 2D space, where the two coordinates are the normalized meridional position x/L and the angular circumferential position q. Fig. 7.3.10 shows the conical surface opened at q ¼ p with the imperfections from cylinders Z23, Z25, and Z26 mapped to cone C02, whose properties are presented in Table 7.3.4. For the cases of Fig. 7.3.10, one can expect the same convergence behavior shown in Figs. 7.3.7e7.3.9 when the measured imperfections are approximated using different values of m0 and n0. The effect of using different values for m0 and n0 on the nonlinear buckling behavior is investigated in Section 7.3.5.6, where the results from the semianalytical models are verified against finite element analyses.

7.3.5.4

Linear buckling analysis of unstiffened cones and cylinders

Castro et al. [1,19] verified the convergence rate of the approximation functions discussed in Section 7.3.5.2 and the recommended number of terms is at least m1 ¼ m2 ¼ n2 ¼ 50 for axial compression, and the analyses presented here are using this level of approximation, when not otherwise specified. The linear buckling results have been verified using converged finite element models with 420 elements around the circumference and 136 elements along the meridian, keeping the element aspect ratio close to 1:1. The shell element chosen is the linear squared with reduced integration, called S4R in Abaqus [45]. The first 50 buckling modes were compared for cylinder Z33 and cone C02 under axial compression, with both structures described in Table 7.3.4. The SS1 type of boundary condition was applied to both the bottom and top edges. All the cases were simulated using m1 ¼ m2 ¼ n2 ¼ 80. A schematic view of the finite element model used for axial

Stability of composite shelletype structures

317

FC Reference point fixed in u, v, w

MPC connecting u, v, w for SS1 u, w for SS3

zrec, wrec yrec, vrec Bottom edge: Fixed u, v, w for SS1 Fixed u, w for SS3 xrec, urec

Figure 7.3.11 Finite element model for linear buckling with axial compression and pressure.

compression and pressure is illustrated in Fig. 7.3.11. The transformation of the outputs from the rectangular coordinate system to the semianalytical model coordinate system is performed using Eq. (7.3.99). qcoord ¼ arctanðycoord =xcoord Þ ucyl ¼ wrec

ucone ¼ wcyl sin a þ ucyl cos a

vcyl ¼ vrec cosqcoord  urec sinqcoord

vcone ¼ vcyl

wcyl ¼ vrec sinqcoord þ urec cosqcoord

wcone ¼ wcyl cos a  ucyl sin a (7.3.99)

Table 7.3.5 shows the linear buckling predictions for cylinder Z33 under axial compression using Abaqus and the CLPT model described here. The percentage difference between the semianalytical models and Abaqus calculated is shown in Table 7.3.6, where a maximum difference of 1.8% can be seen for the 37th mode. The buckling modes are compared in Fig. 7.3.12, showing that the first buckling modes are identical and that the higher buckling modes may be interchanged, especially because the eigenvalues are close to one another. Table 7.3.7 shows the predictions for cone C02, and the percentage difference is shown in Table 7.3.8, where a maximum difference of 0.70% can be seen for the 41st mode. From the buckling modes shown in Fig. 7.3.13, it can be seen that the first modes are identical and that there is an interchange of the higher modes because of the proximity of the eigenvalues. Castro et al. [19] presented a study about linear buckling predictions in which the proposed models are compared to other models available in the literature, showing

318

Stability and Vibrations of Thin-Walled Composite Structures

Table 7.3.5 Linear buckling under axial compression for cylinder Z33 (all in kilonewtons) Mode

Abaqus

CLPT

Mode

Abaqus

CLPT

01

192.860

194.532

26

208.086

210.452

02

192.860

194.532

27

209.122

211.083

03

192.954

194.959

28

209.122

211.083

04

192.954

194.959

29

209.689

212.733

05

195.748

197.200

30

209.689

212.733

06

195.748

197.200

31

212.495

214.531

07

196.635

199.039

32

212.495

214.531

08

196.635

199.039

33

213.636

216.114

09

198.100

200.489

34

213.636

216.114

10

198.100

200.489

35

214.204

216.200

11

199.808

201.950

36

214.204

216.200

12

199.808

201.950

37

214.241

218.094

13

200.422

202.580

38

214.241

218.094

14

200.422

202.580

39

215.959

219.395

15

201.244

203.305

40

215.959

219.395

16

201.244

203.305

41

217.062

219.694

17

204.686

206.745

42

217.062

219.694

18

204.686

206.745

43

218.305

220.642

19

205.569

208.826

44

218.305

220.642

20

205.569

208.826

45

219.216

222.367

21

207.015

209.402

46

219.216

222.367

22

207.015

209.402

47

220.115

224.053

23

207.821

210.169

48

220.115

224.053

24

207.821

210.169

49

220.609

224.166

25

208.086

210.452

50

220.609

224.166

that many available models assume the orthotropic laminate approximation, i.e., the terms A16, A26, B16, B26, D16, D26, A45 of the laminate stiffness matrix are assumed to be zero, such that the torsionlike buckling shapes (found in Figs. 7.3.12 and 7.3.13) cannot be obtained.

Stability of composite shelletype structures

319

Linear buckling under axial compression for cylinder Z33, percentage differences

Table 7.3.6

7.3.5.5

Mode

CLPT (%)

Mode

CLPT (%)

01

0.87

27

0.94

03

1.04

29

1.45

05

0.74

31

0.96

07

1.22

33

1.16

09

1.21

35

0.93

11

1.07

37

1.80

13

1.08

39

1.59

15

1.02

41

1.21

17

1.01

43

1.07

19

1.58

45

1.44

21

1.15

47

1.79

23

1.13

49

1.61

25

1.14

Nonlinear analysis using the single perturbation load approach

The single perturbation load approach (SPLA) is introduced in Section 7.2.2.1 as a potential method to calculate the knockdown factor of imperfection-sensitive shells. The full NewtoneRaphson nonlinear algorithm, presented in Section 7.3.3.1, has been used to calculate the knockdown curve and the results are compared to those presented by Castro et al. [51] for cylinder Z33 and with Abaqus for cone C02, as shown in Figs. 7.3.14 and 7.3.15, respectively. The first observation is that the correct imperfection sensitivity was obtained using the semianalytical models. In Fig. 7.3.15 a small convergence analysis using different sets of integration points nx and nq is presented, where a higher influence of nx and nq can be seen for small imperfection amplitudes and the results are converged for these parameters for SPL > 5 N. The numerical integration adopted here is detailed in the work by Castro et al. [1,2,42]. Castro et al. [1,2] observed that the full NewtoneRaphson method stops in the local snap through, i.e., the first instability, whereas the finite element results go further up to the global buckling. Both the local snap through and the global buckling are explained in Ref. [51]. To verify if the local snap through is a limit point (where all the eigenvalues of the tangent stiffness matrix KT tend to zero) or a bifurcation point (where at least two eigenvalues of KT tend to zero), the eigenvalue analysis of Eq. (7.3.73) has been performed to verify if the eigenvalues are approaching 1 as the nonlinear

320

Stability and Vibrations of Thin-Walled Composite Structures

Mode

Abaqus

CLPT

1

3

5

7

9

11

13

15

17

19

21

23

25

27

Figure 7.3.12 Linear buckling under axial compression for cylinder Z33, modes 1e27.

Stability of composite shelletype structures

321

Linear buckling under axial compression for cone C02 (all in kilonewtons)

Table 7.3.7

Mode

Abaqus

CLPT

Mode

Abaqus

CLPT

01

122.626

123.285

26

128.695

129.469

02

122.626

123.285

27

130.186

130.592

03

122.676

123.311

28

130.186

130.592

04

122.676

123.311

29

130.214

130.980

05

122.871

123.541

30

130.214

130.980

06

122.871

123.541

31

131.742

132.109

07

122.972

123.669

32

131.742

132.408

08

123.064

123.669

33

131.902

132.408

09

123.064

123.800

34

132.193

132.587

10

123.805

124.375

35

132.193

132.587

11

123.805

124.375

36

132.570

132.893

12

124.259

125.067

37

132.570

132.893

13

124.836

125.432

38

133.059

133.304

14

124.836

125.432

39

133.059

133.304

15

124.901

125.622

40

133.358

134.291

16

124.901

125.622

41

133.358

134.291

17

125.953

126.730

42

134.503

134.793

18

125.953

126.730

43

134.503

134.793

19

126.346

126.833

44

135.046

135.391

20

126.346

126.833

45

135.046

135.391

21

127.273

128.045

46

135.118

136.100

22

127.273

128.045

47

135.118

136.100

23

128.126

128.562

48

136.517

136.860

24

128.126

128.562

49

136.517

136.860

25

128.695

129.469

50

136.910

137.817

analyses approach the critical loads shown in Fig. 7.3.15. The value “1” means that multiplying the current geometric stiffness matrix KG by l ¼ 1 will lead to the instability condition for which the neutral equilibrium criterion applies: det(KL þ lKG) ¼ 0. Three single perturbation load (SPL) values chosen were 24, 30, and 40 N, with the first two falling below P1 and the third above P1. The FCcrit values for the three cases

322

Stability and Vibrations of Thin-Walled Composite Structures

Linear buckling under axial compression for cone C02, percentage differences

Table 7.3.8

Mode

CLPT (%)

Mode

CLPT (%)

01

0.54

27

0.31

03

0.52

29

0.59

05

0.55

31

0.28

07

0.57

33

0.38

09

0.60

35

0.30

11

0.46

37

0.24

13

0.48

39

0.18

15

0.58

41

0.70

17

0.62

43

0.22

19

0.39

45

0.26

21

0.61

47

0.73

23

0.34

49

0.25

25

0.60

are 97.52, 88.52, and 75.55 kN, respectively. As shown in Fig. 7.3.16, the first four eigenvalues approach 1 when FC approaches FCcrit , and this behavior is the same for all the three cases. Recalling from Fig. 7.3.15 that for SPL ¼ 24 N and SPL ¼ 30 N the FCcrit corresponds to a global buckling and that for SPL ¼ 40 N the FCcrit corresponds to a local snap through, one concludes that the local snap through can also be interpreted as a limit point, and therefore, the full NewtoneRaphson method is not capable of going beyond this point to find the global buckling load represented by the filled lines in Figs. 7.3.14 and 7.3.15 in the region SPL > P1.

7.3.5.6

Nonlinear analysis using initial imperfections

In the previous sections, it was shown how the measured imperfection pattern is progressively better approximated using higher values of m0 and n0 in the half-cosine function of Eq. (7.3.94). In this section the effect of m0 and n0 on the nonlinear buckling response for axially compressed structures is investigated. Cylinders Z23, Z25, and Z26, detailed in Table 7.3.4, are evaluated using their respective geometric imperfection measurements obtained by Degenhardt et al. [44], given in Figs. 7.3.7e7.3.9. Material properties for all study cases herein presented are detailed in Table 7.3.3.. Scaling factors in the range between 0.1 and 4.0 are applied to the original imperfection amplitudes by directly multiplying vector c0 (cf. Eq. (7.3.96)), and this helped obtain a range of imperfection amplitudes up to 2h, where h is the laminate thickness (cf. Fig. 7.3.1).

Stability of composite shelletype structures

Mode

Abaqus

CLPT

323

Mode

1

15

3

17

5

19

7

21

9

23

11

25

13

27

Abaqus

CLPT

Figure 7.3.13 Linear buckling under axial compression for cone C02, modes 1e27.

Finite element simulations were used to verify the semianalytical predictions, and in the following discussion, all the semianalytical results are obtained using the model BC1 of Table 7.3.2 with all simulations applying the SS1 type of boundary conditions. Two methods for applying the measured imperfections to the finite element models have been investigated: an inverse-weighted (IW) interpolation and the half-cosine

324

Stability and Vibrations of Thin-Walled Composite Structures Donnell

Fc, kN

190

Castro et al., snap-through Castro et al., global buckling

150 110 P1 = 46.5 N

70 30 0

20

40 60 Perturbation load, N

80

Figure 7.3.14 Knock-down curve: cylinder Z33.

Donnell, nx = 50, nθ = 100 Donnell, nx = 100, nθ = 100 Donnell, nx = 75, nθ = 150 Abaqus, snap-through Abaqus, global buckling

125

Fc, kN

105 85 65

P1=32.1 N

45 0

10

20

30 40 Perturbation load, N

50

60

(a) 10

(b) 10

Eigenvalue

Eigenvalue

Figure 7.3.15 Knockdown curve: cone C02.

7 4

7 4 1

1

0

1

(c) 10

(d) 10 Eigenvalue

0.5 FC/FCcrit

Eigenvalue

0

7 4 1

0.5 FC/FCcrit

1

SPL = 24 N SPL = 30 N SPL = 40 N

7 4 1

0

0.5 FC/FCcrit

1

0

0.5 FC/FCcrit

1

Figure 7.3.16 Eigenvalue analysis for cone C02: (a) first eigenvalue, (b) second eigenvalue, (c) third eigenvalue, and (d) fourth eigenvalue.

Stability of composite shelletype structures

325

function of Eq. (7.3.94), for which the imperfection at each node w0node is directly obtained when the nodal coordinates are inputted using the cylindrical coordinate system of Fig. 7.3.1. In both the methods the initial nodal positions are changed according to the translations Dxnode, Dynode, and Dznode calculated with the relations 

ynode q ¼ arctan xnode



Dxnode ¼ SF$w0node $cos a$cos q

(7.3.100)

Dynode ¼ SF$w0node $cos a$sin q Dznode ¼ SF$w0node sin a where w0node is the nodal imperfection computed using one of the two methods, a is the cone semivertex angle defined in Fig. 7.3.1, and xnode and ynode are the coordinates of the node in the rectangular coordinate system of Fig. 7.3.11. In the IW interpolation algorithm for each node in the finite element model, a set of n closest points belonging to the measured data is selected and used to compute the imperfection value of the node. Eq. (7.3.101) is used for this purpose and this approach has been presented by Castro et al. [52]. w0node ¼

n X i

h

1 w0i wi

!,

n X 1 wi i

wi ¼ ðxnode  xi Þ2 þ ðynode  yi Þ2 þ ðznode  yi Þ2

ip

(7.3.101)

where wi is the weight of one closest point wi, w0i is the measured imperfection corresponding to the point i, and p is the power parameter set to 2 in the current study. Increasing the power parameter will increase the relative weight of the closest points. The results plotted in Fig. 7.3.17 present how the number of terms m2 and n2 affect the buckling response using m1 ¼ 120. The imperfection amplitude is normalized by the shell thickness, giving the abscissa parameter x/h. The approximation field is described using m0 ¼ 20, n0 ¼ 30. A good balance between computational cost and accuracy is achieved using m2 ¼ 25, n2 ¼ 45, and this set of parameters is used to further investigate the effect of an imperfection field described with a different number of terms m0, n0. Table 7.3.9 shows the nonlinear buckling loads obtained with finite elements and the proposed model for the three imperfect cylinders Z23, Z25, and Z26 by using six different approximation levels. From the relative errors, it can be seen that the proposed models achieve a high accuracy for all cases, with the maximum error of 3.74% and an average error of 1.43%. In Fig. 7.3.18 the cone C02 was simulated using the imperfections from cylinders Z23 and Z25 mapped as illustrated in Fig. 7.3.10. An approximated imperfection field using m0 ¼ 20, n0 ¼ 30 was selected and the difference between the Ritz method and the finite element predictions is small up to x/h ¼ 2.

326

Stability and Vibrations of Thin-Walled Composite Structures

Abaqus m2 = 15, n2 = 45 m2 = 20, n2 = 45 m2 = 25, n2 = 35 m2 = 25, n2 = 45 m2 = 30, n2 = 55

35

FC crit, (kN)

30 25 20 15 10 0.0

0.5

1.0

ξ /h

1.5

2.0

2.5

Figure 7.3.17 Cylinder Z23 with geometric imperfection convergence for m2 and n2.

7.3.6

Semianalytical approach based on the assembly of domains

Section 7.3.5 was fully dedicated to a semianalytical approach that could perform a diversity of analyses on monolithic cylinders and cones. Castro et al. [1,2,42] demonstrated that for linear static analysis with homogeneous laminate properties the singledomain approach performs more than two orders of magnitude faster than the counterpart finite element models, whereas for linear buckling analysis, it performs between one and two orders of magnitude faster. However, for nonlinear analysis the computational efficiency of the proposed single-domain models is significantly reduced, being one or even two orders of magnitude slower than the finite element counterparts. This is attributed to the numerical integration of these single-domain models, for which a huge amount of integration points is needed (hundreds for each dimension) to achieve a satisfactory result, adding to the fact that the evaluation of each integration point is expensive because the stiffness matrix of the whole domain must be computed during the evaluation of each integration point. Linear analyses using variable laminate properties over the domain would also require this slow numerical integration and the single-domain models become no longer the best option. Additional limitations of single-domain models appear when one needs to model stiffening structural elements using 2D panels, e.g., stiffened panels, isogrid structures, or when discontinuous domains exist, which is the case of any structure with cutouts or with manufacturing defects such as debonding of stiffeners or any loss of connection between parts. To overcome the aforementioned limitations, a strategy to assemble semianalytical models is discussed here and some of the results are verified against the single-domain approach previously presented, where the computational cost for some of the nonlinear analyses will be compared.

Z23

Z25

Z26

Abaqus

Ritz

Error (%)

Abaqus

Ritz

Error (%)

Abaqus

Ritz

Error (%)

m0 ¼ 10, n0 ¼ 15

28.2662

28.0928

0.61

27.5364

27.4670

0.25

28.4355

28.1704

0.93

m0 ¼ 20, n0 ¼ 30

25.1809

25.2666

0.34

24.2768

24.8645

2.42

24.4703

24.4893

0.08

m0 ¼ 30, n0 ¼ 45

24.6906

25.0311

1.38

24.3387

24.9011

2.31

25.2465

25.5161

1.07

m0 ¼ 40, n0 ¼ 60

24.8207

25.0319

0.85

24.5905

25.1889

2.43

25.0483

25.2667

0.87

m0 ¼ 50, n0 ¼ 75

24.5939

24.9160

1.31

24.1308

25.0336

3.74

25.1253

25.5419

1.66

m0 ¼ 60, n0 ¼ 90

24.6139

24.9890

1.52

25.4377

25.2263

0.83

24.9266

25.7106

3.15

Stability of composite shelletype structures

Critical axial buckling loads (kN) for the imperfect cylinders Z23, Z25, and Z26 using the measured imperfection amplitude (m1 [ 120, m2 [ 30, n2 [ 55) Table 7.3.9

327

328

Stability and Vibrations of Thin-Walled Composite Structures

120

120

100

C02 with w0 from Z25 FC crit, (kN)

FC crit, (kN)

C02 with w0 from Z23

80

60

100

80

60 0

2

1

3

0

1

2

ξ /h

3

ξ /h Abaqus IW Abaqus, m0 = 20, n0 = 30 m0 = 20, n0 = 30 (m2 = 25, n2 = 45)

Figure 7.3.18 Cone C02 using the imperfection of cylinders (a) Z23 and (b) Z25.

7.3.6.1

Assembly for unstiffened cylinders

Fig. 7.3.19 demonstrates how a series of cylindrical panels can be assembled to obtain a closed cylindrical shell with a global coordinate system. Fig. 7.3.20 depicts the local coordinate system of two adjacent panels that is used to illustrate their connectivity. Only the compatibility relation shown in Eq. (7.3.102) is necessary for the assembly of Fig. 7.3.19.



pi 4 pj



8 u¼u > > > > > < v¼v ycte>

w¼w > > > > : wy ¼ wy

P03

ðat constant yÞ

P02

(7.3.102)

P01 x

z y

Figure 7.3.19 Assembly of panels to form an unstiffened cylinder.

Stability of composite shelletype structures

329

P01

P02

x1 x2 z1 y1

y2

z2

Figure 7.3.20 Coordinate systems to evaluate the connectivity between two panels.

bj

bi

pj

pi xj yj

xi yi

Figure 7.3.21 Connection (pi 4 pj)ycte.

where u, v, w form the displacement field and pi and pj represent any pair of adjacent panels in Fig. 7.3.19. The connection at constant y (ycte) is achieved by applying the condition of Eq. (7.3.102) at yi ¼ bi for panel pi and at yj ¼ 0 for panel pj, as illustrated in Fig. 7.3.21.

7.3.6.2

Assembly for blade-stiffened cylinders

Fig. 7.3.22 shows an assembly of cylindrical panels, similar to the one for unstiffened cylinder that will form the cylinder’s skin, and a flat panel that represents a blade-type stiffener.

P02 Ps

P01

xs

x1 z1

z2

x2

zs y2

y1

ys

Figure 7.3.22 Assembly of panels to form a blade-stiffened cylinder.

330

Stability and Vibrations of Thin-Walled Composite Structures

For the assembly in Fig. 7.3.22, two compatibility relations are required, as shown in the following: 8 u¼u > > > > >

< v¼v ðat constant yÞ (7.3.103) pi 4 pj ycte > w¼w > > > > : wy ¼ wy

skin 4 flange

8 u ¼ u0 > > > > > > < v ¼ w0 > w ¼ v0 > > > > > : w ¼ w0 0 y y

ðat constant yÞ

(7.3.104)

where u, v, w represent the skin and u0 , v0 , w0 represent the flange displacements. Compatibility (pi 4 pj)ycte is the same as already discussed for the unstiffened case, whereas for the compatibility skin 4 flange a connection at the midsurface is assumed between the skin and the stiffener flange. The connection is performed by applying Eq. (7.3.104) for each stiffener (Fig. 7.3.23).

7.3.6.3

Approximation functions and kinematic equations in the natural coordinate system

A great challenge to use assembled domains is to find a set of approximation functions that allow a large flexibility of boundary conditions for each of the panel edges. Vescovini and Bisagni [53] presented a semianalytical model for omega-stiffened panels using trigonometric approximation functions that is valid for local buckling predictions where no displacement occurs at the connection and only the rotations must be coupled. The use of more-flexible approximation functions such as the Legendre

x′

x y

z

z′ y′ bf

Figure 7.3.23 Connection skin 4 flange.

Stability of composite shelletype structures

331

polynomials explored here permits the generalization of the modeling approach presented by Vescovini and Bisagni, allowing any displacement and rotation coupling at the interfaces between adjacent panels, which are necessary to achieve the assemblies proposed in Figs. 7.3.19 and 7.3.22. Rodrigues derived a form of the Legendre hierarchic orthogonal polynomials [54,55] largely applied by Bardell et al. [56e58] on the vibration problems. In this form the first four terms i ¼ 1, 2, 3, 4 consist of the Hermite cubic polynomials:   1 3 1 3  x þ x flagt1 si¼1 ðx or hÞ ¼ 2 4 4   1 1 1 1  x  x2 þ x3 flagr1 si¼2 ðx or hÞ ¼ 8 8 8 8   1 3 1 þ x  x3 flagt2 si¼3 ðx or hÞ ¼ 2 4 4   1 1 1 2 1 3 si¼4 ðx or hÞ ¼   x þ x þ x flagr2 8 8 8 8

(7.3.105)

Flags flagt1, flagr1, flagt2, and flagr2 will always be 0 or 1. Using these flags the first four terms of the Rodrigues polynomials can be used to enable/disable the translation and rotation of each domain boundary. Flag flagt1 is used to control the translation at boundary 1 (x ¼ 1), which is possible because using the Rodrigues polynomials, this is the only term among all the terms in the approximation function that produces si(x ¼ 1) ¼ 1. Similarly, flagt2 is used to control the translation at boundary 2 (x ¼ þ1). The rotation at x ¼ 1 and x ¼ þ1 is controlled using flagr1 and flagr2, respectively, as they are the only terms that produce a non-null rotation vs/vx at each respective domain boundary. A detailed overview of the produced shape functions is given by Bardell [56]. All terms with i > 4 are higher-order K-orthogonal hierarchic polynomials that always generate null translation (si ¼ 0) and null rotation (vsi/vx ¼ 0) at both extremities (x ¼ 1 or x ¼ þ1), defined as si>4 ðx or hÞ ¼

i=2 X ð1Þp ð2i  2p  7Þ!! p¼0

2p p!ði  2p  1Þ!

xi2p1

(7.3.106)

where q!! ¼ q(q  2).(2 or 1), 0!! ¼ (1)!! ¼ 1, and i/2 in the summation is an integer division. The Rodrigues form of Legendre’s polynomials requires each dimension varying within the natural range between 1 and þ1. Therefore, the cylindrical panel kinematic equations should be derived for a natural coordinate system, as the one illustrated in Fig. 7.3.24, which is valid for every panel in the assemblies in Figs. 7.3.19 and 7.3.22.

332

Stability and Vibrations of Thin-Walled Composite Structures

bp

ξ xp

η

zp

yp

ap r

Figure 7.3.24 Curved panel using natural coordinates.

Applying the variable transformation x¼

2xp 1 ap

(7.3.107)

2yp h¼ 1 bp to the kinematic relations for a cylindrical shell given in Eq. (7.3.22) and keeping only the terms belonging to Donnell’s equations, a new set of kinematic relations expressed in terms of the natural coordinate system is obtained, as given in Eq. (7.3.108). εTp ¼

εð0Þ xx

  2 ¼ ux þ ap

gð0Þ xy

n

o

εð0Þ xx

εð0Þ yy

gð0Þ xy

kTp ¼ f kxx

kyy

kxy g

! 2 w2x a2p

εð0Þ yy

  2 1 ¼ vh þ flagcyl w þ bp r

      2 2 4 ¼ uh þ vx þ wx wh bp ap ap bp ! 4 kyy ¼  2 whh bp

! 2 w2h b2p

! 4 kxx ¼  2 wxx ap



 4 kxy ¼ 2 wxh ap bp (7.3.108)

where flagcyl ¼ 1 for the skin panels and the stiffener’s base, both cylindrical panels, and flagcyl ¼ 0 for the stiffener’s flange, modeled as a flat panel. All structural matrices introduced in Section 7.3.2 can be readily derived using the kinematic relations expressed in terms of the natural coordinate system in Eq. (7.3.108) and all linear and nonlinear systems can be similarly solved, once the compatibility equations between all the panels in the assembly is enforced.

Stability of composite shelletype structures

7.3.6.4

333

Enforcing compatibility using a penalty approach

The compatibilities of Eqs. (7.3.103) and (7.3.104) are imposed using a penalty approach on the internal energy of the system, as given by Eq. (7.3.109). Upi pjycte ¼ Usf ¼

aZ

2 h 2 . i 2 2 2 2 kc1  u Þ þ ðv  v Þ þ ðw þ w Þ þ w  w k c1 ðu kc1 i j i j i j h h t r t dx b pi i b pj j x

2 aZ 2

x

" k c2 t



2 2 wh  w0h0 ðu  u Þ þ ðv  w Þ þ ðw þ v Þ þ bs bf 0 2

0 2

0 2

2 k c2 r

.

# kc2 t

dx

(7.3.109)

where bs is the width of one of the skin panel attached to the flange, bf is the flange width, and a is the panel length, which in fact equals the cylinder length for the cases ci investigated here. Constants with k ci t and k r are translational and rotational penalty stiffnesses, respectively. ci Theoretically, the penalty stiffnesses kci t and k r can be arbitrarily high to impose the energy penalty. However, the use of high values is associated with numerical instabilities such that one should choose the penalty stiffnesses that are just high enough to impose the proper penalties, but not excessively high. In the current study, it is proci posed to calculate kci t and k r based on laminate properties of the panels being connected, instead of using fixed high values, which is common practice in the literature. All penalty stiffnesses are calculated assuming that the membrane and bending loads Nxx, Nyy, Mxx, and Myy are continuous from one panel to another and that the compatibility of strains is such that the strain at the connection is assumed to be the average between the strains at the adjacent panels. Two strain compatibility scenarios are used, as presented in Eq. (7.3.110). p

ðaÞ

pi εyy þ εyyj

2

p

¼ εconn yy

ðbÞ

pi kyy þ kyyj

2

¼ kconn yy

(7.3.110)

The compatibility of Eq. (7.3.110(a)) is used to compute k c1 t . For both panels the simplification εyy ¼ Nyy/A22 is assumed, with A22 taken from the laminate ABD matrix. 2 From a dimensional analysis of Eq. (7.3.109) the units of k c1 t must be (N/m ), such that 

¼ Nyy kc1 the strain at the connection can be stated as εconn t h , where h will be taken yy as the average thickness of both panels being connected. Holding these assumptions, the compatibility of Eq. (7.3.110(a)) gives Nyy Nyy pi þ pj ¼ 2A22 2A22

Nyy h þ h  pi pj kc1 t 2 p

kc1 t

pi 4A22 A j  22 ¼ p pi þ A22j ðh pi þ h pj Þ A22

(7.3.111)

334

Stability and Vibrations of Thin-Walled Composite Structures

 With similar assumptions and using Eq. (7.3.110(b)) with kconn ¼ Myy kc1 r h and yy for both panels using kyy ¼ Myy/D22, with D22 taken from the laminate ABD matrix, k c1 r can be computed as kc1 r

¼

p

pi 4D22 D j  22 p pi þ D22j ðh pi þ h pj Þ D22

(7.3.112)

c2 Finally, the penalty constants kc2 t and k r used to connect the skin to the stiffener’s c1 flange are assumed to be the same as kt and k c1 r , respectively, giving c1 kc2 t ¼ kt

(7.3.113)

c1 kc2 r ¼ kr

(7.3.114)

and

ci The penalty constants kci t and k r calculated as suggested here showed to provide numerically stable results for a wide range of inputs, with the Legendre polynomial orders up to 25th, which is the highest order tested by the authors.

7.3.6.5

Linear buckling analysis of unstiffened cylinders using the multidomain approach

Fig. 7.3.25 shows a convergence analysis using the multidomain approach described here for cylinder Z33. A different number of panels along the circumference were tested to generate various convergence curves. The computational cost for each case is shown in Fig. 7.3.26, where the computational cost of the single-domain model presented in Section 7.3.5 is also added. Both single- and multidomain approaches converge to the same result. The multidomain approach already converges with npanels ¼ 6 and m ¼ n ¼ 10 for the first eigenmode. The single-domain approach is already proved to be highly efficient for predicting the linear buckling of homogeneous cylinders and cones [19], and considering the multidomain model convergence with m ¼ n ¼ 10 for npanels ¼ 6, the multidomain strategy becomes only twice as slow as the single-domain approach for homogeneous cylinders, which is reasonably good considering that the finite element approach showed to be as much as 10e100 times slower than the single-domain approach for linear buckling analyses. For higher modes, more terms may be needed and the single-domain approach showed a convergence with about m2 ¼ n2 ¼ 50, as mentioned earlier, whereas the multidomain approach converged with npanels ¼ 8 and m ¼ n ¼ 14, being about 50% slower than the single-domain approach. The higher eigenvectors obtained with the multidomain approach as showed in Fig. 7.3.27, where a better correlation with the finite element model is seen, when compared to the singledomain model.

Stability of composite shelletype structures

335

1000 npanels = 2 npanels = 3 npanels = 4 npanels = 5 npanels = 6 npanels = 7

900 800

FCCR (kN)

700 600

npanels = 8 npanels = 9 npanels = 10 Abaqus

500 400 300 200 100 6

8

10 12 m = n (for each domain)

14

16

Figure 7.3.25 Convergence for linear buckling using multidomains.

Computational cost (s)

101

100

npanels = 2 npanels = 3 npanels = 4 npanels = 5 npanels = 6 npanels = 7

10–1

10

npanels = 8 npanels = 9 npanels = 10 SD, m2 = n2 = 20 SD, m2 = n2 = 50

–2

6

8

10 12 m = n (for each domain)

14

16

Figure 7.3.26 Computational cost for linear buckling using multidomains.

It is important to emphasize the aforementioned argument that the single-domain approach will always be the best cost-efficient solution for shells with constant laminates over the domain where no discontinuities such as cutouts or debonding defects take place. For cases with nonhomogeneous properties or where discontinuities take

336

Stability and Vibrations of Thin-Walled Composite Structures

Mode

Assembled Model

Mode

1

15

3

17

5

19

7

21

9

23

11

25

13

27

Assembled Model

Figure 7.3.27 Linear buckling using assembled model for cylinder Z33, modes 1e27.

place, the multidomain approach is the best choice because of the simplicity of describing the discontinuous domain as an assembly of continuous subdomains. Regarding nonlinear analyses, the multidomain approach allows for a better distribution of the integration points at each subdomain, preventing the calculation of huge matrices during the evaluation of each integration point. The following section deals with the nonlinear analyses of imperfect cylinders where the imperfection is induced by SPLs.

7.3.6.6

Linear buckling analysis of stiffened cylinders using multidomain approach

This section will compare finite element results with the proposed approach using an assembly of semianalytical models. Cylinder Z33 was used adding 10 equally spaced stiffeners using the same laminate properties of Tables 7.3.3 and 7.3.4. Each stiffener has 20 mm width and a 15-ply laminate, whose stacking sequence is [0, 90, 0]5.

Finite element result

Assembly of semi-analytical models

Figure 7.3.28 Normal displacements for a stiffened cylinder under a constant Nxx load.

Finite element result –0.821 –0.832 –0.843 –0.855 –0.866 –0.877 –0.888 –0.899 –0.91 –0.921 –0.932 –0.943 –0.955 –0.966 –0.977 –0.988 –0.999

Assembly of semi-analytical models –0.7997

–1.0141

Figure 7.3.29 Membrane stress Nxx field for the stiffened cylinder.

338

Stability and Vibrations of Thin-Walled Composite Structures

Finite element model

Semi-analytical model

1st mode

NxxCR = 111.719 N / mm

Nxx

NxxCR = 114.9236 N / mm

Nxx

NxxCR = 116.536 N / mm

NxxCR = 116.518 N / mm

CR

= 112.748 N / mm

2nd mode

CR

= 115.011 N / mm

3rd mode

Figure 7.3.30 Linear buckling modes for the stiffened cylinder.

The stiffened cylinder is simply supported with a constant Nxx load applied along the whole skin and the upper top edges of the stiffeners. The obtained displacement field is shown in Fig. 7.3.28, in which a very close correlation between the finite element result and the semianalytical model can be seen.

Stability of composite shelletype structures

339

Fig. 7.3.29 shows the membrane stress Nxx distribution for both the finite element and the semianalytical model, in which again a very good correlation is verified with a relative difference of 2.5% for Nxxmax and þ1.5% for Nxxmin . The possibility to represent complex membrane stress states is crucial for linear buckling analyses, as the geometric stiffness matrix depends on this membrane stress state. Fig. 7.3.30 shows linear buckling results for the stiffened cylinder under evaluation, with the first three eigenmodes and eigenvalues compared. A very good correlation was found between finite elements and the assembly of semianalytical models proposed here.

References [1] S.G.P. Castro, Semi-Analytical Tools for the Analysis of Laminated Composite Cylindrical and Conical Imperfect Shells under Various Loading and Boundary Conditions, Clausthal University of Technology, Clausthal, Germany, 2014. [2] S.G.P. Castro, C. Mittelstedt, F.A.C. Monteiro, M.A. Arbelo, R. Degenhardt, A semianalytical approach for the linear and non-linear buckling analysis of imperfect unstiffened laminated composite cylinders and cones under axial, torsion and pressure loads, Thin-Walled Structures 90 (May 2015) 61e73. [3] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells, Theory and Analysis, second ed., CRC Press, Boca Raton, 2004. [4] S. Timoshenko, Strength of Materials, Part I, D. Van Nostrand Company, Inc., Lancaster, PA, 1948. [5] S. Timoshenko, Strength of Materials, Part II, D. Van Nostrand Company, Inc., Lancaster, PA, 1948. [6] S. Timoshenko, J.N. Goodier, Theory of Elasticity, second ed., McGraw-Hill, 1951. [7] S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, second ed., McGrawHill, 1959. [8] J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, second ed., John Wiley & Sons, New Jersey, 2002. [9] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, Volume 2: Solid Mechanics, fifth ed., Butterworth-Heinemann, Oxford, 2000. [10] G.-Q. Zhang, Stability Analysis of Anisotropic Conical Shells, Technical University Delft e Faculty of Aerospace Engineering, 1993. [11] N.H. Hadi, K.A. Ameen, Nonlinear free vibration of cylindrical shells with delamination using high order shear deformation theory: a finite element approach, American Journal of Scientific and Industrial Research 2 (2) (2011) 251e277. [12] V.V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity, Graylock Press, Rochester, 1953. [13] F. Shadmehri, S.V. Hoa, M. Hojjati, Buckling of conical composite shells, Composite Structures 94 (2012) 787e792. [14] F. Shadmehri, Buckling of Laminated Composite Conical Shells; Theory and Experiment (Ph.D. thesis), Concordia University, Montreal, Quebec, Canada, 2012. [15] E.J. Barbero, J.N. Reddy, J.L. Teply, General two-dimensional theory of laminated cylindrical shells, AIAA Journal 28 (3) (1990) 544e553. [16] L. Tong, T.K. Wang, Simple solutions for buckling of laminated conical shells, International Journal of Mechanical Sciences 34 (2) (1992) 93e111.

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Stability and Vibrations of Thin-Walled Composite Structures

[17] J.N. Reddy, A general non-linear third-order theory of plates with moderate thickness, International Journal of Non-Linear Mechanics 25 (6) (1990) 677e686. [18] J.N. Reddy, A simple higher-order theory of plates with moderate thickness, Journal of Applied Mechanics 51 (1984) 745e752. [19] S.G.P. Castro, C. Mittelstedt, F.A.C. Monteiro, M.A. Arbelo, G. Ziegmann, R. Degenhardt, Linear buckling predictions of unstiffened laminated composite cylinders and cones under various loading and boundary conditions using semi-analytical models, Composite Structures 118 (December 2014) 303e315. [20] L.H. Donnell, A new theory for the buckling of thin cylinders under axial compression and bending, ASME Transactions 56 (1934) 795e806. [21] J.L. Sanders, Nonlinear theories of thin shells, Quarterly of Applied Mathematics 21 (1963) 21e36. [22] G.J. Simitses, D. Shaw, I. Sheinman, J. Giri, Imperfection sensitivity of fiber-reinforced, composite, thin cylinders, Composite Science and Technology 22 (1985) 259e276. [23] G.J. Simitses, I. Sheinman, D. Shaw, The accuracy of the Donnell’s equations for axiallyloaded, imperfect orthotropic cylinders, Computers and Structures 20 (6) (1985) 939e945. [24] Y. Goldfeld, I. Sheinman, M. Baruch, Imperfection sensitivity of conical shells, AIAA Journal 4 (3) (2003) 517e524. [25] Y. Goldfeld, Imperfection sensitivity of laminated conical shells, International Journal of Solids and Structures 44 (2007) 1221e1241. [26] J. Arbocz, The Effect of Initial Imperfections on Shell Stability e An Updated Review, TU Delft Report LR-695, Faculty of Aerospace Engineering, The Netherlands, 1992. [27] K.Y. Yeh, B.H. Sun, F.P.J. Rimrott, Buckling of imperfect sandwich cones under axial compression e equivalent-cylinder approach. Part I, Technische Mechanik 14 (3/4) (1994) 239e248. [28] B.O. Almroth, Influence of Imperfections and Edge Restraint on the Buckling of Axially Compressed Cylinders, NASA CR-432, Lockheed Missiles and Space Company, Sunnyvale, California, 1966. [29] S. Yamada, J.G.A. Croll, N. Yamamoto, Nonlinear buckling of compressed FRP cylindrical shells and their imperfection sensitivity, Journal of Applied Mechanics 75 (July) (2008) 41005-1e41005-10. [30] K.-J. Bathe, Finite Element Procedures, Prentice Hall, New Jersey, 1996. [31] M.A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures e Volume 1, John Wiley & Sons, London, UK, 2000. [32] A.J.M. Ferreira, J.T. Barbosa, Buckling behaviour of composite shells, Composite Structures 50 (1) (2000) 93e98. [33] S.G.P. Castro, Computational Mechanics Tools, Version 0.7.1, November 2016 [Online]. Available: http://compmech.github.io/compmech/. [34] R.M. Jones, Buckling of Bars, Plates and Shells, Bull Ridge Publishing, Blacksburg, Virginia, USA, 2006. [35] A. Jennings, J. Halliday, M.J. Cole, Solution of linear generalized eigenvalue problems containing singular matrices, Journal of the Institute of Mathematics and its Applications 22 (1978) 401e410. [36] T.E. Oliphant, Python for scientific computing, Computing in Science and Engineering 9 (3) (2007) 10e20. [37] E. Jones, T. Oliphant, P. Peterson, et al., Scipy: open source scientific tools for Python, 2001 [Online]. Available: http://www.scipy.org/.

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341

[38] P. Som, A. Deb, A generalized Ritz-based method for nonlinear buckling of thin cylindrical shells, Thin-Walled Structures 76 (2014) 14e27. [39] E.W. Weisstein, Generalized Fourier Series, From MathWorldeA Wolfram Web Resource, 2013 [Online]. Available: http://mathworld.wolfram.com/GeneralizedFourier Series.html. [40] P. B€urmann, R. Rolfes, J. Tessmer, M. Schagerl, A semi-analytical model for local postbuckling analysis of stringer- and frame-stiffened cylindrical panels, Thin-Walled Structures 44 (2006) 102e114. [41] J. Arbocz, The imperfection data bank, a mean to obtain realistic buckling loads, Journal of Applied Mechanics (1968) 535e567. [42] S.G.P. Castro, C. Mittelstedt, F.A.C. Monteiro, R. Degenhardt, G. Ziegmann, Evaluation of non-linear buckling loads of geometrically imperfect composite cylinders and cones with the Ritz method, Composite Structures 122 (April 2015) 284e299. [43] R. Degenhardt, A. Kling, H. Klein, W. Hillger, H.C. Goetting, R. Zimmermann, K. Rohwer, Experiments on buckling and postbuckling of thin-walled CFRP structures using advanced measurement systems, International Journal of Structural Stability and Dynamics 7 (2) (2007) 337e358. [44] R. Degenhardt, A. Kling, A. Bethge, J. Orf, L. K€arger, R. Zimmermann, K. Rohwer, A. Calvi, Investigations on imperfection sensitivity and deduction of improved knockdown factors for unstiffened CFRP cylindrical shells, Composite Structures 92 (8) (2010) 1939e1946. [45] Abaqus-6.11, Analysis User’s Manual. Volume II: Analysis, Dassault Systemes, 2011. [46] B. Geier, G. Singh, Some simple solutions for buckling loads of thin and more date thick cylindrical shells and panels made of laminated composite material, Aerospace Science and Technology 1 (1997) 47e63. [47] B. Geier, H. Meyer-Piening, R. Zimmermann, On the influence of laminate stacking on buckling of composite cylindrical shells subjected to axial compression, Composite Structures 55 (2002) 467e474. [48] C. H€uhne, R. Rolfes, E. Breitbach, J. Teßmer, Robust design of composite cylindrical shells under axial compression e simulation and validation, Thin-Walled Structures 46 (2008) 947e962. [49] H.-R. Meyer-Piening, M. Farshad, B. Geier, R. Zimmermann, Buckling loads of CFRP composite cylinders under combined axial and torsion loading e experiment and computations, Composite Structures 53 (2001) 427e435. [50] R. Zimmermann, Optimierung Axial Gedr€uckter CFK-zylinderschalen, Fortschrittsberichte VDI-Reihe 1, Nr.207, D€usseldorf VDI Verlag, 1992. [51] S.G.P. Castro, M.A. Arbelo, R. Zimmermann, R. Degenhardt, Exploring the constancy of the global buckling load after a critical geometric imperfection level in thin-walled cylindrical shells for less conservative knock-down factors, Thin-Walled Structures 72 (2013) 76e87. [52] S.G.P. Castro, R. Zimmermann, M.A. Arbelo, R. Khakimova, M.W. Hilburger, R. Degenhardt, Geometric imperfections and lower-bound methods used to calculate knock-down factors for axially compressed composite cylindrical shells, Thin-Walled Structures 74 (January) (2014) 118e132. [53] R. Vescovini, C. Bisagni, Semi-analytical buckling analysis of omega stiffened panels under multi-axial loads, Composite Structures 120 (2015) 285e299. [54] A.G. Peano, Hierarchies of conforming finite elements for plate elasticity and plate bending, Computers and Mathematics with Applications 2 (1976) 211e224.

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Stability and Vibrations of Thin-Walled Composite Structures

[55] D.C. Zhu, Development of hierarchical finite element methods at BIA, in: Proceedings of the International Conference on Computational Mechanics, May 1985, Tokyo, 1986. [56] N.S. Bardell, Free vibration analysis of a flat plate using the hierarchical finite element method, Journal of Sound and Vibration 151 (2) (1991) 263e289. [57] N.S. Bardell, J.M. Dunsdon, R.S. Langley, Free and forced vibration analysis of thin, laminated, cylindrically curved panels, Composite Structures 38 (1e4) (1997) 453e462. [58] N.S. Bardell, J.M. Dunsdon, R.S. Langley, On the free vibration of completely free, open, cylindrically curved, isotropic shell panels, Journal of Sound and Vibration 207 (5) (1997) 647e669.

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7.4 Composite spheroidal shells under external pressure Jan Błachut The University of Liverpool, Liverpool, United Kingdom

7.4.1

Introduction

A spheroid is formed by rotating an ellipse about its axis (Fig. 7.4.1(a)). The shape of a football (A/B < 1) is called a prolate spheroid. The doorknob shape (A/B > 1) is called an oblate spheroid, and the transition geometry (A ¼ B) gives a sphere. Spheroids, or their parts, subjected to external pressure are found in a wide range of engineering applications. They have initially been considered as shapes for the pressure hulls [1,2], but there are several other reasons for the continuing interest in their performance under external pressure. One of them is related to a new breed of space launchers (common bulkhead separating LOX and LH2). Another is associated with the quest for oil/gas and other natural resources in deeper environments. Here special interests are associated with depths greater than 3 km. Several manned vehicles capable of exploring the deep sea (6 km) have been developed within the past few decades and a new addition to this list is the Jiaolong vehicle rated at 7 km depth [3,4]. Pulse waves generated by controlled implosion of buoyancy units, across a range of depths, are also actively studied as a part of new warfare (assault and/or protection of existing subsea assets) [5,6]. Spheroidal shells made from ceramics are of major interest here. Buoyancy aids used in the oil industry provide uplift, therefore reducing the total net weight of large submerged structures, e.g., risers, pipelines. Again, spheroids are the forefront of interest here. Some buoyancy systems have relied on the use of foams, which have a specific gravity less than water. As the operational depth increases the compressive strength of the material has to increase to oppose the pressure created

p

t

C L Spherical cap

B

t

A

t

B

(c)

p

B

C L

A

(c) t

t

p p

Knuckle s

C L

C L

R

Pole

t

A Rs

(b)

D

D Equator (a)

(d)

(e)

Figure 7.4.1 Geometry of spheroidal, elliptical (prolate/oblate), hemispherical, and torispherical shells.

r

344

Stability and Vibrations of Thin-Walled Composite Structures

by the weight of the water. The standard technique for increasing the compressive strength of the material, while maintaining a specific gravity less than water, is to include microscopic hollow glass spheres in the matrix, known as syntactic foams. These types of materials have been extensively used where the ultimate design depth is less than 3000 m. One significant drawback in the use of syntactic foams for underwater applications is the low compressive modulus, which results in a significant reduction in buoyancy with operational depth. Where deeper design depths are required, carbon fiber macrospheres are incorporated into the syntactic foams. Availability of a new generation of block-type syntactic foam buoyancy units has facilitated a range of underwater activities otherwise deemed as impossible, e.g., diving into the bottom of the oceans in 2012 [7]. A potential competitor for syntactic foam in deepwater application is the use of a composite vessel of the form of ellipsoid or more classical form, i.e., cylinder capped with hemispherical, elliptical, or torispherical domed closures. The load-carrying capacity of either geometry of the vessel, when subjected to external pressure, can be affected by static stability loss (buckling), which remains one of fundamental design limitations, and this is the subject of this section.

7.4.2 7.4.2.1

Brief review of major works to date on metallic and composite spheroids Background

The magnitude of buckling pressure of spheroidal shells depends on various factors and one of the major ones is the influence of initial geometric imperfections. Buckling of geometrically imperfect shells, in general, has been a subject of major research effort when it was discovered that the initial geometric imperfections are behind the large discrepancy between experiment and theory, especially for spheroids (hemispheres, spheres, torispheres, etc.). To ascertain the sensitivity of buckling load to initial deviations from perfect shape, one has to decide what are shape deviations, i.e., how they are defined, where they are positioned, what their maximum amplitude is, etc. These questions still remain unanswered. Over the years a number of approaches to model the shape of initial geometric imperfections have been tried. The shape that reduces the buckling strength the most has always been sought, as this would allow the designer to plan for the worst-possible scenario. Irrespective of the approach, it is imperative to understand the mechanisms of stability loss in spheroids or their parts before considering the sensitivity of buckling loads to shape deviations from perfect geometry. Aerospace-oriented research on buckling of spheroids includes Refs. [8e14]. Buckling of rib-reinforced hemispherical shells aimed at separating LOX and LH2 is discussed in Ref. [8]. The main thrust of this analytical work is devoted to the estimation of knockdown factors being the function of imperfection amplitude. Practicalities of spin forming of nonsegmented 5.4-m-diameter

Stability of composite shelletype structures

345

hemispherical bulkheads manufactured from aluminum AA 2219 are given in Ref. [9]. As a part of the next generation of launchers, research has been carried out on the common bulkhead between LOX and LH2 in the form of sandwich construction [10]. The hemispherical shell of 5 m diameter has two aluminum faces and foam core. It aims to not only withstand buckling but also insulate two cryoliquids that differ in temperature by about 70K. It is worth noting here the use of composites to repair large sonar dome [11] and in tests on laboratory-scale glass-reinforced plastic (GRP) isotropic ellipsoidal shells [12]. A numerical study of elastic buckling of isotropic ellipsoids is given in Ref. [13]. Review of published papers on stress and buckling of isotropic elliptical dome closures can be found in Ref. [14]. For spheroids made from composite materials the subject of imperfections has gained importance because of their perceived use as primary and secondary structures in aerospace as well as their use in underwater structures. The imperfections in fabricated domes are likely to be distributed randomly and will normally consist of dimples and increased-radius flat spots of various sizes. It has been shown that for metallic heads the force-induced dimple (FID) in doubly curved shells has a number of advantages over the lower-bound concept, which otherwise has been successfully used for approximately two decades [15]. These include much faster computing of sensitivity response and being able to trace physics in the case of compound shells, e.g., in torispherical heads. In the latter case, small-amplitude shape deviations do not dissipate the buckling strength. This explicitly contradicts an entrenched belief in the eigenmode modelization of imperfections as leading to the worst buckling performance scenario. Questions have always been raised about how realistic are deviations from perfect geometry in the form of eigenmode(s), especially when taken as a string of regular ineout deformations. Ref. [16], for example, shows that if one extracts a half wave from the eigenshape as the new shape imperfection then the sensitivity of the buckling pressure literally follows the same response as for the whole eigenshape imperfection pattern. This obviously does not answer the question what will happen when this “single eigen-dimple” is positioned elsewhere at a shell’s surface. An answer to this question is the creation of a localized dimple by a concentrated force that is much simple to create and easier to move around the shell’s surface. Also, occurrence of such localized inward dimple is more likely in practice than that of “eigen-dimple.” As mentioned earlier, the inward dimples created by a concentrated force, in metallic shells, can lower the buckling strength well below the other commonly used imperfection patterns. But the response of composite domes to this pattern of imperfection remains largely unknown. Despite a lot of effort to reduce the disparity between theoretical predictions of buckling pressure and test data, the design of externally pressurized heads still relies on empirical results and on the concept of “lower bound,” see, for example, Ref. [15]. It is worth noting here that decades of accumulated know-how on the sensitivity of buckling pressure to the initial shape imperfections for metallic shells have been transferred to the following practical design recommendations: PD 5500, Ref. [17]; ASME, Ref. [18]; ECCS, Ref. [19]; and NASA, Ref. [20].

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Stability and Vibrations of Thin-Walled Composite Structures

7.4.2.2

Composite domed closuresda brief outline of manufacturing, modeling, and analysis

This section is concerned with domed closures. It is assumed that there are no openings in these components and that they are subjected to quasistatic external pressure of follower nature. Primarily composite components are discussed, but references to metallic heads are also provided. The following mechanisms of structural integrity loss, due to external pressure, are considered: (1) bifurcation buckling, (2) snap through, (3) first-ply failure (FPF), and (4) last-ply failure (LPF). On the manufacturing side, polar filament winding has in the past been used for winding vessels able to carry internally pressurized media. But as seen in Fig. 7.4.2 the method can also be used to manufacture shells aimed at external pressure. “Straightforward” polar winding, seen in Fig. 7.4.2(a), produces a dome (torispherical in this case, Fig. 7.4.2(c)) with an apex hole and substantial accumulation of wall thickness around it. But as shown, the latter can be appropriately redistributed by adjusting the cover sequencing and the inclination angle of the shaft. The second approach in filament winding (Fig. 7.4.2(b)) produces a dome without the apex hole, as seen in Fig. 7.4.2(d). The shell is axisymmetric geometrically, but its material properties are not. Hence the structural analysis would require in this case full two-dimensional modeling. A different approach to manufacture composite dome is based on draping woven cloth into either female or male molding tool. Carbon-fiber-reinforced plastic (CFRP) dome shown in Fig. 7.4.3(a) is just such a case. Again, geometry is axisymmetric here but the material properties are not. This is best seen by comparing

(a)

(c)

(b)

(d)

Figure 7.4.2 (a, b) Two possible ways of filament winding of domes. (c, d) Resulting heads.

Stability of composite shelletype structures

347

(a) (c)

C L

Flat cloth

(Square)

(b)

Draped cloth

Figure 7.4.3 (a, b) View of draped torisphere with visible pattern of distorted fiber paths. (c) Illustration of draped cloth onto torisphere. The FE grid is also shown.

Fig. 7.4.3(a) and (b) where the trajectories of woven fabric are distinctly different. Each ply will therefore require the application of the draping algorithm predicting local fiber orientation together with local wall thickness, and these quantities are to be transferred to the finite element (FE) model, as illustrated in Fig. 7.4.3(c). Further details, together with a list of references, on the above information can be found in Ref. [15].

7.4.2.3

Bifurcation and first-ply failure analyses

For axisymmetric cases (e.g., filament wound), the wall of a shell is to be made from N plies stacked at angles q1, ., qN with respect to the meridian, as illustrated in Fig. 7.4.4(a). Ply number “1” is the innermost ply, whereas ply number “N” is the most outer one. The stacking sequence may not necessarily be symmetric with respect to the midsurface. In the current study, the load-carrying capacity of shells subjected to external hydrostatic pressure is to be found numerically using BOSOR4 (for axisymmetric modeling), Ref. [21] and ABAQUS/Standard, Ref. [22]. The classical lamination theory is to be adopted in all the ensuing analyses. A composite dome when subjected to external pressure can fail through bifurcation buckling, collapse, or material failure. Obviously, only the lowest of the three pressures is of practical relevance. The FPF pressure is to be based on the TsaieWu criterion written in the stress space, Fijsij ¼ 1.0, where the coefficients Fij depend on the tensile/compression and shear ultimate strengths, and their exact form can be found, for example, in Ref. [23]. All the required stresses, sij, are computed at the top and bottom of every layer throughout the wall thickness for all meridional mesh points. Typical material data for CFRP is provided in Table 7.4.1. Consider two types

348

S

(a) y Pl

.1

C L A/B=0.5 p=0.80 MPa

no

Z 0 r

er e Inn rfac u s

1

)

an arc idi er ular M irc (c

N

p

(b)

(d)

D/to=100 C L A/B=1.50 p=9.7 MPa (h)

C L

t

Mid-s

A/B=1.15 p=12.4 MPa (f)

2

e urfac

C L

A/B=2 C L p=1.04 MPa

N=11

N=2

C L

C L FPF

(c)

(e)

(g)

C L FPF (i)

Figure 7.4.4 (a) Arrangements for ply stacking. (bee) Prebuckling and buckling shapes for prolate and oblate heads . (fei) The position of first-ply failure in thicker ellipsoids.

Stability and Vibrations of Thin-Walled Composite Structures

N-1

D/to=250

Stability of composite shelletype structures

349

Table 7.4.1 Elastic constants E11, E22, G12, n12 and strength allowables Xc, Xt, Yc, Yt, and S for unidirectional and woven carbon fiber (the subscripts “c” and “t” mean compression and tension, respectively) E11 Material

E22

G12

Xc

Xt

n12

GPa

Yc

Yt

S

GPa

Unidirectional

140.0

10.0

5.0

0.214

6000

1200

250

50

70

Woven

70.0

70.0

5.0

0.10

570

679.5

570

679.5

93.24

of six-ply [0 /60 /60 ]s half-ellipsoids having the diameter-to-wall thickness ratio equivalent to hemisphere’s D/t0 ¼ 250 and 100 (assuming that the mass of hemisphere and half-ellipsoid are the same, and both heads have the same diameter, D). Computations show that asymmetric bifurcation buckling will determine the load-carrying capacity of the thinner heads, i.e., those with D/t0 ¼ 250. In Fig. 7.4.4(b) and (c) the prolate head with A/B ¼ 0.5 bifurcates at pbif ¼ 0.8 MPa with n ¼ 11 circumferential waves. In Fig. 7.4.4(d) and (e), the oblate ellipsoid also bifurcates at pbif ¼ 1.04 MPa with n ¼ 2 circumferential waves. There is a different positioning of the eigenshape’s maximum at the bifurcation pressure for oblate and prolate geometries, i.e., at the base for the prolate profile and at the apex for the oblate profile. These facts will influence the sensitivity of buckling pressures to the initial shape imperfections, as discussed later. Thicker ellipsoids, D/t0 ¼ 100, on the other hand, fail through the FPF. Fig. 7.4.4(f)e(i) shows positions where the failure index reaches unity for A/B ¼ 1.15 (inner layer, close to the base), and for A/B ¼ 1.50 (outer ply, close to the clamped edge). The values of computed FPF pressures for these heads were pFPF ¼ 12.4 MPa and pFPF ¼ 9.7 MPa, respectively. For these cases, there is very little postbifurcation strength and internally unsupported shells will normally shutter. It is rare to see the buckling pattern in reality, as exemplified in Fig. 7.4.5(a) and (b), in the case of two machined spheroids from 7075-T6 aluminum (A ¼ 25 mm, B ¼ 75 mm, t ¼ 0.76 mm) [15]. It is seen here that different boundary conditions at the equatorial plane did not affect the buckling strength to any greater amount (10% difference). Also the thicker shells shuttered in experimentation Fig. 7.4.5(c) and (d). Prolate ellipsoids clamped at the equatorial plane can bifurcate into the shape showed in Fig. 7.4.6(a). But in reality the shapes seen in Fig. 7.4.6(b) and (c) are to be a likely experimental outcome (two nominally identical steel ellipsoids, Ref. [15]). Let us extend the procedure outlined earlier to [0 /60 /60 ]s, six-ply hemispheres with 100  D/t  1000. Assume that the wall thickness, t, is constant, and material properties remain axisymmetric. Let us also assume that the hemisphere is fully clamped at the base, i.e., around the equatorial plane. The FPF pressures will, as mentioned earlier, be based on stresses calculated at the top and bottom of each ply. The results obtained for a range of the (D/t) ratios are tabulated in Table 7.4.3. To test the capabilities of both codes (Bosor4, ABAQUS), comparisons of bifurcation buckling pressure obtained from both codes are given in Ref. [24]. The estimated

350

Stability and Vibrations of Thin-Walled Composite Structures

C L

C L

(a)

(b)

(d)

(c)

Figure 7.4.5 Photographs of failed machined metal ellipsoids. From J. Błachut, Experimental perspective on the buckling of pressure vessel components, Applied Mechanics Reviews, Transactions of the ASME, 66 (2014), 011003-1e011003-24.

(b)

(a)

(c)

Figure 7.4.6 The FE-computed eigenshape and experimental view of two steel elliptical domes after test.

Table 7.4.2 Various pressures in externally pressurized torispheres Composite shell

Mild steel shell

pFPF (MPa)

2.22

pyp (MPa)

2.22

pLPF (MPa)

2.82

pcoll (MPa)

3.98

t (mm)

3.75

t (mm)

3.03

Note the same value of first-ply failure pressure, pFPF, and first yield pressure, pyp.

Stability of composite shelletype structures

351

Table 7.4.3 Bifurcation buckling (p1), axisymmetric collapse (p2), and FPF (p3) pressures for CFRP hemispheres Bosor4 D/t

p1

p2

p3

100

19.89 (10)

26.37

9.084

150

8.86 (13)

11.076

6.057

200

5.01 (15)

6.671

4.541

250

3.219 (17)

4.133

3.627

300

2.241 (18)

2.981

e

400

1.264 (21)

1.684

e

500

0.824 (24)

e

e

750

0.359 (29)

e

e

1000

0.204 (34)

e

e

All results are in megapascals. The numbers in brackets indicate the number of waves in the circumferential direction in the bifurcation mode.

magnitudes of asymmetric bifurcation buckling pressure given by both codes are nearly the same. In all analyzed cases the buckling mode remains the same. As an example, Fig. 7.4.7(a) depicts deformed hemisphere (D/t ¼ 150) just before buckling. The eigenmode corresponding to, n ¼ 13, hoop waves close to the base is shown in Fig. 7.4.7(b). All hemispheres considered so far were clamped at the equatorial plane. The complete sphere is likely to be assembled from two equal-sized parts connected at the equatorial plane. There are a number of jointing possibilities and each translates itself to a different set of boundary conditions in a numerical modeling. With this in

(a) Pre-buckling

(b) Eigenmode

Figure 7.4.7 (a) View of CFRP deformed hemisphere (D/t ¼ 150) just before buckling, and (b) the corresponding eigenmode (n ¼ 13).

352

Stability and Vibrations of Thin-Walled Composite Structures

(b)

(a) 20.0

Pressure (MPa)

Hemisphere; (0°/60°/–60°)s

to

10.0

10.0 D

5.0

0.0 0.0

Pressure (MPa)

Bifurcation buckling First ply failure to

15.0

D to 100

200

300

D

5.0

400

500

0.0 0.0

D to 100

200

300

400

500

Figure 7.4.8 Bifurcation buckling and first-ply failure pressures versus the diameter-to-thickness ratio, D/t0, for (a) clamped and (b) pivoted/roller-type boundary conditions.

view, unrestrained boundary conditions at the equator were considered. The effect of boundary conditions on the load-carrying capacity is shown in Fig. 7.4.8(a) and in Fig. 7.4.8(b). One obvious observation is that the clamped shell is able to support larger pressures than the shell without any restraints at the equator. Also, for a fully clamped shell, bifurcation controls the failure of thinner shells, i.e., for D/t > 220, and the FPF remains to be the controlling failure for thicker shells (see Fig. 7.4.8(a)). For rollertype boundary conditions, on the other hand, the controlling mode of failure is the FPF over the whole range of 100  D/t  1000 (Fig. 7.4.8(b)). The practical realization of a specific set of boundary conditions might be difficult to achieve; therefore, the whole spectrum of boundary conditions needs to be assessed for buckling strength during the design stage. Ref. [24] contains a set of results that were obtained for several types of boundary conditions applied at the shell midlength. The wide range of loadcarrying capacities obtained there and reflecting different boundary conditions will require a careful consideration of jointing technique of two segments. Let us now consider an ellipsoidal shell made from six plies of woven fabric and stacked symmetrically, i.e., [0 /60 /60 ]s. The load-carrying capacity of prolate, spheroidal shells has been computed for four types of boundary conditions given in Table 7.4.4. Magnitudes of asymmetric bifurcation buckling and FPF pressure are provided in Table 7.4.5. For fully clamped shells along the equator, bifurcation buckling remains the controlling mode of failure for the whole range of the 2A/t) ratio, i.e., for 100  2A/t  500. It is worth noting here that for hemispheres, both modes can control the failure. The mode of failure depends on the (D/t0) ratio (compare Fig. 7.4.8(a) with Fig. 7.4.9(a)). These results have been obtained for prolate ellipsoids with A/B ¼ 0.5, and it is seen from Fig. 7.4.9(b) that bifurcation buckling is of concern when the equator is clamped. Similar situations exist for more slender ellipsoids, i.e., for A/B ¼ 0.33. Results for the latter geometry are depicted in Fig. 7.4.9(a), in which the magnitude of load-carrying capacity is associated with bifurcation buckling only and FPF plays no role in the shell’s failure.

Stability of composite shelletype structures

353

Table 7.4.4 Boundary conditions applied at the equatorial plane, e.g., type “1” means fully clamped support Type of Boundary Conditions

v

w

b

“1”

c

c

c

“2”

c

c

f

“3”

c

f

c

“4”

c

f

f

c, zero; f, free; v, hoop; w, normal; b, rotation.

The effect of boundary conditions on the magnitude of bifurcation and first-ply failure pressures in six-ply externally pressurized CFRP prolate ellipsoid with A/B [ 0.5 Table 7.4.5

“1”

“2”

“3”

“4”

D/t0

pbif

pFPF

pbif

pFPF

pbif

pFPF

pbif

pFPF

500

0.186(15)

1.09

0.183(15)

1.05

0.186(15)

2.53

0.185(15)

0.372

400

0.297(14)

1.39

0.291(11)

1.32

0.298(14)

3.16

0.296(14)

0.533

300

0.543(12)

1.87

1.24(10)

1.76

0.546(12)

4.21

0.541(12)

0.827

200

1.28(9)

2.83

1.24(10)

2.63

1.29(10)

6.32

1.28(10)

1.46

100

5.66(7)

5.66

5.41(7)

5.26

5.76(7)

12.65

5.68(7)

3.40

All results are in megapascals and for woven fabric.

Another frequently used profile of a dome is called torisphere (see Fig. 7.4.1(e) for nomenclature). There has been a large amount of research on the buckling of torispheres made from metals. But there has been relatively little experience with externally pressurized torispheres made from composites. To fill this gap, a number of torispherical domes have been manufactured and collapsed by external pressure as a part of research at the University of Liverpool [25]. One particular head was made by draping 3 k, prepreg, 4  4 woven twill of nominal 0.31 mm ply thickness into torispherical female molding tool of 800 mm diameter characterized by r/D ¼ 0.24, Rs/D ¼ 0.60 and (D/t)avg ¼ 90. The woven fabric was assumed to be a solid continuum sheet with linear-elastic anisotropic properties. The local fiber orientation and local wall thickness were determined by the draping analysis module. The initial

354

Stability and Vibrations of Thin-Walled Composite Structures

(b)

(a) Pressure (MPa) 3.0

CL

5.0

t

A =0.33 B

Pressure (MPa)

CL

A =0.5 B

B 4.0

t B

2.0 3.0 2.0

A

1.0

A

1.0 0.0 0.0

2A t 100

200

300

400

500

Bifurcation buckling First ply failure

0.0 0.0

100

200

2A t 300

400

500

Figure 7.4.9 Bifurcation and first-ply failure pressures for ellipsoidal shells having (a) A/B ¼ 0.33 and (b) A/B ¼ 0.5. Both shells are clamped at the equatorial plane and made from six-ply woven material stacked at [0 /60 /60 ]s.

computing trials described here are made for perfect geometry of domes. For 800-mmdiameter domes, the stress allowables are given in Table 7.4.1. Fig. 7.4.10(a) shows a section cut through collapsed 30-ply torisphere. The magnified area through which the thickness cracking happened is depicted in Fig. 7.4.10(c). The plot of failure index (FI) at FPF pressure is showed in Fig. 7.4.10(b), in which the most stressed layer is the innermost ply, close to the spherical cap, i.e., the knuckle junction. Two observations are worth to be made here. First, most of the spherical cap remains understressed. The magnitude of the FI at the FPF level remains around 0.20. This clearly indicates how underused the material strength is. In situations like this, the draping algorithm allows the rearrangement of stacking to increase utilization of the material’s strength.

4

1.0

5

6

2

3

(0°/0°/15°/30°/45°/45°/60°/75°/90°/ Apex 105°/120°/135°/150°/165°/180°)s

0.7 0.4

Inner ply

0.1

1

7

0

FI

(a)

0.25 0.50 0

(c)

Outer ply

0.75 1 Clamped edge

(b)

Figure 7.4.10 (a) Cut-through 30-ply torisphere after collapse. (b) Plot of the failure index (FI). (c) Magnified view of through-the-thickness cracking at LPF.

Stability of composite shelletype structures

355

This has already been done as a part of research and industrial practices. The second observation, which is to be addressed later, is the fact of progression of ply-by-ply failure up to the LPFdall occurring at more or less the same place where the FPF emerges. The search for the ultimate load-carrying capacity of externally pressurized domes is described in the following because the FPF does not mean their imminent loss of structural integrity.

7.4.2.4

Progressive failure: last-ply failure

The progressive failure is carried out as a multistage reanalysis process. After the structure reaches the FPF level the pressure is increased by Dp and the Gauss points (GPs) in the model that are overstressed are noted. The stiffness matrix is then reduced at these points according to a selected criterion outlined below. The rest of the model remains unchanged and the reanalysis, at a given load, is repeated until there are no overstressed GPs in the entire model. Only then the next pressure increment is applied. This process continues until no further increase in the load-carrying capacity could be obtained. The TsaieWu interactive failure criterion, written in the stress space, has been adopted as a measure of the FPF pressure. This criterion does not provide information on which is the most stressed direction under the combined action of stresses. Hence the following four different expressions are introduced: •

tensile failure in material direction 1

 2   s1 s12 2 F1t ¼ þ ; Xt S •

for s2  0

(7.4.2)

compressive failure in material direction 1

F1c ¼ •

(7.4.1)

tensile failure in material direction 2

 2   s2 s12 2 F2t ¼ þ ; Yt S •

for s1  0

js1 j ; Xc

for s1 < 0

(7.4.3)

compressive failure in material direction 2

F2c ¼

js2 j ; Yc

for s2 < 0

(7.4.4)

Various computing schemes were adopted for the degradation of the stiffness matrix at points that have become overstressed in terms of Eqs. (7.4.1e7.4.4). Details about the utilization of Eqs. (7.4.1e7.4.4) are given in the next section.

356

Stability and Vibrations of Thin-Walled Composite Structures

7.4.2.4.1

Stiffness matrix degradation

Once the type of failure and its direction are known, the stiffness matrix can be reduced in the appropriate way. Eight noded shell elements S8R, which are used in the analysis, have four Gauss integration points. The TsaieWu FI is evaluated at these points and the largest magnitudes of F1t, F2t, F1c, and F2c, as specified by Eqs. (7.4.1e7.4.4), are found. This enables identification of the direction and type of local overstressing. The following numerical scheme has been adopted for degradation of E1, E2, G12, and n12 within a given shell element: 4

4 4

4 P m¼1

¼

(7.4.5)

E2

(7.4.6)

ðim þ jm Þ 8

8 ¼

4 P m¼1

Gd12 ¼

E1

jm

4 8

nd12

im

m¼1

Ed1 ¼

Ed2

4 P

4 P m¼1

G12

(7.4.7)

n12

(7.4.8)

ðim þ jm Þ 8

where im ¼ 1, 2, ..; jm ¼ 1, 2, . and the letters “i” and “j” refer to the material directions “1” and “2,” respectively. The subscript “m” indicates the GP. Although the information about overstressing is gathered at every GP in each ply, the subsequent degradation is based on an element basis. This scheme, which is given by Eqs. (7.4.5e7.4.8), reflects this necessity. If any of the numerators in Eqs. (7.4.5e7.4.8) becomes negative then the corresponding quantity is assigned a value of zero for that particular ply and the shell element. After each stress evaluation the necessary degradation is carried out and a new FE input is prepared for a subsequent FE reanalysis. If the same GP is overstressed for the second time then the values of “im” and/or “jm” in Eqs. (7.4.5e7.4.8) take the value 2 (hoverstressed for the second time, etc.). The looping continues at a given pressure until all overstressed points in the model disappear in accordance with the TsaieWu criterion [f(p) h FI  1.0 ¼ 0]. It is only then that the next increment of pressure is made.

7.4.2.4.2

Arbitrary incrementing of pressure

Trials were performed to assess the post-FPF path as a function of pressure increments and as a function of number of plies for a given meridional geometry and the same FE meshing. A trial geometry was taken as r/D ¼ 0.1, Rs/D ¼ 1.0, L/D ¼ 0.05, D/t ¼ 80, and six

Stability of composite shelletype structures

357

External pressure (MPa)

4.0

30

Degradation count

3.0

2000

1500 ‘f’

‘d’‘e’ ‘c’ ‘b’ ‘a’

29

2.0

1000 28 27 26 25 23 19 16 12

1.0

6 0

0

0.2

500 30 = Total no. of abaqus runs δ/ t

7 0.4

0.6

0.8

1.0

0

Figure 7.4.11 Scheme for computing the last-ply failure (LPF) pressure in torispherical shell.

plies stacked as [0 /60 /60 ]s. Fig. 7.4.11 shows the history of pressure versus apex deflection for this configuration. Point “a” refers to the occurrence of the FPF at p ¼ 2.27 MPa. The most stressed point, corresponding to the FPF pressure, is located on the inside of the torisphere. An arbitrary increment of Dp ¼ 0.1 MPa was then applied and the process of stiffness degradation and reanalyses continued until all GPs satisfied conditions (1)e(4) and f(p) ¼ 0. The pressure versus apex deflection curve moves to point “b” in Fig. 7.4.11. Six ABAQUS reanalyses were needed to move from point “a” to “b” and this is also depicted in Fig. 7.4.11. An accompanying curve in Fig. 7.4.11 shows a measure of the degradation/damage sustained by the shell in the course of pressure increments. The path from “a” to “b”, for example, required six ABAQUS runs and degradation on 78 occasions. Subsequent pressure increase led to the path “b / c / d / e” with a total number of 16 ABAQUS reanalyses. The final pressure increment, from “e” to “f”, did not result in a converged and damage-free configuration. Once the damage penetrated the thickness, the calculations for point “f” were terminated. The pressure corresponding to the point “e” can therefore be regarded as the LPFand its value is pLPF ¼ 2.67 MPa. Computations show that the FPF is not very sensitive to the step size for a six-ply dome. Additional calculations were performed for {[0 /60 / 60 ]4}s, {[0 /60 /60 ]8}s, {[0 /60 /60 ]16}s, i.e., for 24-, 48-, and 96-ply domes, respectively, while the torispherical geometry was kept the same. Some numerical trials were also carried out for Dp ¼ 0.04 MPa once the final increment Dp ¼ 0.1 MPa led to the lack of convergence. Such small pressure steps allowed stiffness degradation to progress through the entire wall thickness in 6-, 24-, and 48-ply domes. But it proved to be a slow process. The abovementioned calculations showed that there is some residual strength beyond the LPF level but it is usually not greater than 5% of the FPF

358

Stability and Vibrations of Thin-Walled Composite Structures

(a)

(b)

y

FPF

y

LPF x

(c) y

x

Through Thickness failure

x

(d) C L

6-Ply p = 2.67 MPa LPF

C L

C L

Through thickness failure

48-Ply p = 2.83 MPa LPF

12-Ply p = 2.83 MPa LPF

L

L

C L

L

96-Ply p = 2.82 MPa LPF

L

Figure 7.4.12 Spread of damage at various pressure levels: (a) FPF, (b) LPF, and (c) point “f.” (d) Penetration of the failure through the wall thickness at the most stressed point at last-ply failure pressure.

pressure. The ratio of the number of failed plies to the total number of plies at the LPF is 0.67, 0.67, 0.54, and 0.45 for 6-, 24-, 48-, and 96-ply wall, respectively. It is worth noting here that FPF occurs on the inside of the torisphere and its position, for the six-ply dome, is shown in Fig. 7.4.12(a). Increase of pressure beyond the FPF leads to accumulation of damage and Fig. 7.4.12(b) shows contours of the accumulated damage at the LPFpressure, i.e., at point “e”. Fig. 7.4.12(c) shows the same quantity, once through the thickness propagation of the damage occurred and calculations were stopped (point “f” in Fig. 7.4.11). Spread of damage through the thickness is shown in Fig. 7.4.12(d) for a shell section passing through the most stressed point in the model.

7.4.2.4.3

FPF-controlled incrementing of pressure

One way in which the arbitrary load incrementing could be removed is to rely on fresh FPF reanalysis after each degradation of the wall properties. This procedure requires

Stability of composite shelletype structures

359

several reanalyses to obtain an exact value of the FPF (through any method of rootfinding). There is however an approximate technique that allows evaluation of the FPF through scaling of stresses and this method has also been explored here for progressive failure analysis. The algorithm consists of the following steps: • •

degradation of stiffness, as specified by Eqs. (7.4.5e7.4.8), takes place at each FPF pressure for a range of GPs satisfying 0.01  f(pFPF)  0.01; a new FE analysis is carried out at the same loading, pFPF, but with the modified stiffness. The next pressure, pnext FPF , is obtained through the scaling of obtained stresses. The scaled stresses are also used to identify which GPs are

overstressed. Stiffness is reduced again for all GPs falling inside the range 0:01  f pnext FPF  0:01 and a fresh FE analysis is carried out.

This algorithm was applied to a six-ply dome. The magnitudes of changes in FPF pressures proved to be small and hence, 162 FE reanalyses were needed to reach the LPFpressure. The difference between both predictions of the LPF varies from 12% for Dp ¼ 0.1 MPa to 5% for Dp ¼ 0.01 MPa. Comparing 30 versus 162 FE reanalyses, which were required to reach the ultimate strength of a CFRP torisphere, it was decided to adopt the first approach and use Dp ¼ 0.05pFPF in subsequent calculations.

7.4.2.5

Some of the numerical results for draped domes

At the next stage of evaluation of the abovementioned algorithm, calculations were performed for randomly stacked plies in 6-, 24-, 48-, and 96-ply torispheres. It was found that the maximum difference between pFPF and pLPF was 23%. Also the level of damage penetration through the thickness at LPF pressure was found to vary between 0.17 and 0.44. Influence of geometry on the LPF in the 96-ply dome was evaluated and the results are shown in Fig. 7.4.13(a) for different (r/D) ratios. The stacking sequence was kept constant as was the total number of plies N ¼ 96. It is seen from Fig. 7.4.13(a) that there is a greater difference between pFPF and pLPF for shallower torispheres. Location of the FPF and then spread of failed plies resemble, in general, the position and spread of plastic zones in metallic torispheres. Results for a torispherical dome made from mild steel are shown in Fig. 7.4.13(b)e(d). Calculations were carried out for a Young’s modulus E ¼ 210 GPa, the yield point syp ¼ 400 MPa, and Poisson’s ratio n ¼ 0.3, while the geometry was defined by r/D ¼ 0.1, Rs/D ¼ 1.0, L/D ¼ 0.05, and D/t ¼ 99.0. Comparison of pyp, pFPF, pcoll, and pLPF is provided in Table 7.4.2. The ratio of pcoll/pyp is 1.79 and the corresponding ratio for a composite dome, i.e., pLPF/pFPF, is 1.27. At the same time the ratio of wall thicknesses tcomposite/tsteel ¼ 1.24, and this results in the weight ratio of 0.25. It is worth noting here that plastic deformations in steel domes are axisymmetric. Damage in the composite wall appears broadly at the same meridional position, i.e., around the spherical cap/knuckle junction, but the damage is highly localized and it is not axisymmetric, as illustrated in Fig. 7.4.12(b).

7.4.2.6

Comparison of numerical and experimental results

Details about 26 domed end closures manufactured from prepreg carbon fibers and tested at University of Liverpool are summarized in Table 3 of Ref. [24]. The first

360

(a)

(0.47)

(c)

(b) p (MPa)

4.0

3.0

No. of damaged plies = Total no. of plies p = 0.05pFPF

pyp

2.0

L

0

0

0.1

0.2

Plastic deformation (axisymmetric)

pFPF Steel p = 3.98 MPa coll

δ/ t

r/D 0

(d) C L

96-Ply CFRP

1.0

First ply failure

5

r

Steel pLPF

(0.49)

6

p = 3.02 MPa LPF

pcoll

(0.49)

Last ply failure

7

Composite

0

0.2

0.4

r

L

0.6

Figure 7.4.13 (a) The difference between first-ply failure (FPF) and last-ply failure (LPF). (bed) Comparison between composite and steel torispheres.

Stability and Vibrations of Thin-Walled Composite Structures

9

Ply damage (non-axisymmetric)

s

10

8

C L

R

Rs L =0.05; D =40 =1; D t D [(0°/60°/–60°)16]6

s

11

(0.43)

R

p (MPa)

12

Stability of composite shelletype structures

(a)

(b)

361

(c)

Figure 7.4.14 Collapsed (a) torisphere, (b) hemisphere (petaled), and (c) 300-mm-diameter torisphere.

11 shells in the table are torispheres with nominal 800 mm diameter, four are hemispheres with 700 mm diameter, and the remaining are 300-mm-diameter torispheres. The first two domes in the table are filament wound with the driving shaft passing through the apex, and further two models were filament wound without the apex opening (see Fig. 7.4.2(c) and (d)). Numerical algorithms outlined earlier were used to predict FPF pressures in “thin heads” (models 1e16 in Table 3 of Ref. [24]). Application of detailed modeling (draping, variable wall thickness) resulted in a good correlation between the predicted LPF pressures and the experimental collapse pressures for a range of tested torispheres made from carbon/epoxy woven cloth. For 300-mm domes, both FPF and LPF pressures were computed. It is obvious from Table 7.4.3 that material failure associated with LPF is close to experimental collapse pressures. The ratio of (pexpt/pnumerical) varied between 0.90 and 1.10. The ratio of LPF pressure to FPF pressure, on the other hand, varied from 1.28 to 1.44. The large ratio of the LPF to the FPF pressure indicates that it is important for the progressive failure analyses to be carried out to assess the margin for the ultimate loadbearing capacity of domed shells made from composites. Typical photographs of collapsed CFRP domes are seen in Fig. 7.4.14. Fig. 7.4.14(a) shows 30-ply 800-mm-diameter torisphere after collapse by external pressure. All 30 plies were made from single (noncut) prepreg fabric draped into female mold. It is seen here that through-thickness crack runs in hoop direction at spherical capeknuckle junction. This agrees well with numerically predicted region of material failure seen, for example, in Fig. 7.4.10. Fig. 7.4.14(b) depicts 700-mm diameter, 20-ply hemisphere made from petaled and butt-jointed pieces of carbonwoven prepreg cloth. Finally, Fig. 7.4.14(c) illustrates circumferential, throughthickness crack in 300-mm-diameter torisphere after the collapse test. This particular dome was draped from noncut fabric and with off-axis placement of the focal point.

7.4.3

Imperfect composite spheroids (hemispheres, torispheres, ellipsoids)

It is widely accepted that buckling pressures are affected by the presence of initial geometric imperfections. The imperfections are unavoidable because of manufacturing

362

Stability and Vibrations of Thin-Walled Composite Structures

process, exploitation, or accidental damage. In metallic shells, the existing design codes made provisions for their anticipated presence by the application of knockdown factors. These reduction factors are being based on the usually known, worst scenario (the lower-bound approach, eigenmode imperfections, local dimple, etc.). It has to be said that for composite domed closures, there is relatively little data on imperfection sensitivity of buckling pressures to initial shape imperfections. The following sections provide numerical results for various types of shape distortions in doubly curved closures manufactured from composites.

7.4.3.1 7.4.3.1.1

Imperfect hemispheres Localized flattening in hemispheres

Among the many initial shape imperfections, local flat spots of increased-radius type have been chosen first to probe the sensitivity of buckling pressure to deviations from perfect shape in clamped hemispherical shells. It is argued here that if imperfection patch is away from the spherical capeknuckle junction, it is immaterial whether one considers hemisphere or torisphere. Also, some experimentation was carried out for torispheres, hence an additional reason for mentioning a torisphere. Geometry of the flat patch positioned at the apex of a torisphere is shown in Fig. 8 in Ref. [24]. The patch is uniquely defined by the amplitude, d0, and the semiangle, a, (or alternatively by the magnitude of the increased-radius, Rimp). For a given amplitude, d0, one can have an infinite number of Rimp values. Each of these geometries is likely to respond differently in terms of load-carrying capacity. By varying, Rimp, for given amplitude, d0, one can compute the largest possible reduction of collapse pressure. In the past, this approach has been satisfactorily used for metallic shells. It has also been benchmarked against experimental data, see, for example, Ref. [15]. As an illustration, computations have been carried out for six-ply, [0 /60 /60 ]s, hemispherical shell with local, increased-radius patch positioned at the apex (D/t ¼ 500). The dome was clamped at the equatorial plane. The festooned curve to the lowest buckling pressures for each fixed amplitude of imperfection, d0/t, is plotted in Fig. 7.4.15 as the curve number

1.0

pp

Hemisphere; D= 500, [0°/60°/–60°] s t

bif

A

B

0.5

(i) A (ii)

s/stot = 0.50

(iii)

B 0.0 0.0

0.4

0.8

δ°/t

1.2

1.6

2.0

Figure 7.4.15 Comparison of imperfection sensitivities [(i) h eigenshape, (ii) h lower bound, and (iii) h inward dimple].

Stability of composite shelletype structures

363

(ii), with further details in Fig. 9, Ref. [24]. In this case, “small-amplitude” imperfections at the apex did not reduce the buckling strength.

7.4.3.1.2

Eigenmode affine imperfections

One of the frequently used shape deviations from perfect geometry has been associated with the eigenmode corresponding to the asymmetric bifurcation buckling. It is customary here to superimpose the eigenshape onto the perfect geometry with a scaling factor, usually as a fraction of wall thickness and expressed as the (d0/t) ratio. Next the load-carrying capacity of the shell with modified geometry is computed. This is then repeated for a range of values of the (d0/t) ratio. In the case of composite hemisphere defined by D/t ¼ 500, the eigenshape had n ¼ 24 circumferential waves spread around the base. When these are superimposed on the perfect geometry with the scaling factor, d0/t, the response obtained is shown in Fig. 7.4.15 as curve number (i). The load-carrying capacity of imperfect hemisphere is governed by the collapse mechanism. The latter remained true for the whole range of imposed imperfection, i.e., for 0.0 < d0/t  2.0. The collapse loads were computed using the Riks algorithm implemented in the FE code ABAQUS. The shape at collapse, and corresponding to d0/t ¼ 1.0, is shown as inset “A” in Fig. 7.4.15. It clearly remains debatable how likely in practice is the occurrence of such imperfection pattern.

7.4.3.1.3

Force-induced inward dimple

Localized perturbation of perfect geometry can be obtained once a concentrated force, F0, is applied at a given position. Here, inward dimple was created by the radially acting force, F0. Once the dimple had been created, incremental external pressure was applied and the load-carrying capacity was traced using the Riks algorithm. The response curve was obtained for the position of the dimple at s/stot ¼ 0.50. It is plotted in Fig. 7.4.15 as curve number (iii). For d0/t  0.37 the largest reduction in buckling strength was obtained for eigenmode imperfection. For larger amplitude of the (d0/t) ratio the worst imperfection profile is associated with FID. The insets “A” and “B” in Fig. 7.4.15 illustrate deformed shapes just before collapse. It can be argued here that a single indent is more likely to be found in practice than 24 evenly spaced dimples associated with the eigenmode. Also it is seen that at, d0/t ¼ 1.0, the eigen imperfection is not as dangerous as the FID imperfection [e.g., p/pbif (FID) ¼ 0.18 vs. p/pbif (eigenmode) ¼ 0.41].

7.4.3.2

Imperfect torispheres

Computations have also been carried out for six-ply torispherical shells given by D/t ¼ 500, Rs/D ¼ 1.0, and r/D ¼ 0.2, with [0 /60 /60 ]s lamination sequence. Under external pressure, and in the case of perfect geometry, the torisphere’s bifurcation pressure is 0.144 MPa, with n ¼ 10 circumferential waves (Bosor4 prediction). The FE ABAQUS gave pbif ¼ 0.147 MPa and n ¼ 10.

364

Stability and Vibrations of Thin-Walled Composite Structures

pp 1.0

bif

R Torisphere; D = 500, s = 1; r = 0.2; [0°/60°/–60°] s t D D A

B

(i)

0.5

A (ii) s/stot = 0.49

(iii)

(iv)

= 0.70 B

0.0 0.0

δ°/t

0.4

0.8

1.2

1.6

2.0

Figure 7.4.16 Comparison of imperfection sensitivities [(i) h eigenshape, (ii) and (iii) h inward dimple, and (iv) h lower bound].

As for hemispherical heads, three forms of imperfection profiles were investigated for torispherical domed-end closures, i.e., increased-radius flattening at the apex, eigenmode-type perturbation, and FID. Details are given in the next three subsections.

7.4.3.2.1

Increased-radius flattening

The increased-radius flat patch was introduced to the geometry at the apex. For each value of 0  d0/t  2.0, the worst extent of the flattening was identified numerically. The festooned curve is plotted in Fig. 7.4.16 as curve number (iv). The sensitivity here is similar to the response curve obtained for the case of hemispherical shell. One noted difference is the threshold value, d0/t z 0.2, below which shell remains insensitive to initial deviations from perfect geometry.

7.4.3.2.2

Eigenmode affine imperfections

As mentioned earlier the eigenmode has n ¼ 10 circumferential waves equally spaced in the hoop direction and positioned at the junction between the knuckle and spherical portion of the dome. Fig. 7.4.16 shows the sensitivity of buckling pressure to eigenmode affine shape perturbation within the scaling factor, 0.0 < d0/t  2.0. Both hemispherical and torispherical domes show similar sensitivity of buckling pressure to the initial eigenmode-type imperfections. After a quick drop in the load-carrying capacity for smaller magnitudes of imperfections, say for d0/t  0.4, the sensitivity levels off (i.e., it remains nearly constant for d0/t > 0.4). But at d0/t ¼ 2.0 the torisphere is slightly less sensitive than hemisphere, i.e., p/pbif ¼ 0.5 versus p/pbif ¼ 0.4 for hemisphere. The inset “A” in Fig. 7.4.16 depicts the deformed torisphere at collapse for d0/t ¼ 1.0. Again the question is how realistic are these equally spaced dimples around the spherical capeknuckle junction?

7.4.3.2.3

Force-induced inward dimple

Localized inward dimple can be seen as a realistic deviation from the perfect shape, and this imperfection profile can be generated by a concentrated force, F0, acting

Stability of composite shelletype structures

365

radially. The response of buckling pressure to the FID was computed for two positions of the dimple along the shell’s generator. The two positions were measured by the arc length, s, from the base. The first set of results was obtained for s/stot ¼ 0.70. The second position corresponded to the maximum deflection at bifurcation buckling. From the FE results the maximum corresponded to s/stot ¼ 0.49. The curves that were obtained for both the positions of the dimple are shown in Fig. 7.4.16. It is clearly seen that the single dimple positioned at s/stot ¼ 0.49 represents the worst scenario nearly for the whole range of (d0/t) scaling factor. For example, at d0/t ¼ 2.0 the eigenmode imperfection reduces the buckling strength to p/pbif ¼ 0.5, whereas for the FID profile the same ratio is 0.17. This means (0.5/0.17) times weaker domed head (i.e., nearly three times).

7.4.3.3

Imperfect ellipsoidal shells

Consider two typical ellipsoidal shells with the semiaxis ratio of A/B ¼ 0.5 (prolate) and A/B ¼ 2.0 (oblate), see Fig. 7.4.17 for notation of geometry. Let both heads be clamped at the equator; have constant wall thickness, t, given by the ratio 2a/t ¼ 500; and with the stacking sequence [0 /60 /60 ]s. Under external pressure, both shells can buckle into distinctly different modes. At bifurcation, the prolate ellipsoidal dome, a/b ¼ 0.5, has n ¼ 15 circumferential waves and pbif ¼ 0.185 MPa (pbif ¼ 0.185 MPa, n ¼ 15 from Bosor4). The corresponding eigenmode is depicted in Fig. 7.4.17(b). The oblate ellipsoid with the semiaxes ratio, a/b ¼ 2.0, fails at pbif ¼ 0.256 MPa and with n ¼ 2 circumferential waves (pbif ¼ 0.258 MPa, n ¼ 2, from Bosor4). The shape at buckling is shown in Fig. 7.4.17(c). The response of buckling pressure to different forms of initial geometric imperfections is assessed next.

(a)

(c) S = Stot

b

(b) t p

(d) S

C

0 a

Figure 7.4.17 (a) Geometry of half-ellipsoid. (b, c) FE-generated eigenshapes for prolate and oblate ellipsoids. (d) Eigenshape for a torisphere.

366

Stability and Vibrations of Thin-Walled Composite Structures

PP

Force induced dimple, s = 100% s tot

bif

1.0

= 75%

0.5

Eigenmode

= 50% a = 0.5; 2a = 500;(0°/60°/–60°) b s t

0.0 0.0

0.4

0.8

= 25% & 14%

1.2

1.6

δ°/t

2.0

Figure 7.4.18 Comparison of imperfection sensitivities obtained for different positions of inward dimple with the response obtained for eigenmode shape deviations.

7.4.3.3.1

Eigenmode affine imperfections

Sensitivity of buckling pressure to eigenmode imperfections is shown by dotted line in Figs. 7.4.18 and 7.4.19 for prolate and oblate domes, respectively. When comparing these two geometries, buckling loads for prolate head appear to be more sensitive to small-amplitude imperfections (d0/t < 0.5). For d0/t ¼ 2.0, both heads show leveled imperfection sensitivity. But oblate head (a/b ¼ 2.0) is 40% [(0.42e0.30)/0.30] weaker than the prolate dome when the amplitude of imperfection is d0/t ¼ 2.0. As in the previous cases the following question still remains valid, i.e., how realistic are eigenmode deviations from perfect shape? The next section examines the case when the shape imperfection is generated by a concentrated force acting inward and perpendicularly to the shell’s surface.

PP 1.0

bif

= 25%

a = 2.0; 2a = 500; [0°/60°/–60°] b s t

0.5 Eigenmode = 50%

0.0 0.0

Force induced dimple, s = 75% s tot

0.4

0.8

δ°/t

1.2

1.6

2.0

Figure 7.4.19 Comparison of reduction in buckling pressure caused by different deviations from perfect geometry.

Stability of composite shelletype structures

7.4.3.3.2

367

Force-induced inward dimple

Concentrated force, F0, has been applied at different locations along the shell generator. Dome’s geometry was such that 2a/t ¼ 500 and the stacking sequence was [0 /60 /60 ]s. For prolate ellipsoid (A/B ¼ 0.5), the following five positions were examined: s/stot ¼ 100% (apex), 75%, 50%, 25%, and 14%. The last position corresponds to the meridional position of maximum deformation in the eigenmode. The magnitude of force, F0, was chosen in such a way that the magnitude of inward dimple, d0, covered the range 0  d0/t  2.0. For each depth of the dimple, d0/t, the FE code Abaqus was used for computing the buckling strength. Results for all the abovementioned positions of the dimple along the shell generator are depicted in Fig. 7.4.18. It is seen that buckling strength is not at all affected for dimples at the apex. As the dimple moves toward the clamped edge, the buckling strength begins to decrease. For each analyzed configuration, there is a threshold value of the indentation below which the magnitude of buckling load is not affected. It is worth noting here that a similar situation has been encountered in externally pressurized, isotropic torispheres with localized flattening. When the dimple is positioned at the maximum associated with buckling mode (s/stot ¼ 14%) the buckling strength is significantly lower than that associated with eigen-type imperfections. The percentage difference, at d0/t ¼ 2.0, reaches 130% [(0.42e0.18)/0.18]. Similar computations have been carried out for oblate ellipsoid (a/b ¼ 2.0), sometimes also called 2:1 ellipsoid. Meridional positions of the dimple were placed arbitrarily at s/stot ¼ 75%, 50%, and 25%. The obtained sensitivities are plotted in Fig. 7.4.19 where it is seen that small-amplitude dents do not affect buckling strength, especially when the dent is away from the edge (e.g., the case of s/stot ¼ 25%, Fig. 7.4.19). Comparing results shown in Fig. 7.4.18 with those in Fig. 7.4.19, it is seen that response to a dimple rapidly increases as it approaches the area of the maximum of the eigenshape (apex for A/B ¼ 0.5, Fig. 7.4.18, and equatorial plane for A/B ¼ 2.0, Fig. 7.4.19). Again, dimpled oblate ellipsoid is significantly weaker than the eigen-imperfect counterpart. This time the difference at d0/t ¼ 2.0 amounts to 170% [(0.3e0.11)/0.11] Fig. 7.4.19. It is clear that a single dimple in oblate head lowers the buckling strength for the whole 0  d0/t  2.0 range, and it remains well below that of the eigenmode response. For prolate ellipsoid, a/b ¼ 2.0, this happens only for d0/t > 0.7. Finally, it is worth noting that a wider discussion of these results is available in Ref. [24], with different types of inititial imperfections being also analyzed in Ref. [26].

7.4.4

Closure

Geometrically axisymmetric components may not necessarily be axisymmetric in terms of material properties, and this applies to both filament winding and draping of a woven cloth. In turn, a nonaxisymmetric modelization is required. It has been illustrated how both manufacturing routes can enhance load-carrying capacity by suitably adjusting parameters of their manufacture.

368

Stability and Vibrations of Thin-Walled Composite Structures

Buckling loads of all discussed geometries suffer from strong loss of buckling strength when their shape is not perfect. The role of recently researched imperfection in the form of inward dimple has been confronted with the more traditional approach to initial shape imperfections based on eigenmode affine, or on flat spots. It is shown that “the dimple approach” has a lot to offer when imperfection sensitivity of buckling load to the geometric deviations from perfect shape is investigated.

References [1] D.A. Danielson, Buckling and initial postbuckling behaviour of spheroidal shells under pressure, AIAA Journal 7 (1969) 936e944. [2] W.A. Nash, Hydrostatically Loaded Structures, Pergamon, New York, 1995, ISBN 0-08037876-5, 183 p. [3] B.B. Pan, W.C. Cui, Y.S. Shen, Experimental verification of the new ultimate strength equation of spherical pressure hull, Marine Structures 29 (2012) 169e176. [4] Q. Du, W. Cui, Stability and experiment of large scale spherical models built by high strength steel, in: Proc. of the ASME 23rd Intl Conf. on Ocean, Offshore and Arctic Engineering, San Francisco, USA, OMAE2014-24056, 2014, pp. 1e8. [5] K. Asakawa, S. Takagawa, New design method of ceramics pressure housings for deep ocean applications, in: Proc. of Oceans 2009 e Europe, Bremen, Germany, vols. 1 & 2, IEEE, 2009, pp. 1487e1489. [6] C. Farhat, K.G. Wang, A. Main, S. Kyriakides, L.-H. Lee, K. Ravi-Chandar, T. Belytschko, Dynamic implosion of underwater cylindrical shells: experiments and computations, International Journal of Solids and Structure 50 (2013) 2943e2961. [7] DeepSea Challenger, www.deepseachallenger.com. [8] H. Ory, H.-G. Reimerdes, T. Schmid, A. Rittweger, J. Gomez Garcia, Imperfection sensitivity of an orthotropic spherical shell under external pressure, International Journal of Non-Linear Mechanics 37 (2002) 669e686. [9] A. Trenkler, U. Glaser, J. Hegels, M. Dogigli, Spinforming of XXL bulkheads for large cryo tanks, in: Proc. of 56th International Astronautical Congress, Oct 17e21, 2005, Fukuoka, Japan, IAC-05C2.1.A.03, 2005, pp. 1e7. [10] B. Szelinski, H. Lange, C. Rottger, H. Sacher, S. Weinland, D. Zell, Development of an innovative sandwich common bulkhead for cryogenic upper stage propellant tank, in: 62nd International Astronautical Congress, 2011 Cape Town, South Africa, IAC-11-C2.4.3, 2011, pp. 1e11. [11] T.S. Koko, M.J. Connor, G.V. Corbett, Composite repair of a stainless steel-GRP sonar dome, Journal of Composite Technology and Research 19 (1997) 228e234. [12] C.T.F. Ross, B.H. Huat, T.B. Chei, C.M. Chong, M.D.A. Mackney, The buckling of GRP hemi-ellipsoidal dome shells under hydrostatic pressure, Ocean Engineering 30 (2003) 691e705. [13] Y.Q. Ma, C.M. Wang, K.K. Ang, Buckling of super ellipsoidal shells under uniform pressure, Thin-Walled Structures 46 (2008) 584e591. [14] S.N. Krivoshapko, Research on general and axisymmetric ellipsoidal shells used as domes, pressure vessels, and tanks, Applied Mechanics Reviews e Transactions of the ASME 60 (2007) 336e355.

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[15] J. Błachut, Experimental perspective on the buckling of pressure vessel components, Applied Mechanics Reviews, Transactions of the ASME 66 (2014) 011003-1e011003-24. [16] J. Błachut, O.R. Jaiswal, On the choice of initial geometric imperfections in externally pressurised shells, Journal of Pressure Vessel Technology, Transactions of the ASME 121 (1999) 71e76. [17] London, UK, PD 5500, Published Document, Specification for Unfired Fusion Welded Pressure Vessels, British Standards, 2005. [18] ASME, Code Case 2286-2, Alternative Rules for Determining Allowable External Pressure and Compressive Stresses for Cylinders, Cones, Sphere and Formed Heads, Section VIII, Divisions 1 and 20 , Cases of the ASME Boiler and Pressure Vessel Code, ASME, New York, 2008, pp. 1e13. [19] ECCS, TC8 TWG 8.4 Shells. ‘Buckling of Steel Shells e European Design Recommendations’, No. 125, fifth ed., ECCS, Multicomp Lda, Portugal, 2013, ISBN 978-92-9147116-4, 398 p. (revised). [20] NASA, Buckling of Thin-Walled Doubly Curved Shells,” NASA, Space Vehicle Design Criteria (Structures), Report No. NASA SP-8032, 1969, pp. 1e33. [21] D. Bushnell, Bosor4: program for stress, buckling and vibration of complex shells of revolution, in: N. Perrone, W. Pilkey (Eds.), ‘Structural Mechanics Software Series’, 1, University Press of Virginia, Charlottesville, 1977, pp. 11e143. [22] ABAQUS Inc., Theory and Standard User’s Manual Version 6.3, 2006. Pawtucket, 028604847, RI, USA. [23] S.W. Tsai, H.T. Hahn, Introduction to Composite Materials, Technomic Publ. Co., 1980, pp. 277e327. [24] J. Błachut, Buckling of composite domes with localised imperfections and subjected to external pressure, Composite Structures 153 (2016) 746e754. [25] J. Błachut, The use of composites in underwater pressure hull components, in: B. Falzon (Ed.), Buckling and Postbuckling Structures II: Experimental, Analytical and Numerical Studies, Imperial College Press/World Scientific, 2017, 50 p. [26] J. Błachut, Composite spheroidal shells under external pressure, International Journal for Computational Methods in Engineering Science and Mechanics 18 (2017) 2e12.

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7.5 Vibration correlation technique for the estimation of real boundary conditions and buckling load of unstiffened plates and cylindrical shells Mariano Arbelo 1 , Richard Degenhardt 2, 3 ITA, Aeronautics Institute of Technology, S~ao José dos Campos, Brazil; German Aerospace Center (DLR), Institute for Composite Structures and Adaptive Systems, Braunschweig, Germany; 3PFH, Private University of Applied Sciences G€ottingen, Composite Engineering Campus Stade, Germany

1 2

7.5.1

Introduction

Southwell proposed one of the first nondestructive methods to predict the buckling load of simple structures such as slender beams [1], which was modified by Galletly and Reynolds in order to be applicable for stiffened cylindrical shells. The main disadvantage of the latter is the need for high applied loads, close to the onset of buckling, to provide a reliable prediction of the buckling load [2]. Vibration correlation methods can also be used as a nondestructive technique. The concept of relating vibration characteristics to buckling loads was considered at the beginning of the 20th century for Sommerfeld [3], but only in the 1950s, some experimental investigations were conducted by Chu, Lurie, and Meier, among others (see Refs. [4e6]). A very detailed review of the theory, application, experimental setup, and results of the vibration correlation technique (VCT) approach on different structures can be found in Ref. [7] (Chapter 15). For a better understanding of the applications of VCT on plates and shells, it is important to classify the method according to its purpose: (1) determination of actual boundary conditions for numerical calculation purposes and (2) direct estimation of buckling load. This chapter will present the findings from the authors on these two matters.

7.5.2

Vibration correlation technique applied for the determination of boundary conditions

The VCT for the determination of boundary conditions consists essentially of an experimental determination of the lower natural frequencies for a loaded shell and evaluation of equivalent elastic restraints representing the actual boundary conditions. As an example, a compilation of test results carried out by Technion was presented by Singer. The studies performed in 35 shells showed a significant reduction of the

Stability of composite shelletype structures

371

knockdown factor scatter as a result of including the experimental determination of the boundary conditions the prediction of the buckling load [8]. Technion studies were not limited only to laboratory boundary conditions. To establish a reliable methodology, VCT was applied to structures with realistic boundary conditions, such as component joints commonly used in aerospace, taking into account the introduction of load eccentricity. Ref. [9] presented a detailed modified VCT to define the boundaries once the load eccentricity has been identified, showing a good correlation with experimental results.

7.5.2.1

Benchmark case of an isotropic plate

To study the advantages and limitation of VCT applied to thin-walled structures, it is required to establish a benchmark case to provide reliable experimental data for further investigations. In this case, a 2-mm-thick rectangular aluminum plate, with 355 mm width and 355 mm height, is used. The top and bottom edges are clamped and the lateral edges are simply supported, as shown in Fig. 7.5.1. The aluminum plate is loaded under longitudinal compression using a fixture frame attached to a testing machine, as shown in Fig. 7.5.2. For testing purposes, the compressive load is applied with 2 kN increments, up to 20 kN, and the natural frequencies and vibration modes are measured in each step. The results presented in the following show the typical plate behavior before and after buckling. in Fig. 7.5.3, all vibration frequencies decrease for load increments far below buckling (up to 6 kN). When buckling takes place, the frequency of the first

Load

355 mm

Simply Support

325 mm

Clamped

Clamped

340 mm 355 mm

Figure 7.5.1 Geometry and boundary conditions of aluminum plate for benchmark case.

372

Stability and Vibrations of Thin-Walled Composite Structures

Top cross-head

Knife-end plates for simply support B.C.

Aluminum plate

Bottom frame

Figure 7.5.2 Experimental fixture frame for aluminum plate loaded in compression.

buckling load reach its minimum value and then starts to increase again. After buckling (above 10 kN), the dynamic behavior becomes highly nonlinear and two well-defined characteristics can be pointed out: (1) the frequency of the first and second vibration modes start to increase and (2) the natural frequencies of the third and fourth vibration modes become nearly constant. The four vibration mode shapes are presented in Fig. 7.5.4, where index m corresponds to the number of half waves in the longitudinal direction and n indicates the half waves in the transverse direction. 400 350

Frequency (Hz)

300 250 200 150 1st mode 2nd mode 3rd mode 4th mode

100 50 0 0

2

4

6 8 10 12 14 Axial compression (kN)

16

18

20

Figure 7.5.3 Evolution of the four vibration modes for the isotropic benchmark case with axial compression load.

Stability of composite shelletype structures

1st mode: m = 1; n = 1

3rd mode: m = 2; n = 1

373

2nd mode: m = 1; n = 2

4th mode: m = 2; n = 2

Figure 7.5.4 Characteristics of the first four vibration mode shapes for the isotropic benchmark case.

The result obtained herein can be used as a benchmark case for establishing the finite element procedures and understanding the concept of VCT.

7.5.2.2

Finite element modeling considering vibration correlation technique inputs

The results provided by VCT on the numerical modeling of thin-walled structures allows one to replace ideal boundary condition, e.g., clamped or simply supported, by more realistic elastic restraints. This approach improves the correlation between the predicted buckling load and the experimental results. On the other hand, initial geometric imperfections play an important role in the finite element simulation to calculate the buckling load of plates and shells. Initial geometric imperfections must also be taken into account with elastic boundary conditions to assure a good correlation between the finite element model (FEM) and test results. This section presents different FEM approaches and the advantages to take into account both realistic boundary conditions and initial geometric imperfections. Abaqus is used for modeling and analysis. The benchmark plate is modeled using a converged mesh of four node S4R elements [10]. The isotropic material properties for aluminum are elastic modulus E ¼ 70 GPa, Poisson n ¼ 0.33, and density r ¼ 2780 kg m3. Initially an eigenvalue analysis without compressive load is performed to characterize the vibration modes considering two different sets of boundary conditions: clamped and simply supported at the top and bottom edges. Lateral edges are assumed simply supported for both cases. The third approach is to represent the experimental boundary conditions as equivalent elastic restraints, modeled by rotational springs, attached to the top and bottom edges. A rotational stiffness of 22.5 N m/rad is selected for both the top and bottom boundary conditions, after an iterative analysis, to have the best fit with the experimental results.

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Stability and Vibrations of Thin-Walled Composite Structures

Table 7.5.1 Comparison of vibration frequencies obtained using FE model with ideal and elastic restraint boundary conditions and experimental results from the isotropic benchmark case (without compressive load) Vibration mode

Experimental test (Hz)

FE model with clamped BC (Hz)

FE model with simply supported BC (Hz)

FE model with elastic restraint BC (Hz)

1st

118

131

87

114.4

2nd

230.5

239

212

226.8

3rd

269.2

319

225

276.5

4th

380

425

350

387.3

BC, boundary condition; FE, finite element.

The results presented in Table 7.5.1 clearly show that the real boundary conditions lie between clamped and simply supported boundary conditions. Elastic restraint boundary conditions offer the best agreement with the experimental results, with a deviation lower than 3% on each vibration mode. The use of VCT allows one to estimate the real boundary conditions of the proposed FEM to obtain a better correlation with experimental results. The next step is to study the correlation of this new model taking into account a compressive axial load in the pre- and postbuckling regime. The results of this approach and a comparison with the experimental results are presented in Fig. 7.5.5(a) and (b). The results presented in Fig. 7.5.5(a) with a perfect plate shows that this approach allows one to characterize, with a good correlation, the frequency variation of the benchmark case at lower levels of compressive load. The difference between the numerical predictions and experimental results is due to the fact that this model does not take into account initial geometry imperfections. The imperfection pattern can be extracted from real measured geometric imperfections, but this information is not always available. To overcome this problem, several researchers use linear combination of buckling modes to generate a geometric imperfection pattern (see Refs. [11e13]). For the present study an initial geometric imperfection is introduced using the pattern extracted from the first buckling mode obtained from a linear buckling analysis of the perfect plate. The scaling factor used is the maximum initial geometric imperfection measured in the benchmark case before testing (approximately 0.8 mm). The results of the FEM considering VCT and initial geometric imperfections are presented in Fig. 7.5.5(b). The results of the FEM considering VCT and the initial geometric imperfections present a remarkable agreement with the experimental results. The behavior of the natural frequencies is well characterized for compressive loads in the pre- or postbuckling

Stability of composite shelletype structures

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(a) 400 350 Frequency (Hz)

300 250

1st mode (FEM) 1st mode (EXP)

200

2nd mode (FEM) 2nd mode (EXP) 3rd mode (FEM) 3rd mode (EXP) 4th mode (FEM) 4th mode (EXP)

150 100 50 0 0

(b)

5

10 15 Axial compression (kN)

20

400 350

Frequency (Hz)

300 250

1st mode (FEM) 1st mode (EXP)

200

2nd mode (FEM) 2nd mode (EXP) 3rd mode (FEM) 3rd mode (EXP) 4th mode (FEM) 4th mode (EXP)

150 100 50 0 0

5

10 15 Axial compression (kN)

20

Figure 7.5.5 Variation of the natural frequencies of vibration for (a) perfect and (b) imperfect plates loaded in compression; comparison with experimental results from the isotropic benchmark case. EXP, experimental results; FEM, finite element model.

regime. The minimum frequency value of the first vibration mode is also observed when buckling is predicted by the FEM. As a conclusion of the results presented in Fig. 7.5.5, it becomes clear that this approach can be used as an accurate nondestructive method to estimate the buckling load of thin-walled structures, as the benchmark case. The prediction of the buckling load (located at the point of minimum frequency of the first vibration mode) is reached by the combination of the experimental measurements of vibration modes (VCT) and initial geometric imperfection of the structure, without any compressive load applied on it.

376

Stability and Vibrations of Thin-Walled Composite Structures

7.5.3

Vibration correlation technique applied to unstiffened cylindrical shells

There is no established procedure about how to apply the VCT for unstiffened cylindrical shells commonly used in space applications for launcher structures. This type of structure is usually associated with a high imperfection sensitivity, requiring the application of empirical guidelines to calculate the design buckling load, currently leading to conservative estimations [14]. Skukis presented a preliminary assessment correlating the vibration modes with the buckling load of stainless steel cylinders [15]. If a relationship between the buckling load and the variation of the natural frequencies of vibration exists, it is possible to use the VCT as a nondestructive technique for estimating the real knockdown factor of space structures. Moreover, for this type of structures, there is a remarkable influence of the boundary conditions on the buckling load (see Refs. [16,17]), where the VCT could be used for a better characterization of the actual boundary conditions to provide reliable data for numerical simulation, such as FEM s (see Refs. [18e20]). Recent efforts to improve the work done so far on the VCT field are presented by Jansen et al., where new semianalytical tools are introduced to extend the existing semiempirical VCT for shells, considering the nonlinear effect of both the static state and the geometric imperfections [20]. Abramovich et al. carried out new experimental tests on different thin-walled structures, yielding areas of applicability of VCT to efficiently predict the buckling load of such structures [21]. This section will present and discuss a numerical and experimental verification of a new VCT approach presented by Arbelo et al. [22]. This approach is based on the observations made by Souza et al. [23]. The original approach proposed by Souza is a linear fit between (1  p)2 and (1  f4), where p ¼ (P/Pcr), f ¼ (fm/f0); P is the applied axial load, Pcr is the critical buckling load for a perfect shell, fm is the measured frequency at P load, and f0 is the natural frequency of the unloaded shell. Souza states that the value of (1  p)2 corresponding to (1  f4) ¼ 1 would represent the square of the drop of the load-carrying capacity (x2), due to the initial imperfections. However, if this approach is applied to unstiffened cylindrical shells the results will be negative values of the drop of the loadcarrying capacity (x2), which does not have a coherent physical meaning (see Ref. [22]). Instead of plotting (1  p)2 versus (1  f4), Arbelo proposed to plot (1  p)2 versus (1  f2) and represented the points by a second-order fitting curve. Therefore, the minimum value of (1  p)2 obtained using this approximation represents the square of the knockdown of the load-carrying capacity (x2) for unstiffened cylindrical shells. Then the buckling load can be estimated by using Eq. (7.5.1)  Pimperfect ¼ Pcr

7.5.3.1

qffiffiffiffiffi  1  x2

(7.5.1)

Vibration correlation technique applied to cylindrical shells, numerical models

The benchmark test chosen from the open literature is the so-called “cylinder Z15.” This test article is taken from the results of the European Space Agency (ESA) study,

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Table 7.5.2

Material properties of Z15 cylinder

E1 (GPa)

157.4

E2 (GPa)

8.6

G12 (GPa)

5.3

n12

0.28

Table 7.5.3 Geometric parameters of Z15 cylinder Free length (mm)

500.0

Radius (mm)

250.27

Thickness (mm)

0.463

Layup (ineout)

[24/41]

conducted at DLR and published by Degenhardt et al. [14]. The material properties [unidirectional (UD) carbon fiber prepreg with epoxy matrix] and geometry are detailed in Tables 7.5.2 and 7.5.3, respectively. A nonlinear analysis considering midsurface imperfections, followed by a frequency analysis, is used to characterize the variation of the vibration frequencies when the cylinder is loaded under axial compression. The midsurface imperfection is imported into the FEM by shifting the radial position of each node by using the measured imperfection after fabrication. Plotting the natural frequencies of vibration versus applied load as proposed by Souza’s approach will result in negative values of the drop of the load-carrying capacity (x2), which does not have a physical meaning (see Fig. 7.5.6(a)). Instead of plotting (1  p)2 versus (1  f4), Arbelo suggested plotting (1  p)2 versus (1  f2) [22] (see example for Z15 cylinder in Fig. 7.5.6(b)). Then the minimum value of (1  p)2 obtained using this approximation represents the square of the drop of the loadcarrying capacity (x2) for unstiffened cylindrical shells, due to the initial imperfections. This proposed approach is still valid even when vibration frequencies calculated for lower compressive loads (far below buckling) are used to estimate the buckling load through the second-order fit. Table 7.5.4 shows the results of the buckling load estimation using frequencies obtained for compressive loads below 50% of the buckling load for Z15. Very good correlation is observed between the proposed VCT approach and the nonlinear buckling load of the imperfect shell (Pimperfect), obtained from a nonlinear analysis considering midsurface imperfection up to buckling. More details and numerical examples are reported in Ref. [22]. The same methodology can be used for experimental testing. The vibration frequencies at different compression load levels can be measured experimentally and the linear buckling load can be estimated through an eigenvalue analysis using a

378

Stability and Vibrations of Thin-Walled Composite Structures

(a) 1.2

(1 – p)2

1 0.8

1st mode (Z15)

0.6

Linear fit (Z15)

0.4 0.2 0 0.2

0

0.4

0.6

0.8

1

1.2

–0.2 ξ2 –0.4

–0.6

(1 – f 4)

(b) 1.2 1 1st mode Z15

(1 – p)2

0.8

2nd order fit (Z15)

0.6 0.4 0.2 ξ2

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2

(1 – f )

Figure 7.5.6 VCT methodology proposed by (a) Souza et al. [23] and (b) Arbelo et al. [22] applied to unstiffened cylindrical shell (cylinder Z15).

Table 7.5.4 Buckling load predicted using the proposed VCT approach for Z15 cylinder Estimated buckling load (kN)

Deviation (%)

Pimperfect

23.7

e

P with VCT up to 31% load

21.8

7.5

P with VCT up to 46% load

22.1

6.5

Stability of composite shelletype structures

379

FEM of a perfect cylindrical shell. Then the real buckling load can be estimated without reaching the instability point. The next section will deal with an experimental verification of this novel approach.

7.5.3.2

Vibration correlation technique applied to cylindrical shells, experimental verification

For the experimental validation of the proposed empirical VCT approach, the testing results of three different composite laminated cylindrical shells with clamped boundary conditions are used. The applied load and the first natural frequency of vibration and mode shapes are measured and correlated. The main goal of this work is to compare the predicted buckling load and the real buckling load measured on samples with different materials, geometries (radius, thickness, height), and fabrication technologies. More details about each study case can be found in the original publications of Arbelo et al. [24] and Kalnins et al. [25].

7.5.3.2.1

Composite cylindrical shells R07, R08, and R09, with R/t ¼ 399

Three identical unstiffened cylindrical shells (named R07, R08, and R09) are fabricated by hand-layup using six plies of UD carbon fiber Unipreg 100 g/m2. The geometry and layup are presented in Table 7.5.5. The material properties are presented in j Table 7.5.6, in which Ei is the elastic modulus along the fiber direction (i ¼ 1) or matrix direction (i ¼ 2) in tension ( j ¼ T) or compression ( j ¼ C), G12 is the shear modulus, n12 is the Poisson ratio, and t is the ply thickness. The top and bottom cylinder edges are clamped using a 25-mm-high resin potting from the outside and metallic rings from the inside, as presented in Fig. 7.5.7. For testing, each cylinder is placed between two metallic plates and glued with epoxy resin to them. Moreover, a spherical joint is placed between the crosshead of the testing machine and the lower plate to assure axial loading only. The compressive load is measured using a load cell and it is applied using displacement control at a constant velocity of 1.5 mm/min. Two types of tests are carried out to validate the proposed VCT approach. First the cylinder is loaded in compression up to buckling (notice that the cylinders are designed to buckle in the elastic regime to avoid damage accumulated during repeated tests).

Geometric parameters for R07, R08, and R09 cylinders Table 7.5.5

Length (mm)

500  1

Radius (mm)

250  1

Thickness (mm)

0.6264  0.11

Layup (ineout)

[0 2/(45 )2]  1

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Stability and Vibrations of Thin-Walled Composite Structures

Table 7.5.6 Material properties of UD prepreg Unipreg 100 g/m2 ET1

116.44  8.71

GPa

EC1

91.65  7.58

GPa

ET2

6.73  0.23

GPa

EC2

6.39  0.81

GPa

G12

3.63  0.2

GPa

v12

0.34  0.04

t

0.1044  0.0015

mm

Once the onset of buckling is found, a second test is performed using a Polytec laser vibrometer to characterize the first natural frequency of vibration and mode shape for different values of applied compressive load, below the onset of buckling. The experimental buckling load obtained for the R07, R08, and R09 cylinders is 22.44, 22.74, and 21.55 kN, respectively. The average buckling load then is 22.24 kN. The variation of the first natural frequency of vibration measured on each cylinder, when loaded in compression, is shown in Fig. 7.5.8, plotting (1  p)2 versus (1  f2).

Figure 7.5.7 Experimental test setup for compressive loading on cylindrical shells.

Stability of composite shelletype structures

381

1

2nd order fit Exp 1st mode R07 Exp 1st mode R08 Exp 1st mode R09

(1 – p)2

0.8 0.6 0.4 0.2 ξ2 0 0

0.1

0.2

0.3

0.4 1 – f2

0.5

0.6

0.7

Figure 7.5.8 Plot of (1  p)2 versus (1  f2) for R07, R08, and R09 cylinders and second-order fit based on all the experimental data.

As all three cylinders have the same characteristics a second-order fit can be done using all the experimental data. To predict the onset of buckling following the proposed VCT approach, it is needed to know the following: 1. The evolution of the first vibration mode on the cylindrical shell with the applied load on the prebuckling regime. 2. The critical buckling load of the perfect cylinder. This value can be obtained by a FEM. An eigenvalue analysis of a perfect cylindrical shell is fast, simple and provides such information. For the study case, the critical buckling load obtained for a perfect cylindrical shell with the layup, dimensions, material properties, and boundary conditions of the study case is Pcr ¼ 34.17 kN. More details about the modeling tools and techniques are provided in Ref. [24].

The minimum value of (1  p)2 obtained using this approximation represents the square of the drop of the load-carrying capacity (x2). Using Eq. (7.5.1), the predicted buckling load obtained from the proposed VCT approach is 22.84 kN, which correlates very well with the average value of the experimental buckling load measured during testing, with less than 3% of deviation.

7.5.3.2.2

Composite cylindrical shells R15 and R16, with R/t ¼ 478

The R15 and R16 cylinders were fabricated by hand layup by using four plies of UD carbon fiber prepreg Hexcel IM7/8552 and cured in an autoclave. The geometry and layup are presented in Table 7.5.7. The material properties are presented in Table 7.5.8. The same experimental setup adopted for cylinders R07eR09 is used for these two specimens. Both the top and bottom edges are trimmed and clamped using a 25-mmhigh resin potting and metallic rings. The final radius to thickness (R/t) ratio is about 478.

382

Stability and Vibrations of Thin-Walled Composite Structures

Table 7.5.7 Geometric parameters for R15 and R16 cylinders Free length (L) (mm)

500

Radius (R) (mm)

250

Thickness (t) (mm)

0.523

Layup (ineout)

[(24 )/(41 )]  1

Measured material properties of UD prepreg Hexcel IM7/8552 Table 7.5.8

ET1

171.5  4.45

GPa

EC1

150.2  6.9

GPa

ET2

8.9  0.37

GPa

EC2

9.4  1.02

GPa

G12

5.1  0.39

GPa

v12

0.32  0.04

t-ply

0.125

mm

For testing, each cylinder is placed between two metallic plates and glued with epoxy resin. A spherical joint is placed between the crosshead of the testing machine and the loading plate to avoid bending moments (see Fig. 7.5.9). The compressive load is measured using a load cell and it is applied using displacement control at a constant velocity of 0.5 mm/min. A more detailed insight of testing parameters and results is given in the original publication by Kalnins et al. [25]. The experimental buckling load obtained for the R15 and R16 cylinders is 25.04 and 25.2 kN, respectively. The average buckling load then is 25.12 kN. The variation of the first natural frequency of vibration measured on each cylinder, when loaded in compression, is shown in Fig. 7.5.10, plotting (1  p)2 versus (1  f2). As both cylinders have the same characteristics a second-order fit can be done using all the experimental data. To estimate the buckling load of the real structure, the critical buckling load of the perfect cylinder (Pcr) is required. The latter can be obtained by a linear eigenvalue analysis of a simple FEM. For the present study, the FEM is generated using Abaqus. The critical buckling load obtained for a perfect cylindrical shell with the layup, dimensions, material properties, and boundary conditions used on the studied cases is 35.1 kN. Using Eq. (7.5.1), the predicted buckling load obtained from the proposed VCT approach, for R15 and R16 cylinders, is 25.8 kN. There was very good correlation

Stability of composite shelletype structures

383

Figure 7.5.9 Experimental test setup for compressive loading in R15 and R16 cylindrical shells.

1

(1 – p)2

0.8

2nd order fit Exp 1st mode R15 Exp 1st mode R16

0.6

0.4

0.2 ξ2 0

0

0.1

0.2

0.3

0.4 0.5 1 – f2

0.6

0.7

0.8

Figure 7.5.10 Plot of (1  p)2 versus (1  f2) for R15 and R16 cylinders and second-order fit based on all the experimental data.

384

Stability and Vibrations of Thin-Walled Composite Structures

Table 7.5.9

Geometric parameters for Z37

cylinder Free length (L) (mm)

800

Radius (R) (mm)

400

Thickness (t) (mm)

0.785

Layup (ineout)

[(34 )/0 2/(53 )]  1

with the experimental buckling load measured during testing (the average buckling load was 25.12 kN), with less than 3% of deviation.

7.5.3.2.3

Composite cylindrical shell Z37, with R/t ¼ 510

The Z37 cylinder is fabricated at DLR-Braunschweig by hand layup, using plies of the same UD carbon fiber prepreg Hexcel IM7/8552, and cured in an autoclave. The geometry and layup are presented in Table 7.5.9. This cylinder is clamped on both ends using a sanderesin concrete and the final R/t radius is about 510. As the other study cases, the top and bottom cylinder edges are clamped, using a 20-mm-high resin potting, together with metallic rings placed at the inner face. DLR test setup for Z37 cylinder uses special loading tables that can be adjusted to avoid loading imperfection and bending introduction along the cylinder edges (see DLR’s hydraulic testing machine in Fig. 7.5.11). The compressive load is measured using a load cell and it is applied using displacement control at a constant velocity of 0.5 mm/min. The experimental buckling load obtained for the Z37 cylinder is 59 kN. The variation of the first natural frequency of vibration measured on the cylinder, when loaded in compression, is shown in Fig. 7.5.12, plotting (1  p)2 versus (1  f2).

Figure 7.5.11 Experimental test setup for compressive loading in Z37 cylindrical shell (DLR).

Stability of composite shelletype structures

385

1

(1 – p)2

0.8

Exp 1st mode Z37 2nd order fit

0.6

0.4

0.2 ξ2 0 0

0.1

0.2

0.3

0.4 0.5 2 1–f

0.6

0.7

0.8

Figure 7.5.12 Plot of (1  p)2 versus (1  f2) for Z37 cylinder and second-order fit.

The critical buckling load (Pcr) obtained for a perfect cylindrical shell with the layup, dimensions, material properties, and boundary conditions of Z37 can be obtained through a linear eigenvalue analysis of a simple FEM, as explained for the other study cases. For Z37 cylinder, the critical buckling load is 89.8 kN (more detailed explanation of the finite element simulation can be found in the original publication by Kalnins et al. [25]). Using Eq. (7.5.1), the predicted buckling load obtained from the proposed VCT approach for this case is 58.41 kN, which agrees remarkably well with the experimental buckling load measured during testing, with a deviation lower than 2%. Based on the test results carried out from each specimen the authors conclude that the present approach could be applied as an experimental nondestructive method to estimate the buckling load on unstiffened cylindrical shells loaded in compression. Furthermore, one can apply the VCT approach on each study case to investigate the minimum threshold of compressive load needed to achieve a good correlation, with less than 10% deviation from the experimental buckling load. In general, very good results are obtained for all the studied cases when the maximum applied compression load is equal to or higher than 80% of the buckling load, always starting from the unloaded state (See Refs. [22,24,25]).

7.5.4

Summary and concluding remarks

In this chapter the implementation of VCT, on plates and shells, for the determination of realistic boundary conditions and direct estimation of buckling load is discussed. •

An experimental benchmark case on imperfect plates is detailed and the results are used to include realistic boundary conditions on an FEM to predict the buckling load of a plate

386

•

Stability and Vibrations of Thin-Walled Composite Structures

loaded under longitudinal compression. The results show very good correlation if the initial geometric imperfections are taken into account. At this point the authors remark that all the required data for the estimation of the buckling load can be obtained for experimental tests of the completely unloaded structure, which characterizes a truly nondestructive methodology. A validation of a novel empirical approach using the VCT as a nondestructive method to estimate the buckling load of unstiffened cylindrical shells is presented. Three different study cases are tested to verify the robustness of the proposed approach for different structures under different test setups. The proposed approach presents a very good correlation when the maximum load adopted in the VCT is higher than 80% of the buckling load obtained with tests. If no failure occurs at this maximum load, the proposed approach characterizes a truly nondestructive methodology.

References [1] R.V. Southwell, On the analysis of experimental observations in problems of elastic stability, Proceedings of the Royal Society, (London), Series A 135 (1932) 601e616. [2] G.D. Galletly, T.E. Reynolds, A simple extension of Southwell’s method for determining the elastic general instability pressure of ring stiffened cylinders subjected to external pressure, SESA Proceedings 12 (2) (1955) 141e153. [3] A. Sommerfeld, Eine einfache Vorrichtung zur Veranschaulichung des Knickungsvorganges, Zeitschrift des Verein Deutscher Ingenieure (ZVDI) (1905) 1320e1323. [4] T.H. Chu, Determination of Buckling Loads by Frequency Measurements (Thesis), California Institute of Technology, Pasadena, Calif., 1949. [5] H. Lurie, Lateral vibration as related to structural stability, Journal of Applied Mechanics, ASME 19 (June 1952) 195e204. [6] J.H. Meier, Discussion of the paper entitled “the determination of the critical load of a column or stiffened panel in compression by the vibration method”, in: Proceedings of the Society for Experimental Stress Analysis, vol. 11, 1953, pp. 233e234. [7] J. Singer, J. Arbocz, T. Weller, Buckling Experiments, Experimental Methods in Buckling of Thin-walled Structures, Volume 2, John Wiley & Sons, New York, 2002 (Chapter 15). [8] J. Singer, Vibration and buckling of imperfect stiffened shells e recent developments, in: J.M.T. Thompson, G.W. Hunt (Eds.), Collapse: The Buckling of Structures in Theory and Practice, Cambridge University Press, Cambridge, 1983, pp. 443e481. [9] J. Singer, H. Abramovich, Vibration techniques for definition of practical boundary conditions in stiffened shells, AIAA Journal 17 (7) (July 1979) 762e763. [10] ABAQUS User’s Manual, Abaqus Analysis User’s Manual, 2011. [11] C.A. Featherston, Imperfection sensitivity of flat plates under combined compression and shear, International Journal of Non-Linear Mechanics 36 (2001) 249e259. [12] C.A. Featherston, Imperfection sensitivity of curved panels under combined compression and shear, International Journal of Non-Linear Mechanics 38 (2003) 225e238. [13] A. Tafreshi, C.G. Bailey, Instability of imperfect composite cylindrical shells under combined loading, Composite Structures 80 (2007) 49e64. [14] R. Degenhardt, A. Kling, A. Bethge, J. Orf, L. K€arger, K. Rohwer, R. Zimmermann, A. Calvi, Investigations on imperfection sensitivity and deduction of improved knockdown factors for unstiffened CFRP cylindrical shells, Composite Structures 92 (8) (2010) 1939e1946.

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[15] E. Skukis, K. Kalnins, A. Chate, Preliminary Assessment of Correlation between Vibrations and Buckling Load of Stainless Steel Cylinders. Shell Structures Theory and Applications, CRC Press, London, 2013, pp. 325e328. [16] R. Zimmemann, Buckling research for imperfection tolerant fiber composite structures, in: Proceeding of: Conference on Spacecraft Structures, Materials and Mechanical Testing, 27e29 March 1996, Noordwijk, The Netherlands, 1996. [17] C. H€uhne, R. Zimmermann, R. Rolfes, B. Geier, Sensitivities to geometrical and loading imperfections on buckling of composite cylindrical shells, in: Proceeding of: European Conference on Spacecraft Structures, Materials and Mechanical Testing, 11e13 December 2002, Toulouse, 2002. [18] M.W. Hilburger, M.P. Nemeth, J.H. Starnes Jr., Shell Buckling Design Criteria Based on Manufacturing Imperfection Signatures, NASA Report TM-2004-212659, 2004. [19] R. Degenhardt, A. Bethge, A. King, R. Zimmermann, K. Rohwer, J. Teßmer, A. Calvi, Probabilistic approach for improved buckling knock-down factors of CFRP cylindrical shells, in: Proceeding of: First CEAS European Air and Space Conference, 10e13 September 2008, Berlin, Germany, 2008. [20] E.L. Jansen, H. Abramovich, R. Rolfes, The direct prediction of buckling loads of shells under axial compression using VCT e towards an upgraded approach, in: Proceedings of: 27th Congress of the International Council of the Aeronautical Sciences, 7e12 September 2014, St. Petersburg, Russia, 2014. [21] H. Abramovich, D. Govich, A. Grunwald, Buckling prediction of panels using the vibration correlation technique, Progress in Aerospace Sciences 78 (2015) 62e73. [22] M.A. Arbelo, S.F.M. de Almeida, M.V. Donadon, S.R. Rett, R. Degenhardt, S.G.P. Castro, K. Kalnins, O. Ozolins, Vibration correlation technique for the estimation of real boundary conditions and buckling load of unstiffened plates and cylindrical shells, Journal of ThinWalled Structures 79 (2014) 119e128. [23] M.A. Souza, W.C. Fok, A.C. Walker, Review of experimental techniques for thin-walled structures liable to buckling, part I e neutral and unstable buckling, Experimental Techniques 7 (1983) 21e25. [24] M.A. Arbelo, K. Kalnins, O. Ozolins, E. Skukis, S.G.P. Castro, R. Degenhardt, Experimental and numerical estimation of buckling load on unstiffened cylindrical shells using vibration correlation technique, Thin-Walled Structures 94 (2015) 273e279. [25] K. Kalnins, M.A. Arbelo, O. Ozolins, E. Skukis, S.G.P. Castro, R. Degenhardt, Experimental nondestructive test for estimation of buckling load on unstiffened cylindrical shells using vibration correlation technique, Shock and Vibration (2015), 729684, 8 p.

388

Stability and Vibrations of Thin-Walled Composite Structures

7.6 New robust knockdown factors for the design of axially loaded composite shells Ronald Wagner, Christian H€ uhne, Steffen Niemann German Aerospace Center (DLR), Institute for Composite Structures and Adaptive Systems, Braunschweig, Germany The design of cylindrical shells relies till today on the application of empirical knockdown factors (KDFs), which are based on experimental data from the beginning of the 20th century. For the design of composite cylindrical shells, the NASA SP-8007 is used that proposes a KDF for the buckling load, which depends on the radius-to-wall thickness ratio (R/t), but neglects the influence of the cylinder length L. However, a pronounced R/t dependency of the KDF for composite shells is not visible (compare with Fig. 7.6.1 and Appendix A). The illustrated empirical KDFs scatter mainly in the range from 0.6 to 0.9 and are in some cases even below 0.4. Most of the shown empirical data are based on short shells with L/R ¼ 1e2; however, the empirical KDFs from Herakovich [1] are based on long shells (L/R ¼ 4.39e6.74), which have R/t ¼ 60e120. These results indicate that the cylinder length has a significant influence on the lower-bound buckling load. The discussion regarding the influence of the R/t and L/R on the buckling load is based on “Robust design criterion of axially loaded cylindrical shellseSimulation and Validation” [2]. Within this publication a parametric study was performed with the single boundary perturbation approach (SBPA) which was introduced in Ref. [3]. The SBPA is a lower-bound design approach for axially loaded cylindrical 1

Hühne Degenhardt Waters Wilkins

Knockdown factor

0.8

Priyardarsini Hilburger

0.6

Giavotto Bisagni

0.4

iKobayashi Herakovich Hirano Meyer

0.2

Kasuya Tennyson

0 50

150

450 250 350 Radius-to-thickness ratio, R/t

550

650

Sun Khakimova Kalnis Geier

Figure 7.6.1 Distribution of the experimental data of axial compressed composite cylindrical shells for different R/t ratios [2].

Stability of composite shelletype structures

389

shells and considers the influence of geometric and loading imperfections on the lower-bound buckling load. The corresponding KDFs were determined for isotropic shells with L/R ¼ 1e10 and R/t ¼ 50e2000 (compare with Fig. 7.6.2). The results show that KDF for the buckling load depends on the R/t and the L/R ratio. There are two effects that greatly influence the lower-bound buckling load, the slenderness of the shell (represented by the R/t ratio), and the length effect of the shell (represented by the L/R ratio). The influence of the length effect was discussed by both Weingarten et al. [4] and Von Karman and Tsien [5]. It was concluded that the mechanical boundary conditions have a stiffening influence on cylindrical shells, which results in a less imperfectionsensitive structure. As the shell length increases, the stiffening influence of the mechanical boundary conditions decreases, and hence the structure is more imperfection sensitive. It shows that the influence of the length effect on the lower-bound KDF is most pronounced between L/R ¼ 1e5, the KDF reduces from L/R ¼ 1e5 by about 40%. For long shells that have an L/R ¼ 5e10, the reduction is about 20%. The influence of the slenderness was discussed in Ref. [2] and is based on the stabilizing influence of the shell curvature on the lower-bound buckling load. The curvature is the reciprocal of the radius, and for a constant wall thickness (t ¼ 1 mm), but an increasing radius (50e2000 mm), the curvature approaches zero. This means a shell with a low R/t ratio benefits from a small radius because of the increased stabilizing effect of the curvature. This stabilizing influence decreases asymptotically with increasing shell radius. The results indicate in addition that there is a nonlinear interaction between slenderness and length effect because for low L/R ratios a low slenderness increases the lower-bound KDF more than that in case of high L/R ratios. Based on these results, it is concluded that the lower-bound buckling load decreases as the R/t and L/R ratios increase and vice versa.

Threshold - L/R = 1

1

Threshold - L/R = 2 Threshold - L/R = 3

Knockdown factor

0.8

Threshold - L/R = 4

0.6

Threshold - L/R = 5 Threshold - L/R = 7

0.4

Threshold - L/R = 10

0.2

0 50

200 350 500 650 800 950 1100 1250 1400 1550 1700 1850 2000

Radius-to-thickness ratio, R/t

Figure 7.6.2 Lower-bound KDF from Ref. [2] for unstiffened isotropic cylindrical shells.

390

Stability and Vibrations of Thin-Walled Composite Structures

The corresponding lower-bound KDFs can be determined according to Eq. (7.6.1). This equation is based on curve fitting of numerical lower-bound KDFs from Ref. [2] and depends on the R/t ratio and the coefficients U and h. rTH ¼ UTH $ðR=tÞhTH

(7.6.1)

The coefficients U and h for short shells (L/R w 1e3) are given by Eq. (7.6.2) from Ref. [2]: UTH z 0:0196$

 2   L L  0:0635$ þ 1:3212 R R

 2   L L þ 0:061$ þ 0:08 hTH z 0:013$ R R

(7.6.2)

For long shells (L/R w 3e10), Eq. (7.6.3) from Ref. [2] has to be applied: UTH z 0:0118$

 2   L L  0:2247$ þ 1:5285 R R

 2   L L  0:021$ þ 0:2036 hTH z 0:0011$ R R

(7.6.3)

These KDFs are valid for isotropic and quasi-isotropic composite shells with clamped boundary conditions, which buckle in the elastic region. The KDFs were validated for composite shells with L/R w 1e7 and R/t w 50e550 (compare with Figs. 7.6.3 and 7.6.4). As mentioned earlier the lower-bound KDFs (denominated as thresholds) are valid only for isotropic and quasi-isotropic composite shells. But they also give a good approximation for the lower-bound buckling load of axially stiff composite shells because the influence of slenderness and length effect is considered correctly. The results in Fig. 7.6.3 are based on composite shells with L/R ¼ 1.3e2 (compare with Appendix A), and it shows that most of the empirical results are above the L/R ¼ 1 threshold and only some are in between the L/R ¼ 1e2 thresholds. Kalnis [6] tested quasi-isotropic shells with [0, 60, 60], and Waters [7], Bisagni [8], and Priyardarsini [9] also performed buckling experiments with nearly quasi-isotropic composite shells. All the corresponding results are in the expected range between the L/R ¼ 1e2 thresholds. Most of the long and thick shells shown in Fig. 7.6.4 are either quasi-isotropic or have a laminate stacking sequence which consists of alternating [45, 45] layers. The results from Herakovich [1] are compared with the threshold KDFs for L/R ¼ 4e7 and it shows that the threshold KDFs for thick and long shells are even lower than the NASA SP-8007 KDFs, but they approximate the corresponding experimental results really well. The NASA SP-8007 was developed for launch-vehicle primary structures; the corresponding shells have L/Re1 and the threshold KDFs for the buckling load for those

Stability of composite shelletype structures

391

1

Hühne Degenhardt Waters

Knockdown factor

0.8

Wilkins Priyardarsini

L/R = 1.3

0.6

Hilburger L/R = 1.54

L/R ~2

Giavotto

L/R = 2

L/R = 1.75

0.4

Bisagni Kobayashi Meyer

0.2

Sun Khakimova Kalnis

0 50

150

250 350 450 Radius-to-thickness ratio, R/t

550

650

Geier NASA SP-8007 Threshold - L/R = 1 Threshold - L/R = 2

Figure 7.6.3 Comparison of the SBPA thresholds for L/R ¼ 1e2 and empirical data with for L/R ¼ 1.3e2.0 [2]. Herakovich NASA SP-8007 Threshold - L/R = 4 Threshold - L/R = 5 Threshold - L/R = 6 Threshold - L/R = 7

Knockdown factor

0.8

0.6 L/R = 6.62 L/R = 4.39

0.4 L/R = 6.72

0.2

0 50

150

250 350 450 Radius-to-thickness ratio, R/t

550

650

Figure 7.6.4 Comparison of the SBPA thresholds for L/R ¼ 4e7 and empirical data with for L/R ¼ 4.39e6.74 [2].

short shells are significantly higher. For very long shells with low R/t ratios; the threshold KDFs are even more conservative than the NASA SP-8007. From this section the following conclusions are drawn: 1. The lower-bound buckling load depends on the slenderness and the length effect. 2. The KDFs rTH can be used to define a robust design load for an isotropic, axially stiff, and quasi-isotropic composite cylindrical shell. 3. The KDFs rTH for short and thin shells (L/R w 1e2 and R/t > 200) are significantly higher than the KDFs of the NASA SP-8007. 4. The KDFs rTH for long and thick shells (L/R > 3 and R/t < 250) are more conservative than the KDFs of the NASA SP-8007.

392

Stability and Vibrations of Thin-Walled Composite Structures

References [1] C.T. Herakovich, Theoretical-Experimental Correlation for Buckling of Composite Cylinders under Combined Compression and Torsion, NASA-CR-157358, 1978. [2] H.N.R. Wagner, et al., Robust design criterion for axially loaded cylindrical shells e simulation and validation, Thin-Walled Structures 115 (2017) 154e162. [3] H. Wagner, C. H€uhne, K. Rohwer, S. Niemann, M. Wiedemann, Stimulating the realistic worst case buckling scenario of axially compressed cylindrical composite shells, Composite Structures 160 (2017) 1095e1104. [4] V.I. Weingarten, E.J. Morgan, P. Seide, Elastic stability of thin-walled cylindrical and conical shells under axial compression, AIAA Journal 3 (1965) 500e505. [5] T. Von Karman, H. Tsien, The buckling of thin cylindrical shells under axial compression, Journal of the Aeronautical Sciences 8 (1941). [6] K. Kalnis, M. Arbelo, O. Ozolins, S. Castro, R. Degenhardt, Numerical characterization of the knock-down factor on unstiffened cylindrical shells with initial geometric imperfections, in: 20th Int. Conf. On Composite Materials (ICCM20), 19e24 July 2015, Copenhagen, Denmark, 2015. [7] W. Waters, Effects of initial geometric imperfections on the behavior of graphite-epocy cylinders loaded in compression (MS thesis), in: Engineering Mechanics, Old Dominion University, Norfolk, VA, 1996. [8] C. Bisagni, P. Cordisco, An experimental investigation into the buckling and post-buckling of CFRP shells under combined axial and torsion loading, Composite Structures 60 (4) (2003) 391e402. [9] R. Priyadarsini, V. Kalyanaraman, S.M. Srinivasan, Numerical and experimental study of buckling of advanced fibre composite cylinders under axial compression, International Journal of Structural Stability and Dynamics 12 (04) (2012). [10] C. H€uhne, Robuster Entwurf beulgef€ahrdeter, unversteifter Kreiszylinderschalen aus Faserverbund (Ph.D. thesis at Technische Universit€at Carolo-Wilhelmina zu Braunschweig), 2005. [11] R. Degenhardt, A. Bethge, A. Kling, R. Zimmermann, K. Rohwer, Probabilistic approach for improved buckling knock-down factors of CFRP cylindrical shells, in: Proceeding of 18th Engineering Mechanics Division Conference, 2007. [12] D. Wilkins, T. Love, Combined compression-torsion buckling tests of lamintated composite cylindrical shells, Journal of Aircraft 12 (11) (1975) 885e889. [13] M. Hilburger, M. Nemeth, J.J. Starnes, Shell Buckling Design Criteria Based on Manufacturing Imperfection Signatures, NASA/TM-2004-212659, 2004. [14] V. Giavotto, C.C.M. Poggi, P. Dowling, Buckling behaviour of composite shells under combined loading, in: J.F. Julien (Ed.), Buckling of Shell Structures, on Land, in the Sea and in the Air, Elsevier Applied Science, 1991, pp. 53e60. [15] C. Bisagni, Composite cylindrical shells under static and dynamic axial loading: an experimental campaign, Progress in Aerospace Sciences 78 (2015) 107e115 (special issue): DAEDALOS e Dynamics in Aircraft Engineering Design and Analysis for Light Optimized Structures. [16] C. Bisagni, Experimental buckling of thin composite cylinders in compression, AIAA Journal 37 (2) (1999) 276e278. [17] S. Kobayashi, H. Seko, K. Koyama, Compressive buckling of CFRP circular cylinderical shells, part 1, theoretical analysis and experiment, Journal of the Japan Society for Aeronautical and Space Sciences 32 (361) (1984) 111e121.

Stability of composite shelletype structures

393

[18] G. Sun, Optimization of Laminated Cylinders for Buckling, UTIAS Report No. 317, Institute for Aerospace Studies, University of Toronto, 1987. [19] Y. Hirano, Optimization of laminated composite cylindrical shells for axial buckling, Journal of the Japan Society for Aeronautical and Space Sciences 32 (360) (1984) 46e51. [20] H.R. Meyer-Piening, M. Farshad, B. Geier, R. Zimmerman, Buckling loads of CFRP composite cylinders under combined axial and torsion loading e experiments and computations, Composite Structures 53 (4) (2001) 427e435. [21] H. Kasuya, M. Uemura, Coupling effect on axial compressive buckling of laminated composite cylindrical shells, Journal of the Japan Society for Aeronautical and Space Sciences 30 (346) (1982) 664e675. [22] R. Khakimova, D. Wilckens, J. Reichardt, R. Degenhardt, Buckling of axially compressed CFRP truncated cones: experimental and numerical investigation, Composite Structures 146 (2016) 232e247. [23] R.C. Tennyson, J.S. Hansen, Optimum design for buckling of laminated cylinders, in: Collapse: The Buckling of Structures in Theory and Practice, Cambridge University Press, Cambridge, 1983. [24] B. Geier, H.-R. Meyer-Piening, R. Zimmermann, On the influence of laminate stacking on buckling of composite cylindrical shells subjected to axial compression, Composite Structures 55 (2002) 467e474.

Appendix A. Empirical KDFs and geometric properties for composite shells

Ply layup

L (mm)

R (mm)

t (mm)

L/ R

R/t

Nper (kN)

Nexp (kN)

rexp

H€ uhne [10] [24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

33.57

21.80

0.65

[24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

33.57

21.90

0.65

[41, 41, 24, 24]

500.0

250.0

0.5

2.0

500.0

17.48

15.70

0.90

[24, 41, 41, 24]

500.0

250.0

0.5

2.0

500.0

23.91

15.70

0.66

[24, 41, 41, 24]

500.0

250.0

0.5

2.0

500.0

23.91

16.70

0.70

[45, 45, 0, 79]

500.0

250.0

0.5

2.0

500.0

23.19

18.60

0.80

500.0

250.0

0.5

2.0

500.0

38.28

23.36

0.61

Degenhardt [11] [24, 24, 41, 41] [24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

38.28

24.63

0.64

[24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

38.28

21.32

0.56

[24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

38.28

23.08

0.60

[24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

38.28

22.63

0.59

[24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

38.28

23.99

0.63

[24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

38.28

25.02

0.65

[24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

38.28

23.62

0.62 Continued

394

Stability and Vibrations of Thin-Walled Composite Structures

Continued Ply layup

L (mm)

R (mm)

t (mm)

L/ R

R/t

Nper (kN)

Nexp (kN)

rexp

[24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

38.28

25.69

0.67

[24, 24, 41, 41]

500.0

250.0

0.5

2.0

500.0

38.28

22.43

0.59

[45, 45, 0, 90]s

355.6

203.2

1.01

1.75

200

183.63

133.59

0.73

[45, 45, 45, 45]2s

355.6

203.4

2.02

1.75

101

448.05

328.89

0.73

[45, 45, 0, 90]2s

355.6

203.3

2.01

1.75

101

773.05

656.26

0.85

[45, 45, 04, 45, 45]s

355.6

203.0

1.95

1.75

104

619.48

557.63

0.90

[45, 45, 904, 45, 45]s

355.6

203.4

2.02

1.75

101

697.18

408.66

0.59

[0, 45, 45]s

381.0

190.5

0.85

2.0

223.0

102.45

89.74

0.88

[45, 45, 45, 45]

381.0

190.5

0.54

2.0

354.0

30.68

22.30

0.73

[0, 45, 45, 0]s

390.0

300.0

1.0

1.3

300.0

169.88

97.01

0.57

[0, 45, 45, 0]s

390.0

300.0

1.0

1.3

300.0

169.88

98.68

0.58

[0, 45, 45, 0]s

390.0

300.0

1.0

1.3

300.0

169.88

97.06

0.57

[45, 45, 0, 0]s

355.6

200.0

1.016

1.78

197

132.53

123.46

0.93

[45, 45, 90, 90]s

355.6

200.0

1.016

1.78

197

169.17

141.84

0.84

[45, 45, 0, 90]s

355.6

200.0

1.016

1.78

197

180.52

151.42

0.84

[0, 90, 90, 0]

550.0

350.0

1.04

1.57

337

37.92

32.06

0.85

[45, 45, 45, 45]

550.0

350.0

1.04

1.57

337

37.43

29.93

0.80

[0, 45, 45, 0]

540.0

350.0

1.32

1.54

265.0

244.72

172.71

0.71

[0, 45, 45, 0]

540.0

350.0

1.32

1.54

265.0

244.72

151.52

0.62

[0, 45, 45, 0]

540.0

350.0

1.32

1.54

265.0

244.72

155.59

0.64

[0, 45, 45, 0]

540.0

350.0

1.32

1.54

265.0

244.72

164.59

0.67

[0, 45, 45, 0]

540.0

350.0

1.32

1.54

265.0

244.72

140.20

0.57

[45, 45, 45, 45]

540.0

350.0

1.32

1.54

265.0

120.26

120.17

1.00

[45, 45, 45, 45]

540.0

350.0

1.32

1.54

265.0

120.26

116.40

0.97

[45, 45, 45, 45]

540.0

350.0

1.32

1.54

265.0

120.26

102.46

0.85

[45, 45, 45, 45]

540.0

350.0

1.32

1.54

265.0

120.26

111.17

0.92

[45, 45, 45, 45]

540.0

350.0

1.32

1.54

265.0

120.26

97.82

0.81

[45, 45, 45, 45]s

540.0

350.0

1.2

1.54

292.0

119.82

96.32

0.80

Waters [7]

Wilkins [12]

Priyardarsini [9]

Hilburger [13]

Giavotto [14]

Bisagni [8,15,16]

Stability of composite shelletype structures

395

Continued Ply layup

L (mm)

R (mm)

t (mm)

L/ R

R/t

Nper (kN)

Nexp (kN)

rexp

[45, 45, 45, 45]s

540.0

350.0

1.2

1.54

292.0

119.82

92.89

0.78

[45, 45, 45, 45]s

540.0

350.0

1.2

1.54

292.0

119.82

81.99

0.68

[90, 0, 90, 0]s

540.0

350.0

1.2

1.54

292.0

121.18

92.09

0.76

[90, 0, 90, 0]s

540.0

350.0

1.2

1.54

292.0

121.18

99.48

0.82

[90, 0, 90, 0]s

540.0

350.0

1.2

1.54

292.0

121.18

73.99

0.61

[45, 45, 45, 45]

520.0

250.0

0.5

2.08

500.0

23.65

15.35

0.65

[45, 45, 45, 45]

520.0

250.0

0.5

2.08

500.0

23.65

14.33

0.61

[45, 45, 45, 45]

520.0

250.0

0.5

2.08

500.0

23.65

14.41

0.61

[45, 45, 45, 45]

520.0

250.0

0.5

2.08

500.0

23.65

13.01

0.55

[45, 45, 45, 45]

520.0

250.0

0.5

2.08

500.0

23.65

12.75

0.54

[45, 45, 45, 45]

520.0

250.0

0.5

2.08

500.0

23.65

15.79

0.67

[20, 20, 90]

200.0

100.0

0.42

2.0

238.0

14.08

13.27

0.94

[0, 45, 45, 90]

200.0

100.0

0.578

2.0

173.0

30.00

27.31

0.91

[30, 30, 30, 30, 90, 90]

200.0

100.0

0.899

2.0

111.0

105.39

88.34

0.84

[0, 60, 60, 60, 60, 0]

200.0

100.0

1.017

2.0

98.0

121.90

98.15

0.81

Kobayashi [17]

Herakovich [1] [0, 0, 0, 0]s

355.6

75.36

0.935

4.72

81.0

101.88

55.55

0.55

[45, 45, 45, 45]s

507.2

76.58

1.118

6.62

69.0

153.16

61.59

0.40

[0, 45, 45, 90]s

508.0

75.59

1.016

6.72

74.0

216.92

136.89

0.63

[82.5, 20, 30, 82.5]

508.8

76.25

0.549

6.67

139.0

21.67

25.80

1.19

[45, 45, 45, 45]

333.4

75.87

0.660

4.39

115.0

42.06

18.18

0.43

[45, 45, 45, 45]s

508.0

75.59

1.270

6.72

60.0

155.08

54.28

0.35

[0, 45, 45, 90]s

428.6

75.54

1.016

5.67

74.0

157.30

99.91

0.64

[82.5, 20, 30, 82.5]

508.8

76.38

0.579

6.66

132.0

19.30

20.08

1.04

[0, 45, 45, 90]s

509.6

75.54

1.041

6.75

73.0

158.10

85.52

0.54

[90, 0, 0, 90]

152.4

83.31

0.49

1.83

169

20.04

18.26

0.91

[0, 90, 90, 0]

152.4

83.31

0.51

1.83

165

21.92

22.37

1.02

[20, 20, 0, 0, 40, 40]

300.0

100.0

0.814

3.0

123.0

84.47

61.78

0.73

[20, 20, 0, 0, 40, 40]

300.0

100.0

0.814

3.0

123.0

84.47

63.67

0.75

Sun [18]

Hirano [19]

Continued

396

Stability and Vibrations of Thin-Walled Composite Structures

Continued L (mm)

R (mm)

t (mm)

L/ R

R/t

Nper (kN)

Nexp (kN)

rexp

[20, 20, 0, 0, 40, 40]

300.0

100.0

0.814

3.0

123.0

84.47

61.78

0.73

[20, 20, 40, 40, 0, 0]

300.0

100.0

0.814

3.0

123.0

70.45

48.58

0.69

[20, 20, 40, 40, 0, 0]

300.0

100.0

0.814

3.0

123.0

70.45

51.24

0.73

[20, 20, 40, 40, 0, 0]

300.0

100.0

0.814

3.0

123.0

70.45

43.88

0.62

[40, 40, 20, 20, 0, 0]

300.0

100.0

0.814

3.0

123.0

41.08

33.60

0.82

[40, 40, 20, 20, 0, 0]

300.0

100.0

0.814

3.0

123.0

41.08

34.77

0.85

[40, 40, 20, 20, 0, 0]

300.0

100.0

0.814

3.0

123.0

41.08

32.68

0.80

[60, 60, 0, 0, 68, 68, 52, 52, 37, 37]

510.0

250.0

1.25

2.04

200.0

261.68

228.00

0.87

[60, 60, 0, 0, 68, 68, 52, 52, 37, 37]

510.0

250.0

1.25

2.04

200.0

261.68

221.70

0.85

[51, 51, 45, 45, 37, 37, 19, 19, 0, 0]

510.0

250.0

1.25

2.04

200.0

98.13

93.50

0.95

[51, 51, 45, 45, 37, 37, 19, 19, 0, 0]

510.0

250.0

1.25

2.04

200.0

98.13

90.20

0.92

[51, 51, 45, 45, 37, 37, 19, 19, 0, 0]

510.0

250.0

1.25

2.04

200.0

98.13

92.40

0.94

[30, 30, 90, 90, 22, 22, 38, 38, 53, 53]

510.0

250.0

1.25

2.04

200.0

265.07

278.50

1.05

[30, 30, 90, 90, 22, 22, 38, 38, 53, 53]

510.0

250.0

1.25

2.04

200.0

265.07

227.90

0.86

[30, 30, 90, 90, 22, 22, 38, 38, 53, 53]

510.0

250.0

1.25

2.04

200.0

265.07

249.70

0.94

[30, 30, 90, 90, 22, 22, 38, 38, 53, 53]

510.0

250.0

1.25

2.04

200.0

265.07

210.80

0.80

Ply layup

Meyer [20]

Stability of composite shelletype structures

397

Continued L (mm)

R (mm)

t (mm)

L/ R

R/t

Nper (kN)

Nexp (kN)

rexp

[37, 37, 52, 52, 68, 68, 0, 0, 60, 60]

510.0

250.0

1.25

2.04

200.0

219.45

212.60

0.97

[38, 38, 68, 68, 90, 90, 8, 8, 53, 53]

510.0

250.0

1.25

2.04

200.0

258.37

206.60

0.80

[38, 38, 68, 68, 90, 90, 8, 8, 53, 53]

510.0

250.0

1.25

2.04

200.0

258.37

228.20

0.88

[0, 0, 19, 19, 37, 37, 45, 45, 51, 51]

510.0

250.0

1.25

2.04

200.0

199.23

172.80

0.87

[0, 90, 0, 90]s

300.0

100.0

1.0

3.0

100.0

95.58

84.38

0.88

[0, 90, 0, 90]s

300.0

100.0

1.0

3.0

100.0

95.58

82.50

0.86

[0, 0, 0, 0, 90, 90, 90, 90]

300.0

100.0

1.0

3.0

100.0

66.67

64.65

0.97

[0, 0, 0, 0, 90, 90, 90, 90]

300.0

100.0

1.0

3.0

100.0

66.67

65.91

0.99

[20, 20, 20, 20]

300.0

100.0

0.5

3.0

200.0

16.45

12.16

0.74

[20, 20, 20, 20, 20, 20, 20, 20]

300.0

100.0

1.0

3.0

100.0

65.44

56.86

0.87

[20, 20, 20, 20, 20, 20, 20, 20]

300.0

100.0

1.0

3.0

100.0

65.44

55.29

0.84

[20, 20, 20, 20, 20, 20, 20, 20]

300.0

100.0

1.0

3.0

100.0

108.26

82.50

0.76

[45, 45, 45, 45]

300.0

100.0

0.5

3.0

200.0

21.15

13.23

0.63

[45, 45, 45, 45]

300.0

100.0

0.5

3.0

200.0

16.16

13.23

0.82

[45, 45, 45, 45, 45, 45, 45, 45]

300.0

100.0

1.0

3.0

100.0

62.97

49.01

0.78

[45, 45, 45, 45, 45, 45, 45, 45]

300.0

100.0

1.0

3.0

100.0

93.16

56.86

0.61

[45, 45, 45, 45, 45, 45, 45, 45]

300.0

100.0

1.0

3.0

100.0

93.16

56.86

0.61

[70, 70, 70, 70]s

300.0

100.0

1.0

3.0

100.0

110.36

79.42

0.72

[70, 70, 70, 70]s

300.0

100.0

1.0

3.0

100.0

110.36

74.52

0.68

[70, 70, 70, 70]s

300.0

100.0

1.0

3.0

100.0

110.36

78.23

0.71

[70, 70, 70, 70]s

300.0

100.0

1.0

3.0

100.0

110.36

78.23

0.71

Ply layup

Kasuya [21]

Continued

398

Stability and Vibrations of Thin-Walled Composite Structures

Continued Ply layup

L (mm)

R (mm)

t (mm)

L/ R

R/t

Nper (kN)

Nexp (kN)

rexp

[70, 70, 70, 70]

300.0

100.0

0.5

3.0

200.0

26.91

18.63

0.69

[70, 70, 70, 70, 70, 70, 70, 70]

300.0

100.0

1.0

3.0

100.0

70.76

58.81

0.83

[70, 70, 70, 70, 70, 70, 70, 70]

300.0

100.0

1.0

3.0

100.0

70.76

58.81

0.83

[70, 70, 70, 70]as

300.0

100.0

1.0

3.0

100.0

112.52

80.68

0.72

[70, 70, 70, 70]as

300.0

100.0

1.0

3.0

100.0

112.52

78.23

0.70

[30, 0, 30, 30, 0, 30]

366.23

400.0

0.73

0.91

548

47.0

41.0

0.87

[30, 30, 0, 0, 30, 30]

366.23

400.0

0.73

0.91

548

41.0

35.0

0.85

Khakimova [22]

Tennyson [23] [0, 45, 90, 45]as

282.7

83.90

1.12

3.37

75.0

203.39

129.83

0.64

[0, 45, 45, 90]s

287.8

83.82

0.99

3.43

85.0

138.25

124.03

0.90

[0, 0, 45, 45, 45, 45, 90, 90]

284.7

83.85

1.0

3.40

84.0

90.52

89.35

0.99

[0, 0, 45, 45, 45, 45, 0, 0]

269.2

83.82

0.94

3.21

89.0

95.48

96.04

1.01

[0, 90, 90, 0]

282.7

83.57

0.46

3.38

183.0

17.45

15.21

0.87

[90, 0, 0, 90]

267.7

83.57

0.43

3.20

192.0

15.14

17.18

1.13

[24, 24, 41, 41]

500.0

251.13

0.523

1.99

480.17

39.0

25.38

0.65

Kalnis [6] [24, L24, 41, L41]

500.0

251.8

0.523

1.99

481.45

39.0

25.64

0.66

[0, 45]a

300

150.4

0.261

1.99

576.2

4.36

1.6

0.37

[0, 45]

300

150.4

0.261

1.99

576.2

4.36

2.44

0.56

[0, 60, L60]

300.0

150.52

0.392

1.99

383.98

13.66

6.22

0.46

[0, 60, 60]

300.0

150.61

0.392

1.99

384.21

13.66

6.34

0.46

[0, 60, 60]

300.0

150.22

0.392

2.00

383.21

13.66

7.28

0.53

[0, 45, 45]

300.0

150.66

0.392

1.99

384.34

17.26

8.71

0.50

[0, 45, 45]

300.0

150.76

0.392

1.99

384.59

17.26

8.50

0.49

[0, 45, 45]

300.0

150.73

0.392

1.99

384.52

17.26

9.63

0.56

[24, 24, 41, 41]

300.0

151.32

0.523

1.98

289.33

39.48

28.96

0.73

[24, 24, 41, 41]

300.0

150.76

0.523

1.99

288.26

39.48

26.85

0.68

Stability of composite shelletype structures

399

Continued Ply layup

L (mm)

R (mm)

t (mm)

L/ R

R/t

Nper (kN)

Nexp (kN)

rexp

[24, 24, 41, 41]

300.0

151.16

0.523

1.98

289.02

39.48

21.10

0.53

[24, 24, 41, 41]

300.0

151.01

0.523

1.99

288.74

39.48

25.47

0.65

[51, 51, 90, 90, 40, 40]

510.0

250.0

0.75

2.04

333.0

80.65

82.70

1.02

[39, 39, 0, 0, 50, 50]

510.0

250.0

0.75

2.04

333.0

71.28

69.27

0.97

[49, 49, 36, 36, 0, 0]

510.0

250.0

0.75

2.04

333.0

35.62

34.40

0.97

Geier [24]

a

This shell was excluded from the analysis procedures within this paper because of the questionable low buckling load.

400

Stability and Vibrations of Thin-Walled Composite Structures

7.7 Design and manufacture of cones Khakimova Regina INVENT GmbH, Braunschweig, Germany

List of definitions “As-designed” structural model: reference numerical description of the physical structure without any manufacturing imperfections, deviations, mistakes, and repairs. For composite structures, the “as-designed” structure includes the description of the layup and all kinds of details as they are planned to be on the physical object. “As-to-be-built” structural model: “as-designed” structural model enriched by additional data to describe the reference manufacturing process and any physical elements that are added to the “as-designed” structure for manufacturing reasons under standard conditions. For composite structures, this shall include the technology to lay down the plies, the machine programming and also any additional details that are added to the as-designed structure to ensure manufacturability and/or feasibility. “As-built” structural model: numerical representation of the real (physical) structure. It numerically gathers the “as-designed” model, the “as-to-be-built” model, and any numerical information captured during the manufacture and of interest to describe the specific history of this structure manufacturing process. This can include manufacturing parameters recorded for this specific part (autoclave curve), NDT information captured for this part (C-scan with specific defect distribution to be considered for simulation, geometric measurement using photogrammetry, laser shape measurement, etc.).

7.7.1

Introduction

Thin-walled conical structures are widely used in aerospace, offshore, civil, and other engineering fields. Parts of space launcher transport systems are an example of the

Figure 7.7.1 (a) ECA adapter 3936 and (b) equipment bay structure [1].

Stability of composite shelletype structures

401

application of conical shells (Fig. 7.7.1). Conical shells carry heavy payloads in space launch vehicles and are therefore subjected to axial compression, which makes buckling the limiting design constraint. Moreover, such structures are usually imperfection sensitive (geometry, boundary conditions, load introduction, thickness, etc.). Thus it is very important to consider a buckling criterion accounting for imperfections in the design stage of such structures. Unfortunately, despite its wide application, the mechanical behavior of conical shells has been less investigated than that of cylinders. Fiber-reinforced composites in the form of laminates made from single unidirectional prepreg layers are widely used in the space industry. Generally, in the design of composite aerospace structures not only stiffness and strength but also the manufacturing process needs to be taken into account. The reason for this is the fact that different manufacturing methods with their individual effects on the laminate quality have an impact on the mechanical characteristics of composite structures that has to be reflected in the design in addition to simple checks for manufacturability. In this context, particularly truncated carbon-fiber-reinforced plastic (CFRP) cones need a special consideration, as unlike composite cylinders, their manufacturing process is challenging because of the changing radius of the structure (discussed in more detail in Section 7.7.5). Although composite parts made by the latest production technologies, such as ATL and AFP, are used in the space launch vehicles, they are still based on guidelines that were developed for isotropic materials, NASA SP-8007 for cylinders and NASA SP8019 for cones, dated from the late 1960s. One problem is that these guidelines do not appropriately take into account the material characteristics of the composite structures. Besides, for composite cones an impact on the load-carrying characteristics depending on its manufacturing method is expected. Therefore, new design guidelines for composite truncated cones need to take into account the imperfection sensitivity and manufacturing.

7.7.2

Conical shell stability

A truncated cone is a shell of revolution and is similar to a cylindrical shell in its structural behavior [2]. Consider a truncated conical shell with the smallest radius Rtop, the biggest radius Rbot, height H, uniform wall thickness t, semivertex angle a, and the slant length L (Fig. 7.7.2). An analytical formula was derived by Seide [3,4] for the critical buckling load of truncated isotropic cones subjected to axial compression. The formula is similar to the classical buckling load for cylinders but here the influence of the semivertex angle a is taken into account. 2pEt 2 cos2 a Pcrit ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Pcyl cos2 a 3ð1  n2 Þ

(7.7.1)

Many authors consider cones as “degenerated” cylinders. The buckling behavior of a truncated cone is very similar to a cylinder, although cones under axial compression

402

Stability and Vibrations of Thin-Walled Composite Structures

P Rtop t

L

H α

Rbot

Figure 7.7.2 Geometry of a truncated cone.

are believed to be less imperfection sensitive than cylinders. Some authors concluded that the buckling behavior of cones and its imperfection sensitivity is dependent on the semivertex angle a [5,6]. Both the geometry of a cylinder and a cone with the corresponding theoretical buckling load equations for isotropic material are depicted in Fig. 7.7.3. It is important to note that the theoretical buckling load of a cone (Eq. 7.7.1) and a cylinder does not take into account the slenderness and the height of the conical structure. Fig. 7.7.4 shows the dependency of the theoretical buckling load (Eq. 7.7.1) on the semivertex angle a.

(a)

(b)

Rtop t

H

H

t

Rbot

R

Pcyl ═

σcyl ═

α

2πEt2 3 (1 – ν2) P 2πrt

Pcone ═

σcone ═

2πEt2 cos2α 3 (1 – ν2) P 2πrt cos α

Figure 7.7.3 Theoretical buckling load and compressive stress of (a) cylinder and (b) cone.

Stability of composite shelletype structures

403

300

Buckling load (kN)

250

200

150

100

50

0 0

10

20

30 40 50 Semi-vertex angle (°)

60

70

Figure 7.7.4 Theoretical buckling load depending on the semivertex angle.

As the conical radius changes along the axis, the scone is decreasing from the top to the bottom, whereas in the case with cylinders the membrane stress scyl stays constant along the axis because its radius does not change. The buckling of cones under axial compression initiates at the smaller end where the stresses are highest and then propagates to the middle of the cone [7]. In general, the structural behavior of both structures is very similar. Cylinders and cones under pure axial compression when the loading is applied along the shell theoretically exhibit membrane stresses only. To prevent rigid-body motions and to reduce the effect of the Poisson ratio, which causes tension in the shell’s circumferential direction under axial compression, boundary conditions at both ends have to be introduced. The boundary conditions influence the buckling behavior of cylinders depending on the shell’s height [8]. It might be expected that short shells are affected significantly by the end conditions because of induced local stresses in the prebuckling regime associated with the restraint radial displacements at the boundaries. Moreover, the buckling displacements in very short shells are restrained by the boundary conditions. In contrast, longer shells buckle relatively independent of their length and boundary conditions. As stated in Ref. [8], short cylinders have rather stable postbuckling response and are not as much as sensitive to geometric imperfections as long shells. This is an important remark, as the structural behavior of cones and cylinders is similar and in most cases cones that are used in practice are short (see Fig. 7.7.1 as example). Chryssanthopoulos and Spagnoli [9] investigated the influence of the radial edge constraint on the stability response of stiffened cones under axial compression. The “cylinder” boundary conditions were compared with the “ring” boundary conditions (see Fig. 7.7.5). The two boundary conditions cause different deformed prebuckling

404

Stability and Vibrations of Thin-Walled Composite Structures

∆V ∆H = 0 ∆H

Cylindrical shells W=0 V=0

Conical shells with ‘cylinder’ boundary conditions W=0 V=0

Conical shells with ‘ring’ boundary conditions U sin α–W cos α = 0 V=0

Figure 7.7.5 Boundary conditions for cylinders and cones [9].

states that become identical when the semivertex angle is zero (the cone becomes a cylinder). As explained in Ref. [9], cones with the ring conditions exhibit additional radial forces that cause inward bending at the smaller upper end and outward bending at the bigger bottom end of the cone, which are proportional to tan a. It was found out that the buckling loads obtained with the cylinder and ring boundary conditions are virtually identical for the studied cone with L/Rtop ¼ 4.66, Rtop/t ¼ 129.4, and a ¼ 15 degrees. However, the initial axial stiffness with the ring conditions is 15% higher than that with the cylinder condition for the same cone. The difference between the buckling loads obtained with the cylinder and ring boundary conditions might become more significant when considering geometric parameters different from the studied ones. It is recommended in Ref. [9] to use the ring conditions in deriving design recommendations because it is a better representation of the real practical boundary conditions. According to Ref. [7], the cylinder boundary conditions represent bulkheads that are very rigid in the shell’s radial direction and flexible in the direction of the slant length, whereas the ring boundary conditions mimic bulkheads that are rigid in a plane perpendicular to the axis and flexible in the axis direction. The latter case is more representative in reality (industrial use cases with stiffening rings on both ends and buckling test setup configurations). When the load is applied parallel to the axis (ring boundary conditions), the cone exhibits an additional horizontal load component Fr, as shown in Fig. 7.7.6, which is usually carried by stiffening rings in real applications [10]: Fr ¼

P tan a 2pRtop

(7.7.2)

Thus the ring at the top is loaded in compression on the plane perpendicular to the shell’s axis, whereas the bottom one is in tension, which has to be taken into account when designing the stiffening rings.

Stability of composite shelletype structures

405

F

P Rtop

Figure 7.7.6 Load components in a cone with “ring” boundary conditions.

Fs

FΓ

H

t

α

Rbot

Several semianalytical methods for the buckling of truncated cones were developed. Jabareen and Sheinman [11,12] studied the nonlinear buckling behavior of cones, in which the effect of nonlinear prebuckling deformations and initial geometric imperfections in the form of eigenmode shape was taken into account. Shadmehri [13,14] proposed a semianalytical approach (the Ritz method) for linear buckling predictions of composite cones. Castro [15,16] developed a semianalytical procedure also based on the Ritz method for linear and nonlinear buckling of laminated cones. He considered different kinds of geometric and load introduction imperfections in his investigations. The fact that laminated cones have variable shell coordinate dependent stiffness due to the changing thickness and/or fiber angle was not considered in these works.

7.7.3

Lower-bound design for truncated cones

This part considers existing design recommendations that are delivered in the form of design buckling loads, stresses, and imperfection-sensitive KDFs. The reference load for the KDF can be determined with a linear buckling analysis or with any theoretical formula valid for the considered structure. The NASA SP-8019 [17] is a design guideline that was developed and published in 1968 by NASA especially for cones used in the aerospace industry. However, for axially compressed cones, it delivers a single value of the KDF equal to 0.33 for all configurations with the range of the semivertex angle ɑ between 10 and 75 degrees. For truncated cones with the semivertex angle less than 10 degrees the NASA SP-8007 employing the equivalent cylinder concept, which is explained later in the section, is to be used. In 1998 NASA delivered a report [18] that reviewed the existing NASA design codes including the SP-8019. It was discussed that even though the existing NASA guidelines are reliable, they need to be improved and extended because of their conservatism.

406

Stability and Vibrations of Thin-Walled Composite Structures

It is common to use the equivalent cylinder approach for cones employing the existing design methods and formulas for cylinders under axial compression. The equivalent cylinder would have the same wall thickness t as the cone, a length equal to the slant length of the cone L, and a radius equal to the average radius Rm of the cone [3]. Lackman and Penzien in Ref. [19], however, recommended using the big radius of the cone curvature for the equivalent cylinder based on the buckling experiments. Findings of Seide [20] and Weingarten et al. [21] contradict with those in Ref. [19]. After collecting and comparing all available experimental results, Refs. [20,21] show that replacing the cylindrical radius R with the small radius of conical curvature Rtop gives a better agreement with the equivalent cylinder results. As a result, Ref. [21] delivered an empirical dependency C for calculating the buckling load based on the lower-bound curve for cylinders: ( "     #)   Rtop 2 t 1 Rtop 1=2 C ¼ 0:606  0:546 1  exp  þ 0:9 16 Rtop t L (7.7.3) It is worth saying that Eq. (7.7.3) is a function of two variables: top radius-to-thickness ratio Rtop/t and top radius-to-slant length ratio Rtop/L. In Ref. [22,23] an empirical correction factor “g,” or in other words a KDF, which also depends on Rtop/t , is proposed as a design recommendation: Rtop 0:83 g ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for  212 t Rtop 1þ 100t (7.7.4) Rtop 0:7 > 212 g ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for t Rtop 0:1 þ 100t The NASA SP-8007 that takes into account orthotropic properties can be assumed valid for composite cones. In this approach the buckling problem of truncated cones can be transformed to an equivalent cylinder, as sketched in Fig. 7.7.7. Basically, the real cylinder radius is replaced with the equivalent radius in the design load calculations. To calculate the KDF “g,” an empirical correction factor, which takes into account disparities between test and theoretical predictions is employed:

g ¼ 1  0:902 1  e4

(7.7.5)

where 1 4¼ 16

sffiffiffiffiffiffiffi Req teq

(7.7.6)

Stability of composite shelletype structures

Rm

407

α Req

Figure 7.7.7 Cone converted to equivalent cylinder.

The equivalent thickness teq depends on the shell’s bending and extensional stiffness terms of the ABD matrix [24]: teq

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 D11 D22 ¼ 3:4689 A11 A22

(7.7.7)

The equivalent radius of a cone (Fig. 7.7.7) is calculated according to the following formula: Req ¼

Rm cosðaÞ

(7.7.8)

where Rm is the average radius of a cone Rm ¼

Rtop þ Rbot 2

(7.7.9)

where Rtop and Rbot are the radius of the top edge and the bottom edge, respectively. Finally, the design load is computed using the following formula: P ¼ g$Pcr

(7.7.10)

where Pcr is the reference buckling load that can be calculated from the linear buckling analysis. The formula (Eq. 7.7.5) for “g” was originally proposed by Seide in 1960 and improved by Weingarten in 1965, to the form presented in NASA SP-8007. However, the NASA SP-8007 does not consider all the composite material properties appropriately.

408

Stability and Vibrations of Thin-Walled Composite Structures

The equivalent cylinder philosophy was adopted in several design codes [25e27]. These design codes were created mainly for civil and offshore engineering and deal with metallic structures. Some authors argue that it is not appropriate to use the equivalent cylinder approach in the design of cones, as their imperfection sensitivity is different (for example, Ref. [28]). If the equivalent cylinder approach is still employed, the limiting range of semivertex angles in which the design methodology is valid must be specified. For instance, in Ref. [27] the limiting semivertex angle is 65 degrees beyond which the equivalent cylinder assumption is not valid. Finzi and Poggi [29] proposed an improved design procedure for truncated cones under axial compression. They concluded that the design KDFs for cylindrical shells can be adopted and used for conical shells employing the ratio Rtop/cos a. Their recommendations are valid for cones with a semivertex angle less than 50 degrees. In Ref. [28] the authors proposed an improvement of the existing design code ECCS [27] for stiffened and unstiffened metallic cones. Although the equivalent cylinder approach is used for design purposes of truncated cones, especially in the civil and offshore industry, there was no agreement on which radius (small end, big end, or middle) should be employed. The main characteristics of each radius are given in the following: • • •

the small radius has the most highly stressed region the middle radius has the biggest out-of-plane displacements the largest radius has the greatest slenderness ratio, which makes this region sensitive to elastic instabilities.

The most recent shell buckling design recommendations for thin-walled space structures, the so-called European Convention for Constructional Steelwork (ECSS) [30], was issued in 2010. For conical shells in the space industry, the NASA SP-8019 and, alternatively, the KDFs using Eq. (7.7.4) is proposed.

7.7.4

Experiments of conical shell structures

This section revises and summarizes buckling experiments of truncated cones under axial compression. It needs to be noted that there have been considerably less buckling experiments performed on cones than on cylinders. The number of buckling experiments on composite cones is even lower. All the sources on the buckling experiments of cones subjected to axial compression are summarized in chronicle order in Table 7.7.1. The range of the semivertex angle a of all axial compression experiments is quite wide, ranging from 3 to 75 degrees. The values of the Rtop/t ratio that are given in Table 7.7.1 are rounded. The slenderness range is also quite wide and covers both civil engineering and aerospace applications. In the case of Refs. [31,32] the Rtop/ttop is given because the thickness values are given for the top edge and the bottom edge. The thickness in Refs. [31,32] is significantly larger close to the top edge than to the bottom edge. The first buckling experiments on cones were carried out by Lackman and Penzien [19]. They observed that the postbuckling of the test articles has a diamond-shaped pattern with two rows of buckles, similar to cylinders. Thick-walled cones have typically

Stability of composite shelletype structures

Table 7.7.1

409

Buckling experiments of cones under axial compression

Year

References

Material

Rtop/t (rounded)

a (degrees)

1961

[19]

Nickel

170e738

20e40

1962

[40]

Mylar

392e1360

10e30

1965

[21]

Mylar, steel

105e2300

10e75

1967

[41]

Aluminum

1968

[42]

Copper

180e1110

5e25

1977

[43]

Steel, aluminum

14e18

11

1980

[37]

Mylar

e

3e60

1983

[38]

Aluminum

4e40

5e10

1984

[44]

Steel

6e72.5

5e10

1986

[45]

PVC

2e23

5e14.35

1987

[39]

Epoxy

105e230

15

1990

[34]

Steel

33e100

15e30

1997

[31]

Aluminum

5e19

16.5e65

1999

[35]

Aluminum

29

20

2001

[46]

Steel

111e143

15e30

2003

[47]

Aluminum

e

2005

[32]

Aluminum

5e18

44.51e67.1

2006

[48]

Aluminum

30e70

16e29

2011

[33]

Steel

16.3

25.56

axisymmetric collapse mechanism, the so-called elephant-foot-type deformation that was observed in Ref. [33]. Theoretically, buckling of an ideal cone starts close to the smaller edge. This was confirmed by performing buckling experiments [34e36]. Esslinger and Geier [37]concluded that buckling starts in the upper edge for steep cones and in the bottom for flat cones. However, some of the experiments [38,39] showed that buckling can start close to the larger diameter of the cone even in the case of steep cones [38]. This can be explained by the imperfections that are predominant in the vicinity of the larger edge of the cone. A small amount of buckling tests has been done on composite cones, so all the available experiments with different loading conditions are reviewed and summarized in Table 7.7.2. A typical postbuckling mode of a laminated cone under axial compression is depicted in Fig. 7.7.8. Khakimova et al. [53,54] performed buckling experiments on axially compressed, unstiffened CFRP truncated cones needed for validation of high-fidelity numerical

410

Stability and Vibrations of Thin-Walled Composite Structures

Table 7.7.2

Buckling experiments of composite cones

Year

Reference

Material

Fabrication method

a (degrees)

Type of loading

1969

[49]

Sandwich

Tape laying

5

Bending, torsion

1999

[50]

CFRP, GFRP

Filament winding

9

Axial compression

2002

[51]

GFRP

Filament winding

10e15

Axial compression

2012

[13]

Carbon/ Peek

AFP

1e3

Bending

2015

[52]

Sandwich

Tape laying

45

Axial compression

2016

[53]

CFRP

Tape laying

35

Axial compression

GFRP, glass-fiber-reinforced plastic.

models and a new design concept. Three geometrically identical cones named K01, K06, and K08 with different layup were designed, manufactured, and tested. Before testing, the thickness of the cones was measured with ultrasonic inspection and the geometry was measured with photogrammetry. During testing the ARAMIS measurement system was employed, strain gage readings and load-shortening data were extracted. The measured displacement field of cone K06 at several points along the load-shortening curve is shown in Fig. 7.7.9 where automatic scaling for each frame is utilized.

Figure 7.7.8 Typical buckling mode of laminated cone [50].

Stability of composite shelletype structures

411

45 40

Reaction load (kN)

35 30 25 20 15 10 5 0 0.00

0.05

0.10 0.15 0.20 Axial shortening (mm)

0.25

0.30

Figure 7.7.9 Load-shortening curve of cone K06, with the displacement field measured from ARAMIS [53].

Fig. 7.7.10 depicts the test KDF values of the three cones along with the design lower-bound curves considered in Section 7.7.3. As the latest design recommendation for space structures under buckling [30] proposes using two ways of calculation of the KDF for conical structures, both options are considered here and the values are plotted in Fig. 7.7.10. The NASA SP-8019 and the ECCS KDF are too conservative for the test results delivered. Section 7.7.5 summarizes and discusses the manufacturing techniques that were used to produce the cones listed in Table 7.7.2. 1.0 ECCS NASA SP-8019 K01 K06 K08

0.8

KDF

0.6

0.4

0.2

0.0 0

200

400

600

800

1000

Rtop/t

Figure 7.7.10 KDF values of composite cones tested at DLR plotted with the current design recommendations for cones.

412

7.7.5

Stability and Vibrations of Thin-Walled Composite Structures

Manufacturing of laminated cones

There are a number of manufacturing methods for laminated composite structures. The material and manufacturing method of composite structures are usually chosen according to its structural requirements, costs, design, etc. It is important to take into account the manufacturing method limits or special features of laminated structures because they influence the design of the specimen. Manufacturing of laminated cones is especially challenging because the shell’s radius changes along the axis, meaning that it is impossible to obtain homogeneous thickness and constant fiber angle along the shell coordinates. Moreover, the fiber path of the laminated cone can vary depending on the manufacturing method and parameters (see Fig. 7.7.11). The simplest fiber path that is used mostly in manufacturing is geodesic (green line in Fig. 7.7.11). This path is laid down on a surface in a natural way without steering, resulting in the shortest distance between two points on a surface. Note that for composite cylinders, the geodesic, constant angle, and constant curvature (zero curvature) paths are identical. In Section 7.7.5.1, suitable manufacturing methods for laminated composite cones are reviewed based on the information taken from Ref. [56]. It enables the reader to understand the basic principles, advantages, and disadvantages of the manufacturing methods for cones. In Section 7.7.5.2, examples of composite-manufactured cones and their production ways are reviewed. Finally, in Section 7.7.5.3, existing studies on the buckling analysis of laminated cones that consider the manufacturing effects in the form of the thickness and/or fiber angle variations are discussed.

7.7.5.1 7.7.5.1.1

Manufacturing methods for laminated cones Filament winding

Filament winding is an automated process of applying fibers onto a special tool. The process is usually used for manufacturing shells of revolution, such as pressure vessels, pipelines, and other parts. The fibers are wound onto the tool and then the resin is injected to create matrix in the laminate. The part is then cured in an autoclave. Constant curvature Constant angle Geodesic

Constant curvature Constant angle Geodesic

Figure 7.7.11 Examples of different fiber paths on the (a) cone and (b) unfolded cone [55].

Stability of composite shelletype structures

413

When manufacturing cones by the filament winding method, the laid fibers follow the geodesic path (see Fig. 7.7.11). Special characteristics are that the method can be applied only to convex shapes and has a poor external finishing. Fibers cannot be easily laid exactly along the length of the component. Another feature of the method is that the fibers are placed continuously along the length that does not allow it to drop, causing nonuniform thicknesses if the shape has a changing radius, which is the case for conical shells.

7.7.5.1.2

Automated fiber placement

Automated fiber placement (AFP) has a machine configuration similar to filament winding. Here a special robotic placement head is employed to dispense, clamp, cut, and restart each tow during placement. Thermoset prepreg materials, thermoplastic materials, or dry fibers in the form of tows can be used to layup concave and convex surfaces. Several tows (from 1 to 32) with possible tow widths 3.2, 6.4, and 12.7 mm are placed in bands of parallel fibers forming a course. This process is widely used in the aerospace industry, particularly in the production process of some Ariane 5 parts [1]. It is possible to steer the fiber directions and drop the fibers during the AFP process. Apart from the geodesic path, one can lay fibers using the constant curvature and constant-angle fiber paths, as shown in Fig. 7.7.11, when steering the fiber directions. The AFP is prone to generate unavoidable gaps and overlaps already in the “as-to-be built” structural design phase [57]. As soon as the surface has some curvature, material strips in AFP cannot be exactly parallel to previous ones because the surface area changes as the course traverses across it. For example, on a shape that is smaller at one end and bigger at the other, courses have to converge at the smaller end and diverge at the bigger end. As the courses converge, they begin to overlap. If two courses completely overlap then the ply at this location would have two layers instead of one, which is undesirable. The curved surface of a cylinder is an exception because it has a constant radius along the axis; the problem described earlier does not occur in this case. Special characteristics are that there is a certain minimum curvature and minimum length of the course. These limitations need to be considered already at early structural design stages. High requirements for placement tolerances especially when using narrow tow widths make the method application challenging.

7.7.5.1.3

Tape laying

Tape laying is a manufacturing process in which sheets rather than tows are applied onto the tooling. This process is widely used for large parts that have a low curvature. The sheets can be made of prepreg or woven fabric. Unlike the AFP process, by tape laying, it is possible to use not only rectangular ply patches or pieces but also customized shapes including curved pieces or any irregular forms. By controlling the shape, it is possible to obtain continuous laminates with no gaps and overlaps in the “as-to-bebuilt” structure. When the laying process of the sheets onto the mold is done manually, the manufacturing method is called hand layup. The hand layup process can be

414

Stability and Vibrations of Thin-Walled Composite Structures

considered as semiautomated because it is possible to cut the prepreg into required shape using an automated cutter. The limit of the hand layup process is that the speed and the quality of the laying process itself highly depend on the skills of the worker. Moreover, the process is hardly repeatable, which can be improved by automation. The automated tape laying (ATL) process is very similar to the hand layup method with the difference that the process of prepreg laying in ATL is done automatically by a robotically controlled head. This manufacturing method is advantageous over the other ones because of its excellent material layup rates. Sheets of prepreg are placed onto tooling and then the part is cured. The tape widths that are used in ATL are usually 75, 150, and 300 mm. The length is regulated automatically depending on how the process is programmed. Three material delivery methods are used in the ATL technology: “single-phase,” “two-phase,” and “dual-phase” [58]. In the single-phase technology, which is mostly used in the ATL process, a roll of prepreg tape material is loaded into the delivery head. In the delivery head, the courses are cut to shape and size (length) and are placed on the tool surface with compaction pressure. The conventional shape of the prepreg patches in the “single-phase” material delivery method is rectangular and it is prone to generate unavoidable gaps and overlaps [57]. In the “twophase” technology the material is cut and shaped on an offline cutting machine (Fig. 7.7.12). The cassette with the precut tape pieces is loaded into a delivery head and then applied onto the mold. The “dual-phase” material delivery combines both the aforementioned methods. The latest “dual-phase” ATL machines, such as Forest-Line ATLAS [59], are able to work with specific irregular prepreg pieces for laying double-curvature complex parts such as wing skins. Forest-Line ATLAS machine (Fig. 7.7.13) has a double head that allows working with the conventional rectangular tapes as well as offline precut customized irregular prepreg pieces, thus allowing a part to be laid using precut pieces of material and material from the bulk rolls of tape. This two-phased process considerably reduces scrap and, therefore, the material costs. Besides, the layup time and the labor requirement are reduced (although

Spool of prepreg tape materials

Completed tape courses are respooled on a “cassette”

Figure 7.7.12 Offline tape cutting system for two-phase ATL (Forest-Line “ACCESS” cutting machine) [58].

Stability of composite shelletype structures

Singlephase side of the delivery head (12″ wide tape)

415

Machin z-axis

Two-phase side of the delivery head 6″ wide tape)

Figure 7.7.13 ATL “dual-phase” head of Forest-line ATLAS [58].

it is questionable if the latter aspect is an advantage). A limitation in some cases is the fact that the fibers are not able to be steered, meaning that the fiber orientation follows the geodesic path (see Fig. 7.7.11). Using the two-phase tape laying process, dualphase or hand layup, one can avoid gaps and overlaps by cutting the prepreg into pieces with the right shape, referred here as ply pieces. To sum up, it is important to take into account the manufacturing methods of laminated truncated conical structures because they influence the design of the specimen. Unlike cylinders, laminated composite cones are more challenging to design and manufacture because of the changing radius along the axis.

7.7.5.2

Examples of manufactured laminated cones

A short summary and a few comments on the manufacturing of the test composite cones from Table 7.7.2 together with a manufactured cone by tape laying that was not tested [60] are given.

7.7.5.2.1

Filament winding

The conical shells in Ref. [50] were manufactured by filament winding using the horizontal helical winding machine and a stainless steel conical mandrel. The fabrication method, however, delivered specimens with quite a significant thickness deviation. As it is mentioned in Ref. [50], the thickness at the smaller end was thicker by about 20% than in the bigger end in some structures. The glass-fiber-reinforced plastic (GFRP) conical parts were produced by wet filament winding in Ref. [51]. However, it is known that structural geometries that can be produced by filament winding are basically restricted to convex nearly cylindrical shapes. No presence of the fiber angle or thickness deviations and its influence on the crushing behavior of the structure were reported [51].

416

7.7.5.2.2

Stability and Vibrations of Thin-Walled Composite Structures

Automated fiber placement

In Ref. [13] the conical shells that were produced by six-axis gantry-type AFP machine equipped with the thermoplastic fiber placing head were not carefully inspected and measured. Most likely, in that case the problem of the thickness and fiber angle deviation is not faced because of a negligible semivertex angle that makes the structure close to a cylinder. However, normally the AFP causes gaps and/or overlaps that cause thickness deviation structures with a changing radius (like truncated cones).

7.7.5.2.3

Tape laying

Bert et al. [49,61] performed buckling tests of sandwich cylindrical and conical sandwich shells made of aluminum honeycomb core and GFRP face sheets. The face sheets were manufactured by the tape laying method and the manufacturing process is described in Ref. [61]. There were no difficulties met and discussed due to the overlapping/gaps and fiber angle deviation because the semivertex angle of the sandwich cone was quite small (a ¼ 5 degrees), which makes the structures close to cylinders. The ply piece topology design is not considered in the manufacture of cylinders because the thickness and fiber direction and thus the stiffness in the structure do not vary locally with the shell coordinates (Fig. 7.7.14(a)). However, the usage of the conventional rectangular ply pieces in the case of cones causes gaps and/or overlaps that result in the thickness deviation (Fig. 7.7.14(b)). The work of Ahmed et al. [60] deals with manufacturing of a composite engine thrust frame cone, which is a laminated stiffened conical structure. The cone is made of eight individual segments (panels), which is shown in Fig. 7.7.15. The study suggests that for the manufacture of composite laminated cones, the proposed method based on the ATL process is more cost-effective compared to the AFP. The production method, however, delivers cones with the thickness deviation already in the “as-to-bebuilt” design step because the “single-phase” ATL with conventional rectangular tapes

Overlap

Gap

Figure 7.7.14 Conventional tape laying process in (a) cylinders and (b) cones (rectangular ply pieces).

Stability of composite shelletype structures

417

Top

Foot

Figure 7.7.15 Tapes laid in the radial direction (a) without and (b) with a ply drop. Green, no overlap; yellow, double overlap; and red, triple overlap [60].

is used. The tape width of 150 mm with a ply drop close to the small end was used to produce the conical panels. The method improved the overall thickness homogeneity of the structure, meaning that a double overlap (Fig. 7.7.15(a)) instead of the original triple overlap (Fig. 7.7.15(b)) was achieved in the structure. The composite cones in Ref. [60] are neither nondestructively tested nor mechanically tested. Therefore, it is difficult to assess the method suggested in Ref. [60]. Within DESICOS [62], two sandwich cones using tape laying were manufactured by Griphus [63]. The layup of the design sandwich structure that originally included the 60 -degree fiber layers had to be changed to the 40-degree fiber layers by the manufacturer, as the first fiber direction was not continuous along the meridian. Fig. 7.7.16(a) demonstrates the 60-degree geodesic fiber path together with the 11-degree fiber path showing that the first one does not reach both ends of the cone.

(a)

(b)

Figure 7.7.16 Sandwich cone manufactured by tape laying in DESICOS: (a) continuity fiber problem and (b) suggested tailored ply. Courtesy of Griphus.

418

Stability and Vibrations of Thin-Walled Composite Structures

A feasibility study that determines if a single fiber is continuous for the given geometry of tape laid cones is therefore needed. This is essential because a ply piece (patch) consists of parallel single fibers that follow the geodesic path. The nonzero layers were laid without gaps but with unavoidable overlaps with two ply drops along the structure’s meridian so that it resulted in three levels of ply pieces (Fig. 7.7.16(b)). As there was no ply topology construction method available, it was not possible to control the real fiber angle and thickness distribution during the process, which would allow optimizing the fiber and thickness deviation values. Moreover, the manufacturing information was not included in the numerical model of the conical structure, which resulted in the significant difference between the buckling loads of the built cone and the designed cone. Khakimova et al. [64] developed a ply topology construction method for conical shells manufactured by tape laying. The proposed method delivers laminates without gaps and overlaps in the “as-to-be-built” laminate by cutting the prepreg into pieces with the right shape, referred here as ply pieces. The “continuous fiber” or, in other words, the feasibility criterion is defined in their work as a single fiber that has to reach both ends of the conical structure (like the 11-degree fiber direction in Fig. 7.7.16). The ply topology method proposes the use of trapezoidal ply pieces; the quality of these patches can be assessed with a set of evaluation parameters. The proposed ply piece is continuous, meaning that no drops along the meridian direction of the cone are allowed (unlike the ply topology shown in Fig. 7.7.15(b)). The ply topology parameters can be optimized to control and minimize the fiber-angle deviation and thickness imperfections. With the proposed method, it is possible that the “as-to-be-built” structure is close to the nominal “as-designed” model. This can be achieved by minimizing the fiber-angle deviation in the model. The benefit of this is that the structural behavior of the “as-designed” and “as-to-be-built” models does not differ much from each other. Fig. 7.7.17 shows the laminating process with the prepreg pieces of the ply with 30- and 0-degree fiber orientations.

(a)

(b)

Figure 7.7.17 Ply with (a) 30-degree fiber direction and (b) 0-degree fiber direction [64].

Stability of composite shelletype structures

7.7.5.3

419

Consideration of manufacturing method effects in the analysis of cones

The analytical and semianalytical references related to the stability behavior of laminated cones from Section 7.7.2 assume constant stiffness coefficients with changing shell coordinates. However, it is known that the stiffness coefficients of truncated cones depend on the shell coordinates because of unavoidable thickness and/or fiber orientation variation [65]. Laminated cones, due to their geometry, cannot have constant thickness and constant fiber-angle paths at the same time. Moreover, Baruch et al. [66] shows that the fiber-angle deviation on a conical structure depends on the chosen fiber path. Thus this fact needs to be taken into account in the buckling analysis and design procedures of the “as-to-be-built” laminated cones. The local fiber-angle or angle deviation problem corresponding to each of the manufacturing methods (Section 7.7.5), found in the literature, is discussed. For consistency a unified nomenclature is used in this section, although it might be different from the original sources.

7.7.5.3.1

Filament winding

Cones manufactured by filament winding have variable laminate stiffness along the cone’s length as shown by Baruch et al. [66]. They were the first who took into account the variation of stiffness coefficients of laminated cones manufactured by filament winding. Zhang [67,68] investigated initial postbuckling and imperfection sensitivity of anisotropic cones. The study also included modeling of variable-stiffness coefficients due to the thickness variation. Goldfeld et al. [69e71] investigated the buckling behavior of cones manufactured by filament winding. They assumed that the fiberangle orientation changes along with the geodesic path and described the coordinatedependent stiffness. It was shown that consideration of the constant (nominal) stiffness properties of laminated cones leads to invalid results [69]. In Ref. [72], it was concluded that the bigger the semivertex angle a, the larger the fiber inclination and the thickness deviation. When the semivertex angle equals to zero (cylindrical case), no fiber and thickness deviation take place in the laminated structure. Besides, optimization study of laminate configuration for minimum weight subjected to the buckling load constraint has been carried out [70]. Variable ply angle along the axial coordinate (constant in the circumferential direction), the ply angle being larger at the small end, is defined as qðsÞ ¼ arcsin

s  qnom $sinðqnom Þ ; s

(7.7.11)

Where sqnom is the nominal starting position at the top of the cone, qnom is the nominal local fiber angle at sqnom , and q(s) is the local fiber angle at s (see Fig. 7.7.18). The work, however, was validated neither by manufacture nor by buckling experiments.

7.7.5.3.2

Automated fiber placement

Blom [55] devoted her work to the design and manufacture of cylindrical and conical structures fabricated by the AFP. The method however delivers unavoidable gaps or

420

Stability and Vibrations of Thin-Walled Composite Structures

(a) α

α

(b) s z r

sθnom

ζ

ψ

s

φ η

Figure 7.7.18 Parameters of cone’s geometry and unfolded cone: (a) making a cut in the cone and (b) coordinate systems of the unfolded cone.

overlaps, which in case of cones are triangular [55]. There were five types of fiber paths considered because it is possible to steer the fiber direction in the AFP process. The five fiber paths are geodesic path, constant angle path, path with linearly varying fiber angles, constant curvature path, and multiple-segment angle variation. An example of constant curvature (red), constant angle (black), and geodesic (green) fiber paths are shown in Fig. 7.7.11. An example of a tailored cone consisting of two plies with opposite orientation constructed with the developed method is shown in Fig. 7.7.19. The number of layers is more than doubled in some places. The proposed method for tailoring of “as-to-be-built” shells to be manufactured by AFP was validated for cylinders but not for cones [55]. For cylinders, in bending the

Number of layers 4 3 2

Figure 7.7.19 Number of layers of an overlapped laminate consisting of two plies with opposite orientation achieved with constant course width [55].

Stability of composite shelletype structures

421

unconstrained variable-stiffness designs increased the buckling load-carrying capability by about 30% compared to the optimized baseline design consisting of 0-, 90-, and 45-degree plies. However, when imposing manufacturing constraints, the resulting variable-stiffness designs showed improvements of up to 17% compared to the optimized baseline.

7.7.5.3.3

Tape laying

As the tape laying does not cause variable coordinate-dependent stiffness in cylinders, this problem was not considered in those structures. However, for structures with a variable radius, the aforementioned problem occurs. The manufacturing method of composite cones employing tape laying suggested in Ref. [60] does not take into account the stiffness variation of the structure caused by the fiber and thickness deviations. The phenomenon of variable-stiffness coefficients was considered for laminated open conical shell panels in Ref. [73]. When the surface of a conical shell is unfolded, it becomes a conical panel and is therefore reviewed here. The considered structures were to be manufactured by tape laying, where each laminate ply had to be cut to the developed shape of the conical panel from a unidirectional prepreg material. Unlike filament winding, here the stiffness coefficients vary around the circumference as a function of the modified fiber angle. Fig. 7.7.20 shows a single developed ply’s geometry with the depicted nominal fiber angle qnom and local real fiber angle q. Eq. (7.7.12) defines the modified angle q as q ¼ qnom  4

(7.7.12)

A C

ϕ,η X

θnom η = –1

θ

2

η = +1 1

D B

Figure 7.7.20 Plain geometry of a single developed ply. qnom, nominal fiber angle; q, local fiber angle (fiber angle relative to the generator). Modified from K. Khatri, N. Bardell, The variation of the stiffness coefficients for laminated open conical shell panels, Composite Structures 32 (1995) 287e292.

422

Stability and Vibrations of Thin-Walled Composite Structures

Range over which the use of constant Qxx,xx is valid.

(a)

Qxx,xx x 1010 N/m2

1.6

A

1.4 1.2

B

1 0.8 0

15

30

1 45

0.5 0

60 75

–0.5 90 –1

Panel circumference η

Apex angle α

(b)

Qxx,xx x 1010 N/m2

12 10

Range over which the use of constant Qxx,xx is valid.

8 A

6 4 2

B

0 15

1

30 45

0.5 0

60

–0.5

75 90 –1

Panel circumference η

Apex angle α

Figure 7.7.21 Variation of Qxx for the nominal fiber angle: (a) 0 degree and (b) 45 degrees [73].

Fig. 7.7.21 demonstrates that the stiffness coefficient Qxx depends on the nominal fiber angle as well as the semivertex angle a. As expected, for cylinders (semivertex angle a ¼ 0 degree) the fiber angle along the meridian is constant and thus the stiffness coefficient does not vary. It was concluded that for structures with the semivertex angle a < 10 degrees, the constant stiffness coefficients can be assumed. However, if the semivertex angle is more than 10 degrees, this assumption is no longer valid and the stiffness coefficients depend on the circumferential coordinate, which needs to be considered in the lamina constitutive equations.

Stability of composite shelletype structures

423

Figure 7.7.22 Ply piece. Aeff

θnom θ(ϕ)

ϕ θnom ϕ

For the ply topology method developed by Khakimova et al. [64], a set of design and evaluation parameters is suggested. This modeling process generates the ply piece shape according to the design parameters, which allows for the calculation of the resulting fiber-angle deviations. As an outcome of the modeling, the data (e.g., the local fiber angle) can be used to improve the numerical model by updating the stiffness matrices, creating an “as-to-be-built” structure. For fiber patches (pieces) that are used in the context of the tape laying technique the formula (Eq. 7.7.12) is not effective anymore. This procedure considers ply pieces with a finite width and uses the angular coordinate 4 of the conical coordinate system (see Fig. 7.7.18) to calculate the local fiber angle. In a ply piece the fiber angle varies within the s coordinate and the 4 coordinate. This is why it is important to set a point with the nominal fiber angle qnom with the corresponding distance from the cone apex defined as sqnom . qð4Þ ¼ qnom  ð4qnom  4Þ

(7.7.13)

In Eq. (7.7.13), q(4) is the local fiber angle at s (see Fig. 7.7.18) and 4qnom is the angular coordinate at which qnom can be found and can be considered as the reference angle of the ply piece (see Fig. 7.7.22). An example of a 30-degree ply with the actual fiber orientation of each finite element is shown in Fig. 7.7.23. 46.94

42.85

14.55

16.56

Figure 7.7.23 Example of a (a) wide ply piece and (b) thin ply piece with the actual fiber orientation of each finite element of the 30-degree ply (top view).

424

Table 7.7.3

Summary of studies on laminated cones found in the literature

References

Filament winding

[50]

No

Yes

No

Yes

Monolithic cone

Filament winding

[72]

Yes

No

No

No

Monolithic cone

AFP

[55]

Yes

No

No

No

Monolithic cone

Tape laying

[60]

No

Yes

No

No

Stiffened cone

Tape laying

[73]

Yes

No

No

No

Conical panel

Tape laying

[49,61]

No

Yes

No

Yes

Sandwich cone

Tape laying

n/a

No

Yes

No

Yes

Sandwich cone

Tape laying

[53,64]

Yes

Yes

Yes

Yes

Monolithic cones

Manufactured?

Non-destructive inspected?

Tested?

Remark

Stability and Vibrations of Thin-Walled Composite Structures

Type of manufacturing

Varying stiffness coefficients in computational model considered?

Stability of composite shelletype structures

425

Summing up, this section shows that the stiffness coefficients of the conical structure vary along the shell coordinate (meridian or circumferential, depending on the manufacturing method). Therefore, it is important to take the variation of the stiffness coefficients into account in the buckling analysis of laminated conical structures. The ply construction design process and consideration of the changing stiffness coefficients of laminated cones was developed for the AFP process in Ref. [55]. The changing stiffness coefficients in the buckling of filament-wound cones were considered in Ref. [72]. Khakimova et al. developed a ply topology method for laminated cones manufactured by tape laying that allows including the resulting manufacturing effects in the form of the fiber-angle deviation in the finite element analysis. Table 7.7.3 summarizes the studies on laminated cones which consider one or several of the following aspects: manufacturing, nondestructive inspection, test and consideration of manufacturing effects on the material properties in simulation.

References [1] “Ariane 5 manual”. [2] O. Ifayefunmi, A survey of buckling of conical shells subjected to axial compression and external pressure, Journal of Engineering Science and Technology Review 7 (2) (2014) 182e189. [3] P. Seide, Axisymmetrical buckling of circular cones under axial compression, Journal of Applied Mechanics 23 (4) (1956) 625e628. [4] P. Seide, Buckling of circular cones under axial compression, Journal of Applied Mechanics 28 (2) (1961) 315e326. [5] Y. Goldfeld, I. Sheinman, M. Baruch, Imperfection sensitivity of conical shells, AIAA Journal 4 (3) (2003) 517e524. [6] J. Pontow, Imperfektionsempfindlichkeit und Grenzlasten von Schalentragwerken, Institute f€ur Statik, Technische Universit€at Braunschweig, Braunschweig, 2009. [7] J. Singer, J. Arbocz, T. Weller, Buckling Experiments, John Wiley &Sons, 2002. [8] J.G. Teng, J.M. Rotter, Buckling of Thin Metal Shells, Spon Press, London, 2004. [9] M. Chryssanthopoulos, A. Spagnoli, The influence of radial edge constraint on the stability of stiffened conical shells in compression, Thin-Walled Structures 27 (2) (1997) 147e163. [10] L.A. Samuelson, E. Sigge, Shell Stability Handbook, Elsevier Science Publishers, 1992. [11] M. Jabareen, I. Sheinman, Effect of the nonlinear prebuckling state on the bifurcation point of conical shells, International Journal of Solids and Structures 43 (7e8) (2006) 2146e2159. [12] M. Jabareen, I. Sheinman, Postbuckling analysis of geometrically imperfect conical shells, Journal of Engineering Mechanics 132 (2006) 1326e1334. [13] F. Shadmehri, Buckling of Laminated Composite Conical Shells; Theory and Experiment (Ph.D. thesis), Concordia University, Montreal, Quebec, Canada, 2012. [14] F. Shadmehri, S.V. Hoa, M. Hojjati, Buckling of conical composite shells, Composite Structures 94 (2012) 787e792. [15] S.G.P. Castro, C. Mittelstedt, F.A.C. Monteiro, M.A. Arbelo, G. Ziegmann, R. Degenhardt, Linear buckling predictions of unstiffened laminated composite cylinders and cones under various loading and boundary conditions using semi-analytical models, Composite Structures 10 (2014), http://dx.doi.org/10.1016/j.compstruct.2014.07.037.

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[16] S.G.P. Castro, C. Mittelstedt, F.A.C. Monteiro, M.A. Arbelo, R. Degenhardt, A semianalytical approach for the linear and non-linear buckling analysis of imperfect unstiffened laminated composite cylinders and cones under axial, torsion and pressure loads, Thin-Walled Structures (2014) (submitted). [17] V.I. Weingarten, P. Seide, NASA SP-8019-buckling of thin-walled truncated cones. NASA Space Vehicle Design Criteria e Structures. [18] M. Nemeth, J.H. Starnes, The NASA monographs on shell stability design recommendations, NASA/TP-1998-206290, Hampton, Virginia, 1998. [19] L. Lackman, J. Penzien, Buckling of circular cones under axial compression, Journal of Applied Mechanics 27 (3) (1961) 458e460. [20] P. Seide, A Survey of Buckling Theory and Experiment for Circular Conical Shells of Constant Thickness, Space Corporation, 1962. [21] V.I. Weingarten, E.J. Morgan, P. Seide, Elastic stability of thin-walled cylindrical and conical shells under axial compression, AIAA Journal 3 (1965) 500e505. [22] European Convention for Constructional Steelwork, Enhancement of ECCS Design Recommendations and Development of Eurocode 3 Parts Related to Shell Buckling, Office for Official Publications of the European Communities, Luxembourg, 1998. [23] A. Spagnoli, M.K. Chryssanthopoloulos, Buckling design of stringer-stiffened conical shells in compression, Journal of Structural Engineering 125 (1999) 40e48. [24] R.M. Jones, Mechanics of Composite Materials, Taylor & Francis, United States of America, 1999. [25] A.R. 2A, Recommended Practise for Planning, Designing and Constructing Fixed Offshore Platforms, American Petroleum Institute, 2003. [26] D.-R. Recommended Practise, Buckling Strength Analysis, Det Norske Veritas, 2013. [27] ECCS, Buckling of Steel Shells: European Design Recommendations, fifth ed., European Convention for Constructional Steelwork, 2008. [28] M. Chryssanthopoulus, C. Poggi, A. Spagnoli, Buckling design of conical shells based on validated numerical models, Thin-Walled Structures 31 (1e3) (1998) 257e270. [29] L. Finzi, C. Foggi, Approximation formulas for the design of conical shells under various loading conditions, in: ECCS Colloqium on Stability of Plate and Shell Structures, Ghent University, 1987. [30] ECSS, Space Engineering: Buckling of Structures, ESA Requirements and Standards Division, Noordwijk, The Netherlands, 2010. [31] N.K. Gupta, G.L. Easwara Prasda, S.K. Gupta, Plastic collapse of metallic conical frusta of large semi-apical angles, International Journal of Crashworthiness 2 (4) (1997) 349e366. [32] G.L. Easwara Prasad, N.K. Gupta, An experimental study of deformation modes of domes and large-angled frusta at different, International Journal of Impact Engineering 32 (1e4) (2005) 400e415. [33] J. Blachut, On elastic-plastic buckling of cones, Thin-Walled Structures 29 (2011) 45e52. [34] R. Krysik, H. Schmidt, Beulversuche an l€angsnahtgescheißten st€ahlernen Kreiszylinderund Kegelstumpfschalen im elastisch-plastischen Bereich unter Meridiandruck- und innerer Manteldruckbelastung, Universit€at-Gesamthochschule Essen, 1990. [35] H. El-Sobsky, A.A. Singace, An experiment on elastically compressed frusta, Thin-Walled Structures 33 (4) (1999) 231e244. [36] S. Kobayashi, The Influence of Prebuckling Deformation on the Buckling Load of the Truncated Conical Shells under Axial Compression, NASA CR-707, 1967. [37] M. Esslinger, B. Geier, Buckling and postbuckling behaviour of conical shells subjected to axisymmetric loading and of cylinders subjected to bending, in: W.T. Koiter, G.K. Mikhailov (Eds.), Theory of Shells, North Holland Publishing Company, 1980, pp. 263e288.

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[38] A.G. Mamalis, W. Johnson, The quasi-static crumpling of thin-walled circular cylinders and frusta under axial compression, International Journal of Mechanical Sciences 25 (9e10) (1983) 713e732. [39] C.G. Foster, Axial compression buckling of conical and cylindrical shells, Experimental Mechanics 27 (3) (1987) 255e261. [40] W. Schnell, K. Schiffner, Experimentelle Untersuchungen des Stabilit€atsverhaltens von d€unnwandigen Kegelschalen unter Axiallast und Innendruck, DFL, 1962. [41] A. Berkovits, J. Singer, T. Weller, Buckling of unstiffened conical shells under combined loading, Experimental Mechanics (1967) 458e467. [42] J. Arbocz, Buckling of Conical Shells under Axial Compression, 1968. [43] H. Ramsey, Plastic buckling of conical shells under axial compression, International Journal of Mechanical Sciences 19 (5) (1977) 257e272. [44] A.G. Mamalis, W. Johnson, G. Viegelahn, The crumpling of steel thin-walled tubes and frusta under axial compression at elevated strain-rates: some experimental results, International Journal of Mechanical Sciences 26 (11e12) (1984) 537e547. [45] A. Mamalis, D.E. Manolakos, G. Viegelahn, N. Vaxevanidis, W. Johnson, On the inextensional axial collapse of thin PVC conical shells, International Journal of Mechanical Sciences 28 (5) (1986) 323e335. [46] M. Chryssanthopoulus, C. Poggi, Collapse strength of unstiffened conical shells under axial compression, Journal of Constructional Steel Research 57 (2) (2001) 165e184. [47] C. Thinvongpituk, H. El-Sobsky, Buckling load characteristic of conical shells under various end conditions, in: The 17th Annual Conference of Mechanical Engineering Network of Thailand, Prachinburi, Thailand, 2003. [48] N. Gupta, M.N. Sheriff, R. Velmurugan, A study on buckling of thin conical frusta under axial loads, Thin-Walled Structures 44 (9) (2006) 986e996. [49] C. Bert, W. Crisman, G. Nordby, Buckling of cylindrical and conical sandwich shells with orthotropic facings, AIAA Journal 7 (1) (1969) 250e257. [50] L. Tong, Buckling of filament-wound laminated conical shells, AIAA Journal 37 (6) (1999) 778e791. [51] E. Mahdi, A. Hamouda, B. Sahari, Y. Khalid, Crushing behaviour of cone-cylinder-cone composite system, Mechanics of Advanced Materials and Structures 9 (2) (2002) 99e117. [52] H. Abramovich, Stability and Vibrations of Thin Walled Composite Structures, Woodhead Publishing Limited, 2017. [53] R. Khakimova, D. Wilckens, J. Reichardt, R. Zimmermann, R. Degenhardt, Buckling of axially compressed CFRP truncated cones: experimental and numerical investigation, Composite Structures (2016). [54] R. Khakimova, R. Zimmermann, D. Wilckens, K. Rohwer, R. Degenhardt, Buckling of axially compressed CFRP truncated cones with additional lateral load: experimental and numerical investigation, Composite Structures 157 (2016) 436e447. [55] A. Blom, Structural Performance of Fiber-Placed, Variable-Stiffness Composite Conical and Cylindrical Shells (Ph.D. thesis), The Netherlands, Delft, 2010. [56] M.G. Bader, Selection of composite materials and manufacturing routes for cost-effective performance, Composites, Part A 33 (2002) 947e962. [57] D. Lukaczevicz, C. Ward, K.D. Potter, The engineering apsects of automated prepreg layup: history, present and future, Composites: Part B 43 (2012) 97e1009. [58] C. Grant, Automated processes for composite aircraft structure, Industrial Robot: An International Journal 33 (2) (2006) 117e121. [59] “Fives’ Metal Cutting j Composites,” Fives’ Metal Cutting j Composites, [Online]. Available: http://metal-cutting-composites.fivesgroup.com/.

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[60] T. Ahmed, A. Brodsjo, A. Kremers, H. Cruijssen, F. van der Bas, D. Spanjer, C. Groenendijk, A composite engine thrust frame cone, made with novel cost-effective manufacturing technology, in: 12th European Conference on Space Structures, Materials and Environmental Testtng, Noordwijk, The Netherlands, 2012. [61] C. Bert, W. Crisman, G. Nordby, Fabrication and full-scale structural evaluation of glass-fabric reinforced plastic shells, Journal of Aircraft 5 (1) (1968) 27e34. [62] DESICOS, New Robust DESign Guideline for Imperfection Sensitive Composite Launcher Structures, 2012 [Online]. Available: http://www.desicos.eu. [63] Griphus e Aeronautical Engineering & Manufacturing Ltd., 2016. [Online]. Available: http://www.griphus-aero.com/. [64] R. Khakimova, F. Burau, R. Degenhardt, M. Siebert, S. Castro, Design and manufacture of conical shell structures using prepreg laminates, Applied Composite Materials (2015) 1e24. [65] Y. Goldfeld, The influence of the stiffness coefficients on the imperfection sensitivity of laminated cylindrical shells, Composite Structures 64 (2004) 243e247. [66] M. Baruch, J. Arbocz, G. Zhang, Laminated conical shells e considerations for the variations of the stiffness coefficients, in: AIAA-94-1634-CP, 1994. [67] G.-Q. Zhang, Stability Analysis of Anisotropic Conical Shells, Technical University Delft e Faculty of Aerospace Engineering, 1993. [68] G.Q. Zhang, J. Arbosz, Stability analysis of anisotropic conical shells, in: 34th AIAA/ ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, USA, 1993. [69] Y. Goldfeld, J. Arbocz, A. Rothwell, Design and optimization of laminated conical shells for buckling, Thin-Walled Structures 43 (2005) 107e133. [70] Y. Goldfeld, K. Vervenne, J. Arbocz, F. van Keulen, Multi-fidelity optimization of laminated conical shells for buckling, Structural Multidisciplenary Optimisation 30 (2005) 128e141. [71] Y. Goldfeld, “Imperfection sensitivity of laminated conical shells, International Journal of Solids and Structures 44 (2007) 1221e1241. [72] Y. Goldfeld, J. Arbocz, Buckling of laminated conical shells given the variations of the stiffness coefficients, AIAA Journal 42 (3) (2004) 642e649. [73] K. Khatri, N. Bardell, The variation of the stiffness coefficients for laminated open conical shell panels, Composite Structures 32 (1995) 287e292.

Vibrations of composite shell-type structures

8

Eelco Jansen Leibniz Universit€at Hannover, Hannover, Germany

8.1

Introduction

In this chapter, characteristics of the vibration behavior of composite shell-type structures are illustrated through the treatment and application of analysis models with different levels of complexity, with emphasis on large amplitude vibrations. Because of their favorable strength-over-weight and stiffness-over-weight ratio, stiffened and unstiffened composite shells are widely used as primary components in (weightcritical) structural applications in aerospace and other fields of engineering. These thin-walled structures are prone to static and dynamic buckling instabilities and can be excited to vibrate at large amplitudes. The topic of large amplitude, nonlinear vibrations has a special significance in the vibration and dynamic stability analysis of shells. Reviews on the nonlinear vibrations of shells can be found in Refs. [1,2]. The circular cylindrical shell is of special interest for several reasons. First, this configuration is significant from a practical viewpoint because it is commonly applied. The results of an analysis for cylindrical shells may in certain cases also be characteristic for a more general class of shell structures, such as shells of revolution. Second, the severe instability of a cylinder under axial compressive loading may result in a catastrophic failure. Finally, its basic, simple geometry makes the cylindrical shell extremely suitable for theoretical analysis. For these reasons, in this chapter the cylindrical shell will be used to demonstrate the characteristics of the vibration behavior of composite shells. The description in this chapter closely follows the treatment in Refs. [3e7]. The vibration analysis of structures is not only a relevant topic in itself, it is well known that there are also various connections between the vibration behavior and the buckling behavior of a structure. The formal analogy between buckling and vibration has stimulated the use of vibration tests to obtain information that is important to assess the buckling behavior. First, it has been suggested that one could use vibration tests to establish the actual boundary conditions, the so-called vibration correlation technique [8]. Another possibility is to estimate the buckling load from vibration tests as a way of nondestructive testing (e.g., Ref. [9]). A good understanding of nonlinear effects (due to imperfections, large amplitudes, etc.) on the vibration behavior is indispensable when such methods are applied. There is also an analogy between postbuckling and nonlinear vibrations. The typically unstable postbuckling behavior of shells, or in other words, the negative slope of the postbuckling path, corresponds with the softening vibration behavior of shells which is often observed, i.e., the vibration frequency decreases with increasing vibration amplitude. Stability and Vibrations of Thin-Walled Composite Structures. http://dx.doi.org/10.1016/B978-0-08-100410-4.00008-9 Copyright © 2017 Elsevier Ltd. All rights reserved.

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Stability and Vibrations of Thin-Walled Composite Structures

Cylindrical shells form the primary structure of missiles and launch vehicles. During powered flight, these thin-walled structures are often forced to vibrate at large amplitudes by their environment. This observation provided a direct reason to study this problem in the beginning of the space age (about 1960). If the vibration amplitude is sufficiently small, the dynamic behavior of a shell may be described adequately by a linear analysis, but when the amplitude is of the order of the shell thickness, nonlinear effects should be taken into account. These large-amplitude vibrations are therefore generally referred to as nonlinear vibrations. The nonlinearity is clearly demonstrated by two phenomena: (1) the shape of the responseefrequency relationship in the vicinity of a resonant frequency and (2) the occurrence of a traveling wave response in the circumferential direction of the shell (Fig. 8.1). The early developments in the research field of nonlinear vibrations of cylindrical shells (and related work) can be found in a review by Evensen [10]. More recent reviews on the nonlinear vibrations of shells can be found in Refs. [1,2]. The second reason to study the nonlinear vibrations of shell structures is that the subject is important from a theoretical viewpoint. The nonlinear (free or forced) vibration problem of shell structures is one of the fundamental topics in the dynamic stability analysis of shells. The term “dynamic stability” here denotes a number of different phenomena in the dynamics of shells that are related to the structural stability [11] and which have in common that they are investigated by means of Newton’s law or equivalent methods (i.e., inertia is taken into account in the formulation). Important fields in the dynamic stability of shell structures include dynamic buckling (buckling under step loading or impulsive loading), parametric excitation (vibration buckling under pulsating loads), and flutter (instability induced by a gas flow). It should be noted that a discrepancy between theory and experiment has occupied not only researchers working on shell buckling but also investigators in these areas of dynamic analysis, in particular, in the field of shell flutter [12,13] and nonlinear vibrations [10]. A long-debated question concerned the type of nonlinearity of the flexural vibrations of cylindrical shells. The earliest theoretical investigations predicted a hardening nonlinearity, that is, the frequency increases with the amplitude of vibration, whereas experiments showed the contrary, a softening nonlinearity (i.e., the frequency decreases with amplitude). The omission of the appropriate axisymmetric deflection functions in the analysis (which may lead to not satisfying the circumferential periodicity condition) was identified as the principal shortcoming of these theoretical investigations, and a softening nonlinearity was obtained for the practically important cases [14,15]. The work in Ref. [16] was motivated by the lack of a clear and consistent model to describe the nonlinear vibrations of axisymmetric structures and this paper gives insight into the background of the type of nonlinearity of the vibrations of these structures and highlights the importance of the median surface curvature in particular. The reasons stated earlier to study besides the linear vibrations also the nonlinear vibrations of shells are strongly related to structural stability problems, i.e., to problems involving large amplitude responses which might directly lead to structural failure. If sonic fatigue is a design consideration, one may also need a nonlinear analysis rather than a linear one. In the analysis of composite plates and panels subjected to

Vibrations of composite shell-type structures

431

(a) A

Region 1 6.0

2

3

4

Region 5

Bifurcation point

Stable Unstable

5.0 Backbone curve 4.0

3.0

2.0

Driven mode response Bifurcation point Single-mode response (at starting up)

Single-mode response

1.0

0.9680

0.9840

1.0000

1.0160

1.0320

>

0.0

Ω

(b) B 3.0

2.0

0.0

0.9680

0.9840

1.0000

1.0160

>

1.0

Ω

Figure 8.1 (a) Single- and driven-mode responses and (b) companion mode response of b isotropic cylindrical shell. Average amplitude A; B versus normalized frequency U. From D.K. Liu, Nonlinear Vibrations of Imperfect Thin-Walled Cylindrical Shells (Ph.D. thesis), Faculty of Aerospace Engineering, Delft University of Technology, The Netherlands, 1988.

high-intensity acoustic fatigue loading, the effect of large amplitudes can be taken into account (e.g., Ref. [17]). Another example of an application considering large amplitude vibrations not directly related to structural stability is the nonlinear acoustic response in fluidestructure interactions on a thin cylindrical shell, see Ref. [18].

432

Stability and Vibrations of Thin-Walled Composite Structures

Practical problems in the stability and vibration analysis of shell structures can be very complex due to geometry, anisotropy, nonlinearity, and number of degrees of freedom, and their solution may require a powerful discretization method, a Level 3 analysis in the terminology used by Arbocz [19]. The finite element method can be used for the spatial discretization, in combination with numerical time integration to determine the temporal behavior. A prerequisite for the application of a Level 3 analysis is to have insight into the structural behavior. The mechanisms that play a role must be understood in advance for the model definition and for the interpretation of the results. For the cylindrical shell (an important complicated problem with a simple geometry), it is possible to devise semianalytical (i.e., analytical/numerical) methods that emphasize characteristic aspects of the structural behavior. These models can provide reference solutions, and as they are, in general, efficient with respect to computational time, they are suited for parametric studies. Therefore, such models are indispensable for a good understanding of the structural behavior. Methods with different levels of complexity are complementary in the analysis of complicated structural behavior. To support the Level 3 analysis, analogous to the analysis strategy for buckling problems proposed in Ref. [19], for nonlinear shell vibration problems the availability of the following methods for shell vibration analysis is recommended: •

•

In the so-called Level 1 analysis or Simplified Analysis the system is modeled with a small number of degrees of freedom using the Galerkin or variational approach. The assumed deflection modes approximately satisfy “simply supported” boundary conditions at the shell edges. In this method, the effect of imperfections is included. The term “simplified” refers to the use of a limited number of assumed modes that (approximately) satisfy simply supported boundary conditions and to the use of a simple imperfection shape. A Level 1 analysis forms the first step in the characterization of the nonlinear behavior of the shell and should, in general, be accompanied by more accurate analysis methods. In a Level 2 analysis or Extended Analysis, a Fourier decomposition of the dependent variables in the circumferential direction of the shell is used to eliminate the dependence of the circumferential coordinate. The problem is formulated as a set of 2-point boundary value problems for the axial direction, which is solved numerically via an accurate numerical method, such as the finite difference method or as is done in the approach presented in this chapter the parallel shooting method. In the Level 2 analysis, the specified boundary conditions can be satisfied rigorously. This chapter presents various characteristics of the large amplitude vibration behavior of composite shells using, in particular, the Level 1 and Level 2 analyses to obtain insight in the basic mechanisms of the nonlinear vibration behavior.

8.1.1

Problem definition

In this chapter, the nonlinear vibration problem of a laminated (composite) circular cylindrical shell is considered to illustrate several characteristics of the behavior of composite shell-type structures. The approach is shown for laminated shells but the

Vibrations of composite shell-type structures

433

T P R

θk h x

L

p, q x,u z,W y,v

Figure 8.2 Shell geometry, coordinate system, and applied loading.

extension to shells stiffened additionally by closely spaced rings and stringers can be carried out with relative ease. The shell geometry is characterized by the shell radius R, the length L, and the thickness h (see Fig. 8.2). The static loading consists of the three basic (axisymmetric) loads, axial comprese uniform radial pressure e e (Fig. 8.2). In addition, the shell can sion P, p, and torsion T b b ðtÞ, specified b also be loaded dynamically by the three basic loads PðtÞ, p ðtÞ, and T as functions of time. Furthermore, the shell is subjected to a (spatially varying) radial harmonic loading q and to the radial aerodynamic loading due to a supersonic flow pae. The shell loading will be discussed in more detail in section Boundary conditions and applied loading. A class of deformations is considered, which is intermediate in the sense that the geometric nonlinearity is limited to moderately small rotations [20e22]. In the present analysis, the usual Kirchhoff assumptions are employed. The nonlinear equations used in the present work are based on a Donnell-type thin shell theory. The basic assumptions are [23] • • • • •

the shell is thin, i.e., h/R  1, h/L  1, strains are small (of the order ε, where ε  1), displacements u, v are infinitesimal and W is of the order of the shell thickness, flexural rotations of shell elements are moderately small (W,2x and W,2y are of the order ε), the Kirchhoff assumptions are used, that is, • the transverse normal stress is small compared to the other normal stress components and may be neglected, • normals to the undeformed middle-surface remain straight and normal to the deformed middle surface and suffer no extension.

For shells consisting of composite laminae, it may be necessary to include the effect of transverse shear deformation because the transverse shear stiffness is usually relatively small as compared to the in-plane stiffness [24]. The possibility of incorporating transverse shear deformation within the framework of the present approach is outlined in Ref. [3].

434

Stability and Vibrations of Thin-Walled Composite Structures

In addition, the governing equations are based on the assumptions made in Ref. [25]: •

the term containing the transverse shear stress resultant Qy is neglected in the force equilibrium equation in y-direction or equivalently,

•

the displacement term v is neglected in the expression for the rotation by.

For static problems, Ref. [21] has shown that under the Donnell assumptions, terms containing quadratic displacements can be omitted in the potential energy expression of the applied load, i.e., the applied loads can be treated as dead loads. For buckling problems, it is well known that if the number of circumferential waves n of the buckling pattern becomes small (say n  4), while also the circumferential bending contribution in the strain energy is important as compared to the axial bending contribution [26], the Donnell equations lose accuracy. The restrictions of the Donnell equations in the nonlinear range have been discussed for static (buckling) problems in Ref. [21] (see also Ref. [27]). Using the Donnell equations, one may miss certain relevant quartic terms in the potential energy expression, which contribute to the stabilizing behavior at large deflections. The error in the linear frequency due to the use of the Donnell equations in dynamic problems is small for high circumferential wave numbers n. It should be noted that for dynamic problems the maximum error does not necessarily occur for the lowest n [28]. The Donnell-type equations for both linear and nonlinear dynamic problems have been discussed in Ref. [29]. Other assumptions used in the analysis are listed in the following: •

•

• •

In-plane inertia of the (predominantly) radial modes is neglected in the governing equations. The error in the frequency due to neglecting in-plane inertia is proportional to 1/n2 for high n. Rotatory inertia is also neglected in the analysis. Rotatory inertia may be important for deformations with small wavelengths, i.e., in the cases in which transverse shear deformation may also be significant. The classical lamination theory is employed [30]. The constitutive equations for a layered anisotropic shell are used, in which the layers are assumed to be orthotropic and their principal axes can be oriented in arbitrary directions. The effect of stiffening elements, rings in circumferential direction and stringers in axial direction, can be included via a smeared stiffener approach [31]. Linearly elastic material behavior is assumed. Thermal and hygrothermal effects, i.e., moisture effects at high temperatures [32], are not considered.

The basic Donnell-type governing equations for a layered anisotropic shell, namely, • • •

the nonlinear strainedisplacement relations, the constitutive equations, the equations of motion,

can be found in Chapter 10. The equations of motion can be derived from the straine displacement relations and the appropriate energy and work expressions by the

Vibrations of composite shell-type structures

435

application of Hamilton’s principle. The equations obtained are reduced to a set of governing equations in terms of the radial displacement W and a stress function F. These equations, together with the corresponding boundary conditions, form the basis of the present analyses.

8.1.2

Governing equations

The equations governing the nonlinear dynamic behavior of a cylindrical shell vibrating about a nonlinear static state will be presented in this section. It is assumed that the radial displacement W is positive inward (see Fig. 8.2). By introducing an Airy stress function F as Nx ¼ F,yy, Ny ¼ F,xx, and Nxy ¼ F,xy (see Fig. 8.3), the Donnelltype nonlinear imperfect shell equations for a general anisotropic material can be written as [3]   1 1 LA ðFÞ  LB ðWÞ ¼  W;xx  LNL W; W þ 2W R 2 LB ðF Þ þ LD ðW Þ ¼

(8.1)

  1 F;xx þ LNL F; W þ W þ p  rhW;tt R

(8.2)

where the variables W and F depend on x, y and the time t; R is the shell radius; rhW;tt is the radial inertia term; r is the (averaged) specific mass of the laminate; h is the (reference) shell thickness; and p is the (effective) radial pressure (positive inward), which can depend on time. The fourth-order linear differential operators   LA ðÞ ¼ A22 ðÞ;xxxx  2A26 ðÞ;xxxy þ 2A12 þ A66 ðÞ;xxyy  2A16 ðÞ;xyyy þ A11 ðÞ;yyyy

(8.3)

    LB ðÞ ¼ B21 ðÞ;xxxx þ 2B26  B61 ðÞ;xxxy þ B11 þ B22  2B66 ðÞ;xxyy    2B16  B62 ðÞ;xyyy þ B12 ðÞ;yyyy x,u

x,u

Nx

Mxy

Qx

Nxy

Mx

Nyx Ny

y,v

My

Qy

z,W

Figure 8.3 Definition of stress and moment resultants.

y,v z,W

Myx

(8.4)

436

Stability and Vibrations of Thin-Walled Composite Structures

  LD ðÞ ¼ D11 ðÞ;xxxx þ 4D16 ðÞ;xxxy þ 2 D12 þ 2D66 ðÞ;xxyy þ 4D26 ðÞ;xyyy þ D22 ðÞ;yyyy

(8.5)

depend on the stiffness properties of the laminate, and the nonlinear operator defined by LNL ðS; T Þ ¼ S;xx T;yy  2S;xy T;xy þ S;yy T;xx

(8.6)

reflects the geometric nonlinearity. The stiffness parameters Aij , Bij , and Dij (i, j ¼ 1, 2, 6) are given in Chapter 10. Eq. (8.1) guarantees the compatibility of the strains and the radial displacement field. Eq. (8.2) is the equation of motion (dynamic equilibrium equation) in radial direction. The shell can be loaded both statically and dynamically by the basic loads axial compression, radial pressure, and torsion. Both W and F are expressed as a superposition of the two states of displacement and stress: e þW b W ¼W

(8.7)

eþF b F¼F

(8.8)

e and W e are the stress function and normal displacement, respectively, of the where F static geometrically nonlinear state that develops during the application of a static load b and W b are the stress function and normal on the imperfect shell, whereas F displacement, respectively, of the dynamic state corresponding to the large-amplitude vibration about the static state. The Donnell-type equations governing the nonlinear static state of an imperfect anisotropic cylindrical shell become       e xx  1LNL W; e  LB W e ¼  1 W; e W e þ 2W LA F R 2

(8.9)

    1   e þ LNL F; e þ L D W e ¼ F; e W e þW þe LB F p R xx

(8.10)

where e p is the static radial loading, and the equations governing the nonlinear dynamic state can be written as         b ; xx  1LNL W; b ;W e þ 2W b  LB  W b ¼ 1 W e W b  1LNL W LA F R 2  2  1 b;W b (8.11)  LNL W 2     1       b xx þ LNL F; b þ L D W b ¼ F; e W b þ LNL F; b W e þ W þ LNL F; b W b LB F R b ; tt þ b rh W p (8.12)

Vibrations of composite shell-type structures

437

where b p is the dynamic radial loading. This may be a specified load that is explicitly given as a function of time or a load that implicitly depends on time, as in the case of flutter (see section Boundary conditions and applied loading). In the dynamic (i.e., time-dependent, in general, nonconservative) case, one can distinguish between nonstationary loads (loads that are specified functions of time) and stationary loads (loads that do not explicitly depend on time). Damping terms have not been included in the dynamic equations presented. To solve the aforementioned sets of partial differential equations, in section Simplified analysis for vibrations of composite cylindrical shells the differential equations will be reduced to a system with a finite number of degrees of freedom by using a number of assumed spatial modes in a Galerkin or variational procedure. Damping is introduced via viscous modal damping terms in the equations of motion of the discretized system.

8.1.3

Boundary conditions and applied loading

In this section, the following cases are distinguished: • • • •

static state (prebuckling, previbration, preflutter) dynamic buckling (buckling under step loading) parametric excitation (vibration buckling under pulsating loads) nonlinear vibrations

8.1.3.1

Pressure loading

In the static state, i.e., the time-independent case, the radial pressure load is conservative. The pressure loading is assumed to be constant over the shell surface and is therefore axisymmetric. The net pressure (positive inward) is the difference between the external and internal pressures and will be denoted by pe, e p ¼ pe

(8.13)

If the net pressure is directed outward (i.e., is negative) then pe ¼ pi, where pi is the net internal pressure, The nonstationary load in the case of nonlinear vibrations is given by b p¼q¼b q ðx; yÞcos ut

(8.14)

where q is the specified radial loading (with spatial distribution of the vibration mode). In the case of dynamic buckling and parametric excitation the nonstationary pressure (constant over the shell surface) is given by b p¼b p 0 f ðtÞ

(8.15)

where b p 0 is a constant and f(t) a specified function of time. For step loading (the dynamic buckling case), f(t) ¼ u(t), where u(t) is the unit step function, and for a

438

Stability and Vibrations of Thin-Walled Composite Structures

Table 8.1

Summary of loads [u(t) [ unit step function]

Case

N0

T0

p

Static state

e0 N

e0 T

pe e pae þ pe

Preflutter Dynamic buckling

b 0 uðtÞ N

b 0 uðtÞ T

b p 0 uðtÞ

Parametric excitation

b 0 cos Ue t N

b 0 cos Ue t T

b p 0 cos Ue t

Nonlinear vibration

b q ðx; yÞcos ut

Flutter

b p ae

pulsating load (the parametric excitation case), f(t) ¼ cos Uet, where Ue is the excitation frequency. The possibilities in the different cases for the radial loading p are summarized in Table 8.1. It is possible to extend the models used in the present chapter to include the flutter behavior in an external supersonic flow. In this flutter analysis case, the outer surface of the shell is exposed to a high Mach number supersonic flow directed parallel to its axis, as is illustrated in Ref. [5]. In Ref. [5], the aerodynamic pressure is obtained from linear piston theory, which is often applied and the simplest theory available for modeling this problem. For completeness, the corresponding aerodynamic loads are also included in Table 8.1.

8.1.3.1

In-plane loading and boundary conditions

In the static state, at the shell edges the basic axisymmetric in-plane loads, axial compression and torsion, are applied. They are assumed to be uniform, i.e., axisymmetric in the sense that they do not depend on the circumferential coordinate, e0 N0 ¼ N

(8.16)

e0 T0 ¼ T

(8.17)

In the case of dynamic buckling and parametric excitation the nonstationary loading consists of the basic loads, axial compression and torsion, specified as functions of time, b 0 fðtÞ N0 ¼ N

(8.18)

b 0 fðtÞ T0 ¼ T

(8.19)

Vibrations of composite shell-type structures

439

b 0 and T b 0 are constants and f(t) is the specified function of time. For step where N loading (the dynamic buckling case), f(t) ¼ u(t), where u(t) is the unit step function, and for a pulsating load (the parametric excitation case), f(t) ¼ cos Uet, where Ue is the excitation frequency. The different possibilities for the in-plane loading are summarized in Table 8.1. The in-plane displacements ue and ve at the shell edges can be written in the following form: "

ue

#

ve

" ¼

ue0

# þ

ve0

N X n¼1

("

ue1n ve1n

#

" cos nq þ

ue2n ve2n

#

) sin nq

(8.20)

e at the shell edges can be written as where the stress resultants Nxe and Nxy

"

Nxe e Nxy

#

"

e Nx0

¼

#

e Nxy0

8" 9 # " e # N < Ne = Nx2n X x1n þ cos nq þ sin nq e e : Nxy1n ; Nxy2n n¼1

(8.21)

By definition, the applied compressive force and torque at the shell edges correspond to in-plane stress resultants at the edges, which have been averaged over the edge plane, N0 ¼ 

1 2pR

1 T0 ¼ 2pR

Z

2pR

Nx dy ¼ specified

(8.22)

0

Z

2pR

Nxy dy ¼ specified

(8.23)

0

In the present work, these stress resultants are assumed to be prescribed. They correspond with the constant parts (uniform in the circumferential direction) in the general expressions for the stress resultants (Eq. 8.21). There will be an averaged displacement u0 and an averaged twist v0/R of the edge planes x ¼ 0 and x ¼ L relative to each other: u0 ¼

1 2pR

1 v0 ¼ 2pR

Z

2pR

Z

0

Z

u;x dxdy

(8.24)

v;x dxdy

(8.25)

0 2pR

0

L

Z

L

0

These displacements correspond with the constant parts (uniform in the circumferential direction) of the in-plane displacements in Eq. (8.20). The associated boundary conditions are often referred to in the literature as the “movability” conditions of the

440

Table 8.2

Stability and Vibrations of Thin-Walled Composite Structures

Standard boundary conditions

SS-1

Nx ¼ N0

Nxy ¼ T0

W¼0

Mx ¼ N0q

SS-2

u¼0

Nxy ¼ T0

W¼0

Mx ¼ N0q

SS-3

Nx ¼ N0

v¼0

W¼0

Mx ¼ N0q

SS-4

u¼0

v¼0

W¼0

Mx ¼ N0q

C-1

Nx ¼ N0

Nxy ¼ T0

W¼0

W,x ¼ 0

C-2

u¼0

Nxy ¼ T0

W¼0

W,x ¼ 0

C-3

Nx ¼ N0

v¼0

W¼0

W,x ¼ 0

C-4

u¼0

v¼0

W¼0

W,x ¼ 0

C, clamped; SS, simply supported.

edge planes. As the averaged in-plane membrane stresses are prescribed, the average displacements of the shell edges, relative to each other, are not constrained. For definiteness, the loads are assumed to be applied at x ¼ L, and the circumferentially averaged edge plane displacements at x ¼ 0 are assumed to be zero. The in-plane loading is assumed to be specified. If the value of the axial compression or torsional load is not given, it is implicitly assumed to be zero. The standard boundary conditions are denoted as in Table 8.2, where q is the axial load eccentricity, measured from the shell midsurface (positive inward). For the firstorder state problems treated in section Vibration analysis of cylindrical shells including boundary conditions, the (linear) buckling or linear vibration problem, the boundary conditions in Table 8.2 become homogeneous. For the in-plane boundary conditions a distinction is made between circumferentially constant and circumferentially varying in-plane displacements. The zero displacements in the standard boundary conditions are not to be interpreted as “immovability” conditions in the aforementioned sense, but as constraints for the circumferentially varying displacements, relative to the average displacement of the “moving” edge plane. The conditions u ¼ 0 and v ¼ 0 in the standard boundary conditions are equivalent to u,yy ¼ 0 and v,y ¼ 0, respectively, which can be used to express these conditions in terms of the variables W and F (the so-called reduced boundary conditions). Average in-plane loads are assumed to be specified in the fundamental state and those that correspond to an incremental (first-order) or higher-order state are implicitly assumed to vanish. Two consequences of the distinction between constant and circumferentially varying displacements deserve particular attention. First, when axisymmetric deformation occurs, for example, for the zeroth-order and the second-order states in a nonlinear (perturbation) analysis (see section Vibration analysis of cylindrical shells including boundary conditions), or in the linearized vibration of a cylinder with asymmetric imperfection with the shape of the vibration mode, this will involve either in-plane displacements or in-plane stresses at the shell edges. Second, for anisotropic shells, there may be a coupling between the axial and torsional

Vibrations of composite shell-type structures

441

deformation. For both cases, it should be realized that in the present formulation the circumferentially constant displacements of the edge planes are assumed to be free. The more general case of elastic edge restraint is modeled by introducing the elastic stiffness parameters ku, kv, kw, and kwx in the boundary conditions (for circumferentially varying displacements) as follows: Nx þ ku u ¼ 0

(8.26)

Nxy þ kv v ¼ 0

(8.27)

    Mx;x þ ðMxy þ Myx Þ;y þ Nx W; x þ W; x þ Nxy W; y þ W ; y þ kw W ¼ 0

(8.28)

Mx þ kwx W;x ¼ 0

(8.29)

where the sign of the stiffness parameters depends on the shell edge (x ¼ 0 or x ¼ L). Finally, it should be noted that the cylindrical shell geometry requires the periodicity of all variables in the circumferential direction of the shell.

8.2 8.2.1

Simplified analysis for vibrations of composite cylindrical shells Introduction

In this section, the steady-state nonlinear flexural vibration behavior of imperfect anisotropic cylindrical shells under harmonic lateral excitation is analyzed via the Level 1 analysis or Simplified Analysis (cf. section Introduction). The early investigations of the nonlinear vibration behavior of shells belong to the class of Level 1 analyses. The present treatment can be seen as an extension of these investigations from isotropic and orthotropic shells to laminated (anisotropic) shells. Deviations from the perfect cylindrical form due to the fabrication process (the so-called geometric imperfections) can have a considerable influence on the dynamic behavior of cylindrical shells. The same holds for a stress in the previbration state due to an applied (static) loading. In Ref. [33] the effect of asymmetric imperfections on the nonlinear vibrations of isotropic cylinders was studied. In Ref. [34] a model to investigate the effect of axisymmetric and asymmetric imperfections on the linearized vibrations of axially loaded ring- and stringer-stiffened cylindrical shells was developed, while in Ref. [35] the nonlinear vibrations of such shells was studied. To study the characteristic aspects of the nonlinear behavior, a small number of deflection modes is used in the Galerkin procedure. Two asymmetric modes are included in the deflection function, which are circumferentially 90 degrees out of phase between each other, the directly excited “driven mode” and its “companion mode.” The coupled-mode response of these modes can be interpreted as a travelling

442

Stability and Vibrations of Thin-Walled Composite Structures

wave pattern in the circumferential direction of the shell. The two assumed asymmetric modes have m axial waves. The axisymmetric mode C1 cos ipx L , which satisfies the strong coupling condition i ¼ 2m with the two asymmetric modes [35], is included in the assumed deflection function. The axisymmetric mode plays an essential role in the nonlinear behavior. The shell is statically loaded by axial compression, radial pressure, and torsion. The static state response is assumed to be affine to the given two-mode imperfection, which consists of an axisymmetric and an asymmetric mode. The nonlinear Donnell-type governing equations are used, and the classical lamination theory is employed. Khot’s formulation [36] is used to account for a possible skewedness of the asymmetric modes. The modes approximately satisfy “simply supported” boundary conditions. Galerkin’s method is applied to solve for the static state. Galerkin’s method and the method of averaging are used in sequence to obtain frequencyeamplitude curves for free and forced nonlinear vibrations.

8.2.2

Galerkin’s method and the method of averaging

In this section, the governing equations for the nonlinear dynamic state will be presented. The vibration behavior is modeled by assuming an axisymmetric mode and two asymmetric vibration modes. The static state response is assumed to be affine to the given two-mode imperfection. Application of Galerkin’s procedure to eliminate the spatial dependence and the method of averaging to eliminate the time dependence results in two coupled nonlinear algebraic equations for the average vibration amplitudes A and B. To investigate the important phenomena of the nonlinear (large amplitude) vibrations of statically loaded imperfect anisotropic cylindrical shells, the following expressions for the imperfection and response modes are used: Imperfection  2mpx mpx n þ x2 sin cos ðy  sK xÞ W h ¼ x1 cos L L R Static state

(8.30)

 2mpx e mpx n e h¼e þ x2 sin cos ðy  sK xÞ x1 cos W x0 þ e L L R Dynamic state

(8.31)

 b h ¼ C0 ðtÞ þ C1 ðtÞcos 2mpx þ AðtÞsin mpx cos ‘ ðy  sK xÞ W L L R mpx ‘ þ BðtÞsin sin ðy  sK xÞ L R

(8.32)

where C0(t), C1(t), A(t), and B(t) are the unknown time-dependent coefficients of the displacement modes and n and ‘ are the number of full waves in the circumferential direction of the imperfection and the vibration mode, respectively. Notice that as for the static case, the response modes of the dynamic state do not satisfy the classical

Vibrations of composite shell-type structures

443

simply supported boundary conditions exactly. The imperfection and the static and dynamic responses are assumed to be affine, and hence, the skewedness parameter of the imperfection is taken to be equal to the skewedness parameter of the response, sK ¼ sK . The radial displacement Eq. (8.32) contains two asymmetric modes. The mode with time-dependent coefficient A(t) (driven mode) is excited directly by the external excitation, which has the same spatial distribution and is assumed to be harmonic in time, q ¼ Qm‘s sin

mpx ‘ cos ðy  sK xÞcos ut L R

(8.33)

where u is the excitation frequency and Qm‘s is the excitation amplitude (a constant). The mode with time-dependent coefficient B(t) (companion mode) can respond because of a nonlinear coupling with the driven mode. This coupling is classified in nonlinear vibration terminology as a one-to-one internal (autoparametric) resonance [37]. The physical background is that the stress in the circumferential direction due to the large-amplitude motion (parametrically) excites the companion mode. The driven mode and the companion mode together give a coupled-mode response, which can be interpreted as a traveling wave pattern in the circumferential direction of the shell [14,15]. e (Eq. By substituting the given imperfection W (Eq. 8.30), the static solution W b 8.31) obtained earlier, and the dynamic response W (Eq. 8.32) into the dynamic compatibility equation (Eq. 8.11), one obtains an inhomogeneous linear partial difb A particular solution can be ferential equation for the dynamic stress function F. b denoted as F b p , into the compatobtained by substituting a particular solution for F, ibility equation and equating coefficients of corresponding goniometric terms. The resulting linear equations for the unknown coefficients of the stress function, which contain terms that are linear and quadratic in the time-dependent parts of the displacement modes, can be solved routinely. The coefficients are listed in Ref. [3]. Substitution of the given imperfection mode, the assumed radial deflections of the e and F b into the dynamic static state, and the solutions obtained for the stress functions F out-of-plane equilibrium Eq. (8.12) yields a “residual” equation, and application of Galerkin’s method leads to a coupled set of nonlinear ordinary differential equations in the time-dependent parts A and B of the asymmetric vibration modes. The weighting functions used in the Galerkin method are  

   b   vW mpx n 2R 2 mpx e ¼ h sin cos ðy  sK xÞ þ h‘‘ A þ dn;‘ x2 þ x2 sin vA L R 2 L (8.34)     b vW mpx n 2R 2 mpx ¼ h sin sin ðy  sK xÞ þ h‘‘ B sin vB L R 2 L

(8.35)

444

Stability and Vibrations of Thin-Walled Composite Structures

The resulting set of differential equations is of the following form:  2  2 d2 A d2 A dA d2 B dB g0 2 þ g11 A 2 þ g12 þ g13 B 2 þ g14 dt dt dt dt dt  2  2 d2 A dA d2 B dB þ g11 A2 2 þ g12 A þ g13 AB 2 þ g14 A dt dt dt dt þ c10 A þ c20 A þ c02 B þ c30 A þ c12 AB 2

2

3

(8.36)

2

þ c40 A4 þ c22 A2 B2 þ c04 B4 þ c50 A5 þ c32 A3 B2 þ c14 AB4 ¼ cexc Qm‘s cos ut  2  2 2 d2 B d2 A dA dB d2 A 2d B d0 2 þ d11 BA 2 þ d12 B þ d13 B 2 þ d14 B þ d15 B 2 dt dt dt dt dt dt þ d01 B þ d11 AB þ d21 A2 B þ d03 B3 þ d31 A3 B þ d13 AB3 þ d41 A4 B þ d23 A2 B3 þ d05 B5 ¼ 0 (8.37) where the coefficients g0, gij, cij, cexc, d0, dij, and dij are constant coefficients depending on the geometry and stiffness parameters and on the wave numbers of the imperfection and deflection modes. The coefficients are listed in Ref. [3]. These equations include Liu’s equations for orthotropic shells [35] as a special case. The asymmetry for the two equations with respect to A and B is because of the assumed imperfection shape. In the present approach the equations contain nonlinear inertia 2 terms (terms of the form A ddtA2 , etc. dthe terminology is from Ref. [38]). For steadystate vibrations, the time dependence can be eliminated by the method of averaging [14]. It is well known (see, e.g., Ref. [39]) that the first-order approximation of averaging is not consistent when even-order (quadratic, quartic, etc.) terms are present. This is the case for shells with asymmetric imperfections. In this case, to obtain a consistent approximation of the higher-order corrections to the amplitudeefrequency relation, a higher-order approximation or numerical time integration should be employed. The higher-order corrections to the amplitudee frequency  relation  due to the effect of asymmetric imperfections are expected to be 2 2

small, O x2 A

[3].

Finally, it should be noted that in the present approach the higher-order nonlinear effects are taken into account only approximately. For a consistent approximation of the higher-order terms in Eqs. (8.36) and (8.37), corresponding to the shift to hardening behavior for larger-vibration amplitudes, higher-order harmonics in the temporal description should also be included [40]. A more accurate approximation may also require more terms in the assumed spatial mode.

Vibrations of composite shell-type structures

445

To apply the first-order approximation of the method of averaging, we assume A ¼ At ðtÞcos ut

(8.38)

B ¼ Bt ðtÞsin ut

(8.39)

By substituting into the governing Eqs. (8.36) and (8.37) and applying the averaging procedure (for details of this procedure the reader is referred to Ref. [14] or [35]), we obtain two coupled nonlinear algebraic equations of the following form: 

  3   2  a10  a10 U2 A þ a31  a31 U2 A þ a12  a12 U2 A B 5

3 2

4

(8.40)

þ a50 A þ a32 A B þ a14 A B ¼ Gm‘s 

  2  3   b01  b01 U2 B þ b21  b21 U2 A B þ b03  b03 U2 B 4

2 3

5

(8.41)

þ b41 A B þ b23 A B þ b05 B ¼ 0 where A and B are average values (over one period) of At and Bt; aij, aij, bij, and bij are constant coefficients depending on the geometry, stiffness parameters, etc.; and Gm‘s is the generalized dynamic excitation. The coefficients are listed in Ref. [3]. The normalized frequency parameter U is defined by U¼

u ulin

(8.42)

where ulin ¼

rffiffiffiffiffiffiffi a10 a10

(8.43)

is the small amplitude (“linearized”) frequency for the given shell properties, namely, imperfection, vibration mode, and applied loading. Eqs. (8.40) and (8.41) can be used to calculate the amplitudeefrequency curves for nonlinear free or forced vibrations of statically loaded imperfect anisotropic cylindrical shells. If B ¼ 0, it is possible to solve Eq. (8.40) for the unknown averaged amplitude A. This corresponds to the case that only the directly driven mode responds, whereas the companion mode is quiescent. However, if the single-mode response becomes unstable with respect to perturbations in the companion mode, it is necessary to compute both A and B by solving Eqs. (8.40) and (8.41) simultaneously. The equations can be used to determine the nonlinear free vibration behavior by setting Gm‘s equal to zero in Eq. (8.40).

446

Stability and Vibrations of Thin-Walled Composite Structures

8.2.3

Results and discussion

To be able to perform a parametric investigation of the linear and nonlinear vibrations of statically loaded imperfect laminated (anisotropic) shells, the analysis presented was implemented in a FORTRAN program. The symbolic manipulation program REDUCE [41] was extensively used to execute the detailed derivations described in Ref. [3]. Characteristic results of the Simplified Analysis have been presented in Refs. [3,4,6]. To illustrate several characteristics of the linearized and nonlinear vibration behavior of composite shell using the Simplified Analysis, results for a specific anisotropic composite cylindrical shell will be presented [3,4,6]. The data of this shell are given in Table 8.3. This shell has been used earlier in static stability investigations, e.g., in Refs. [42,43]. In this case the shell length is L ¼ 3.776 in., the shell radius R ¼ 2.67 in., and the total shell thickness h ¼ 0.0267 in. The layup of Booton’s shell is [q1, 0, q1] ¼ [30, 0, 30]. Notice that because the laminate of this shell is unbalanced, torsionebending coupling exists. This is reflected by the nonvanishing terms B16 and B26 in the ABD matrix. Results will be presented illustrating the effect of axial loading and imperfections on the linearized and nonlinear vibrations of this anisotropic shell. The effect of varying the layer orientation q1 of Booton’s shell on the lowest natural frequency and the corresponding vibration mode is shown in Fig. 8.4. The frequency qffiffiffiffiffiffiffiffi E , where E is a reference value, in this has been normalized with respect to uref ¼ 2rR 2 case E ¼ E11 is the Young’s modulus of a layer in the 1-direction. The vibration mode i2 2h  b h ¼ ‘ AðtÞsin px þ AðtÞsin px cos ‘ ðy  sK xÞ W L L R 4R Table 8.3

Booton’s shell

Shell geometry

Radius R ¼ 2.67 in. Length L ¼ 3.776 in.

Laminate geometry

Three layers (the numbering starts outside the shell) Layer thickness h1 ¼ h2 ¼ h3 ¼ 0.0089 in. Layer orientation q1 ¼ 30 , q2 ¼ 0 , q3 ¼ 30 (Fig. 8.2)

Layer properties

Composite material: Glasseepoxy Modulus of elasticity 1-direction E11 ¼ 5.83  106 psi Modulus of elasticity 2-direction E22 ¼ 2.42  106 psi Major Poisson’s ratio n12 ¼ 0.363 Shear modulus 12-plane G12 ¼ 6.68  105 psi

Vibrations of composite shell-type structures

447

0.25 l=5

l=6

0.2

ω/ωref

ω/ωref , τK

0.15

τK

0.1 0.05 0

–0.05 –0.1

0

10

20

30 40 50 60 Orientation angle, θ1

70

80

90

Figure 8.4 Effect of layer orientation on lowest natural frequency and the corresponding vibration mode of anisotropic shell in a Booton-type shell layup [q1, 0, q1].

has one half wave in the axial direction and five or six full waves in the circumferential direction and Khot’s parameter sK is close to zero. For the layup with [q1, 0, q1] ¼ [30, 0, 30], the mode corresponding to the lowest natural frequency has the wave number parameters m ¼ 1, ‘ ¼ 6, and sK ¼ 0.002. Khot’s parameter sK is very close to zero, which means that the vibration mode, in contrast to the (lowest) buckling mode under axial compression, exhibits very little skewedness. The effect of axisymmetric imperfections of the form  2mpx W h ¼ x1 cos L on the lowest natural frequency is depicted in Fig. 8.5. The frequency is normalized with respect to the (linear) frequency of the unloaded perfect shell um‘s, which can be obtained from the equation for the linearized frequency ulin (Eq. 8.43), evaluated for the case that x1 ¼ x2 ¼ 0. The axisymmetric imperfection mode satisfies a strong coupling condition with the asymmetric vibration mode [35], the number of

(b)

1.05

Frequency, ω/ωmlT

Frequency, ω/ω mlT

(a)

0.95 0.85 0.75 0.65 –2.5

–2

–1.5

–1

–0.5 Imperfection amplitude, ξ¯1

0

1.05 0.95 0.85 0.75 0.65

0

0.4

0.8

1.2

1.6

Imperfection amplitude, ξ¯2

Figure 8.5 Influence of imperfection amplitude on linear frequency of anisotropic shell (Booton’s shell): (a) axisymmetric imperfection and (b) asymmetric imperfection.

2

448

Stability and Vibrations of Thin-Walled Composite Structures

axisymmetric half waves i in the C1cos(ipx/L) mode is equal to 2m. For small values of the imperfection amplitude the frequency decreases with increasing amplitude. The membrane stresses which correspond to the given deflection mode can be obtained directly from the dynamic state compatibility equation. The stabilizing membrane stress with a spatial distribution of the first-order asymmetric deflection mode (m, ‘) is decreased by the axisymmetric imperfections (inward at the shell midlength). This effect is initially predominant. At a certain imperfection amplitude the frequency begins to increase with growing imperfection amplitude due to the stabilizing effect of the different geometry of the imperfect shell. This stabilizing curvature effect (corresponding to the membrane stress contribution Nx W xx in the dynamic equilibrium equation) predominates for larger imperfection amplitudes. The trends observed for axisymmetric imperfections were reported earlier for isotropic and orthotropic shells in Ref. [44]. The effect of asymmetric imperfections in the form of the vibration mode (n ¼ ‘)  mpx ‘ cos ðy  sK xÞ W h ¼ x2 sin L R is also shown in Fig. 8.5. The frequency decreases with the imperfection amplitude. The interaction of the asymmetric mode with the accompanying axisymmetric mode contributes to a moderate reduction in the frequency with increasing imperfection amplitude. For increasing imperfection amplitudes, the effect of the other terms in the equilibrium equation involving curvature (corresponding to the membrane stress  contribution LNL F; W ) comes into play. This shift to an increasing  frequency  occurs at a larger value of the imperfection amplitude than the maximum x2 ¼ 2:0 shown in the figure. The discrepancy for isotropic shells between the trends predicted by different analyses (Refs. [35,45]) has been explained by pointing out that satisfaction of the circumferential periodicity condition is necesssary. For n s ‘, both the present analysis and Ref. [44], in general, predict an increase in frequency with increasing imperfection amplitude. The effect of axisymmetric imperfections on the nonlinear vibrations in the mode with m ¼ 1, ‘ ¼ 6, and sK ¼ 0.002 is depicted in Fig. 8.6. For this case, ε ¼ 0.1296 and x ¼ pR=‘ L=m ¼ 0:3702. The frequency has been normalized by ulin   U ¼ uulin , the linearized frequency of the imperfect shell. The vibration of the perfect shell shows a softening behavior for small vibration amplitudes and shifts to a hardening behavior for larger amplitudes. For larger imperfection amplitudes, the softening nonlinearity decreases (the nonlinear stiffness increases) because of the stabilizing curvature effect of the imperfections. It should be noted that there is a scaling effect because the linear frequency decreases with increasing imperfection amplitude and may be considerably lower than the frequency of the perfect shell. The effect of asymmetric imperfections on the nonlinear vibrations in the mode with m ¼ 1, ‘ ¼ 6, and sK ¼ 0.002 is depicted in Fig. 8.7. Also in this case the frequency has been normalized by ulin, the linearized frequency of the imperfect shell. As stated earlier, when

Vibrations of composite shell-type structures

449

6

Amplitude, |A¯|

5 4 3

ξ¯1 = 0 ξ¯1 = –0.25 ξ¯1 = –0.5 ξ¯1 = –1.0

2 1 0

0.6

0.8

1 1.2 Normalized frequency, Ω

1.4

Figure 8.6 Amplitudeefrequency curves of anisotropic shell (Booton’s shell) for various   2 ‘2 px þ AðtÞ b axisymmetric imperfection amplitudes. W=h ¼ x1 cos 2px L , W h ¼ 4R AðtÞsin L ‘ sin px L cos R ðy  sK xÞ, sK ¼ 0.002, and ‘ ¼ 6.

6

Amplitude, |A¯|

5 4 3

ξ¯2 = 0 ξ¯2 = 0.25 ξ¯2 = 0.5 ξ¯2 = 1.0

2 1 0

0.6

0.8 1 1.2 Normalized frequency, Ω

1.4

Figure 8.7 Amplitudeefrequency curves of anisotropic shell (Booton’s shell) for various   ‘ b asymmetric imperfection amplitudes. W h ¼ x2 sin px L cos R ðy  sK xÞ, W h ¼  2 ‘2 px þ AðtÞsin px cos ‘ ðy  s xÞ, s ¼ 0.002, and ‘ ¼ 6. K K L R 4R AðtÞsin L

discussing the accuracy of Eqs. (8.36) and (8.37), the higher-order effect of asymmetric imperfections on the frequencyeamplitude relations is generally small,   2 2

O x2 A

[3]. The zeroth-order effect is predominant.

The vibrations of composite shells were studied in Ref. [46], in which a multimode analysis for the nonlinear vibration and postbuckling of unsymmetric cross-ply cylindrical shells was used. The considerable discrepancy between the results in this work and those in Ref. [47], in which also a Simplified Analysis was used, was attributed to

450

Stability and Vibrations of Thin-Walled Composite Structures

the fact that the latter analysis does not satisfy simply supported boundary conditions and to the constraining of the axisymmetric mode in Ref. [47].

8.3 8.3.1

Vibration analysis of cylindrical shells including boundary conditions Two-point boundary value problem and perturbation approach

In this section, the effect of boundary conditions at the shell edges (including the effect of a nonlinear prebuckling or previbration state) on the buckling and vibration of an anisotropic shell is considered. The method can be classified as a Level 2 analysis or Extended Analysis (cf. section Introduction). The underlying theory will be explained more in detail in Chapter 10. It is well known that boundary conditions influence the restoring (incremental) bending moments and the stabilizing (incremental) membrane stresses during buckling and vibration. Moreover, the prebuckling or previbration bending deformation due to the edge restraint will be accompanied by destabilizing compressive membrane stresses in the circumferential direction of the shell. For shorter shells, the fundamental state (prebuckling or previbration state) may affect the buckling or vibration behavior considerably. In the following, a formulation will be presented for the nonlinear vibration analysis of composite cylindrical shells, in which the (uniform) elastic boundary conditions at the shell edge are rigorously satisfied. A method often applied for the analysis of buckling and vibration problems of shells of revolution under axisymmetric loads (including torsion) is to use a Fourier decomposition in the circumferential direction of the shell for the dependent variables to eliminate the dependence of the solution on the circumferential coordinate. The resulting boundary value problem of ordinary differential equations in the meridional direction is solved numerically. Initial value (shooting) techniques have often been applied for this purpose [48e51] and also in problems directly related to the present work [35,42]. Via shooting methods, accurate solutions can be obtained by the numerical integration of the differential equations using an initial value solver. In the present study the vibration behavior of composite cylindrical shells is investigated. In combination with a perturbation method to describe the temporal behavior, the parallel shooting method [52e54] is employed to solve the spatial two-point boundary value problems resulting from the Fourier decomposition of the dependent variables. In Refs. [3] and [7], (see also Chapter 10), a general theory is given for the vibration of structures about a static nonlinear fundamental state, based on a perturbation expansion for both the frequency parameter and the dependent variables. The theory includes the effects of finite amplitudes, imperfections, and a nonlinear static deformation (orthogonal to the fundamental state). The dependence of the frequency on these parameters is given for free vibrations. This theory is applied in this chapter to the nonlinear analysis of composite shells. In section Problem definition, the equations

Vibrations of composite shell-type structures

451

governing the static and dynamic states of composite cylindrical shells have been presented. By means of separation of variables, the problems governing the various states are reduced to a two-point boundary value problem (with the axial coordinate as the independent variable). The static and dynamic first-order states form eigenvalue problems. Values of the eigenvalue parameter are sought, for which a nontrivial solution exists. The formulae derived in Ref. [3] (see also Chapter 10) are generally applicable. They can also be used in a Level 1 (“Simplified”) Analysis, i.e., an analysis in which the displacement satisfies “simply supported” boundary conditions (cf. the Level 1 buckling “b-factor” analysis for composite cylinders in Ref. [55]). In the present analysis, this perturbation theory is applied to the nonlinear vibration problem of composite cylindrical shells including edge effects. It should be noted that the starting point differs from that in the previous section. The starting point of the present analysis is the Donnell-type differential equations of a shell with an axisymmetric imperfection only, and not the governing nonlinear equations including imperfections. The firstorder state problem is an eigenvalue problem for the unknown eigenfrequency and vibration modes. The associated higher-order state problems are response problems that depend on the solution of the corresponding first-order state problem. The perturbation procedure leads to boundary value problems for partial differential equations with the two spatial coordinates as independent variables. A Fourier decomposition in the circumferential direction of the shell is used for the dependent variables to reduce these problems to sets of ordinary differential equations for the axial direction. The specified boundary conditions are satisfied rigorously by solving the resulting two-point boundary value problems numerically via the parallel shooting method. Nonlinear Donnell-type equations formulated in terms of the radial displacement W and an Airy stress function F are used, and the classical lamination theory is employed. Numerical results are presented in Results and discussion section. The perturbation theory used is exact in an asymptotic sense [56]. In its original form, only the lowest-order effects of the nonlinearity are taken into account. In Ref. [3] (see Chapter 10), the theory is extended to a higher-order analysis of the nonlinear vibrations of (perfect) structures. The theory developed in Ref. [3] is related to the initial postbuckling and imperfection sensitivity theory for composite shells [42], in which the dependence of the load parameter on the deflection and imperfection amplitudes is obtained. The initial postbuckling theory can be extended to deal with dynamic buckling problems [57,58]. Furthermore, the linearized vibration problem forms the basis of the analysis to determine the dynamic instability regions under parametric axisymmetric or torsional loading. It can be noted, that the theory presented for free vibrations can be extended to forced vibrations [56,59]. In Ref. [59] the effect of viscous damping is also included. In Ref. [60] the stability of the response was investigated via a perturbation analysis.

8.3.2

Results and discussion

In this section, numerical results of the Level 2 analyses are presented. The data of the shells used in the calculations are given in Ref. [3]. The results of the Extended

452

Stability and Vibrations of Thin-Walled Composite Structures

Analysis have been presented in Refs. [3,4,7]. The notation used to specify the boundary conditions is given in Boundary conditions and applied loading section. They correspond to displacements relative to the “moving” edge plane. In all the examples to be presented, the boundary conditions are symmetric with respect to the shell midlength. The mode shapes are plotted for 0 < x < L=ð2RÞ. It should be noted that the nondimensional displacement w ¼ W/h is used. The (nondimensional) linear buckling and vibration modes are normalized by setting the maximum displacement equal to 1.

8.3.2.1

Buckling and linear vibration of anisotropic shells

In Table 8.4, buckling loads are tabulated for Booton’s anisotropic shell (with L/R ¼ 2) for eight standard boundary conditions. The following (loading) cases are considered: 1. axial compression (l ¼ (cR/Eh2)N0), where c and E are the reference values given in the nomenclature;      2. hydrostatic pressure p ¼ cR2 Eh2 p and l ¼ 12 p , i.e., a uniform pressure applied to the lateral surface   the ends of the cylinder;  as well as to 3. torsion s ¼ cR Eh2 T0 , both counterclockwise (corresponding to a positive sign) and clockwise; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   4. the lowest natural frequency u ¼ R ðrh=A22 Þ u has been tabulated.

For the weak boundary support Nxy ¼ 0 (the circumferential displacement v is unconstrained) the buckling load under axial compression is drastically reduced [61,62]. This support condition, however, is not likely to be encountered in practical applications [63]. In case of both hydrostatic pressure and natural frequency, the axial restraint u ¼ 0 has a strong influence on the incremental in-plane displacements u(1) and v(1), ð1Þ and thereby on the important (stabilizing) incremental membrane stress Ny .

Buckling loads and natural frequency of anisotropic shell for different boundary conditions. Number of circumferential waves between parentheses; s > 0 is counterclockwise and s < 0 is clockwise torque. Booton’s shell, L/R [ 2 [43]

Table 8.4

l

p, l [ 12 p

s>0

s> b), and thickness t and with elastic modulus E and Poisson’s ratio y is loaded in compression (see Fig. A.1) and has the following elastic critical stress (see, for example, Ref. [A1]) Edge A a

σ

b

b a Edge B

Figure A.1 A long flat plate having an AR ¼ a/b >> 1.

σ

610

Stability and Vibrations of Thin-Walled Composite Structures

Table A.1 The value of the plate buckling coefficient k for various boundary conditions Case no.

Edge A

Edge B

k

1

Simply supported

Simply supported

4.000

2

Simply supported

Clamped

5.420

3

Clamped

Simply supported

6.970

4

Simply supported

Free

0.425

5

Clamped

Free

1.277

Adapted from C. Collier, P. Yarrington, B. Van West, Composite, grid-stiffened panel design for post buckling using hypersizer®, AIAA-2002-1222, 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference: 10th AIAA/ASME/AHS Adaptive Structures Forum: 4th AIAA Non-Deterministic Approaches Forum: 3rd AIAA Gossamer Spacecraft Forum, Denver, Colorado, USA, 22e25 April 2002, 2002.

 2 p2 E t scr ¼ k 12ð1  y2 Þ b

(A.1)

where k is the buckling coefficient of the plate and resumes various values as presented in Table A.1. As the plate is long, the influence of the boundary conditions at x ¼ 0,a (assumed to be simply supported) on the buckling of the plate is negligible in relation with those at y ¼ 0,b. Before presenting the concept of “effective width,” it will be instructive to understand the buckling and postbuckling behaviors of a plate and a shell, as presented in Fig. A.2(a) and (b) from Ref. [2]. As known, the plate has a stable postbuckling behavior, enabling it to reach the plastic region of an isotropic material and collapsing at the ultimate load (see Fig. A.2(a)), in comparison with a shell (Fig. A.2(b)), which has a nonstable postbuckling behavior and is buckling at its bifurcation load, leading to a collapse load (ultimate load) at a lower value. In what follows, it is assumed that a plate is made of an isotropic material, having elastic and plastic regions. As the applied axial compression load is increasing (after the buckling of the plate), the stress distribution across the plate is no longer uniform leading to the transfer of the main axial load to the vicinity of the plate’s edges, as schematically depicted in Fig. A.3 from Ref. [A1] for a flat plate or in Fig. A.4 for a stringer-stiffened plate from Ref. [A3]. As stated in Ref. [A1], many effective width formulas have been derived, some empirical, based on approximate analyses, and some based on the large-deflection plate-bending theory, employing varying degrees of rigor. We will present some of the most important ones, such as the one developed by von Karman in 1932 [1] for a plate under uniform compression that is stiffened along the two edges parallel to

(b) Ultimate load

Load

(a)

Load

Test results on stability and vibrations of stringer-stiffened composite panels

611

Bifurcation level critical load

Load at plasticity Largest tolerable deformation Bifurcation level critical load

Load at plasticity Ultimate load

Largest tolerable deformation Initial imperfection

Initial imperfection w

w

Figure A.2 Buckling and postbuckling behaviors: (a) a plate-type structure and (b) a shell-type structure. Adapted from J.P. Martins, L.S. da Silva, A. Reis, Ultimate load of cylindrically curved panels under in-plane compression and bending-extension of rules from EN 1993-1-5, Thin-Walled Structures 77 (2014) 36e47. Actual stress distribution

σe

Region assumed to have a zero stress due to buckling 0.5be

0.5be

Actual stress distribution

σe

b t

Figure A.3 A flat plate: a schematic view of the stress distribution after buckling and effective width definition, be. Adapted from Guide to stability design criteria for metal structures, sixth ed., R.D. Ziemian (Ed.), Inelastic Buckling, Postbuckling and Strength of Flat Plates, John Wiley & Sons, Inc., 2010, pp. 145e163, (Chapter 4.3).

612

Stability and Vibrations of Thin-Walled Composite Structures

(a)

be Load

(b) More applied load leads to a narrow be

be

be1 be2 be3

Figure A.4 A stringer-stiffened plate: a schematic view of the stress distribution after buckling and effective width definition, be. Adapted from C. Collier, P. Yarrington, B. Van West, Composite, grid-stiffened panel design for post buckling using hypersizer®, AIAA-2002-1222, 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference : 10th AIAA/ASME/AHS Adaptive Structures Forum : 4th AIAA Non-Deterministic Approaches Forum : 3rd AIAA Gossamer Spacecraft Forum, Denver, Colorado, USA, 22e25 April 2002, 2002.

the direction of the applied compression load, assuming that the two strips along the edges carry the entire applied load " rffiffiffiffiffi # p E be ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t (A.2) 2 3ð1  y Þ se

Test results on stability and vibrations of stringer-stiffened composite panels

613

where se is defined in Fig. A.3. Another formula was proposed by Ramberg et al. [A4] in 1939 and is written as be ¼ b

rffiffiffiffiffiffi scr se

(A.3)

where se was defined in Eq. (A.1).9 Defining the average stress, sav, as sav ¼

be se b

(A.4)

and assuming se ¼ sy (where sy is the yield point stress), the average stress can be written as a function of the yield point stress sav ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi scr $sy

(A.5)

It is clear from Eq. (A.5) that for a given plate configuration the average stress is limited by the yield point stress, and therefore, based on Eq. (A.4), the effective width is to be determined using an iterative procedure. First the effective width is assumed to be a fraction of the width, b. Then the value of se is determined, using Eq. (A.3), and the average stress is calculated (Eq. A.4). Knowing the left-hand side of Eq. (A.5), while its right-hand side is a priori known, the validity of the equation is checked. If the equation does not hold, then another value of the effective width is taken till the equation is found to be true. Based on many experiments conducted on the postbuckling region, slightly different expressions for the effective width were suggested by Winter [A5] and Winter et al. [A6], which are written as be ¼ 1:9 t

rffiffiffiffiffi rffiffiffiffiffi  rffiffiffiffiffiffi rffiffiffiffiffiffi  E E t be scr scr ¼ 1  0:25 or 1  0:475 se se b b se se

(A.6)

The left-hand side of the expression in Eq. (A.6) can be rewritten as be 1:90 0:90 b  2 ; where B ¼ ¼ B B t b

rffiffiffiffiffi se E

(A.7)

A similar formula was developed and proposed by Conley et al. [A7] having the following form: be 1:82 0:82  2 ¼ B B b

9

By assuming k ¼ 4 and combining Eqs. (A.1) and (A.2), one can obtain Eq. (A.3) (see Ref. [1]).

(A.8)

614

Stability and Vibrations of Thin-Walled Composite Structures

_ B_ = 3.5 B= 3 _ .0 B= 2.5 _ B= 2.0 _ B = 1. 5

1.0 .0

_ =1 B

0.8 _ b B= t

0.6

σy E

σe σy

σaυ = Average stress σe = Maximum edge stress σy = Yield stress

0.4

σaυ

0.2

σe

t b 0

0

0.2

0.4

σaυ σy

0.6

0.8

1.0

Figure A.5 A chart to determine se/sy as a function of sav/sy. From Guide to stability design criteria for metal structures, sixth ed., R.D. Ziemian (Ed.), Inelastic Buckling, Postbuckling and Strength of Flat Plates, John Wiley & Sons, Inc., 2010, pp. 145e163, (Chapter 4.3).

Eq. (A.8) can be rewritten by including in it the stress at yield point, yielding sav 1:82 ¼ B sy

rffiffiffiffiffi rffiffiffiffiffi rffiffiffiffiffi sy b sy se 0:82 sav be and B h B  2 ; where ¼ ¼ sy se b se t E B

(A.10)

Eq. (A.10) is depicted in Fig. A.5 for values of B to obtain the ratio se/sy as a function of sav/sy. One should note that the expressions in Eq. (A.6) were slightly changes in the present American Iron and Steel Institute specifications for cold-formed steel member, as reported in Ref. [A1], to yield be ¼ 1:9 t

rffiffiffiffiffi rffiffiffiffiffi  rffiffiffiffiffiffi rffiffiffiffiffiffi  E E t be scr scr ¼ 1  0:22 or 1  0:415 se se b b se se

(A.11)

As reported in Refs. [A3,A8eA10], the industry method for calculating the effective width is to use the following equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be KE kp2 ; where K h ¼ ¼ 0:904k; for y ¼ 0:3 (A.12) scripple; stringer t 12ð1  y2 Þ

Test results on stability and vibrations of stringer-stiffened composite panels

615

where scripple, stringer is the crippling10 stress of the stiffener and k is the buckling coefficient (see Table A.1). For laminated composite plates, the effective width is calculated using the following expression (see Refs. [6,10]): be ¼ 3:96t t

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11 $D22 $Ex; c.; skin t 3 scripple; stringer $Ex; c.; stringer

(A.13)

where D11 and D22 are the laminate stiffness in the longitudinal and perpendicular directions, respectively; Ex, c., skin is the skin effective compressive modulus; and Ex, c., stringer represents the stringer effective compressive modulus. For a structure constructed from isotropic material, one obtains the following compact equation, which should give identical results to those using Eq. (A.12): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be D11 (A.14) ¼ 3:96 t tscripple; stringer Another approach to calculate the collapse of a stringer-stiffened curved and flat panels made from laminated composite materials is described in Ref. [A11]. The model of the cross section, including the effective width skin contribution to the cross-section of the stringer, is presented in Fig. A.6. The model adopted to calculate the effective width is the Marguerre [A12eA14] formula given by rffiffiffiffiffiffiffiffiffiffiffiffiffi scr; flat be ¼ 0:5 3 (A.15) bpanel sav where scr, flat is the critical buckling stress of a flat composite panel, whose width and laminate construction are equal to those of the real composite cylindrical panel (see Ref. [A12]). The expression for the average stress is given by (see Fig. A.6 for the notation of the variables) sav h

Pcr Pcr ¼ Panel Area ðb1 þ bdop Þtbf lange þ Atweb þ btupf lange þ 2be tskin

(A.16)

The boundary conditions of that flat panel are taken as CCeSS (clamped at the bottom and at the top and simply supported along its sides). Consequently, a first iteration effective width is determined. Introducing this effective width into Eq. (A.16) and by using Eq. (A.15) a new value of the effective width is obtained. This process is repeated until the effective width, be, obtained by Eq. (A.15) leads to the convergence of Eq. (A.16). As stated in Ref. [A11], it has been found that this process had to be 10

Compressive crippling or local buckling (usually occurring in the web of the stringer) is defined as an inelasticity of the cross section of a structural member in its own plane rather than along its longitudinal axis, as in column buckling. The maximum crippling stress of a member is a function of its cross section rather than its length.

616

Stability and Vibrations of Thin-Walled Composite Structures

(a)

W

R

W/n W/(2n) n -- number of stiffeners

(b)

Effective panel skin

2be t skin

Stringer upper flange b

Stringer web

A t b_flange

R-Average skin radius

t web

a1

t up_flange

Neutral axis

bdop

Stringer bottom flange

b1

Figure A.6 Laminated curved-stringer-stiffened panel: (a) test panel model (n, number of stiffeners) and (b) “equivalent” stringereskin cross section combination. From P. Pevzner, H. Abramovich, T. Weller, Calculation of the collapse load of an axially compressed laminated composite stringer-stiffened curved panelean engineering approach, Composite Structures 83 (2008) 341e353.

repeated three to five times until the value of Pcr converges. Once the average stress, the effective width, and the critical load are obtained, the collapse load of the whole panel and the stringers can be evaluated. One should check if sav < scr-panel, where scr-panel is the critical stress of the panel between two adjacent stringers (Fig. A.7(a)), or if sav > scr-panel (Fig. A.7(b)) for which the following expression can be written (see Ref. [A11] for the case sav < scr-panel): Pcrpanel ¼ n$Pcr þ ðn  1Þðbpanel  2be Þscrc $tskin þ 2be $sav $tskin where scrc


"     # 2 12Dskin tskin 1:6 tskin 1:3 11 1  y12 ¼ þ 0:16 9 3 R L tskin

(A.17)

Test results on stability and vibrations of stringer-stiffened composite panels

(a)

617

(b)

Figure A.7 Laminated curved-stringer-stiffened panel. (a) sav < scr-panel: buckling test of a panel with T-type stringers having a height of 15 mm and (b) sav > scr-panel: buckling test of a panel with T-type stringers having a height of 20 mm. From P. Pevzner, H. Abramovich, T. Weller, Calculation of the collapse load of an axially compressed laminated composite stringer-stiffened curved panelean engineering approach, Composite Structures 83 (2008) 341e353.

Based on the model developed in Ref. [A11] a computer code TEW (Technion Effective Width)11 with a user-friendly interface was developed. This version of the code is working in a MATLAB environment. Summarizing the topic of effective width and its applications, the reader is asked to refer to two extensive reviews on the topic [A15,A16], ways of dealing with nonhomogenous in-plane loading [A17], and how to use the concept of effective width to predict the collapse of panels having cutouts [A18].

References [A1]

[A2]

[A3]

[A4] [A5] 11

Guide to stability design criteria for metal structures, in: R.D. Ziemian (Ed.), Chapter 4.3 Inelastic Buckling, Postbuckling and Strength of Flat Plates, sixth ed., John Wiley & Sons, Inc., 2010, pp. 145e163. J.P. Martins, L.S. da Silva, A. Reis, Ultimate load of cylindrically curved panels under in-plane compression and bending-extension of rules from EN 1993-1-5, Thin-Walled Structures 77 (2014) 36e47. C. Collier, P. Yarrington, B. Van West, Composite, grid-stiffened panel design for post buckling using hypersizer®, in: AIAA-2002-1222, 43rd AIAA/ASME/ASCE/AHS/ ASC Structures, Structural Dynamics, and Materials Conference: 10th AIAA/ASME/ AHS Adaptive Structures Forum: 4th AIAA Non-Deterministic Approaches Forum: 3rd AIAA Gossamer Spacecraft Forum, Denver, Colorado, USA, 22e25 April 2002, 2002. W. Ramberg, A.E. McPherson, S. Levy, Experiments on study of deformation and of effective width in axially loaded sheet-stringer panels, NACA Tech. Note No. 684, 1939. G. Winter, Strength of thin steel compression flanges, Transactions ASCE 112 (1947) 527.

The code can be obtained upon request from the second author of the manuscript in Ref. [A11].

618

[A6]

[A7]

[A8] [A9] [A10]

[A11]

[A12]

[A13] [A14] [A15] [A16]

[A17]

[A18]

Stability and Vibrations of Thin-Walled Composite Structures

G. Winter, W. Lansing, R.B. McCalley, Four papers on the performance of thin walled steel structures, in: Engineering Experimental Strn, Rep. No. 33, Cornell University, Ithaca, NY, 1950, pp. 27e32, 51e57. W.F. Conley, L.A. Becker, R.B. Allnutt, Buckling and Ultimate Strength of Plating Loaded in Edge Compression: Progress Report 2. Unstiffened Panels, David Taylor Model Basin, Rep. No. 1682, 1963. M.C.Y. Niu, Airframe Structural Design, Conmilit Press Ltd., 1988. ISBN:962-7128-04-X. M.C.Y. Niu, Airframe Stress Analysis and Sizing, Conmilit Press Ltd., 1997. ISBN:9627128-07-4. C. Collier, P. Yarrington, P. Gustafson, B. Bednarcyk, Local post buckling: an efficient analysis approach for industry use, in: AIAA-2009-2507, 50th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, California, USA, 4e7 May 2009, 2009. P. Pevzner, H. Abramovich, T. Weller, Calculation of the collapse load of an axially compressed laminated composite stringer-stiffened curved panelean engineering approach, Composite Structures 83 (2008) 341e353. K. Marguerre, Die mittragende Breite der gedr€uckten Platte (The apparent width of the plate in compression), Translated in English as, Luftfahrt-Forschung 14 (3) (1937) 121e1288. NACA TM 833. E.E. Sechler, L.G. Dunn, Airplane Structural Analysis and Design, Dover Publications, Inc., 1963. E.E. Sechler, Elasticity in Engineering, John Wiley & Sons, Inc., 1952. D. Faulkner, A review of effective plating for use in the analysis of stiffened plating in bending and compression, Journal of Ship Research 19 (1) (March 1975) 1e17. Guide for the Buckling and Ultimate Strength Assessment of Offshore Structures, ABS, American Bureau of Shipping Incorporated by Act of Legislature of the State of New York 1862, Copyright © 2004 American Bureau of Shipping ABS Plaza 16855 Northchase Drive Houston, TX 77060 USA, April 2004 (Updated February 2014), 75 pp. O. Bedair, Analytical effective width equations for limit state design of thin plates under non-homogeneous in-plane loading, Archives Applied Mechanics 79 (2009) 1173e1189. M.W. Hilburger, M.P. Nemeth, J.H. Starnes Jr., Effective Widths of CompressionLoaded Plates with a Cutout, NASA TP-2000e210538, October 2000.

Test results on the stability and vibrations of composite shells

13

Haim Abramovich 1 , K. Kalnins 2 , A. Wieder 3 1 Technion, I.I.T., Haifa, Israel; 2Riga Technical University, Riga, Latvia; 3 Griphus e Aerospace Engineering and Manufacturing Ltd., Tel Aviv, Israel

13.1

Introduction

This chapter deals with the presentation of some typical experimental results on buckling and vibrations of composite shells, as well as some metal shells for reference. While the number of studies using numerical analysis and finite element (FE) solutions for the stability and vibration shells made from aluminum, steel, or composite materials is very high (see, for instance, the fundamental study by von Karman and Tsien [1], through the NASA reports on buckling of shells and cones, using both metal and orthotropic materials [2e5]), the number of well-presented experimental studies is much less. It is the aim of this chapter to highlight those experimental studies by pointing to their innovative add-on to the state of the art on stability and vibrations of shells.

13.2

Stability of shells

In what follows, a limited list of studies will be reviewed and their accomplishments will be highlighted. Evensen [6] describes a series of photographic sequences taken during the buckling of metal cylinders, showing the transition from the unbuckled state to the fully buckled pattern, while the loads are axial compression, torsion, external hydrostatic pressure, and a suddenly applied external pressure. Jones and Morgan [7] presented a basic numerical study of the buckling and vibration of simply supported circular cylindrical shells having unsymmetrical laminates, showing a decrease in the both the buckling loads and natural frequencies. From time to time, reviews were published in the literature on the recent developments of stability and vibration of shells, such as in [8e11]. Numerous other studies present experimental results for buckling of composite shells. A thorough investigation on the compressive failure of fiber composites was performed by Fleck [12], showing the various failure mechanisms in composites, such as elastic and plastic microbuckling, fiber crushing, splitting, buckle delamination, shear-band formation, and failure maps. Singer [13] advocates the need of extended experimental data regarding the buckling of shells, with other researchers, namely, Rikards et al. [14], Bisagni [15], Eglῑtis et al. [16], and Chitra and Priyadarsini [17], presenting well-performed tests on buckling of composite shells under static and dynamic loadings. Stability and Vibrations of Thin-Walled Composite Structures. http://dx.doi.org/10.1016/B978-0-08-100410-4.00013-2 Copyright © 2017 Elsevier Ltd. All rights reserved.

620

Stability and Vibrations of Thin-Walled Composite Structures

Another important report is the NASA TP-2009-215778 [18] in which the authors present simplified formulas backed by numerical calculations for buckling and stiffness design of compression loaded laminated composite cylindrical shells. The experimental research reported in Ref. [19] tries to reduce the scatter between tests and numerical predictions of buckling loads. Although the study deals with metal cylindrical shells being axially compressed, the main conclusion of the research was to correctly model the boundary conditions in an FE code to obtain very good buckling predictions between tests and numerical calculations (
1.53% from the test results). The same conclusion was presented in Ref. [20], in which advanced graphiteeepoxy composite cylinders under axial compression were tested to obtain the experimental buckling load (EBL) and to be compared to numerical [finite element method (FEM)] calculations. The authors claim that by introducing the actually measured initial geometric imperfections and the adequate material properties into the numerical code, they were able to accurately predict the EBLs. Besides cylindrical shells, cones were also tested and calculated as can be typically seen in Refs. [3,21]. The experimental approach, presented in Ref. [21], consisted of a composite tube-bending setup that had been designed and developed to study bending and buckling under bending load behavior of composite shells. The setup had been designed to apply equal bending moments at both the ends of the structure, simulating pure bending test conditions. Experimental results had been obtained for buckling under pure bending of composite conical shells, with good matching to the numerical calculations. Most manuscripts published in the literature deal with the buckling of thin-walled structures under axial compression. The study presented in Ref. [22] addresses a new loading case, pressure being applied on shell-type structures, and various tests being performed on those structures. In what follows the experimental campaign of two of the participants of the DESICOS1 project [the Riga Technical University (RTU) and the Technion] will be described and presented, enabling the reader to assess up-to-date experimental test setups, manufacturing of laminated and sandwich cylindrical and cone shells tested under axial compression. The structures involved in the DESICOS campaign were (1) thin-walled shells made from either laminated composite materials (IM7/8552) or steel and (2) cones manufactured from either monolithic composite materials (IM7/8552) or sandwich structures (IM7/8552 face sheets and ROHACELL 200WF core). The structures tested by the Technion were manufactured by GRIPHUS2. Table 13.1 presents data on the shells and cones manufactured by GRIPHUS within the DESICOS program. The layup technology in the DESICOS project was dictated by the material characterization: the IM7/8552 prepreg. A prepreg is a preimpregnated fabric. This means that the fabric arrives from the manufacturer already soaked with the resin. The prepreg 1

2

DESICOS: New Robust Design Guideline for Imperfection Sensitive Composite Launcher Structures, FP&-SPACE-2011-282522 [funded by the European Commission with the 7th Framework Programme (2007e2013)]. GRIPHUS is a small SME (Small, Medium Enterprise) located in Israel, which is specialized in designing and manufacturing composite materials.

Test results on the stability and vibrations of composite shells

621

Table 13.1 Cylindrical shells and truncated cones manufactured by GRIPHUS within the DESICOS program Structure name

ISS

VEB

SYLDA

Design#

D1

G1

D2

Type

Sandwich cylinder

Sandwich cone

Sandwich cylinder

Total no. of specimens

3

2

3

No. of specimens w/holes

0

0

1

R (mm)

350

115/350

350

L (mm)

358

270

700

Skin material

Hexcel IM7/8552

Skin layup

(30 ,

Number of layers/skin

3

Core material


30 ,

0 )

(0 , 0 , 60 ,
60 )

(19 ,
19 , 90 )

4

3

EVONIK Rohacell WF200

Core thickness (mm)

2.6

2.0

1.5

Total thickness [nominal; (mm)]

3.386

3.048

2.286

travels in a deep-freeze state so that the resin will not start curing. Choosing the prepreg material dictates the use of a freezer for storage, a clean room as a layup area, and an autoclave (an oven with pressure) for the curing of the structures. Other technologies and infrastructure were used for pre- and postlayup procedures. Before the beginning of the layup process, all the materials should be cut. These include the prepreg and core materials, as well as auxiliary materials used for curing the structures. All the fabrics are cut by hand tools, scissors and sharp knives, using aluminum templates to provide the correct geometry. The core material is cut by electrical saws. After the structures were demoulded from the layup tool, they were trimmed to final dimensions using pneumatic tools such as pneumatic saw and pneumatic router. All the trimming procedure was performed in a room equipped with ventilation and air filtering system. The drawings of the conical and cylindrical mandrels (the layup tool) are presented in Fig. 13.1. The manufacturing process of the cylinders was relatively simple, whereas the cones’ manufacture was very complicated. Because composite faces of the cone were manufactured by hand layups and not by filament wiring, the predesigned angles of the layup could not be realized. The problem came out for layers in which the fibers were laid in parallel to the longitudinal axis. In these layers the patterns are isosceles trapezes. In these trapezes only the fibers along the smaller base have full length. The fibers from each side are getting shorter and shorter while moving aside toward the trapeze bottom corners. Thus in each borderline between two trapeze sections, one gets an area with noncontinuous fibers. For the layers where the fibers should be

622 Stability and Vibrations of Thin-Walled Composite Structures

Figure 13.1 Typical mandrel drawings: (a) conical mandrel and (b) cylindrical mandrel.

Test results on the stability and vibrations of composite shells

623

Axis line ~11deg.

60deg.

Figure 13.2 The original designed angle of 60 degrees gives a parabolic line, while 11 degrees is the maximal angle before the lines become parabolic.

slanted, the implementation was even more complex. It was found that from a certain banking angle, when layering fibers from bottom (cone base) to the top, the fibers do not wrap around and up the cone, but get a parabolic “trajectory,” meaning they bend backward and return down. It was found that for the VEB (G1) design (see Table 13.1) the boundary angle was about 11 degrees related to meridian, whereas the design stated a 60-degree angle (see Fig. 13.2). The obvious solution for the problem was to keep the angle regarding the meridian constant, as a rhumb line (loxodromic, see Fig. 13.3). This demand clearly defines the use of filament-winding manufacturing technology; however, as the chosen manufacturing technology and materials was the use of plies, the presented solution was not feasible. The applicable method was therefore to split each layer not only to longitude trapezes but also to split each longitude trapeze to three sections (bottom, middle, and top). Moreover, each section would be laid with the same starting angle, which was set on 40 degrees instead of 60 degrees, and change gradually the angle to achieve an “average angle.” The trials to obtain that required angle are presented in Fig. 13.4.

0 ≤ t ≤ 2pi

0 ≤ t ≤ 8pi

x = t cos(6t)

x = 0.5exp(0.15t)cos(2t)

y = t sin(6t) z=t

y = 0.5exp(0.15t)sin(2t) z = 0.5exp(0.15t)

Figure 13.3 An example of a rhumb line (loxodromic) curve.

624

Stability and Vibrations of Thin-Walled Composite Structures

Sectioned slanted trapezes and parallelograms

(a)

(b)

(c)

(d) Layers bend a lot and will not be possible to reach trapeze upper base

Multiple doubling areas

(e) Less doubling areas. No voids in layer

1st & 2nd base lines (they look converging only due to photographing angle. In reality they are parallel)

Third row angle is less steep

5th trial - 40°(trapezes)

Figure 13.4 Various trials with the layups: (a) first trial, 60 degrees (trapezes); (b) second trial, 60 degrees (parallelograms); (c) third trial, 60 degrees (larger trapezes); (d) fourth trial, 40 degrees (trapezes); and (e) fifth trial, 40 degrees (trapezes).

Finally, a model was obtained (see Fig. 13.5) that served as the FE model used by Technion to predict the buckling load of the cone having the previously described layup. Another manufacturer of specimens was RTU (with the DLR being the third manufacturer with the DESICOS campaign), which was required to deliver two prototypesd one from Al/stainless steel and the other one from carbon fiberereinforced plastic. The geometric dimensions had been defined within WP-2 of the DESICOS project for delivering properly scaled-down structures for physical tests. The main challenge in this particular task was to design an appropriate prototyping mandrel for production of

Test results on the stability and vibrations of composite shells

625

Figure 13.5 The flattened final configuration described in a CAD model as a basis for preparation of the FE model prepared by the Technion.

the prepreg carbon fiber cylindrical specimen. Furthermore, a design/upgrade/assembly and verification of various means of nondestructive inspection for geometric imperfection (deflection), layup configuration, and thickness variation control equipment had been extensively elaborated during the project. As an initial step for the design of prototyping test specimens, a metallic mandrel with certain heat conduction capacity was a prerequisite for specimen production. The RTU mandrel had been designed, calculated for heat expansion, machined and delivered to RTU for the manufacturing of the shell. The original design drawing of the mandrel is presented in Fig. 13.6. The cylindrical tooling mandrel was

Figure 13.6 The OD500 RTU mandrel.

626

Stability and Vibrations of Thin-Walled Composite Structures

manufactured from a seamless steel pipe using precision machining and is equipped with four diaphragm internal walls and mounting circular holes at both ends designed for a 200 mm standard lathe chuck. A mandrel revolving machine had been designed and manufactured by RTU on-site. It was equipped with a threephase electric drive, a mechanical reduction to 14 rpm, and manual or foot-operated start/stop and reverse switch. Fig. 13.7 presents the D500mm mandrel installed on the revolving machine. The cylinder end plates were produced by using a computer numerical control (CNC) milling of EN AW-6082 T6 aluminum alloy of 20-mm-thick plate, equipped with stainless steel HeliCoil thread inserts for long-lasting application. Each end plate consists of a base plate, an internal ring, and four external segments (20 degrees per side conical surface) (see Fig. 13.8). The same set of parts made of PE500 plastic was used for the cylinder edge wedge casting in resinesand mixture. A mixture ratio of resin to sand 1:4 parts by weight was used for edge potting. Vinylester or epoxy resin, depending on availability, was used in combination with 0.5e1.0 mm fraction sand. Some corrections depending on final viscosity of the mixture were done to obtain desired workability and to avoid layering of the mixture. One of the innovations of the DESICOS project was the single perturbation load approach (SPLA) method aimed at simulating geometric imperfection and its influence on the buckling load of the tested specimen. A perturbation load application device was designed as a deadweight loading device, consisting of two push rods guided in linear rod bearings and deadweight being connected to the rods by the wire and pulley. The SPLA device was equipped with an HBM U9B 200N load cell for precise load measurement. The deadweight consisted of a series of steel weights attached on a central rod with a hook for suspending a pulling wire, as can be seen in Fig. 13.9(a) and (b). As shown, the SPLA setup consists of two (top and bottom) octagonal aluminum profile frames and vertical bars attached to those at the desired position to allow for perturbation load application in any position of two accessible sectors on both sides of the test frame. Taking into account some space restrictions in the region of the test frame columns and the linear variable differential transducer (LVDT) stands, which might interfere with the SPLA device, it was possible to cover at least two-thirds of the specimen surface, without rotating the specimen itself. Another important device, i.e., a geometry scanner system for measurements of initial geometry, SPLA-introduced imperfections, and buckling mode shapes, was designed and built as displayed in Fig. 13.10. It was based on a Panasonic HLG108 laser distance sensor with two axis software controllable servo drive (orbital rotation and elevation), yielding a measuring range of 20 mm and a resolution of 2.5 mm. The laser scanner mechanical assembly consists of (1) an orbital drive base unit composed of a slip ring bearing with a CNC-machined aluminum mounting plate equipped with a ring gear drive and servo-drive stepping motor; (2) a vertical drive unit composed of a two-column linear screw drive equipped with a belt drive and a servodrive stepping motor; (3) a laser sensor arm composed of a carbon fiber reinforced polymer (CFRP) one-piece laser sensor support arm manufactured from carbon fiber prepreg as a lightweight design to reduce vibrations and a rigid body motion during

Test results on the stability and vibrations of composite shells

627

Figure 13.7 The mandrel revolving machine with the D500 mandrel.

Figure 13.8 The CNC-machined end plates.

scanning; and (4) a control unit composed of a custom-built electronic board for the servo drive and the measurement signal conditioning. A Standard Hilgus USPC 3010 HF was modified to replace the Y-coordinate drive by a rotational drive to accommodate the cylinder’s scanning, as presented in Fig. 13.11. A cylinder support frame was manufactured from aluminum profiles, front and rear axles, lathe chuck on drive end, and a PLA6 cone passive end. To reduce inertia forces during the rotation of the cylindrical shells, the specimens were supported by machined foam end plates.

628

Stability and Vibrations of Thin-Walled Composite Structures

(a)

(b)

D40

D3

4

D40

0.5N 1.0N

2.0N

5.0N

10.0N

4 D6 5.75 11.5

23.0

57.5

110.0

Figure 13.9 (a) The schematic deadweight and (b) view on the SPLA device.

Figure 13.10 The internal content of the geometry scanner based on a laser distance sensor.

Figure 13.11 The ISSþJAVE D500 L1000 specimen installed for nondestructive testing (NDT) inspection.

Test results on the stability and vibrations of composite shells

629

Two basic test setups were used during the buckling experimental campaign: • •

a test setup for cylindrical shells a test setup for D500 ISSþJAVE shells.

The test setup for cylindrical shells consisted of two steel plates each of thickness d ¼ 530 mm and 25 mm attached either to one of the test frames (ZWICK Z100 or INSTRON 8002). The shell, prepared for testing, was mounted between those plates employing resin/sand load equalizing layers on the top and bottom interfaces. The D300 L440 ISSþJAVE shell specimen was mounted on the lower compression plate. The test setup for D500 ISSþJAVE shells was manufactured as a welded steel frame, designed to accommodate shells on three support points with spherical seats and to bypass INSTRON grip/load cell assembly without the need to detach it from the frame (see a detailed description at the end of ISSþJAVE manufacturing paragraph). The photogrammetry setup consisted of a turntable for a precise angular rotation (50 or 100 steps) of the specimen located in a vertical orientation and a digital camera on a tripod equipped with a remote switch, capable of releasing camera shooter at preselected time intervals (5 s). The camera settings were 10e25 MP DSLR, AV mode, f/10 aperture, manual focus mode, ISO 200e400, fixed focal distance lens (preferred) or fixed zoom distance (if zoom lens used), zoom distance as far telephoto as possible to reduce barrel distortion (usually barrel distortion is higher on the wide end but can depend on the type of lens used), remote switch obligatory (with timer to be very helpful), and adequate even natural lighting (f/10 requires a lot). The software used for data processing included the Autodesk Recap 3600, the Autodesk Memento project and the MeshLab. Three specimens representing the ISSþJAVE scaled specimen were manufactured employing different prepreg systems. Specimens nicknamed D300 L440 and D500 L700 were manufactured from Unipreg 100 g/m2, whereas the specimen D500 L1000 was produced from IM7/8552. Both systems were oven cured only. The general layup considered for ISSþJAVE consisted of 18 plies [0 , 45 ,
45 , 90 , 0 , 90 ,
45 , 45 , 0 ]sym for the shell structure with additional 12 plies [60 , 0 ,
60 ,
60 , 0 , 60 ]sym for the reinforcement area under the BLIS structure, see details in Fig. 13.12. The reinforcement transition was realized using three steps, 10 mm each, thus reducing the reinforcement thickness on the sides without a step on the top and the bottom of the patch, as can be seen in Fig. 13.13(a) and (b). The width of individual reinforcement plies was 120 and 140 mm for 60 and 160 and 180 mm for 0 . The height of the reinforcement patch was 200 mm. The lower edge of the shell was not reinforced. The support blocks were CNC-machined from EN AW 6082 aluminum blocks to the desired shape, formed as concave and convex two-piece structure, thus matching the shell geometry, and glued to the shell structure (on both outside and inside) using a polyurethane (PU) two-component glue and also bolted by DIN 912 M6 10.8 class bolts. The ISSþJAVE D300 L440 specimen was manufactured from Unipreg 100 g/m2 employing a seven-stage intermediate debunking, aiming at eliminating the slack in consecutive layers generated by eventually entrapped air. The general geometry of the ISSþJAVE D300 L440 specimen is shown in Fig. 13.14(a), while a view

Reinforced area under BLIS

600 mm

Ø 500 mm

1000 mm

200 mm

BLIS–like structure

Cross section

200 mm 50 mm

Figure 13.12 The geometry of the ISSþJAVE scaled-down specimen.

(a)

(b) Shell ISS + JAVE

Figure 13.13 The realized ISSþJAVE scaled-down specimen: (a) the reinforcement and the BLIS positioning and (b) the reinforcement transition arrangement.

(b) (a)

Shell D300 L440 dimensions (mm)

20

180

418

70

40

16

120

38

40

438 35 50

38

Figure 13.14 The ISSþJAVE D300 L440 shell specimen: (a) the general geometry and (b) the supporting blocks attached to the shell’s skin.

Test results on the stability and vibrations of composite shells

631

presenting the shell specimen equipped with the support elements and the required instrumentation can be seen in Fig. 13.14(b). The second shell, the ISSþJAVE D500 L700 specimen was also manufactured from Unipreg 100 g/m2 employing the same method as for the previous specimen, the ISS-JAVE D300 L440 shell. Three equally spaced reinforcement patches consisting of 12 additional layers were prepared and located on the shell before final vacuum bagging and curing. The area designated for the load introduction structure attachment was designed for a better adhesion using peel-off fabric. The cured shell was released from the mandrel and cut at both ends on the mandrel turning machine to the desired length. The ISSþJAVE D500 L700 shell specimen with mounted-on support blocks and casted sand-epoxy boundary box, which is prepared for instrumentation, can be seen in Fig. 13.15. The third shell, the ISSþJAVE D500 L1000 specimen, was manufactured from IM7/8552 using nine-stage intermediate debunking to eliminate slack in consecutive layers generated by entrapped air. Various stages of the manufacturing process are presented in Fig. 13.16. Three equally spaced reinforcement patches consisting of 12 additional layers were prepared and located on the shell before final vacuum bagging and curing. As before, the area designated for the load introduction structure attachment was prepared for better adhesion using peel-off fabric. The final debunking was performed employing perforated release film, breather, and vacuum bag for a duration of 1 h. Both D500 shells shared the same test setup for testing on the INSTRON 8002 test frame. The support ball-bearing joints consisted of 40 mm bearing ball and two conical seats at each side at each support point. Fig. 13.17 presents the support frame mounted on the INSTRON 8002 test frame during pretest assembly. As seen in Fig. 13.17, one of the ball bearings had a fixed length, whereas the other two were adjustable via helical washers to compensate for a possible nonparallelism of the surfaces of the test frame and upper shell edge. Ultrasound inspection of the D500 ISSþJAVE shells was carried out on the modified Hilgus USPC 3010 HF equipment for quality control and thickness measurement reasons. The final bitmap files were stitched from two separate scans because of limited USPC machine travel. The obtained bitmap file for the ISSþJAVE D500 L700 specimen (see Fig. 13.18(a)) can be used for thickness imperfection implementation in the FE model analyses. The thickness measurement histogram was added to obtain the range of the thickness imperfections and an average thickness value of 1.79 mm, which corresponded to a single-ply thickness of 0.994 mm, as displayed in Fig. 13.18(b). Ultrasound inspection of the ISSþJAVE D500 L1000 specimen was also carried out on modified Hilgus USPC 3010 HF equipment. The final bitmap file for the ISSþJAVE D500 L1000 shell is presented in Fig. 13.19(a). Its thickness measurement histogram shows an average thickness value of 2.31 mm, which corresponded to single-ply thickness of 0.128 mm (see Fig. 13.19(b)). The initial geometry imperfections were captured by both photogrammetry and Exascan laser scanner for the ISSþJAVE D500 L1000 shell and by photogrammetry only for the þJAVE D500 L700. The photogrammetry-measured imperfections were filtered with half cosine function to remove barrel distortion. Because of the difficulties

Figure 13.15 The ISSþJAVE D500 L700 specimen with attached support blocks and casted upper boundary.

Figure 13.16 The ISSþJAVE D500 L1000 specimen: various stages of the manufacturing process.

Test results on the stability and vibrations of composite shells

633

Figure 13.17 The support frame for the ISSþJAVE specimens, one of the lower ball bearings has a fixed length (right), the other two are adjustable (left and front) to compensate for nonparallel machine plates.

in processing some large cut-off for support blocks, wires, strain gages (SG), and additional thickness featured by reinforcement patches, only the upper cylindrical part of the ISSþJAVE structure can be represented as an open plot imperfection plot, see Fig. 13.20(a)e(c). The metallic unstiffened shells were manufactured from 0.5-mm-thick AISI 304 stainless steel sheet that is cold-rolled and plasma-welded to form cylindrical shells. Two additional shells of D ¼ 500 mm were manufactured from EN AW 6082 T6 aluminum alloy sheet by an overlap adhesive bonding. Three D ¼ 500 mm and one D ¼ 800 mm cylindrical specimens were manufactured; the former specimen had one weld, whereas the latter specimen had two welds (see Fig. 13.21(a) and (b)). All the steel specimens were manufactured with an inadequate tolerance to be fitted on internal rings, and hence, before casting into the resin/sand mix, all the cylinders were fitted on internal rings by means of cutting edge using short longitudinal cuts (approximately 10 mm each) at equal spacing along the circumference and forced onto the internal aluminum rings by hammering against the ring (thus introducing some plastic deformations to increase the length of the circumference). All the D ¼ 500 mm specimens were fitted that way; however, for the D ¼ 800 mm, the fitting also included threaded bolts forcing both end plates together. This approach considerably increase the initial geometric imperfections at one of the edges, producing “bottleneck”-type geometric imperfections near the casted edge. Initial geometric imperfection measurements for two cylindrical shells (SST_1 and SST_2) were carried out by internal laser scanner, whereas the Exascan external laser scan was also used for the SST_2 shell. Fig. 13.22 presents a typical unfiltered

634

Stability and Vibrations of Thin-Walled Composite Structures

Figure 13.18 Shell ISSþJAVE D500 L700: (a) ultrasound thickness imperfection and (b) thickness measurement histogram.

imperfection pattern for the SST_1 shell, captured by the internal laser scanner and plotted as an open plot graph. The measured imperfection amplitude-to-thickness ratio was a/t ¼ 1.86. Fig. 13.23 shows an unfiltered imperfection pattern for the SST_2 shell, captured by an internal laser scanner and plotted as open plot graph. The measured imperfection amplitude-to-thickness ratio was a/t ¼ 3.18. Fig. 13.24 presents the unfiltered imperfection pattern for the SST_2 shell, captured by Exascan laser scanner, converted to x, y, z format by the Autodesk Memento project and MeshLab software, and plotted as an open plot graph. The measured imperfection amplitude-to-thickness ratio was a/t ¼ 3.22 and has the same shape as in Fig. 13.23. Two more D ¼ 500 mm aluminum shells (nicknamed R29AL and R30AL) were manufactured as one piece, featuring longitudinal adhesive-bonded overlap joint (25 mm), because of the lack of possibility to weld a 0.5-mm-thick aluminum sheet.

Test results on the stability and vibrations of composite shells

635

Figure 13.19 Shell ISSþJAVE D500 L1000: (a) ultrasound thickness imperfection and (b) thickness measurement histogram.

EN AW 6082 T6 (0.5 mm thick) alloy sheet was used for the production of the cylindrical specimens. Overlap joint sections of both ends were machine-tapered to minimize joint thickness increment. The realized thickness increment as measured at the edges of the shell was observed to be about 50% (measured average thickness in the region of overlap joint was 0.8 mm). Huntsman ARALDITE 2011 general purpose structural two-component epoxy adhesive was used for the bonding process. A preabrasive treatment was carried out for the bonded surfaces. Adhesive bonding was carried out on a D ¼ 500 mm steel mandrel (used for the CFRP shell manufacturing, described earlier) with the assistance of a vacuum bag to press the aluminum sheet around the mandrel. The adhesive-bonded joint was pressed with clamps against the mandrel surface to facilitate uniformly compressed overlap bonded joint (Fig. 13.25). Both shell edges were preabrasive-treated on the mandrel for a 20 mm depth and casted in resin-epoxy mixture exploiting the same procedures as for other shells to form a rigid boundary ring. The initial geometric imperfection measurements for the two shells (R29AL and R30AL) were carried out by an internal laser scanner. Fig. 13.26 shows unfiltered imperfection pattern for the R29AL shell when measured by an internal laser

636

Stability and Vibrations of Thin-Walled Composite Structures

(a) 340

x, mm

0.784

0 –π

θ, rad

π

(b) 600

–0.753

x, mm

0.624

0 –π

θ, rad

π

(c) 600

–0.472

x, mm

1.327

0–π

θ, rad

π

–1.171

Figure 13.20 Initial geometric imperfection scans: (a) shell ISSþJAVE D500 L700 photogrammetry, (b) shell ISSþJAVE D500 L1000 photogrammetry, and (c) shell ISSþJAVE D500 L1000 Exascan laser.

(a)

(b)

Figure 13.21 (a) Shell D500 L500, t ¼ 0.5 mm, and (b) shell SST_2 D800 L800, t ¼ 0.5 mm.

scanner. The measured imperfection amplitude-to-thickness ratio was found to be a/t ¼ 0.96. Similar plots were also obtained for the other shell, R30AL. To deepen the understanding, additionally 14 composite shells with various layups (see Table 13.2) were manufactured from IM7/8552 prepreg. The manufacturing and

Test results on the stability and vibrations of composite shells

637

Y

Y X

X

Z

Figure 13.22 SST_1 shell: typical imperfection scan obtained using the internal laser scan system.

Y

Y X

X

Z

Figure 13.23 SST_2 shell: typical imperfection scan obtained using the internal laser scan system.

Y

Y X

Z

X

Figure 13.24 SST_2 shell: typical imperfection scan obtained using the Exascan laser scanner.

curing technology was the same as described earlier for the ISSþJAVE shells made of the IM7/8552 prepreg system. Note that shells R15 and R16 had a diameter of D ¼ 500 mm and a length of H ¼ 500 mm, whereas for shells R17eR28, D ¼ 300 mm and H ¼ 300 mm.

638

Stability and Vibrations of Thin-Walled Composite Structures

Figure 13.25 Aluminum shell edges casted in sand-resin rings.

Y

Y X

Z

X

Figure 13.26 R29Al shell: typical imperfection scan obtained using the internal laser scanner.

13.2.1

The test campaign during the DESICOS project

The Technion lab performed compression tests on four specimens: two sandwich cylinders, nicknamed SH-1 and SH-2, and two cones, C-1 and C-2. Their nominal dimensions are given in Table 13.3 and their associated photos are presented in Fig. 13.27. The layup of the cylindrical shells was [þ30 ,
30 , 0 , core, 0 ,
30 , þ30 ], while the layup of the cones was supposed to be [0 , 0 , þ60 ,
60 , core,
60 , þ60 , 0 , 0 ]. The core for the cylindrical shells was polyvinyl chloride (PVC) closed-cell foam ROHACELL WF200-HT 2.6 mm, whereas for the cones, it was

Test results on the stability and vibrations of composite shells

Table 13.2

639

The 14 additionally laminated composite cylindrical shells

#

Nickname

Layup

Total thickness (mm)

Ply thickness (mm)

1

R15

(24 ,
24 , 41 ,
41 )

0.5685

0.142125

2

R16

(24 ,
24 , 41 ,
41 )

0.5711

0.142775

3

R17

(0 , 45 )

0.3269

0.163450

4

R18

(0 , 45 )

0.3273

0.163650

5

R19

(0 , 60 ,
60 )

0.4274

0.142470

6

R20

(0 , 60 ,
60 )

0.4306

0.143530

7

R21

(0 , 60 ,
60 )

0.4491

0.149700

8

R22

(0 , 45 ,
45 )

0.4341

0.144700

9

R23

(0 , 45 ,
45 )

0.4492

0.149730

10

R24

(0 , 45 ,
45 )

0.4484

0.149470

11

R25

(24 ,
24 , 41 ,
41 )

0.5922

0.148050

R26

(24 ,


24 ,

41 ,


41 )

0.5685

0.142125

R27

(24 ,


24 ,

41 ,


41 )

0.5711

0.142775

R28

(24 ,


24 ,

41 ,


41 )

0.5580

0.139500

12 13 14

Nominal dimensions for the four tested specimens by Technion Table 13.3

Nominal length (height) (mm)

Nominal diametersa (mm)

Nominal core thickness (mm)

Nominal total thickness (mm)

C-1

310

700 (230)

2.0

3.048

C-2

310

700 (230)

2.0

3.048

SH-1

398

700

2.6

3.386

SH-2

398

700

2.6

3.386

Specimen Cones

Cylindrical shells a

For cones, the value in parentheses is the value of the small diameter.

PVC closed-cell foam ROHACELL WF200-HT 2.0 mm. The laminated composite material for both the cones and the cylindrical shells was HexPly 8552, an amine-cured toughened epoxy resin supplied with unidirectional IM7 carbon fiber, produced by HEXCELL. The measured properties (by Technion) are presented in Table 13.4.

640

Stability and Vibrations of Thin-Walled Composite Structures

(a)

(b)

Figure 13.27 The specimens tested by Technion: (a) VEB-type two sandwich cones (C-1 and C-2) and (b) ISS type two sandwich cylindrical shells (SH-1 and SH-2).

Technion measured properties of HexPly IM7/8552 UD carbon prepreg Table 13.4

Measured Mechanical property

Symbol

Unit

Average

Standard deviation

0 Tensile modulus

(E11)t

GPa

168.30

4.52

90 Tensile modulus

(E22)t

GPa

8.93

0.67

0 Compression modulus

(E11)c

GPa

154.3

6.86

90

(E11)c

GPa

8.08

0.72

In-plane shear modulus

G12

GPa

5.08

0.71

Major Poisson’s ratio

n12

e

0.303

0.02

0

(s11)t

MPa

2585

30.1

(s22)t

MPa

58

3.2

(s11)c

MPa

1052

12.8

(s22)c

MPa

244

43.4

s12

MPa

125

22.5

Tensile strength

90 0

Compression modulus

Tensile strength

Compression strength

90

Compression strength

In-plane shear strength

Test results on the stability and vibrations of composite shells

Table 13.5

641

Measured properties of HexPly IM7/8552 UD carbon

prepreg Measured (average values) Mechanical property

Present

COCOMATa

POSICOS

COCOMAT

ESA study

0 Tensile modulus

168.30

142.5

192.3

164.1

175.3

90

8.93

8.7

10.6

8.7

8.6

0 Compression modulus

154.3

e

146.5

142.5

157.4

90 Compression modulus

8.08

e

9.7

9.7

10.1

In-plane shear modulus

5.08

5.1

6.1

5.1

5.3

Major Poisson’s ratio

0.303

0.28

0.31

0.28

e

0 Tensile strength

2585

1741

2714

1741

2440

90

58

28.8

55.6

28.8

42

0 Compression strength

1052

854.7

1399.8

854.7

1332

90 Compression strength

244

282.5

250.15

282.5

269

Tensile modulus

Tensile strength

a

From C. Bisagni, R. Vescovini, DESICOS Technical Report, Deliverable 2.2: Design and analysis of test structures, 31/10/2014.

These properties were compared with other published properties of the same material, as presented in Table 13.5. The thickness distribution for the shells and cones, as measured by a point laser device, is presented in Figs. 13.28 and 13.29, respectively. As one can see from Figs. 13.28 and 13.29, the thickness distribution of the cylindrical shells SH-1 and SH-2 is evenly distributed, whereas for the cones C-1 and C-2 the variation is higher and without a defined pattern. The average thicknesses for SH-1 and SH-2 are 3.251 and 3.204 mm, respectively. For the cones C-1 and C-2 the average thicknesses are 2.768 and 3.860 mm, respectively, and this larger difference between the two specimens can be attributed to the way the two cones were manufactured. After measuring the thickness distributions, the specimens were equipped with strain gages, bonded back to back. Fig 13.30 presents the locations of the various strain gages for all the four tested specimens.

642

Stability and Vibrations of Thin-Walled Composite Structures

(a)

Shell SH-1 Point A (60 (mm) from the upper end)

(b)

Shell SH-2 Point A (60 (mm) from the upper end)

Point B (205 (mm) from the upper end)

Point B (200 (mm) from the upper end)

Point C (310 (mm) from the upper end)

Point C (340 (mm) from the upper end)

Figure 13.28 Shells (a) SH-1 and (b) SH-2: measured circumferential thickness distributions.

(a)

Cone C–1 BS

MS

(b)

Cone C–2 BS

MS

TS TS

Figure 13.29 Cones (a) C-1 and (b) C-2: measured circumferential thickness distributions.

All the specimens were casted in two end plates using epoxy and aluminum spheres to simulate clamped boundary conditions (see Fig. 13.31 for two of the tested specimens). The first shell to be tested was shell SH-1. It was tested on a Beckwood Press Company3 loading rig capable of delivering up to 2 MN. Fig. 13.32 presents the shell inside the testing rig and after its collapse. The shell was first loaded up to 150 kN and

3

www.backwoodpress.com.

Test results on the stability and vibrations of composite shells

Cylindrical shell SH-1

14 (4)

11 (1)

C 120°

12 (2)

18 (8)

13 (3) B 16 (6)

19 (9) C

A

A

(c)

11 (1)

C

B

15 (5)

16 (6)

120°

17 (7) C

15 (5)

14 (4)

120°

120°

18 (8)

17 (7)

B

120°

Locations of back-to-back strain gages (values in parentheses represent inner bonded strain gage)

120°

19 (9)

Cylindrical shell SH-2

(b)

Locations of back-to-back strain gages (values in parentheses represent inner bonded strain gage)

12 (2) 13 (3) B

A

A

(d)

Conical shell C-1 Locations of back-to-back strain gages (values in parentheses represent inner bonded strain gage)

C 120°

C 120°

B

120°

B 120°

12 (2) 13 (3) 11 (1) C B A

120°

12 (2) 13 (3) 11 (1) B C A

Conical shell C-2

Locations of back-to-back strain gages (values in parentheses represent inner bonded strain gage)

120°

(a)

643

A

A A

A

Figure 13.30 Shells (a) SH-1 and (b) SH-2 and cones (c) C-1 and (d) C-2: the various locations of the bonded strain gages (values in parentheses represent inner bonded strain gage). SH-1

C-1

End plates

Figure 13.31 Realization of clamped boundary conditions in shell SH-1 and cone C-1.

(a)

(b)

(c)

(d)

Figure 13.32 Shell SH-1: (a) the loading rig, (b) close view of the shell in the loading rig, (c) back of the shell, and (d) the shell after its collapse at 294.3 kN.

644

Stability and Vibrations of Thin-Walled Composite Structures

then unloaded. Then the shell was loaded till its maximal carrying load at 294.3 kN (30 ton) when it collapsed (see Fig. 13.32(d)) with a loud noise. The collapse was characterized by cracks and breaking of the sandwich skin both inside and outside the shell, with some of the cracks covering almost half of the circumference of the shell. Typical damage can be seen in Fig. 13.33. The experimental load versus end shortening is presented in Fig. 13.34. The uploading curve is characterized by a nonsmoothing behavior, probably due to noncontinuous-type loading. Typical readings of the strain gages are presented in Figs. 13.35e13.37 for compression, bending, and bending versus compression, respectively. The second sandwich shell to be tested was the SH-2 specimen. It was tested on a Beckwood Press Company loading rig, capable of delivering up to 2 MN. Fig. 13.38 presents various views of the shell inside the testing rig. The shell was first loaded up to 150 kN and then unloaded. Then the shell was loaded till its maximal carrying load at 267.25 kN where it collapsed with a loud noise. The collapse was again characterized by cracks and breaking of the sandwich skin both inside and outside the shell, with some of the cracks covering almost half of the circumference of the shell. Typical damage is shown in Fig. 13.39. The load versus end shortening is presented in Fig. 13.40. The uploading curve is characterized by a smoothing behavior. Typical readings of the strain gages are presented in Figs. 13.41e13.43 for compression, bending, and bending versus compression, respectively. One should note that shells SH-1 and SH-2 were externally painted in nonuniform white and then photographed from its entire circumference to enable the photogrammetry method, which will yield the initial geometric imperfections of the shells. Cone C-1 was the next specimen to be tested. Before testing, its load-carrying capability was calculated using the ANSYS FE code. Although this type of cone was already calculated during the preliminary phase of the design, the actual layup of this cone and its “twin” C-2 had to be changed because of the manufacturing techniques, described in detail in this chapter. The layup scheme presented earlier for the sandwich cones was approximated to eliminate overlaps of the layers, thus enabling to prepare a suitable FE model. Using eigenvalue analysis, the calculated buckling load was found to be 135.64 kN and its buckling mode is presented in Fig. 13.44. Note that the failure is expected to be on the top part of the cone, near its upper boundary conditions. Because of their predicted relatively low buckling load, the two cones were tested on an MTS loading rig, capable of delivering up to 100 kN. Fig. 13.45 presents the cone C-1 inside the testing rig. The cone was first loaded up to 20 kN and then unloaded. Then the cone was loaded till its maximal carrying load at 40.894 kN. During the loading, noise of formation of cracks could be heard. No buckling was observed, and the failure was due to the squashing of the top part of the cone. This squashing (crushing) was characterized by cracks and breaking of the sandwich skin both inside and outside the cone only at its upper part near the top end plate, similar with what had been numerically predicted. The bottom part of the cone remained intact without any visible damage. Typical damage on the top part of the cone while the bottom part remains intact is shown in Fig. 13.46.

Test results on the stability and vibrations of composite shells

(a)

645

Shell SH-1 postbuckling damage

External damage

(b)

Internal damage

Axial compression (kN)

Figure 13.33 Shell SH-1: typical (a) external and (b) internal damage after collapse at 294.3 kN. Shell SH1

350 300 250 200 150 100 50 0 0

–0.5

–1 –1.5 –2 End shortening (mm)

–2.5

–3

Figure 13.34 Shell SH-1: end shortening versus axial compression load.

The load versus end shortening is presented in Fig. 13.47. The uploading curve is characterized by a smoothing behavior. Note that the end shortening was taken from the loading rig machine, which might explain its shape at the starting of the loading (the machine closes gaps). Typical readings of the strain gages are presented in Figs. 13.48e13.50 for compression, bending, and bending versus compression, respectively. Note the clearly defined maximal compression load as captured by the strain gage readings. The cone C-2 was also tested on the MTS loading rig that is capable of delivering up to 100 kN. Fig. 13.51 presents the cone C-2 inside the testing rig. The cone was first loaded up to 20 kN and then unloaded. Then the cone was loaded till its maximal carrying load at 32.125 kN. During the loading, noise of formation of cracks could be heard. No buckling was observed, and the failure was due to the squashing of

646

Shell SH-1 SGs 3 & 13-compression 50

Shell SH-1 SGs 5 & 15-compression

500

100 150 200 250 300 350

–2000 –3000 –4000

0

50

–1000 –1500

–5000

–2000 Force (kN)

Shell SH-1 SGs 6 & 16-compression 50

0

100 150 200 250 300 350 Strain (μs)

Strain (μs)

Force (kN)

0

50

–1000

–1000 –1500

50 100 150 200 250 300 350

Shell SH-1 SGs 9 & 19-compression 0

100 150 200 250 300 350

–500

Shell SH-1 SGs 6 & 16-compression

Force (kN)

Shell SH-1 SGs 8 & 18-compression

–2000

0

50 100 150 200 250 300 350

–2000 –3000 –4000 –5000

–2500 Force (kN)

0 0 –1000 –2000 –3000 –4000 –5000 –6000 –7000

Force (kN)

Figure 13.35 Shell SH-1: strain gage readings for load versus compression strains.

Force (kN)

Stability and Vibrations of Thin-Walled Composite Structures

–6000

0 0 –1000 –2000 –3000 –4000 –5000 –6000 –7000

100 150 200 250 300 350

–500

Strain (μs)

0

Strain (μs)

Strain (μs)

–1000

0

Strain (μs)

0

1500 1000

Shell SH-1 SGs 7 & 17-bending

50

100 150 200 250 300 350

Force (kN)

0

50

100 150 200 250 300 350

0 –2000 –400 –600 –800 –1000 –1200 –1400 –1600 –1800

Shell SH-1 SGs 6 & 16-bending 50

100 150 200 250 300 350

Force (kN)

Force (kN)

Strain (μs)

Strain (μs)

200 100 0 –100 0 –200 –300 –400 –500 –600 –700 –800

3500 3000 2500 2000

500 0

Force (kN)

Shell SH-1 SGs 5 & 15-bending Strain (μs)

100 150 200 250 300 350

800 700 600 500 400 300 200 100 0 –1000 –200

Shell SH-1 SGs 8 & 18-bending

Strain (μs)

50

Strain (μs)

Strain (μs)

Shell SH-1 SGs 3 & 13-bending

50

100

150 200 250 300 350

Force (kN)

400 200 0 –2000 –400 –600 –800 –1000 –1200 –1400 –1600

Shell SH-1 SGs 9 & 19-bending

50

Test results on the stability and vibrations of composite shells

0 0 –200 –400 –600 –800 –1000 –1200 –1400 –1600 –1800 –2000

100 150 200 250 300 350

Force (kN)

Figure 13.36 Shell SH-1: strain gage readings for load versus bending strains.

647

648

–5000

–4000

–3000

–2000

–1000

0

–500 –1000 –1500

Shell SH-1 SGs 7 & 17-bending vs. compression

–2000

–1500

–1000

0

–500 –200 –400 –600

Bending strain (μs)

Bending strain (μs)

0 –2500

Compression strain (μs)

–800

–2000

Shell SH-1 SGs 9 & 19-bending vs. compression 500

400 200 0 –2000

–1500

–1000

–500

Compression strain (μs)

Figure 13.37 Shell SH-1: strain gage readings for bending versus compression strains.

–1500

Compression strain (μs)

600

–2500

–1000

500

Shell SH-1 SGs 8 & 18-bending vs. compression 800

200

0 –7000 –6000 –5000 –4000 –3000 –2000 –1000 0 –500

0 –5000

–4000

–3000

–2000

–1000

0 –500

–1000

0 –200

Compression strain (μs)

–1500

Stability and Vibrations of Thin-Walled Composite Structures

–2000

Compression strain (μs)

Shell SH-1 SGs 6 & 16-bending vs. compression Bending strain (μs)

–6000

Shell SH-1 SGs 5 & 15-bending vs. compression 3500 3000 2500 2000 1500 1000 500 0 –2000 –1500 –1000 –500 0 Compression strain (μs)

Bending strain (μs)

Bending strain (μs)

0

Bending strain (μs)

Shell SH-1 SGs 3 & 13-bending vs. compression

Test results on the stability and vibrations of composite shells

(b)

649

Shell SH-2

(c)

(a)

Figure 13.38 Shell SH-2: (a) in the loading rig, left view before loading; (b) the shell in the loading rig under loading, and (c) in the loading rig, right view before loading.

(a)

(b)

Shell SH-2

External damage

Internal damage

Figure 13.39 Shell SH-2: typical (a) external and (b) internal damage after collapse at 267.25 kN.

Stability and Vibrations of Thin-Walled Composite Structures

Axial compression (kN)

650

Shell SH2

300 200 100 0 0

–1 –2 End shortening (mm)

–3

Figure 13.40 Shell SH-2: end shortening versus axial compression load.

the top part of the cone. This squashing (crushing) was characterized by cracks and breaking of the sandwich skin both inside and outside the cone only at its upper part near the top end plate, similar with what had been numerically predicted. The bottom part of the cone remained intact without any visible damage. Typical damage occurred on the top part of the cone, whereas the bottom part remained intact, as shown in Fig. 13.52. The load versus end shortening is presented in Fig. 13.53. The uploading curve is characterized by a smooth behavior. Typical readings of the strain gages are presented in Figs. 13.54e13.56 for compression, bending, and bending versus compression, respectively. Note that cone C-2 was externally painted in nonuniform white and then photographed from its entire circumference to enable the photogrammetry method, which will yield the initial geometric imperfections of the cone. The summary of the test results and their comparison with the numerical predictions are presented in Table 13.6. As one can see, while the experimental loads for the sandwich cylindrical shells are lower than the predictions (0.605e0.666 of the predicted loads) with an acceptable knockdown factor (KDF) of 0.6e0.67, the cones present very low results (0.2368e0.3015 of the predicted loads), leading to very low KDFs of 0.24e0.30. This can be explained by a not enough and precise FE modeling of the composite faces of the sandwich structure for the cones and not enough protection of the lower diameter of the cone against crushing. The same behavior experienced by other partners of the DESICOS project during the testing of sandwich shells, namely, a total collapse of the shell or the cone with debonding between the skin and the core and breakage of the skin along the circumference of the tested specimen, was also found during the tests being performed at the Technion. The complete destruction of the specimen after the application of the compressive load prevented the reuse of the shell after removing the applied load, which is normally characteristic to monolithic composite cylindrical shells. The experimental activity of the RTU within the DESICOS project will now be outlined. As an initial step for designing prototyping test articles of the ISSþJAVE structure, a smaller version of D ¼ 300 mm specimen was manufactured and experimentally tested aiming at investigating the behavior of the specific boundary conditions experimentally realized as three-point support featuring spherical joints at each post, see Fig. 13.57.

2000

2000

1500 1000

Strain (μs)

2000

Strain (μs)

Strain (μs)

2500

1500 1000

500

500

0 50

100

150

200

250

300

0

50

Force (kN)

2000

2500

Strain (μs)

3000

1500 1000

200

250

0

300

0

50

100

0 200

Force (kN)

250

300

Shell SH-2 SGs 7 & 17-compression

–50

200

250

300

Shell SH-2 SGs 5 & 15-compression 3500

1000

0

150

Force (kN)

1500

500 150

150

2000

500 100

100

Strain (μs)

Shell SH-2 SGs 2 & 12-compression

50

1000

Force (kN)

2500

0

1500

500

0 0

Strain (μs)

Shell SH-2 SGs 9 & 19-compression

2500

3000 2500

Test results on the stability and vibrations of composite shells

Shell SH-2 SGs 3 & 13-compression

Shell SH-2 SGs 1 & 11-compression 2500

2000 1500 1000 500 0

0

50

100

150

200

Force (kN)

250

300

0

50

100

150

200

250

300

Force (kN)

Figure 13.41 Shell SH-2: strain gage readings for load versus compression strains.

651

652

Shell SH-2 SGs 3 & 13-bending 12

5

10 8 0

50

100

150

200

250

300

–10 –15

Strain (μs)

–5

6 4 2 0

–20

–2

0

50

150

200

250

300

6

Shell SH-2 SGs 7 & 17-bending

Strain (μs)

2 0 –2 0

50

100

150

–4 –6

100

150

200

Force (kN)

250

300

–8 –10

Force (kN)

Figure 13.42 Shell SH-2: strain gage readings for load versus bending strains.

100

150

200

250

300

Force (kN)

4

50

50

Force (kN)

Shell SH-2 SGs 2 & 12-bending

0

45 40 35 30 25 20 15 10 5 0 –5 0

200

250

300

5 4 3 2 1 0 –1 0 –2 –3 –4 –5 –6

Shell SH-2 SGs 5 & 15-bending

50

100

150

Force (kN)

200

250

300

Stability and Vibrations of Thin-Walled Composite Structures

Force (kN)

Strain (μs)

100

–4

–25

8 7 6 5 4 3 2 1 0

Shell SH-2 SGs 9 & 19-bending

Strain (μs)

0

Strain (μs)

Strain (μs)

Shell SH-2 SGs 1 & 11-bending 10

1500

2000

2500

–15 –20 –25

6 4 2 0 –2 0 –4 –6 –8 –10

Shell SH-2 SGs 2 & 12 bending vs. compression

500

1000 1500

2000

Compression strain (μs)

2500

3000

6 4 2 0 0 –2 –4 –6 –8 –10

Bending strain (μs)

Bending strain (μs)

Compression strain (μs)

500

1000

1500

2000

2500

Bending strain (μs)

1000

Shell SH-2 SGs 7 & 17 bending vs. compression

1000

1500

2000 2500 3000

40 30 20 10 0 –10

Compression strain (μs)

500

Shell SH-2 SGs 9 & 19 bending vs. compression

50

0

500

1000

1500

2000

2500

Compression strain (μs)

Shell SH-2 SGs 5 & 15 bending vs. compression

6 Bending strain (μs)

500

Bending strain (μs)

Bending strain (μs)

0 –50 –10

12 10 8 6 4 2 0 –2 0 –4

Shell SH-2 SGs 3 & 13 bending vs. compression

4

Test results on the stability and vibrations of composite shells

10 5

Shell SH-2 SGs 1 & 11 bending vs. compression

2 0 –2

0

500 1000 1500 2000 2500 3000 3500

–4 –6

Compression strain (μs)

Compression strain (μs)

Figure 13.43 Shell SH-2: strain gage readings for bending versus compression strains.

653

654

Stability and Vibrations of Thin-Walled Composite Structures

Figure 13.44 ANSYS FE results of cone C-1: theoretical buckling mode and its associated critical load Pcr ¼ 135.64 kN.

(a)

Cone C-1

(b)

Figure 13.45 Cone C-1: (a) in the loading rig under loading and (b) after maximal loading yielding a local damage.

Figure 13.46 Cone C-1: typical external and internal damage after maximal loading.

Test results on the stability and vibrations of composite shells

655

Cone C1

45 40

Axial compression (kN)

35 30 25 20 15 10 5 0.5

0

0

–0.5

–1 –1.5 –2 –2.5 End shortening (mm)

–3

–3.5

Figure 13.47 Cone C-1: end shortening versus axial compression load.

The ISSþJAVE D300 L440 shell features the same instrumentation arrangement as D500 shells, as presented in Fig. 13.27. The experimental tests were carried out on a ZWICK Z100 test frame, considering incremental displacement loading with a rate of 1 mm/min. The vertical translation of all three supports was recorded by three LVDTs. The load distribution along the upper shell edge was captured using nine equally spaced strain gages as depicted in Fig. 13.58, as well as by three pairs of back-toback strain gages arranged at the predicted buckle spots at 45 mm over the reinforcement patch upper edge. The interface between the upper edge of the shell and the machine plate was shimmed to obtain a uniform (as possible) load distribution up to 6 kN, while further loading could involve bending initiation due to the eccentric supports. Later on, it was discovered that such a kind of structure was not sensitive to the geometric and loading imperfections. FE analyses based on perfect geometry model were compared with the experimental curve, as presented in Fig. 13.59, yielding a very good matching up to an axial load of 60 kN. The EBL was 61.8 kN, whereas nonlinear FE (assuming a perfect geometry) calculations predicted a buckling load of 68 kN. The ISSþJAVE D500 L700 shell was tested on an INSTRON 8802 test frame, considering an incremental displacement loading with a rate of 1 mm/min. The shell was equipped with nine strain gages equally spaced along the circumference near the top loading edge used to measure the uniformity of the loading, as shown in Fig. 13.60. Additionally back-to-back strain gages were located at the predicted buckle spots 45 mm above the support reinforcement patch upper edge (see Fig. 13.61) aiming at detecting the buckling of the shell. The shell was also equipped with three LVDTs located in-line with the supports, as presented in Fig. 13.62. Two adjustable support columns allowed the aligning of the shell’s upper edge to the machine plate, facilitating a simple shimming of the interface.

656

0 10

20

30

40

50

–100 –150 SG 1

–200

SG 11

–250 –300

10

20

30

–100 –150

SG 2

–250 Force (kN)

0

50

SG 12

0

–40

10 SG 3

–60

50

50 0 –50 0 –100 –150 –200 –250 –350 –400 –450

20

30

40

50

SG 13

–80 –100 –120 –140

10

20

30

–300

–200

–300

40

Strain (μs)

Strain (μs)

10

40

Force (kN) Cone C-1: SG 3 & 13 - compression

Cone C-1: SG 2 & 12 - compression

0 0

30

Force (kN)

Cone C-1: SG 1 & 11 - compression

–50

20

–20

Force (kN) 50

Cone C-1: SG 3 & 13

20

40

20 0

50

0

10

20

30

–20 –40 –60 –80 –100

Force (kN)

Figure 13.48 Cone C-1: strain gage readings for load versus compression strains.

Force (kN)

40

50

Stability and Vibrations of Thin-Walled Composite Structures

–350

Cone C-1: SG 2 & 12

Strain (μs)

0

Strain (μs)

Strain (μs)

–50

50 0 –50 0 –100 –150 –200 –250 –300 –350 –400 –450 –500

Strain (μs)

Cone C-1: SG 1 & 11

50

Test results on the stability and vibrations of composite shells Cone C-1: SG 1 & 11-bending

10

10

20

30

40

50 Strain (μs)

Strain (μs)

0

Cone C-1: SG 2 & 12-bending

14

0 –10

657

–20 –30

12 10 8 6 4 2

–40

0 0 –2

–50 –60

10

20

30

40

50

Force (kN)

Force (kN) Cone C-1: SG 3 & 13-bending 5

Strain (μs)

0 –5

0

10

20

30

40

50

–10 –15 –20 –25 –30

Force (kN)

Figure 13.49 Cone C-1: strain gage readings for load versus bending strains.

0

Cone C-1: SG 2 & 12-bending vs. compression 14

Cone C-1: SG 1 & 11-bending vs. compression

0

10

20

30

40

50

–10 –20 –30 –40

12

Bending strain (μs)

Bending strain (μs)

10

10 8 6 4 2

–50 –60

–500

–400

Compression strain (μs)

–300

–200

–100

0 –2

0

Compression strain (μs)

Cone C-1: SG 3 & 13-bending vs. compression 5 0 Bending strain (μs)

–100

–80

–60

–40

–20

0

20

–5 –10 –15 –20 –25 –30 Compression strain (μs)

Figure 13.50 Cone C-1: strain gage readings for bending versus compression strains.

100

(a)

Cone C-2

(b)

Figure 13.51 Cone C-2: (a) views in the loading rig before loading and (b) views in the loading rig under loading. Cone C-2

Figure 13.52 Cone C-2: typical damage after maximal loading.

Test results on the stability and vibrations of composite shells

659

Cone C2

35

Axial compression (kN)

30 25 20 15 10 5 0

0

–0.5

–1

–1.5

–2 –2.5 –3 –3.5 End shortening (mm)

–4

–4.5

–5

Figure 13.53 Cone C-2: end shortening versus axial compression load.

Numerical simulations were carried out using the ANSYS FE code yielding a predicted buckling load of 39.96, while assuming a perfect shell. A better numerical prediction was introduced by taking into account the nonuniform load distribution using the actually measured LVDT translations of the corresponding supports as presented in Figs. 13.63 and 13.64, yielding a predicted buckling load of 39.70 kN, as compared to the EBL of 39.84 kN. The next shell, namely, the ISSþJAVE D500 L1000 specimen, was also tested on the INSTRON 8802 test frame by using an incremental displacement loading of 1 mm/ min. The specimen was equipped with nine strain gages (same locations like the previous two shells) equally spaced along the circumference of the shell near the top loading edge for checking the loading uniformity. Additionally back-to-back strain gages were bonded at predicted buckle spots 45 mm above the support reinforcement patch upper edge aimed at detecting the buckling of the shell. The shell was also equipped with three LVDTs located in-line with the supports. Two adjustable support columns allowed the aligning of the shell upper edge to the machine plate to facilitate a simple shimming of the interface. Numerical simulations were carried out using the ANSYS FE code predicting a buckling load of 97.45 kN, while introducing the nonuniform load distribution based on actually measured LVDT translations at the corresponding supports yielded a buckling load of 95.68 kN, as presented in Figs. 13.65 and 13.66. The EBL was found to be 93.57 kN. The SPLA procedure described earlier was applied on the ISSþJAVE D500 L1000 specimen at two positions: directly in the middle of the experimental buckle and 120 mm above it, with the maximum possible perturbation load of P ¼ 200 N. No influence of the perturbation load on the buckling load was observed for both the cases. It was decided that the ISSþJAVE type structures are insensitive to the SPLA procedure. Tests on unstiffened stainless shells were performed next. Three specimens of D ¼ 500 mm were manufactured using the same manufacturing technique as

660

Cone C-2: SG 1 & 11

–50 5

10

SG 1

15

20

25

30

35

SG 11

–150 –200

10

15

20

25

30

35

–100 –150 –200

SG 2

SG 12

–250 –300 –350

–300

–400

Force (kN)

10

15

20

25

30

Cone C-2: SG 2 & 12 - compression 35

–50

–50 Strain (μs)

–150 –200

5

10

15

20

–150 –200

Force (kN)

25

30

20

25

30

35

SG 13

Cone C-2: SG 3 & 13 - compression 35

–50

0

5

10

15

20

–100 –150 –200 –250

–250

–300

–300

–350 –400

–350

–250

SG 3

15

0 0

–100

–100

10

Force (kN)

0 5

Cone C-2: SG 3 & 13 5

Force (kN)

Cone C-2: SG 1 & 11 - compression 0

0 –50 0 –100 –150 –200 –250 –300 –350 –400 –450

Force (kN)

Figure 13.54 Cone C-2: strain gage readings for load versus compression strains.

Force (kN)

25

30

35

Stability and Vibrations of Thin-Walled Composite Structures

–250

0

Strain (μs)

5

Strain (μs)

0

0

Strain (μs)

–50

Strain (μs)

Strain (μs)

0

–100

Cone C-2: SG 2 & 12

0

50

Test results on the stability and vibrations of composite shells Cone C-2: SG 1 & 11 - bending

20

0

5

10

15

20

25

30

35

–20

Strain (μs)

Strain (μs)

–10

0 –10

Cone C-2: SG 2 & 12 - bending

0

10

–30

661

0

5

10

15

20

25

30

35

–20 –30 –40 –50

–40 –50

–60

Force (kN) 0

Strain (μs)

–10

Force (kN)

Cone C-2: SG 3 & 13 - bending 0

5

10

15

20

25

30

35

–20 –30 –40 –50 –60 –70

Force (kN)

Figure 13.55 Cone C-2: strain gage readings for load versus bending strains.

Cone C-2: SG 2 & 12 bending vs. compression 20

–250

–200

–150

–100

0

–50

–10 –20 –30 –40

0

–20 –30 –40 –50 –60 Compression strain (μs)

Bending strain (μs)

Cone C-2: SG 3 & 13 bending vs. compression –400

–300

0 –10

–50

Compression strain (μs)

0

–350 –300 –250 –200 –150 –100 –50

10

Bending strain (μs)

Bending strain (μs)

Cone C-2: SG 1 & 11 bending vs. compression

–200

–100

0 –10

0

–20 –30 –40 –50 –60 Compression strain (μs)

–70

Figure 13.56 Cone C-2: strain gage readings for bending versus compression strains.

Sandwich cylinders and cones: experimental results and numerical predictions Table 13.6

Specimen

Maximal load (kN)

Predicted load (kN)

SH-1

294.300

442.00a

SH-2

267.250

442.00a

C-1

40.894

135.64

C-2

32.125

135.64

a

From C. Bisagni, R. Vescovini, DESICOS Technical Report, Deliverable 2.2: Design and Analysis of Test Structures, 31/10/2014.

Figure 13.57 The experimental setup used to test the ISSþJAVE D300 L440 composite shell.

SUP 3 8

7

6

5

4

3

2

6.

SP3 out/SP3 in

SP2 out/SP2 in

ut 2o SP P2 in S

3

P2 SU

P1

SUP 2

SU

SUP 3

9. SP SP 1 out 1 in

SUP 1

8

Short mid

4

SP1 out/SP1 in

7

SP3 out SP3 in

5

9

1

1

2

Figure 13.58 The strain gages arrangement for the ISSþJAVE D300 L440 specimen.

Test results on the stability and vibrations of composite shells

663

ISS+JAVE D300 L400 specimen

80 70

Load (kN)

60 50 40 Experiment 30 FEM

20 10 0 0.00

0.50

1.00 1.50 2.00 End-shortening (mm)

2.50

3.00

Figure 13.59 Load versus end-shortening curves for the ISSþJAVE D300 L440 specimen.

SUP 2

2

3

4

5

6

7

8

4

SP2 out/SP2 in

SP3 out/SP3 in

ut 3o SP P3 in S

7

P3 SU

P1

SUP 3

SU

SUP 2

1 SP 1 SP out 1 in

SUP 1

2

Short mid

6.

SP1 out/SP1 in

3

SP2 out SP2 in

5

1

9.

9

8

Figure 13.60 The strain gage arrangement for the ISSþJAVE D500 L700 specimen.

described earlier. Two of them were used as training specimens to better understand specimen buckling behavior and to obtain knowledge on expected buckling load, as well as to check and tune-up the experimental setup. As there are no data on initial geometric imperfections for those two specimens, the only output from the performed tests was buckling loads. The third one, designated SST_1 specimen, passed the initial imperfection measurement, as well as its implementation into an FE. The SST_1 shell was tested in the ZWICK Z100 test frame (see Fig. 13.67) under displacement control (at 0.5 mm/min), which was equipped with three equally spaced (1200) LVDTs for end-shortening readings along with independent data from the Zwick system, the load-cell force measurement. Two types of tests were carried out during the test campaign: application of the vibration correlation technique (VCT) test up to first buckling (see more data at the end of this chapter) and buckling tests (application of the SPLA with a perturbation load of Pload ¼ 43 N and buckling test up to failure).

664

Stability and Vibrations of Thin-Walled Composite Structures

Figure 13.61 The instrumentation for the ISSþJAVE D500 L700 specimen.

Figure 13.62 The test setup used for the D500 ISSþJAVE specimens.

Test results on the stability and vibrations of composite shells

665

ISS+JAVE D500 L700 specimen

45 40 35 Load (kN)

30 25

Experiment FEM perfect model

20

FEM + loading imperf.

15 10 5 0 0.00

0.50

1.00

1.50 2.00 End-shortening (mm)

2.50

3.00

Figure 13.63 Load versus end-shortening curves for the ISSþJAVE D500 L700 specimen.

(b)

The three supports translations

45

45

40 Load (kN)

Distribution of the loading edge strains

35 30 25 20 15 10

SUP1

40

SUP2

35 30 25

SUP3

20 15 10

Average support 1,2,3

5 0 0

0.5

1 1.5 2 Displacement (mm)

(c)

2.5

1 2 3 4 5 6 7 8 9

Load (kN)

(a)

5 0 –800

–600

–400 –200 0 Microstrains (m/m)

200

Back-to-back strain distributions 45

35 30 25 20

Load (kN)

40 SP1 out SP1 in SP2 out

15

SP2 in

10

SP3 out

5

SP3 in

0 –5000 –4000 –3000 –2000 –1000 0 Microstrains (m/m)

1000

2000

3000

Figure 13.64 Experimental results of the ISSþJAVE D500 L700 specimen: (a) the three supports translations, (b) the strain gage readings at the loading edge, and (c) back-to-back strain gage readings.

400

666

Stability and Vibrations of Thin-Walled Composite Structures

ISS+JAVE D500 L1000 specimen 100

Load (kN)

80

Experiment FEM perfect model

60

FEM perf + loading imp.

40 20 0 0.00

0.50

1.00 1.50 2.00 End-shortening (mm)

2.50

3.00

Figure 13.65 Load versus end-shortening curves for the ISSþJAVE D500 L1000 specimen.

Load (kN)

(b)

The three supports translations

100 90 80 70 60 50 40 30 20 10 0

Distribution of the loading edge strains 100 90 80 70

SUP1

60 50 40 30

SUP2 SUP3 Average support 1,2,3 0

0.5

1

2.5 1.5 2 Displacement (mm)

(c)

3

3.5

1 2

Load (kN)

(a)

3 4 5 6 7 8

20 10 0 –800

–600

–400 –200 Mictostrains (m/m)

9 0

200

Back-to-back strain distribution 100 90 80

SP1 out Load (kN)

70 60 50 40 30

SP3 out SP3 in

10 0 –4000

–3000

–2000

–1000

SP2 out SP2 in

20

–5000

SP1 in

0

1000

Microstrains (m/m)

Figure 13.66 Experimental results of the ISSþJAVE D500 L1000 specimen: (a) the three supports translations, (b) the strain gage readings at the loading edge, and (c) back-to-back strain gage readings.

Test results on the stability and vibrations of composite shells

667

Figure 13.67 The test setup for the SST_1 specimen.

Three buckling tests were performed on the SST_1 shell: local buckling (reversible) was observed during the VCT experiment loading at 70.2 kN, located near weld line. The second run was with an SPLA Pload ¼ 43 N applied at 180 relative to the imperfection measurement origin. The obtained buckling load of the shell in the presence of the SPLA load was 47.1 kN. The local buckling was observed near the weld line (as before). The SPLA approach does not introduce local buckling at the location of the applied load, Pload. An experiment up to collapse was carried out without the perturbation load. A local buckling was observed at 51.8 kN at the same location, followed by a collapse (global buckling pattern) at 78.2 kN. The use of the NASA SP80074 predicted (KDF ¼ 0.32) a buckling load according to their lower bound at 60 kN. The first buckling (local) predicted by a nonlinear FE analysis, including initial geometric imperfections, was 80 kN with a global collapse at 98 kN (see details in Fig. 13.68). Fig. 13.69 presents the postbuckling mode of the shell, as recorded using the photogrammetry approach versus the real buckling mode as captured photographically. A good matching between the two patterns can be observed. A spring back phenomenon was observed during the experimental tests. Fig. 13.70 presents that although the axial load drops at the buckling point, the translation measurements pass high amount of translation in a relatively short period. These

4

Ref. 3: P. Seide, V.I. Weingarten, J.P. Peterson, Buckling of thin-walled circular cylinders, NASA Technical Report SP-8007, August 1968.

668

Stability and Vibrations of Thin-Walled Composite Structures

100 90 80 Load (kN)

70 SST_1 specimen

60 50

FEM ABAQUS (Geom. Imp)

40

EXP (first buckle)

30

EXP (SPLA 43N)

20

EXP (up to collapse)

10

NASA SP-8007 (KDF = 0.32)

0 0

0.05

0.1

0.15 0.2 0.25 End-shortening (mm)

0.3

0.35

0.4

Figure 13.68 Shell SST_1: experimental curves along the nonlinear imperfect FE prediction and NASA SP-8007 lower bound KDF level.

500

z, mm

5.12

–10.74

0 1.172

3.096

θ, rad

Translation (mm)

Figure 13.69 Shell SST_1: photogrammetry-based buckling pattern versus experimental buckling mode.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

SST_1 specimen: LVDT vs. time LVDT 1

LVDT 2

100

LVDT 3

200 300 Time (sec)

400

500

Figure 13.70 The spring back phenomena found during the tests on the SST_1 shell.

Test results on the stability and vibrations of composite shells

669

phenomena can be explained by the spring back effect of the upper and lower machine steel plates (25 mm thickness), which have 100 mm cantilever behavior relative to the thicker and more stiff spacer rings having a diameter of 300 mm. This phenomenon was not observed for shells having a diameter of 300 mm. As mentioned earlier, three more steel specimens having a diameter of D ¼ 800 mm were manufactured employing the same manufacturing technique. Two of them were manufactured with a reduced height H ¼ 560 mm and were used as “training” specimens. The third one designated SST_2 specimen passed initial geometric imperfection measurements, as well as its implementation into an FE model. The SST_2 shell was tested in the INSTRON SATEC 600 kN test frame under displacement control (0.5 mm/min) and was equipped with three equally spaced (1200) LVDTs for end-shortening readings along with an original INSTRON load cell for force measurements, linked to the MGC plus system as 10VDC sensor (see Fig. 13.71). The SST_2 shell was equipped with six strain gages at each edge equally spaced across the circumference as depicted in Fig. 13.72. Two types of tests were carried out during the test campaign: VCT test up to 45 kN load level (see results below) and a series of buckling tests (SPLA Pload ¼ 43, 38, 33, 27, 22, 16 N and a buckling test up to the failure). A local buckle (reversible) was observed during the VCT experiment at loads higher than 25 kN, located near the weld line. SPLA runs with Pload ¼ 43, 38, 33, 27, 22, 16 N applied at
90 relative to the imperfection measurement origin. The obtained buckling load was 49.9 kN for Pload ¼ 43 N, see Figs. 13 73 and 13.74. As before, a local buckle, located near the weld line, was observed. It was concluded that the SPLA approach does not introduce first local buckling at the location of Pload. An experiment up to collapse was carried out with a perturbation load of Pload ¼ 10 N.

Figure 13.71 The test setup for the SST_2 shell.

Stability and Vibrations of Thin-Walled Composite Structures

2

3

7

8

9

4

5

6

10

11

12

Weld line

1

Weld line

670

Figure 13.72 Shell SST_2: the arrangement of the bonded strain gages. Specimen SST_2 : SPLA tests

60

Load (kN)

50 Pload = 43 N

40

Pload = 38 N 30

Pload = 33 N Pload = 27 N

20

Pload = 22 N Pload = 16 N

10 0

Pload = 10 N 0

0.05

0.1 0.15 End-shortening (mm)

0.2

0.25

Figure 13.73 Shell SST_2: the SPLA experimental curves. Specimen SST_2:experimental KDF using SPLA approach 46

Load (kN)

45 44 43 42 41 40 0

5

10

15

20 Pload (N)

25

30

Figure 13.74 SST_2 shell: experimental KDF curve using the SPLA.

35

40

Test results on the stability and vibrations of composite shells

671

Specimen SST_2 90 80 70 Load (kN)

60 50 40

FEM ABAQUS PERF

30

FEM ABAQUS (Geom. Imp)

20

EXP (SPLA = 10 N up to collapse) EXP (SPLA = 43 N)

10 0 0

NASA SP–8007 (KDF = 0.25)

0.05

0.1

0.15

0.2 0.25 0.3 0.35 End-shortening (mm)

0.4

0.45

0.5

Figure 13.75 SST_2 shell: experimental curves for first buckling at perturbation load Pload ¼ 43 N and test up to collapse in the presence of a perturbation load of Pload ¼ 10 N, along with nonlinear predictions for perfect and imperfect shell (using a laser scan).

A local buckle was observed at 44.6 kN (transition) and 50.2 kN (local buckle) at the same location as described earlier, followed by collapse (global buckling pattern) at 60.3 kN. Using the NASA SP-8007 report, the predicted (KDF ¼ 0.25) buckling load according to the lower bound was at 47 kN. The first buckle (local buckle) predicted by nonlinear FE analyses, including initial geometric imperfections, was at 63 kN with a global collapse at 74 kN (71 kN according to inclusion of the measured imperfections measured by Exascan), as depicted in Fig. 13.75. Note that nonlinear FE simulations were carried out employing measured geometric midsurface imperfections (using Exascan and internal laser scanner). The results of two additional unstiffened isotropic shells, manufactured from EN AW 6082 T6 aluminum alloy 0.5-mm-thick sheet, were produced by adhesive overlap bonding (25 mm). Both shells (nicknamed R29AL and R30AL) had a diameter of D ¼ 500 mm with a free length of 460 mm. Both shells were loaded in steps according to the VCT requirements up to buckling. Because of the plastic deformation at buckling, experimental SPLA was not performed. None of the shells were equipped with strain gages, load equalization was performed by shimming of the only free interface between the metallic faces of the test frame, and all other interfaces were filled with resinealuminum powder mixture. A global buckling (nonreversible) was experienced at the end of the VCT experimental loading for the R29AL shell, whereas the R30AL shell buckled when being left under load during the VCT measurements (postbuckling path not being available). Nonlinear FE analyses including only midsurface imperfections (thickness imperfections of bonded section were already included as a permanent model feature) were carried out for comparison reasons. Both shells did not show local buckling before

672

Stability and Vibrations of Thin-Walled Composite Structures

collapse, featuring uniform buckling patterns distributed along the circumference. A global buckling was observed for the R29AL shell at Pcr ¼ 36.33 kN, nonlinear FE prediction (geometric þ thickness imperfection) for the R29AL shell was Pcr ¼ 36.45 kN, SPLA (perfect shell þ thickness imperfection) (with Pload ¼ 7 N) predicted at 39.5 kN (yielding KDF ¼ 0.58), the NASA SP-8007 predicted buckling load was at 21.4 kN (assuming a low KDF ¼ 0.32), linear buckling was predicted to be at Plb ¼ 66.7 kN, and nonlinear perfect shell (thickness imperfections as a permanent feature) was supposed to be at Pperf ¼ 62.13 kN, while the VCT-predicted buckling load was at 38.2 kN. Like for its “twin” shell, R29AL, the global buckling was observed for R30AL shell at Pcr ¼ 38.32 kN (buckled during the VCT test at 38 kN), nonlinear FE prediction (geometric þ thickness imperfection) for R30AL shell was Pcr ¼ 44.96 kN, SPLA (perfect shell þ thickness imperfection) (at Pload ¼ 7 N) was predicted at 39.5 kN (yielding a KDF ¼ 0.58), the NASA SP-8007 predicted buckling load was at 21.4 kN (assuming a low KDF ¼ 0.32), linear buckling was predicted to be at Plb ¼ 66.7 kN, nonlinear perfect shell (thickness imperfections as permanent feature) was supposed to be at Pperf ¼ 62.13 kN, and the VCT-predicted buckling load was at 38.4 kN. As describe earlier, 14 composite shells with different layups were also manufactured from the IM7/8552 prepreg. Two shells (R15 and R16) were configured to be similar to the DLR-manufactured shell, z15, while the other 12 (R17eR28) had a diameter of D ¼ 300 mm and various different layups (see Table 13.2) and were tested applying the VCT and SPLA. An upgraded test setup was used for the R15 and R16 shells. The shells were glued between steel plates with internal aluminum rings being installed. The only free interface is filled with shims (0.07 mm), together with simultaneous LVDT measurement monitoring to obtain the best possible load distribution among the three measuring LVDTs. The SPLA was applied on R15 shell to investigate relative KDFs caused by the application of additional radial perturbation, realized by applying constant radial load in four equally spaced (90 ) locations (A, B, C, D), as presented in Figs. 13.76e13.79. KDFs experimentally obtained by applying the SPLA method were compared with numerically calculated values using the SPLA for imperfect shells (midsurface þ thickness imperfections). A comparison of the relative KDF* (see Eq. 13.1) calculated for experimental SPLA and numerical SPLA calculated for imperfect shell showed a good agreement, see Table 13.7, despite the fact that the nonlinear numerical model seems to overestimate the buckling load by about 18%. KDF ¼

N1
exp N1
num 0KDF ¼ Pcr
exp Pcr
num

(13.1)

The EBL of the specimen R15 (25.38 kN) is higher than the SPLA-predicted load, N1 ¼ 22.41 kN (yielding KDF ¼ 0.58), based on employing the SPLA method for a perfect geometry shell, as presented in Fig. 13.80. The nonlinear FE calculations predicted a buckling load (taking into account geometry þ thickness imperfection) of

Test results on the stability and vibrations of composite shells Shell R15: experimental SPLA @ position A 30

20 15 10

0N 2N 3N 4N 5N 7N 9N 11 N 13 N 15 N

25 Load (kN)

Load (kN)

Shell R15: experimental SPLA @ position B 30

0N 2N 3N 4N 5N 7N 9N 11 N 13 N 15 N

25

20 15 10

5

5

0 0

0.2

0.4 0.6 0.8 Displacement (mm)

0

1

0

Shell R15: experimental SPLA @ position A

0.2

0.3 0.4 0.5 0.6 Displacement (mm)

0.7

0.8

25

20 15 10

1_A_max 1_A_first

5 0

2

4

6

8 10 Pload (N)

12

14

Buckling load (kN)

Buckling load (kN)

0.1

Shell R15: experimental SPLA @ position B

25

0

673

20 15 10

1_B_max 1_B_first

5 0

16

0

2

4

6

8 10 Pload (N)

12

14

Figure 13.76 Shell R15: the experimental application of SPLA at A and B positions.

Shell R15: experimental SPLA @ position D

Shell R15: experimental SPLA @ position C 0N 2N 3N 4N 5N 7N 9N 11 N 13 N 15 N

Load (kN)

25 20 15 10

0N 2N 3N 4N 5N 7N 9N 11 N 13 N 15 N

30 25 Load (kN)

30

20 15 10 5

5 0 0

0.1

0.2

0.3 0.4 0.5 0.6 Displacement (mm)

0.7

0 0

0.8

Shell R15: experimental SPLA @ position C

0.3 0.4 0.5 0.6 Displacement (mm)

0.7

0.8

Shell R15: experimental SPLA @ position D

20 15 10

1_C_max 1_C_first

5 0

2

4

6 8 Pload (N)

10

12

14

Buckling load (kN)

Buckling load (kN)

0.2

25

25

0

0.1

20 15 1_D_max

10

1_D_first

5 0

0

2

4

6

8 10 Pload (N)

12

Figure 13.77 Shell R15: the experimental application of SPLA at C and D positions.

14

674

Stability and Vibrations of Thin-Walled Composite Structures Shell R15: numerical SPLA @ position B 0.1 N

35

0.1 N

30

1N

30

2N

25

3N

25

4N

20

5N

20

5N

15

9N

15

9N

2N

7N

Load (kN)

Load (kN)

Shell R15: numerical SPLA @ position A 35

11 N

10

0 0

0.1

0.2 0.3 0.4 Displacement (mm)

0.5

15 N

5

Exp

0

30

25

25

15

Max load

10 5 0 0

2

4

6

8 10 Pload (N)

12

14

Buckling load (kN)

Buckling load (kN)

Shell R15: numerical SPLA @ position A 30

20

7N 11 N

10

13 N

5

3N

13 N 15 N

0

0.2 0.3 0.4 Displacement (mm)

Exp

0.5

Shell R15: numerical SPLA @ position B

20 15

Max load

10 5 0 0

16

0.1

2

4

6

8 10 Pload (N)

12

14

16

Figure 13.78 Shell R15: the numerical application of SPLA at A and B positions.

Shell R15: numerical SPLA @ position C

Load (kN)

30 25 20 15 10 5 0

0

0.1

0.2 0.3 0.4 Displacement (mm)

0.5

Shell R15: numerical SPLA @ position D 0.1 N 2N 3N 4N 5N 7N 9N 11 N 13 N 15 N Exp

35

0

First buckling

0

2

4

6

8 10 Pload (N)

12

14

16

Buckling load (kN)

Buckling load (kN)

Max load

5 0

15

0

0.1

0.2 0.3 0.4 Displacement (mm)

0.5

Shell R15: numerical SPLA @ position D

30

20 10

20

5

Shell R15: numerical SPLA @ position C

15

25

10

30 25

0.1 N 2N 3N 4N 5N 7N 9N 11 N 13 N 15 N Exp

30 Load (kN)

35

25 20 15

Max load First...

10 5 0

0

2

4

6

8 10 Pload (N)

12

Figure 13.79 Shell R15: the numerical application of SPLA at C and D positions.

14

16

Test results on the stability and vibrations of composite shells

675

Shell R15: relative KDF* using numerical and experimental SPLA

Table 13.7

Position

Experimental SPLA

Numerical SPLA

A

0.71

0.70

B

0.73

0.71

C

0.67

0.71

D

0.70

0.71

Shell R15: application of SPLA @ perfect shell geometry 0.1 N 1N 2N 3N 5N 7N 9N 11 N 13 N 15 N Exp

45 40

Load (kN)

35 30 25 20 15 10 5 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Displacement (mm)

Load (kN)

Shell R15: KDF* after applying SPLA @ perfect geometry 45 40 35 30 25 20 15 10 5 0

Max load First buckling

0

2

4

6

8 Pload (N)

10

12

14

16

Figure 13.80 Shell R15: numerical application of the SPLA method for a perfect shell geometry versus experimental buckling curve.

676

Stability and Vibrations of Thin-Walled Composite Structures

Pcr ¼ 29.86 kN, whereas the linear buckling load was calculated to be Plb ¼ 38.75 kN, a nonlinear perfect shell model predicted a critical load of Pcr ¼ 38.86 kN, and the application of the VCT approach predicted a buckling load of 29 kN. Photogrammetry-based buckling pattern showed good agreement with the numerical simulations based on the measured thickness and midsurface imperfections, as displayed in Figs. 13.81 and 13.82. The experimental test campaign for shell R16 produced similar diagrams as shown for the previous shell. The results are summarized in Table 13.8. 500

z, mm

4.88

0 0.362

θ, rad

3.234

6.89

Figure 13.81 Shell R15: photogrammetry-based buckling pattern (front).

6.09

0.00

Figure 13.82 Shell R15: buckling pattern for nonlinear buckling analyses (geometry þ thickness imperfection).

Shell R16: relative KDF* using numerical and experimental SPLA

Table 13.8

Position

Experimental SPLA

Numerical SPLA

A

0.71

0.67

B

0.78

0.73

C

0.68

0.69

D

0.72

0.68

Test results on the stability and vibrations of composite shells

677

The EBL of the R16 (25.23 kN) shell is higher than the SPLA-predicted N1 ¼ 22.41 kN load (with KDF ¼ 0.58), while employing SPLA approach for a perfect shell geometry, as depicted in Fig. 13.83. Nonlinear FE predicted a buckling load (assuming geometry þ thickness imperfection) of Pcr ¼ 33.97 kN, linear buckling calculations yielded a buckling load of Plb ¼ 38.75 kN, use of nonlinear perfect shell calculated Pcr ¼ 38.86 kN, and the VCT approach predicted a buckling load of 27 kN. Photogrammetry-based buckling pattern showed good agreement with the numerical simulations based on measured thickness and midsurface imperfections, as shown in Figs. 13.84 and 13.85. The experimental buckling results obtained during the test campaign carried out by RTU within the DESICOS project are summarized in Table 13.9, together with numerical results using various sources.

Shell R16: application of SPLA @ perfect shell geometry 0.1 N 1N 2N 3N 5N 7N 9N 11 N 13 N 15 N Exp

45 40

Load (kN)

35 30 25 20 15 10 5 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Displacement (mm) Shell R16: KDF* after applying SPLA @ perfect geometry 45 40

Max load

35

First buckling

Load (kN)

30 25 20 15 10 5 0

0

2

4

6

8 10 Pload (N)

12

14

16

Figure 13.83 Shell R16: numerical application of the SPLA method for a perfect shell geometry versus experimental buckling curve.

678

Stability and Vibrations of Thin-Walled Composite Structures

500

z, mm

4.76

0 0.471

θ, rad

3.361

6.99

Figure 13.84 81Shell R16: photogrammetry-based buckling pattern (front).

6.01

0.00

Figure 13.85 Shell R16: buckling pattern for nonlinear buckling analyses (geometry þ thickness imperfection).

As can be seen from Table 13.9, the various numerical predictions are close to the actual buckling loads of the tested specimens.

13.3

Vibrations of shells

The dynamic behavior of thin-walled composite shells was mainly experimentally studied to measure the natural frequencies of those structures. Monitoring the reduction of the natural frequencies with the increase of the applied compression load enables to nondestructively predict the actual buckling load of the specimens, as described in Chapter 12. Kalnins et al. [23] and Arbelo et al. [24] are typical to those studies aimed at using the VCT to define the buckling loads of composite shells. Another approach, presented in Ref. [22], is to identify the real boundary conditions of the tested shells, thus enabling a better and more accurate prediction of the buckling load. This was done by monitoring the modal response of the tested shells. The vibrations of the shells tested within the DESICOS project, whose buckling behavior was described in detail in Chapter 12, were mainly measured to apply the VCT to predict the on-site buckling load of the tested shell. In what follows, some typical results will be presented.

Test and numerical campaign carried out by RTU within the DESICOS project ISSDJAVE

Isotropicemetallic

Composite

D300 L440

D300 L440

SST_1

SST_2

R29AL

R29AL

R15

R16

0.41

0.53

1.86

3.18

0.96

0.86

1.08

0.77

Linear eigenvalue

187

187

66.7

66.7

38.8

38.8

NASA SP-8007 (KDF)

60 (0.32)

47 (0.25)

21.4 (0.32)

21.4 (0.32)

13 (0.33)

13 (0.33)

SPLA

110.0

117.0

39.5

39.5

22.4

22.4

D300 L440 a/t ratio

Nonlinear perfect

68.0

Real imperfections Experiment (as-built)

61.6

39.96

97.45

182.0

183.0

62.1

62.1

38.9

38.9

39.70

96.68

80.0/98.0

63.0/74.0

39.5

44.96

29.9

33.97

39.84

95.57

70.2/78.2

50.0/60.0

36.33

38.32

25.38

25.23

17.5

17.3

29

27

Experiment (SPLA) Experiment (VCT)

90.02

70.59

47.0

38.2

38.4

Test results on the stability and vibrations of composite shells

Table 13.9

679

680

Stability and Vibrations of Thin-Walled Composite Structures

Laser doppler vibrometer

Shell Loud speaker

Figure 13.86 DLR-manufactured shell Z36 and the RTU VCT system.

(a)

(b)

1.2

1st mode y = 3.9441x2 – 3.6347x + 0.9389 R2 = 0.9961

(1 – p)2

1

ξ2

0.8 0.6 0.4 0.2 0

0

0.1

0.2

0.3 0.4 1 – f2 VCT Ppredicted buckling load 58.49 kN

0.5 VCT experiment with loudspeaker

Figure 13.87 Experimental results using VCT on DLR shell Z36: (a) prediction of the buckling load and (b) the shell mode shape at 45-degree aperture, as projected by the RTU laser Doppler vibrometer.

A test involving the participation of the Technion and RTU was performed on one of the shells produced and tested by DLR. RTU used a loudspeaker and later on a shaker (only touching the skin of the shell, not bonded) to excite the shell, a laser Doppler vibrometer system (Polytech PI5) to acquire the natural frequencies, and their associated mode shapes of the tested shell. The test setup can be seen in Fig. 13.86. Typical test results are shown in Figs. 13.87e13.90. The graphs presenting the (1
p)2 versus (1
f2) were obtained by measuring the natural frequency and nondimensionalizing it by division of the natural frequency at zero axial load, yielding the variable f. The nondimensional variable p is defined as 5

http://www.polytecpi.com/vib.htm.

1st mode (n = 20)

2nd mode (n = 24)

3rd mode (n = ?)

3 mode (m = 2; n = 32)

4th mode (m = 2; n = 18)

0.0

152.75

158.75

164.75

261.75

295.25

1.0

152

158.75

164.75

261.75

295.25

5.0

151

154

158.25

260.25

294.25

10.0

149

152.25

156

255

290.5

Test results on the stability and vibrations of composite shells

Load

VCT experiment with loudspeaker 681

Figure 13.88 Shell Z36: buckling modes captured by photogrammetry when the shell was excited by a loudspeaker.

682

Stability and Vibrations of Thin-Walled Composite Structures

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1st Mode 2

y = 3.1294x – 3.104x + 0.8605 R2 = 0.9975 (1 – p)2

(1 – p)

2

Figure 13.89 Shell Z36: graphs of (1
p)2 versus (1
f2) at various load stations.

0

0.1

0.2 1–f

2

0.3

0.4

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.5

2nd Mode y = 3.0491x2 – 3.1781x + 0.8797 R2 = 0.9997

0

0.2

0.1

1–f

(1 – p)

2

Ppred = 59.54 kN

2

0.3

0.4

0.5

Ppred = 65.87 kN 2nd Mode

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

2

y = 5.5704x – 4.0705x + 0.8335 R2 = 0.9904

0

0.05

0.1

0.15 0.2 2 1–f

0.25

0.3

0.35

Ppred = 59.22 kN VCT experiment with shaker

Figure 13.90 Shell Z36: graphs of (1
p)2 versus (1
f2) when the specimen was excited by a shaker.

Test results on the stability and vibrations of composite shells

683

y = 6.13764x2 – 3.82389x + 0.98383 R2 = 0.99818

1.2 1

(1 – p)2

0.8

ξ2

0.6 0.4 0.387 0.2 0 0

0.05

0.1

0.15

0.2

0.25

1 – f2 1.00 SST1 0.90 0.80

0.94 0.85 0.72

PPred/PEBL

0.70 0.60 0.50

0.48

0.40 0.30 0.20 0.10 0.00

0.40

0.53

0.67

0.80

Pi/PEBL

Figure 13.91 Shell SST_1: VCT curve for the first mode and the variation of the relative predicted load (PPred/PEBL) as a function of the nondimensional relative load (Pi/PEBL) up to which the natural frequencies were experimentally measured. Inset: The lowest experimental mode shape.

the ratio between the axial load divided by the numerical buckling load. A seconddegree polynomial curve is then fitted to the experimental data. Then the polynomial equation is once derived to yield the value of x2. The predicted value of the VCT approach would then be Pcr (VCT) ¼ x Pcr (numerical buckling load). One should note that the VCT-predicted buckling load of 58.49 kN agrees very well with the actual buckling load of the shell Z36, 57.9 kN. Additional VCT tests were performed by RTU in the framework of the DESICOS project. The tests on the stainless steel shells SST_1 and SST_2 are presented in Figs. 13.91 and 13.92, respectively. For the SST_1 specimen, the VCT-predicted

684

Stability and Vibrations of Thin-Walled Composite Structures

1.2

y = 5.52345x2 – 3.10189x + 0.99617 R2 = 0.99959

(1 – p)2

1

ξ2

0.8 0.6 0.629 0.4 0.2 0 0

0.05

0.1

0.15

0.2

0.25

1 – f2 1.00 SST2 0.90 0.80

0.89

0.86

0.91

0.60

0.70 Pi/PEBL

0.80

0.95

0.78

PPred/PEBL

0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

0.50

0.90

Figure 13.92 Shell SST_2: VCT curve for the first mode and the variation of the relative predicted load (PPred/PEBL) as a function of the nondimensional relative load (Pi/PEBL) up to which the natural frequencies were experimentally measured. Inset: The lowest experimental mode shape.

buckling load was 70.59 kN, with measured values up to 86% of PEBL, while for the SST_2 it was 47.37 kN, with measured values up to 90% of EBL. Similar results were also obtained for the two aluminum shells, R29AL and R30AL. The results of shell R29AL are presented in Fig. 13.93 and of its “twin” shell R30AL in Fig. 13.94. One should note that for these shells the VCT application overpredicts the real buckling loads. Finally, typical results for the application of the VCT approach on the composite shells R15 and R16, whose buckling results were presented above in the present Chapter 12, are presented in Figs. 13.95 and 13.96, respectively. One should note that as shown earlier for the aluminum shells, the application of the VCT approach overestimated the real buckling loads.

Test results on the stability and vibrations of composite shells

1.2

685

y = 3.76597x2 – 3.50621x + 0.99948 R2 = 0.99968

1

(1 – p)2

0.8 0.6 0.4 ξ2

0.2 0

0.183 0

0.1

0.2

0.3

0.4

0.5

1 – f2 1.40 1.30

1.29

1.20

R29AL 1.23 1.05

1.06

1.06

0.70

0.77 0.84 Pi/PEBL

1.09 1.09

1.05

0.88

0.99

PPred/PEBL

1.00 0.80 0.60 0.40 0.20 0.00

0.56

0.61

0.66

0.95

Figure 13.93 Shell R29AL: VCT curve for and the variation of the relative predicted load (PPred/PEBL) as a function of the nondimensional relative load (Pi/PEBL) up to which the natural frequencies were experimentally measured. Inset: The lowest experimental mode shape.

A somehow different way for the application of the VCT method was presented by Abramovich et al. [25] to nondestructively predict the buckling loads of stiffened composite stringer-stiffened cylindrical shells. Typical results for the shell nicknamed SH-1 (whose geometric and material properties are presented in Fig. 13.97) tested within the DAEDALOS6 project [25] are presented. Fig. 13.98 displays the large postcollapse deflections of the specimen, its position in the test rig, and the various locations of the 12 pairs of strain gages, bonded back-to-back to monitor the behavior of

6

DAEDALOS: Dynamics in Aircraft Engineering Design and Analysis for Light Optimized Structures (www.daedalos-fp7.eu). The work was partly supported by the European Commission, Seventh Framework Programme, Contract no. 266411.

686

Stability and Vibrations of Thin-Walled Composite Structures

1.2

y = 4.06952x2 – 3.65796x + 1.00292 R2 = 0.99955

1

(1 – p)2

0.8 0.6 0.4

ξ2

0.2 0.181 0 0

0.1

0.2

0.3

0.4

0.5

1 – f2 1.20

PPred/PEBL

1.00

R30AL 0.96

0.99

1.14 1.12 1.12 1.05 1.07 1.09 1.11

1.01 1.00

0.88

0.80 0.60 0.40 0.20 0.00

0.42 0.47 0.53 0.58 0.63 0.69 0.74 0.79 0.84 0.89 0.94 0.99 Pi/PEBL

Figure 13.94 Shell R30Al: VCT curve for and the variation of the relative predicted load (PPred/PEBL) as a function of the nondimensional relative load (Pi/PEBL) up to which the natural frequencies were experimentally measured. Inset: The lowest experimental mode shape.

the shell during the axial loading, as well as the LVDT used to measure the end shortening of the specimen. The excitation of the specimen was performed by using the model hammer method, and the experimental squared frequencies versus the axial compression are presented in Fig. 13.99 (the data is from Ref. [25]). A clear reduction of the frequency squared with an increase in the axial compression can be observed in Fig. 13.99. Although the graph is monotonically approaching the load axis, near the actual buckling, it will bend sharply to yield the buckling load of the shell (the first buckling load, not the collapse load). The EBL was found to be 19.5 kN [25], while the calculated buckling load was 36.28 kN (see details in Table 13.10). Applying the VCT method based on Eq. (13.2) (where f0 is the natural frequency at zero axial load, Pcr is the numerical predicted buckling load, and f is the natural frequency at

Test results on the stability and vibrations of composite shells

1.2

687

y = 1.99692x2 – 2.64907x + 0.98223 R2 = 0.99900

1 (1 – p)2

0.8 0.6 0.4 0.2

ξ2 –0.1

0

0.104 0

0.1

0.2

0.3

0.4

0.5

0.6

1 – f2 1.20

R15

PPred/PEBL

1.00 0.80

0.80 0.63

0.85 0.89

1.02 1.03 1.04 1.05 0.98 1.01

0.70

0.60 0.40 0.20 0.00

0.48 0.53 0.57 0.62 0.67 0.72 0.77 0.82 0.86 0.91 0.97 Pi/PEBL

Figure 13.95 Shell R15: VCT curve for and the variation of the relative predicted load (PPred/PEBL) as a function of the nondimensional relative load (Pi/PEBL) up to which the natural frequencies were experimentally measured. Inset: The lowest experimental mode shape.

the axial load P), namely, best fitting of a second-order polynomial, would yield the following predicted load: PExtrap. ¼ 36.2 kN (as presented in Fig. 13.99), which seems to be still far from the actual EBL.  2 f P ¼1
f0 Pcr

(13.2)

A modified application of the VCT method was then used according to Eq. (13.3). ð1
pÞ2 ¼ 1
1
z2 where ph

P ; Pcr




1
f

4



(13.3) fh

f . f0

(1 – p)2 ξ2

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

y = 1.90280x2 – 2.55340x + 0.94320 R2 = 0.99985

0.087 0

0.1

0.2

0.3

0.4

0.5

0.6

1 – f2 R16

1.40

1.21 1.26

1.27

0.52

0.62

1.23

1.25

1.21 1.18

1.16

0.67

0.73

0.76

0.86

PPred/PEBL

1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.48

0.58

0.81

Pi/PEBL

Figure 13.96 Shell R16: VCT curve for and the variation of the relative predicted load (PPred/PEBL) as a function of the nondimensional relative load (Pi/PEBL) up to which the natural frequencies were experimentally measured. Inset: The lowest experimental mode shape.

Φ408

A

90° 1

6

1

(0°/+45°/-45°/90°)S

45°

0.5

0° Ply orientation

7.5 (+45°/-45°/+45°/-45°)T

Detail A Scale 5 : 1

640

Figure 13.97 Geometric and material properties of shell SH-1. From H. Abramovich, D. Govich, A. Grunwald, Buckling prediction of panels using the vibration correlation technique, Progress in Aerospace Sciences 78 (2015) 62e73.

Test results on the stability and vibrations of composite shells

689

LVDT

1,2

3,4

Shell SH-1 in the test rig

19,20

15,16

9,10

21,22

17,18

5,6

Post collapse deflections

13,14

7,8

11,12

23,24

Strain gages locations

Figure 13.98 Shell SH-1: (a) postcollapse deflections, (b) shell in the test rig, and (c) locations of the strain gages. From H. Abramovich, D. Govich, A. Grunwald, Buckling prediction of panels using the vibration correlation technique, Progress in Aerospace Sciences 78 (2015) 62e73. Shell SH-1 50,000 45,000

y = –19.32x2 – 520.64x + 44149 R2 = 0.9994

40,000

f2 (Hz2)

35,000 30,000 25,000 20,000 15,000

PPredicted = 22.3 kN

10,000 5000

PEx trap. = 36.2 kN

PExp. = 19.5 kN

0 0

5

10

15

20

25

30

Axial compression load (kN)

Figure 13.99 Shell SH-1: frequency squared versus axial compression load.

Calculated and experimental buckling and collapse loads for shell SH-1

Table 13.10

Calculated

Experiment

Specimen

Pcr (kN)

Pcollapse (kN)

Pcr (kN)

Pcollapse (kN)

SH-1

36.28

42.34

19.5

20.28

35

690

Stability and Vibrations of Thin-Walled Composite Structures

and f0 is the natural frequency at zero axial load, Pcr is the extrapolated buckling load based on the curve f2 versus P, and x2 is the experimental KDF achieved at up to 60%   4 of the predicted buckling load. Preparing a graph of ð1
pÞ2 versus 1
f based on the experimental measured points and best fit experimental linear curve, while     4 4 drawing a parallel line to the 1
f axis at the value of 1
f ¼ 1 would yield the value of x2 on the ð1
pÞ2 axis. Multiplying Pcr by x would provide the PPredicted value using the modified VCT approach. For the present specimen the value of PPredicted was found to be 22.3 kN (see Fig. 13.99), which is very close to the EBL of 19.5 kN [25]. The relatively large discrepancies between the EBL and the numerically predicted value are the initial geometric imperfections experienced by the SH-1 specimen. Measuring those initial geometric imperfections would be one way of reducing the aforementioned discrepancies and improving the accuracy of the predicted buckling loads of the shell using the VCT approach.

Acknowledgments The research leading to the results presented in this chapter had received partial funding from the European Community’s Seventh Framework Programme (FP7/2007e2013) under Priority Space, Grant Agreement # 282522 (www.DESICOS.eu). The information presented in this reflects only the authors’ views and the European Community is not liable for any use that may be made of the information contained herein.

References [1] T. von Karman, H.-S. Tsien, The buckling of thin cylindrical shells under axial compression, Journal of the Aeronautical Sciences 8 (8) (1941) 303e312. [2] P. Seide, V.I. Weingarten, J.P. Peterson, Buckling of Thin-Walled Circular Cylinders, NASA Technical Report SP-8007, August 1968. [3] V.I. Weingarten, P. Seide, Buckling of Thin-Walled Truncated Cones, NASA Technical Report SP-8019, September 1968. [4] A.W. Leissa, Vibration of Shells, NASA SP-288, 1973. [5] A. Leissa, Buckling of Laminated Composite Plates and Shell Panels, AFWAL-TR85e3069 Report, AD-A162 723, June 1985. [6] D.A. Evensen, High-speed photographic observation of the buckling of thin cylinders, Experimental Mechanics 4 (4) (April 1964) 110e117. [7] R.M. Jones, H.S. Morgan, Buckling and vibration of cross-ply laminated circular cylindrical shells, AIAA Journal 13 (5) (1975) 664e671. [8] J.G. Teng, Buckling of thins shells: recent advances and trends, Applied Mechanics Review 49 (4) (April 1996) 263e274. [9] M.P. Nemeth, J.H. Starnes, The NASA Monographs on Shell Stability Design Recommendations, NASA/TP-1998e206290, January 1998.

Test results on the stability and vibrations of composite shells

691

[10] M.S. Qatu, R.W. Sullivan, W. Wang, Recent research advances in the dynamic behavior of composite shells: 2000e2009, Composite Structures 93 (1) (2010) 14e31. [11] M.S. Qatu, E. Asadi, W. Wang, Review of recent literature on static analyses of composite shell: 2000e2010, Open Journal of Composite Materials 2 (2012) 61e86. [12] N.A. Fleck, Compressive failure of fiber composites, in: J.W. Hutchinson, T.Y. Wu (Eds.), Advances in Applied Mechanics, vol. 33, 1997, pp. 43e117. [13] J. Singer, On the importance of shell buckling experiments, Applied Mechanics Reviews 52 (6) (June 1999) 17e25. [14] R. Rikards, A. Chate, O. Ozolin¸s, Analysis for buckling and vibrations of composite stiffened shells and plates, Composite Structures 51 (2001) 361e370. [15] C. Bisagni, Dynamic buckling tests of cylindrical shells in composite materials, in: 24th International Council of the Aeronautical Sciences, ICAS2004, Yokohama, Japan, 29th AugusteSeptember 3rd, 2004, 2004. [16] E. Eglῑtis, K. Kalnin¸s, O. Ozolin¸s, Experimental and numerical study on buckling of axially compressed composite cylinders, Construction Science 10 (10) (January 2009), 16 p. [17] V. Chitra, R.S. Priyadarsini, Dynamic buckling of composite cylindrical shells subjected to axial impulse, International Journal of Scientific and Engineering Research 124 (05) (May 2013) 162e165. [18] M.P. Nemeth, M.M. Mikulas Jr., Simple Formulas and Results for Buckling-resistance and Stiffness Design of Compression-Loaded Laminated-Composite Cylinders, NASA/TP2009e215778, August 2009. [19] R. Sliz, M.-Y. Chang, Reliable and accurate prediction of the experimental buckling of thin-walled cylindrical shell under axial load, Thin Walled Structures 49 (3) (2011) 409e421. [20] R.S. Pryadarsini, V. Kalyanaraman, S.M. Srinivasan, Numerical and experimental study of buckling of advanced fibre composite cylinders under axial compression, International Journal of Structural Stability and Dynamics 12 (04) (July 2012), 1250028, 25 p. [21] F. Shadmehri, Conical Shells; Theory and Experiment (Ph.D. thesis), Mechanical Engineering at Concordia University, Montreal, Quebec, Canada, September 2012. [22] J. Blachut, Experimental perspective on the buckling of pressure vessel components, Applied Mechanics Reviews 66 (January 2014), 01083, 23 p. [23] K. Kalnins, M.A. Arbelo, O. Ozolins, E. Skukis, S.G.P. Castro, R. Degenhardt, Experimental nondestructive test for estimation of buckling load on unstiffened cylindrical shells using vibration correlation technique, Shock and Vibration 2015 (2015), 729684, 8 p. [24] M.A. Arbelo, K. Kalnins, O. Ozolins, E. Skukis, S.G.P. Castro, R. Degenhardt, Experimental and numerical estimation of buckling load on unstiffened cylindrical shells using a vibration correlation technique, Thin-Walled Structures 94 (September 2015) 273e279. [25] H. Abramovich, D. Govich, A. Grunwald, Buckling prediction of panels using the vibration correlation technique, Progress in Aerospace Sciences 78 (2015) 62e73.

Further reading [1] E. Skukis, K. Kalnins, O. Ozolinsh, Assessment of the effect of boundary conditions on cylindrical shell modal response, in: 4th International Conference Civil Engineering’13, Proceedings Part 1 Structural Engineering, Jelgava, Latvia, vol. 4, 2013, pp. 41e45.

Computational aspects for stability and vibrations of thin-walled composite structures

14

Tanvir Rahman 1 , Eelco Jansen 2 1 DIANA FEA BV, Delft, Netherlands; 2Leibniz Universit€at Hannover, Hannover, Germany

14.1

Introduction to finite element buckling and initial postbuckling analysis

Nowadays, the finite element method is commonly used for the buckling and vibration analysis of practical engineering structures. Well-known books on finite element analysis and handbooks such as the ECSS Buckling Handbook [1] provide the theoretical background and guidelines to carry out a nonlinear analysis successfully. This chapter presents various examples of nonlinear buckling and vibration analyses of basic composite shell structures. Specific examples relate, in particular, to composite cylindrical shells because of their theoretical and practical significance. It has been recognized that systematic, hierarchical approaches for nonlinear buckling analysis are essential to attain reliable results when complicated structures are analyzed [1]. Finite element based reduced order modeling corresponds to a way to introduce a hierarchical approach in the context of finite element nonlinear buckling and vibration analysis. In this chapter, an overview of a reduced order modeling technique for finite element buckling and vibration analysis of structures based on a perturbation approach [2,3] is presented. The description closely follows the treatment in Refs. [2,4e6]. This chapter first presents a finite element formulation of Koiter’s perturbation approach for single-mode initial postbuckling analysis. In this approach a perturbation expansion for the load and the displacement field is made around the bifurcation buckling point. The first-order terms in the perturbation expansion of the displacement field are the bifurcation buckling modes, while the second-order terms are known as second-order modes. In many cases a limited number of buckling modes and second-order modes can be used to describe the initial postbuckling behavior. The number of equations in the resulting reduced set of nonlinear algebraic equations is the same as the number of buckling modes chosen in the perturbation expansion, and the perturbation approach can be regarded as the basis of a nonlinear reduced order model. Assuming that a single mode is associated with the lowest bifurcation buckling load, together with the computation of the buckling and second-order mode, the postbuckling slope (a coefficient) and curvature (b coefficient) are computed. These postbuckling coefficients directly give a measure of the stability and imperfection sensitivity of the structure. For instance, in the case of conical and cylindrical shells,

Stability and Vibrations of Thin-Walled Composite Structures. http://dx.doi.org/10.1016/B978-0-08-100410-4.00014-4 Copyright © 2017 Elsevier Ltd. All rights reserved.

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Stability and Vibrations of Thin-Walled Composite Structures

for so-called asymmetric modes one has a zero a coefficient and typically a negative b coefficient indicating unstable postbuckling behavior and imperfection sensitivity. The buckling mode, second-order mode, and the corresponding postbuckling coefficients are characteristics of the perfect structure. Once they have been computed, the effect of geometric imperfections can be evaluated with very little additional computational cost for various imperfection shapes and magnitudes using the reduced order model, which in the single-mode case consists of one nonlinear algebraic equation. In the literature, often the prebuckling state has been assumed to be linear in the finite element formulation of Koiter’s perturbation approach. For an important class of problems, namely cylindrical and conical shells under axial compression, the assumption of a linear prebuckling state often leads to an overestimation of the bifurcation buckling load, and the buckling mode can be strongly different from the one obtained when including a nonlinear prebuckling state. Therefore, the effect of prebuckling nonlinearity has been included in the present treatment. Subsequently a multimode analysis will be presented. A multimode approach is necessary to assess the degrading effect on the load-carrying capacity caused by modal interaction among clustered buckling modes (buckling modes associated with simultaneous or nearly simultaneous buckling loads). Often buckling modes being stable by themselves may exhibit an unstable postbuckling behavior and imperfection sensitivity when modal interaction comes into play. Also, it was observed experimentally that even well-separated buckling modes may interact [7]. In the second part of the chapter, dynamic buckling and nonlinear vibration problems will be discussed using the perturbation approach used for static initial postbuckling analysis. In case of dynamic buckling problem, time-dependent loading is considered, in which with increasing magnitude of the load an increasing displacement of the structure results, and often at a particular load level the displacement exhibits a sharp increase with respect to the load increment. This particular load level can be identified as the dynamic buckling load of the structure. This criterion for the dynamic buckling load is known as the BudianskyeRoth criterion and will be used here. The perturbation approach used for static postbuckling problems can be extended to cover dynamic buckling problems, taking the effect of inertia into account. The extension is done following the approach proposed by Budiansky [8] in a multimode context with the inclusion of prebuckling nonlinearity. The formulation is therefore a direct extension of the static multimode formulation using additional inertial terms. Nonlinear or large-amplitude vibration problems are also considered in the second part of this chapter. At large-vibration amplitude (in the order of shell or plate thickness), because of geometrically nonlinear effects, vibration frequency varies with vibration amplitude. As a result, a softening or a hardening effect is observed (the frequency decreases or increases with increasing amplitude), which is visualized by the so-called backbone curves. Many studies in the field of nonlinear vibration that describe this type of vibration amplitudeefrequency relation are based on semianalytical (Galerkin or RayleigheRitz) formulation. However, such formulations are restricted to structures with a relatively simple geometry [9]. A finite element discretization makes it possible to analyze structures with arbitrary geometry. However, standard finite elemente based transient analysis procedures for nonlinear vibration problems are still highly

Computational aspects for stability and vibrations of thin-walled composite structures

695

time-consuming. In this chapter, a finite elementebased perturbation approach for nonlinear vibrations will be presented, which is analogous to the initial postbuckling analysis described in the previous chapters. Based on Rehfield’s [10] work, the method is formulated in a single-mode context. Necessary extensions for multimode case are also discussed. The perturbation approach yields a dynamic b coefficient bD (analogous to the static b coefficient in case of static postbuckling problems) that corresponds to the initial curvature of the frequencyeamplitude relation and accounts for the most important nonlinear effect.

14.2

Single-mode initial postbuckling analysis

In this section the perturbation method for buckling and postbuckling analysis will be discussed, with the inclusion of prebuckling nonlinearity. A detailed derivation of the equations is available in the report by Arbocz and Hol [11]. Here the basic procedure will be explained, mentioning the essential equations. The functional notation introduced by Budiansky [8] will be used. First, the perturbation approach for the perfect structure will be explained. Next it will be extended for the imperfect structure. In the following, symbols with bold font denote vector and tensor quantities, whereas the scalar symbols are written in normal font. Many of the earlier works on Koiter’s theory are based on the principle of stationary potential energy. However, in this chapter an alternative procedure proposed by Budiansky and Hutchinson [12] will be used, which expresses the field equations directly in variational form using the principle of virtual work. In Budiansky and Hutchinson’s work the prebuckling state was assumed to be linear. Cohen [13] and Fitch [14] and later Arbocz and Hol [11,15] derived the modifications necessary to include prebuckling nonlinearity. The present treatment makes use of the derivations presented by Arbocz and Hol [11,15] and applies them within a finite element context. Two different types of shell elements are used. The first type consists of a class of Mindlin-type (considers transverse shear stress) curved shell elements available in DIANA. Out of that class of elements, the eight-node, quadrilateral element named CQ40L (for laminated composite material) is used in the numerical examples presented in this chapter. The other type of element that is used is a Kirchhoff-type [16,17] (transverse shear deformation is ignored) triangular flat shell element developed by Allman [18e20]. Tiso [3] enhanced this element to alleviate the locking problem related to the convergence of the b coefficient. The same element has been further enhanced for composite material and has been implemented as the DIANA element T18SH [2]. Past research showed that for accurate computation of the b coefficient, particularly with the Bernoulli-type beam and the Kirchhoff-type shell elements, special element formulation is required [3,21e25]. An accurate computation of the b coefficient is possible without any modification in the element formulation in case of the Mindline Reissner [17,26,27] type beam and shell elements. In this chapter, problems with both linear and nonlinear prebuckling behavior are treated and composite plate and shell structures are considered. The postbuckling coefficients obtained from the perturbation approach and from the semianalytical tools

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Stability and Vibrations of Thin-Walled Composite Structures

ANILISA [15] and BAAC [28] are presented and compared with other postbuckling coefficients reported in the literature. The efficiency of the reduced order model that is obtained is also illustrated. The initial postbuckling response and limit-point buckling load are compared with full model finite element analysis carried out using DIANA with the same element.

14.2.1

Functional notation

If u and e denote the displacement and strain fields, respectively, then in functional notation, the strainedisplacement relation is written as 1 e ¼ L1 ðuÞ þ L2 ðuÞ 2

(14.1)

Here L1 is a linear operator and L2 is a quadratic operator. Therefore, L1(u) is a linear functional representing the linear part of strain and L2(u) is a quadratic functional representing the nonlinear part of strain. Furthermore, the bilinear operator L11 is defined such that L2 ðu þ vÞ ¼ L2 ðuÞ þ 2L11 ðu; vÞ þ L2 ðvÞ

(14.2)

From Eq. (14.2), it follows that L11 ðu; vÞ ¼ L11 ðv; uÞ

(14.3)

L11 ðu; uÞ ¼ L2 ðuÞ

(14.4)

In the finite implementation presented in this chapter, the GreeneLagrange strain has been used, and in Ref. [2] an explicit definition of the operators L1, L2, and L11 has been shown with respect to this specific strainedisplacement relation.

14.2.2

Perfect structure

Let u, e, f, and s be the generalized displacement, strain, load, and stress variables, respectively. Then the nonlinear strainedisplacement relation is given by Eq. (14.1), and the linear elastic constitutive relation can be written as s ¼ HðeÞ

(14.5)

where H is a linear operator. The equilibrium equation in variational form is written as s$de  f$du ¼ 0

(14.6)

Here s$de and f$du denote, respectively, the internal virtual work of the stress s through the strain variation de and the external virtual work of the load f through

Computational aspects for stability and vibrations of thin-walled composite structures

697

the displacement variation du, both integrated over the entire structure. Furthermore, it follows from Eqs. (14.1) and (14.2) that the first-order strain variation de produced by du can be written as de ¼ L1 ðduÞ þ L11 ðu; duÞ

(14.7)

In case of linear elasticity the following reciprocity relation holds si $ej ¼ sj $ei

ði; j ¼ 1; 2.Þ

(14.8)

In this treatment, proportional loading is considered, i.e., f ¼ lf0. Now the variables ðu; e; sÞ of the postbuckling equilibrium state can be expanded in the following perturbation expansion series about the prebuckling equilibrium state ðu0 ; e0 ; s0 Þ at the same value of the variable load parameter l. u ¼ u0 ðlÞ þ u1 x þ u2 x2 þ u3 x3 þ . e ¼ e0 ðlÞ þ e1 x þ e2 x2 þ e3 x3 þ .

(14.9)

s ¼ s0 ðlÞ þ s1 x þ s2 x2 þ s3 x3 þ . The variables ðu0 ; e0 ; s0 Þ are assumed to be nonlinear functions of l ¼ l(x), whereas the expansion functions ðuk ; ek ; sk Þ where k ¼ 1; 2; .; are independent of l and x. The perturbation expansions (Eq. 14.9) are assumed to be asymptotically valid about the bifurcation point defined by l ¼ lc and x ¼ 0. Substituting Eq. (14.9) into Eqs. (14.1), (14.5), and (14.6), taking the limit x / 0, and with some further manipulations, one obtains the necessary equations for the bifurcation buckling load lc and the corresponding buckling mode u1: e1 ¼ L1 ðu1 Þ þ L11 ðuc ; u1 Þ

(14.10)

s1 ¼ Hðe1 Þ

(14.11)

s1 $dec þ sc $L11 ðu1 ; duÞ ¼ 0

(14.12)

where the subscript c denotes prebuckling quantities evaluated at l ¼ lc. Next, it is assumed that the prebuckling variables can be expanded in the Taylor series 1 u0 ¼ uc þ ðl  lc Þu_ c þ ðl  lc Þ2 u€c þ . 2 1 €c þ . s0 ¼ sc þ ðl  lc Þs_ c þ ðl  lc Þ2 s 2 v ð Þ. where ð_Þ ¼ vl

(14.13)

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Stability and Vibrations of Thin-Walled Composite Structures

In addition, it will be assumed that (l  lc) admits the asymptotic perturbation expansion. l  lc ¼ alc x þ blc x2 þ .

(14.14)

In view of Eq. (14.14), if a plot of the load parameter l versus the mode amplitude x is made then a and b coefficients indicate the slope and curvature of the postbuckling curve, respectively. In the present treatment, symmetric bifurcation with zero postbuckling slope, a ¼ 0, and typically negative postbuckling curvature, b < 0, indicating unstable postbuckling behavior is considered. Inserting Eqs. (14.13) and (14.14) together with Eq. (14.9) into Eqs. (14.1), (14.5), and (14.6) and equating the coefficients of x2 with the assumption a ¼ 0, one can finally obtain the necessary equations for the determination of the second-order mode u2: 1 e2 ¼ L1 ðu2 Þ þ L11 ðuc ; u2 Þ þ L2 ðu1 Þ (14.15) 2 s2 ¼ Hðe2 Þ

(14.16)

s2 $dec þ sc $L11 ðu2 ; duÞ þ s1 $L11 ðu1 ; duÞ ¼ 0

(14.17)

To obtain the expression for the b coefficient, one can set du ¼ u1 in Eqs. (14.12) and (14.17) and can make use of the reciprocity relation (Eq. 14.8). This gives b Þf2s1 $L11 ðu1 ; u2 Þ þ s2 $L2 ðu1 Þg b ¼ ð1=lc D

(14.18)

where b ¼ 2s1 $L11 ðu_ c ; u1 Þ þ s_ c $L2 ðu1 Þ D

(14.19)

If a s 0, Eq. (14.18) takes a more complicated form b Þf2s1 $L11 ðu1 ; u2 Þ þ s2 $L2 ðu1 Þ þ a D½ b s_ c $L11 ðu1 ; u2 Þ b ¼ ð1=lc D þ s1 $L11 ðu_ c ; u2 Þ þ s2 $L11 ðu_ c ; u1 Þ 2

b Þ ½2s1 $L11 ðu€c ; u1 Þ þ s € c $L2 ðu1 Þg þ ð1=2Þða D

(14.20)

According to Cohen [13], Eq. (14.20) can be simplified by making u2 orthogonal to u1 by applying the following orthogonality condition: s_ c $L11 ðu1 ; u2 Þ þ s1 $L11 ðu_ c ; u2 Þ þ s2 $L11 ðu_ c ; u1 Þ ¼ 0

(14.21)

Computational aspects for stability and vibrations of thin-walled composite structures

699

Therefore, Eq. (14.20) simplifies to  b Þf2s1 $L11 ðu1 ; u2 Þ þ s2 $L2 ðu1 Þ b ¼ ð1 lc D 2

b Þ ½2s1 $L11 ðu€c ; u1 Þ þ s € c $L2 ðu1 Þg þ ð1=2Þða D

(14.22)

14.2.3 Imperfect structure This section shows how the behavior of the imperfect structure can be derived from the properties of the perfect structure. If the initial geometric imperfection is denoted by xb u , where x is the imperfection amplitude and b u is any arbitrary geometric imperfection pattern, then the strainedisplacement equation (Eq. 14.1) can be modified as   1 e ¼ L1 ðuÞ þ L2 ðuÞ þ xL11 b u; u 2

(14.23)

Furthermore, the asymptotic expansion as defined by Eq. (14.14) is also modified as xðl  lc Þ ¼ alc x2 þ blc x3  alc x  bðl  lc Þx þ .

(14.24)

where the coefficients a and b are the first and second imperfection form factors, respectively. This equation and Eq. (14.14) are the reduced order model for the imperfect and perfect structures, respectively. Using the same approach as explained in section Perfect structure the expressions for a and b are obtained as      b Þ s1 $L11 b a ¼ ð1=lc D u ; uc þ sc $L11 b u ; u1

(14.25)

       b Þfs1 $L11 b u ; u_ c þ s_ c $L11 b u ; u1 þ H½L11 ðu_ c ; u1 Þ$L11 b u ; uc b ¼ ð1 D  alc ½s1 $L11 ðu€c ; u1 Þ þ ð1=2Þ€ sc $L11 ðu1 ; u1 Þ þ H½L11 ðu_ c ; u1 Þ$L11 ðu_ c ; u1 Þg (14.26) b is defined by Eq. (14.19). where D Now, with the background established so far in this section, one can do the postbuckling analysis by computing up to the second-order terms in the perturbation expansion of the displacement field u in Eq. (14.9). It can be achieved by the following step-by-step procedure: 1. computation of the prebuckling state uc at l ¼ lc, 2. computation of the buckling load lc and the corresponding buckling mode u1, 3. computation of the second-order mode u2,

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Stability and Vibrations of Thin-Walled Composite Structures

4. computation of the coefficients a and b , 5. computation of the first and second imperfection form factors a and b, respectively, 6. computation of the mode amplitude x corresponding to the applied load parameter l using Eq. (14.24), 7. recovering the displacement field by substituting the already computed terms in Eq. (14.9).

14.3 14.3.1

Finite element implementation Nonlinear buckling analysis

According to Eq. (14.7) the strain variation at the critical point ðec Þ can be written as dec ¼ L1 ðduÞ þ L11 ðuc ; duÞ

(14.27)

Insertion of Eqs. (14.10) and (14.11) into Eq. (14.12) together with Eq. (14.27) gives   H L1 ðu1 Þ þ L11 ðuc ; u1 Þ $½L1 ðduÞ þ L11 ðuc ; duÞ þ sc $L11 ðu1 ; duÞ ¼ 0 (14.28) After some algebraic manipulation of Eq. (14.28) and replacing the L1 and L11 operators and the continuous displacement fields u1, uc, du, respectively, with the finite element matrices BL, BNL and nodal displacements q1, qc, dq, the discretized form of Eq. (14.28) can be obtained: h dqT BTL HBL q1 þ BTNL ðqc ÞHBL q1 þ BTL HBNL ðqc Þq1 þ BTNL ðqc ÞHBNL ðqc Þq1 i þ BTNL ðq1 Þsc ¼ 0 (14.29) Because dq is an arbitrary displacement vector, the remaining quantities inside the bracket are zero. BTL HBL q1 þ BTNL ðqc ÞHBL q1 þ BTL HBNL ðqc Þq1 þ BTNL ðqc ÞHBNL ðqc Þq1 þ BTNL ðq1 Þsc ¼ 0 (14.30) Now one can rewrite Eq. (14.30) after element level integration and after the assembly process as ½KM þ KD ðqc Þ þ KG ðsc Þq1 ¼ 0

(14.31)

Computational aspects for stability and vibrations of thin-walled composite structures

701

where KM, KD(qc), and KG(sc) are the material, initial displacement, and geometric stiffness matrices, respectively. They are defined at the element level as Z KMe ¼ BTL HBL dv v

Z KDe ðqc Þ ¼

 v

Z KGe ðsc Þ ¼

v

 BTNL ðqc ÞHBL þ BTL HBNL ðqc Þ þ BTNL ðqc ÞHBNL ðqc Þ dv

  sxxc Kxx þ syyc Kyy þ szzc Kzz þ sxyc Kxy þ syzc Kyz þ szxc Kzx dv

where v is the element volume; sxxc , syyc , szzc , sxyc , syzc , and szxc are the stress components; and Kxx, Kyy, Kzz, Kxy, Kyz, and Kzx are the derivatives of the interpolation polynomial functions defined in Ref. [2] (not to be confused with stiffness matrices). The sum of KMe , KDe , and KGe gives the element tangent stiffness matrix ðKte Þ at the critical point. Therefore, for the buckling problem, Eq. (14.31) can be written after the assembly process as Ktc q1 ¼ 0

(14.32)

Eq. (14.32) can be solved in the following way. First, a standard nonlinear analysis is performed to reach as close as possible to the critical point without encountering any negative diagonal term in the system stiffness matrix. Let that state be defined as the base state, which occurs at l ¼ lb with the corresponding displacement and stress states qb, sb and tangent stiffness matrix Ktb . One can now linearize Kt around l ¼ lb and write Eq. (14.32) as a linear eigenvalue problem:      _ D qb ; q_ b þ K _ G ðs_ b Þ q1 ¼ 0 Ktb þ ðlc  lb Þ K

(14.33)

where lc is the buckling load and q1 is the buckling mode. For determination of q_ b , one can proceed by considering proportional loading (f ¼ lf0) as  1     vq vq vf vf q_ b ¼ ¼ ¼ f 0 ¼ K1 (14.34) tb f 0 vl b vf b vl vq b Therefore, q_ b can be obtained from the linear solution of Ktb q_ b ¼ f 0 and s_ b can be evaluated as

1 vH BL þ BNL ðqb Þ qb vsb 2 s_ b ¼ ¼ ¼ H½BL þ BNL ðqb Þq_ b vl vl

(14.35)

(14.36)

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Stability and Vibrations of Thin-Walled Composite Structures

14.3.2

Postbuckling analysis

For postbuckling analysis, one needs to first compute the second-order mode u2. Insertion of Eqs. (14.10), (14.11), (14.15), (14.16), and (14.27) into Eq. (14.17) and some algebraic manipulations finally give H½L1 ðu2 Þ þ L11 ðuc ; u2 Þ$½L1 ðduÞ þ L11 ðuc ; duÞ þ sc $L11 ðu2 ; duÞ ¼ 1  H½L2 ðu1 Þ$½L1 ðduÞ þ L11 ðuc ; duÞ 2

(14.37)

 H½L1 ðu1 Þ þ L11 ðu1 ; uc Þ$L11 ðu1 ; duÞ In terms of finite element matrices, Eq. (14.37) can be written as  dqT BTL HBL q2 þ BTNL ðqc ÞHBL q2 þ BTL HBNL ðqc Þq2  þ BTNL ðqc ÞHBNL ðqc Þq2 þ BTNL ðq2 Þsc ¼  dq

T

1 ½BL þ BNL ðqc ÞT HBNL ðq1 Þq1 þ BTNL ðq1 ÞH½BL þ BNL ðqc Þq1 2

(14.38)

Again, as dq is arbitrary, Eq. (14.38) can be written as BTL HBL q2 þ BTNL ðqc ÞHBL q2 þ BTL HBNL ðqc Þq2 þ BTNL ðqc ÞHBNL ðqc Þq2 þ BTNL ðq2 Þsc ¼

(14.39)

1  ½BL þ BNL ðqc ÞT HBNL ðq1 Þq1 þ BTNL ðq1 ÞH½BL þ BNL ðqc Þq1 2 It should be noted that the left-hand side of Eq. (14.39) is identical to that of Eq. (14.30), with the only difference being q2 in place of q1. Furthermore, one can identify the right-hand side of Eq. (14.39) as a force vector denoted by g as g¼

i 1h ½BL þ BNL ðqc ÞT HBNL ðq1 Þq1 þ 2BTNL ðq1 ÞH½BL þ BNL ðqc Þq1 2 (14.40)

Under these considerations and with the same manipulations as used for Eq. (14.33), one can write Eq. (14.39) in a compact form as      _ G ðs_ b Þ q2 ¼ g _ D qb ; q_ b þ K Ktb þ 4ðlc  lb Þ K

(14.41)

It can be noticed that in the left-hand side of Eq. (14.41) a factor f has been introduced such that f z 1 but f < 1. Without this factor the left-hand side of Eq. (14.41)

Computational aspects for stability and vibrations of thin-walled composite structures

703

is identical to that of Eq. (14.33) and therefore becomes singular. In the implementation of the approach used in this chapter, f ¼ 0.99 is considered. Furthermore, qc, q_ c and sc, s_ c are also approximated as qc z qb ; q_ c z q_ b sc z sb ; s_ c z s_ b

(14.42)

With these approximations, one can evaluate both g and the orthogonality constraint at l ¼ lb. The orthogonality constraint as defined by Eq. (14.21) can be translated to a finite element context as       1 qT1 KD qb ; q_ b þ KG ðs_ b Þ q2 þ ½HBNL ðq1 Þq1 T BNL ðq1 Þq_ b ¼ 0 2

(14.43)

Eqs. (14.41) and (14.43) are solved together to obtain the second-order mode q2. To determine the postbuckling coefficient b and the imperfection form factors a and b, the same approximations of Eq. (14.42) are used. In case of the second imperfection form factor b defined in Eq. (14.26), it can be noticed that it requires the computation € c , which can also be approximated as of s €b €c z s s

(14.44)

€ b can be evaluated as while s

1 2 B v H B þ ðq Þ L NL b qb     v2 sb 2 €b ¼ ¼ ¼ H BL q€b þ BNL ðqb Þq€b þ BNL q_ b q_ b s 2 2 vl vl (14.45) and q€b can be approximated as q€b ¼

q_ b  q_ b1 Dl

(14.46)

where Dl ¼ lb  lb1. Here lb1 is a load step preceding the final load step lb in the nonlinear prebuckling analysis, and q_ b and q_ b1 are available from the first linear solution during the NewtoneRaphson iteration process at each load step. By making lb1 and lb sufficiently close, a reasonable estimation of q€b is possible.

14.4

Numerical examples

In this section, the initial postbuckling coefficients will be presented for illustrative cases of specific composite cylindrical shells [29] in a single-mode context: 1. Booton’s anisotropic shell under external pressure, 2. Booton-type cylindrical and conical shells under axial loading.

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Stability and Vibrations of Thin-Walled Composite Structures

In case of the first example, the prebuckling state is linear. The second example involves composite cylindrical and conical shells under axial loading in which the prebuckling state is nonlinear. In the second example the importance of the inclusion of prebuckling nonlinearity is demonstrated.

14.4.1

Booton’s anisotropic shell under external pressure

In this example, two composite cylindrical shells having different lengths are analyzed. These cylindrical shells are known as Booton’s shell [29]. The composite layup is [30/ 0/30]. The material and geometric properties are reported in Tables 14.1 and 14.2, respectively. The shells are loaded by external pressure and the edges are simply supported at two ends. The buckling loads are normalized with respect to a reference buckling pressure, which is given by the following formula E11 t 2 (14.47) cR2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   where c ¼ 3 1  n212 . For the present example, this gives Pcr ¼ 0.36124  103 psi. Pcr ¼

The results are compared in Table 14.3 with semianalytical approach and a reasonable agreement is achieved.

14.4.2

The Booton-type cylindrical and conical shells under axial loading

In this example, Booton-type [29] anisotropic cylindrical and conical shells are considered. Analysis of the perfect shells is done, including the effect of prebuckling nonlinearity. Table 14.1

Material properties of Booton’s shell

Young’s modulus, E1

5.83  106 psi

Young’s modulus, E2

2.42  106 psi

Poisson’s ratio, n12

0.36

Shear modulus, G12

6.68  105 psi

Density, r

2.6  104 lb s2/in.4

Geometric properties of Booton’s shell Table 14.2

Radius, R

2.67 in.

Thickness, t

0.0267 in.

Length, L

3.776, 5.34 in.

Stacking sequence

[30/0/30]

Computational aspects for stability and vibrations of thin-walled composite structures

705

Table 14.3 Normalized buckling pressure and b coefficients (Booton’s shell) b

Buckling load L/R

a

Na

Semianalytical 1

DIANA

Semianalytical 1

2

DIANA

1.41

8

0.5656  10

0.5697  10

6.3289  10

6.2664  102

2.0

6

0.3921  101

0.4007  101

5.0141  102

4.9544  102

The number of circumferential waves in the buckling mode.

14.4.2.1 Cylindrical shells This shell was also used earlier in static stability investigation [15,29]. The cylindrical shell is loaded under axial compression. The edges are subject to “simply supported” boundary conditions SS-3 and SS-4. The geometric and material properties of the shell are the same as those in Tables 14.1 and 14.2, respectively. For this example, L ¼ 5.34 inch is chosen. Table 14.4 shows the comparison of the buckling loads and b coefficients between ANILISA and DIANA. Fig. 14.1(a) and (b) show the prebuckling and the first

Comparison of buckling load and b coefficients of Booton’s anisotropic shell

Table 14.4

b coefficient

Buckling load intensity (lb/in.) Boundary condition SS-4 SS-3

(a)

N

ANILISA

8

3.87419  10

7

3.79076  10

DIANA 2 2

(b)

ANILISA

DIANA

3.85395  10

0.31403

0.33193

3.73286  10

0.36632

0.37445

2 2

(c)

Figure 14.1 Deformation modes of Booton’s anisotropic shell, SS-4: (a) prebuckling mode, (b) buckling mode, and (c) second-order mode.

706

Stability and Vibrations of Thin-Walled Composite Structures

buckling mode, respectively, for SS-4 boundary condition. The buckling mode contains eight circumferential full waves. The layup of Booton’s shell is [30/0/30]; therefore, it is not symmetric. The resulting B matrix in the constitutive equation contains nonzero B16 and B26 terms that cause a coupling between bending moments and in-plane shear force. The skewness of the buckling mode is due to this coupling. Corresponding second-order modes are shown in Fig. 14.1(c). As observed in other studies [2] the same pattern in the second-order mode is also noticeable here, i.e., it contains twice the number of circumferential full waves of the buckling mode plus an axisymmetric contraction.

14.4.2.2 Conical shells Zhang [28] considered Booton-type composite conical shells for initial postbuckling analysis. In the following example, some of those results will be compared. The height and the smaller radius of the conical shell remains the same as the height and the radius of the Booton’s shell in the last example. The larger radius of the conical shell is adjusted such that the semivertex angle as ¼ 30 . In this case, clamped boundary condition MC4 [28] is used. The rotational degree of freedom around the circumferential direction is restrained both at the bottom and top edges. The stacking sequence is taken of the type [q/0/q] and several cases, e.g., q ¼ 30 , 40 , 60 , 65 , are considered. In Table 14.5 the normalized buckling loads and b coefficients are compared between the results obtained by Zhang and the present finite element approach. The normalizing factor used for the buckling load is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2pE1t2cos2as/c, where c ¼ 3 1  n212 . It can be noticed that in some cases the level of agreement in b coefficients is higher than in other cases. A possible reason for this discrepancy is the difference between the kinematic model used in DIANA and the Donnell-type kinematics used in the semianalytical tool.

Comparison of bifurcation buckling loads and b coefficients of the Booton-type conical shells

Table 14.5

b coefficient

Normalized buckling load Stacking sequence

N

Zhang (Ref. [28])

DIANA

Zhang (Ref. [28])

DIANA

[30/0/30]

7

0.414

0.411

0.050

0.067

[40/0/40]

8

0.425

0.421

0.338

0.368

[60/0/60]

9

0.453

0.447

0.665

0.656

[65/0/65]

10

0.450

0.442

0.220

0.174

Computational aspects for stability and vibrations of thin-walled composite structures

14.5

707

Multimode initial postbuckling analysis

14.5.1 Introduction In this section the single-mode perturbation approach discussed in section Singlemode initial postbuckling analysis is extended for the multimode case. The extension is based on the formulation by Byskov and Hutchinson [30] that is valid for simultaneous, nearly simultaneous, and well-separated modes and is therefore very suitable for studying modal interactions. The contribution of the prebuckling nonlinearity is included in the formulation in a similar way as in the single-mode case.

14.5.2 Perturbation method The variables ðu; e; sÞ of the postbuckling equilibrium state can be expanded in multimode form in a similar fashion as Eq. (14.9) about the prebuckling equilibrium state ðu0 ; e0 ; s0 Þ: u ¼ u0 ðlÞ þ ui xi þ uij xi xj þ . e ¼ e0 ðlÞ þ ei xi þ eij xi xj þ .

(14.48)

s ¼ s0 ðlÞ þ si xi þ sij xi xj þ . where i, j ¼ 1, 2.m. Here m is the number of buckling modes considered for the multimode analysis. In Eq. (14.48), lowercase repeated indices imply summation, whereas uppercase indices do not imply summation unless otherwise stated. The necessary equations for the bifurcation buckling loads li and the corresponding buckling modes ui take the same form as Eqs. (14.10)e(14.12): ei ¼ L1 ðui Þ þ L11 ðuc ; ui Þ

(14.49)

si ¼ Hðei Þ

(14.50)

sc $dec þ sc $L11 ðui ; duÞ ¼ 0

(14.51)

where the prebuckling quantities with the subscript c are evaluated at the lowest critical or bifurcation load. In addition, it will be assumed that (l  lI) admits the same asymptotic perturbation expansion as Eq. (14.14): xI ðl  lI Þ ¼ aIjk lI xj xk þ bIjkl lI xj xk xl þ .

(14.52)

In view of Eq. (14.52), if a plot of load parameter (l) versus the mode amplitudes (xi) is made then the aIjk and bIjkl coefficients, respectively, indicate the slopes and curvatures of the postbuckling curve. In the present treatment, symmetric structures

708

Stability and Vibrations of Thin-Walled Composite Structures

with postbuckling slopes, aIjk ¼ 0, and typically negative dominant postbuckling curvatures, bIjkl, indicating unstable postbuckling behavior are considered. With the same manipulation as in section Perfect structure, one can finally obtain the necessary equations for the determination of the second-order modes uij: 1 eij ¼ L1 ðuij Þ þ L11 ðuc ; uij Þ þ L11 ðui ; uj Þ 2

(14.53)

sij ¼ Hðeij Þ

(14.54)

1 sij $dec þ sc $L11 ðuij ; duÞ þ ½si $L11 ðuj ; duÞ þ sj $L11 ðui ; duÞ ¼ 0 2

(14.55)

The second-order modes uij are further subject to the following orthogonality condition:   s_ c $L11 ðui ; uij Þ þ s1 $L11 u_ c ; uij þ s2 $L11 ðu_ c ; ui Þ ¼ 0 (14.56) To obtain the expression for bIjkl the following expansion of the total potential energy, P(x, l) of the structure in the postbuckling regime by Byskov and Hutchinson [30] and Van Erp [31] will be used. Although this expansion was derived based on linear prebuckling state, here the additional terms resulting from nonlinearity of the prebuckling state will be accounted for.   m  1X l b I x2 þ Aijk xi xj xk þ Aijkl xi xj xk xl Pðx; lÞ ¼ (14.57) 1 lI D I 2 I¼1 lI b I , Aijk, and Aijkl are defined as where D b I ¼ 2sI $L11 ðu_ c ; uI Þ þ s_ c $L2 ðuI Þ D 1 Aijk ¼ L1 ðui Þ$HL11 ðuj ; uk Þ 2 1 Aijkl ¼ L1 ðuk Þ$HðL11 ðul ; uij ÞÞ þ L1 ðul Þ$HðL11 ðuk ; uij ÞÞ 4

1 þ L1 ðuij Þ$HðL11 ðuk ; ul ÞÞ þ L11 ðui ; uj Þ$HðL11 ðuk ; ul ÞÞ 2

(14.58) (14.59)

(14.60)

dP ¼ 0 The reduced set of Eq. (14.52) can be obtained from Eq. (14.57) by setting dx I and noting that the lowercase indices denote summation and that Aijk is symmetric between (j, k) and Aijkl is symmetric between (i, j) and (k, l). Finally, one obtains the expression for bIjkl as shown in the following:

bIjkl ¼

2 ðA þ AjklI Þ b I Ijkl lI D

(14.61)

Computational aspects for stability and vibrations of thin-walled composite structures

709

For asymmetric buckling modes, in case of the axially loaded cylindrical shells considered in the present application of the approach, Aijk ¼ 0 and consequently aIjk ¼ 0. As bIjkl coefficients have four indices, recalling the permutation formula with repetition, the possible number of bIjkl coefficients would be m4. However, it is possible to express bIjkl in such a way that it becomes symmetric with respect to all indices (i.e., if any two of the indices are swapped, the resulting coefficient remains unchanged). A careful look at Eq. (14.57) reveals that the order of the indices of coefficients Aijkl is immaterial because they are subject to the summation convention of repeated indices. Rather the sum of Aijkl coefficients for the possible permutations of a specific set of indices {i, j, k, l} is decisive. In Eq. (14.60), Aijkl are not symmetric with respect to all indices but in view of the argument made earlier, if one takes an arithmetic average of Aijkl coefficients for the possible permutations of a specific set {i, j, k, l}, then that averaged Aijkl coefficient represents all the Aijkl coefficients related to that specific set {i, j, k, l}. Therefore, a symmetric output of Aijkl with respect to all indices is possible where the order of the indices is immaterial. If Aijkl are made symmetric in this way, bIjkl also becomes symmetric according to Eq. (14.61), provided the buckling modes are scaled such that bI ¼ 1 lI D

(14.62)

b I is defined in Eq. (14.58). Recalling the combination formula with repetition, where D the possible number of symmetric bIjkl, Nbsymm will be Nbsymm ¼ mðm þ 1Þðm þ 2Þðm þ 3Þ=24

(14.63)

In case of imperfect structure, as in section Perfect structure, the asymptotic expansion as defined by Eq. (14.52) is modified to xI ðl  lI Þ ¼ aIjk lI xj xk þ bIjkl lI xj xk xl  aI lI x  bI ðl  lI Þx þ .

(14.64)

where the coefficients aI and bI are known as the first and second imperfection form factors, respectively. Using the same approach as explained in section Perfect structure the expressions for aI and bI are obtained as       b I Þ sI $L11 b aI ¼ ð1 lI D u ; uc þ sc $L11 b u ; uI

(14.65)

       b I ÞfsI $L11 b bI ¼ ð1 D u ; u_ c þ s_ c $L11 b u ; uI þ H½L11 ðu_ c ; uI Þ$L11 b u ; uc  aI lI ½sI $L11 ðu€c ; uI Þ þ ð1=2Þ€ sc $L11 ðuI ; uI Þ þ H½L11 ðu_ c ; uI Þ$L11 ðu_ c ; uI Þg (14.66) Eq. (14.64) is a small set of m nonlinear algebraic equation. Once the coefficients aIjk, bIjkl, aI, and bI are computed, one can carry out imperfection sensitivity analysis

710

Stability and Vibrations of Thin-Walled Composite Structures

by solving Eq. (14.64) with varying imperfection amplitude x at very little additional computational expense. The reduced set of nonlinear equation (Eq. 14.64) is solved using the software package HOMPACK77, a suite of FORTRAN 77 subroutines for solving the nonlinear system of equations using homotopy methods [32].

14.5.3

Finite element implementation

The finite element implementation of the buckling analysis remains the same as in the single-mode case and can be written as      _ G ðs_ b Þ qi ¼ 0 _ D qb ; q_ b þ K Ktb þ ðli  lb Þ K

(14.67)

where li and qi are the buckling loads and modes, respectively. The scaling of the buckling modes according to Eq. (14.62) can be done using the geometric and initial displacement stiffness matrices as     _ D qb ; q_ b þ K _ G ðs_ b Þ qI ¼ 1 lI qTI K

(14.68)

Eqs. (14.53)e(14.55) are the equations for the determination of the second-order modes in functional notation. In terms of finite element matrices, they can be written as      _ G ðs_ b Þ qij ¼ gij _ D qb ; q_ b þ K Ktb þ fðlc  lb Þ K

(14.69)

where lc is the first (lowest) buckling load. The right-hand side force vector gij is defined as 1h gij ¼  ½BL þ BNL ðqc ÞT HBNL ðqi Þqj þ BTNL ðqi ÞH½BL þ BNL ðqc Þqj 2 i   þ BTNL qj H½BL þ BNL ðqc Þqi (14.70) The associated orthogonality constraint as defined by Eq. (14.56) can be translated to finite element context as       1 qTi KD qb ; q_ b þ KG ðs_ b Þ qij þ ½HBNL ðqi Þqi T BNL ðqi Þq_ b ¼ 0 2

(14.71)

Solution of Eq. (14.69) together with Eq. (14.71) gives the second-order modes qij. The postbuckling coefficients bIjkl are computed according to Eq. (14.61). A concise expression for AIjkl defined in Eq. (14.60) can be obtained as AIjkl ¼

1 8

Z

 T 1 HBNL ðqi Þqj ½BNL ðqk Þql dv  qTij gkl 2 v

(14.72)

Computational aspects for stability and vibrations of thin-walled composite structures

711

where H is the stressestrain matrix. In Eq. (14.72) the integration sign implies integration over the entire structure. The imperfection form factors (aI, bI) are evaluated at each integration point using Eqs. (14.65) and (14.66) and summed up over the entire structure. To compute aI, bI, one needs to first compute q_ c , q€c , which can be done in the same way as in Postbuckling analysis section.

14.6

Numerical examples

14.6.1 Waters’ shell under axial loading In this section, modal interactions in the buckling behavior of a composite cylindrical shell, previously studied in Ref. [33], are illustrated using the multimode approach. The geometric and material properties of the eight-ply composite shell are listed in Table 14.6 (also see Fig. 14.2). The cylindrical shell is axially loaded and both ends are subject to the classical simply supported boundary condition SS-3 [33]. In the finite element model, the SS-3 boundary condition is applied by restraining radial displacements at both edges and applying kinematic constraints such that the relative circumferential displacements of the nodes at an edge are the same. The rotational degrees of freedom at the edges remain free. Axially directed equal and opposite distributed shell edge loads are applied at both ends and the displacement degrees of freedom of one of the nodes at one end of the cylinder is fixed to suppress the rigid body mode. Two analysis cases are considered: 1. In the first case, the analysis is carried out without considering any axisymmetric imperfection. 2. In the second case, axisymmetric imperfection is considered. In both cases, asymmetric imperfections as a linear combination of the first few buckling modes are considered. Table 14.6 Geometric and material properties of Water’s composite cylindrical shell (a) Material properties E11 (N/mm2)

12.7629  104

E22 (N/mm2)

1.13074  104

G12 (N/mm2)

6.00257  103

n12

0.300235

(b) Geometric properties Cylinder length (mm)

355.6

Cylinder radius (mm)

203.18603

Layup

[45/0/90]s

Total thickness (mm)

1.01539

712

Stability and Vibrations of Thin-Walled Composite Structures

(a)

(b)

(c)

(d)

(e)

(f)

Figure 14.2 Deformation modes for multimode analysis; imperfection  two asymmetric  modes are included without axisymmetric imperfection xaxi ¼ 0:0 . (a) q0, (b) q1 (N ¼ 11), (c) q2 (N ¼ 10), (d) q11, (e) q12, and (f) q22.

In both cases the analysis is carried out in the following steps: 1. A nonlinear buckling analysis is done. 2. A single-mode analysis of the perfect structure is done followed by an imperfection sensitivity analysis consisting of a comparison between the full model and the reduced order model analysis in terms of prediction of the limit-point buckling load with varying imperfection amplitudes. 3. A multimode analysis of the perfect structure is done. The minimum direction in the space of the buckling modes is computed and an imperfection sensitivity analysis is done in the minimum direction and compared with full model analysis. In all cases the buckling modes are scaled according to Eq. (14.68), unless otherwise stated.

14.6.1.1 Without axisymmetric imperfection The analysis is started by carrying out bifurcation buckling calculations. In Table 14.7 the first five buckling loads are reported and compared with the semianalytical tool ANILISA [11,15]. The buckling loads are normalized with respect to the reference classical buckling load of the same cylindrical shell defined as E11 t 2 (14.73) cR qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   where c ¼ 3 1  n212 . For the present example, this gives Ncl ¼ 391.9888 N/mm. Ncl ¼

Computational aspects for stability and vibrations of thin-walled composite structures

713

Normalized bifurcation buckling loads without  axisymmetric imperfection  xaxi [ 0:0

Table 14.7

Bifurcation buckling load N

ANILISA

DIANA

9

0.33096

0.32615

10

0.32916

0.32466

11 (minimum)

0.32859

0.32442

12

0.33027

0.32635

13

0.33404

0.33057

a

a

The number of circumferential full waves in the buckling mode.

Next the effect of a single asymmetric imperfection of the same shape as the first buckling mode is studied. The imperfection is defined as b u ¼ xui

(14.74)

where x is the imperfection amplitude and ui is a specific buckling mode. Such an asymmetric imperfection will cause buckling to occur at a limit point instead of a bifurcation point. In Table 14.8 the b coefficient corresponding to the first bifurcation buckling mode (N ¼ 11) is reported. The b coefficient compares well with the results reported in Ref. [33], which were obtained using ANILISA. Because the b coefficient depends on the scaling of the corresponding buckling mode, for the sake of comparison, in this case the buckling mode is scaled in the same way as in Ref. [33], where the scaling is made such that the maximum out-of-plane displacement of the buckling mode is equal to the shell thickness. The negative b coefficient indicates an unstable postbuckling response and high imperfection sensitivity of the structure. Imperfection amplitudes x ¼ 7:371; 14:742; and 29:484 are chosen such that they correspond, respectively, to 5%, 10%, and 20% of the shell thickness. In Table 14.9 the buckling loads are compared both with ANILISA and a full model analysis. As anticipated a sharp decrease in the failure load due to the negative b coefficient is noticeable.

Comparison of b coefficient corresponding to the lowest buckling mode (N [ 11) for the case without axisymmetric  imperfection xaxi [ 0:0

Table 14.8

b coefficient N

ANILISA

DIANA

11

0.37605

0.37445

714

Stability and Vibrations of Thin-Walled Composite Structures

Table 14.9 Normalized limit-point buckling loads for  the case without axisymmetric imperfection xaxi [ 0:0 and asymmetric mode (N [ 11) Buckling load Full model

Reduced order model

x

DIANA

ANILISA

DIANA

0.0

0.32442

0.32859

0.32442

7.371

0.28317

0.29065

0.28620

14.742

0.25766

0.26708

0.26272

29.484

0.21939

0.22758

0.22366

Next a multimode analysis is carried out. In the multimode analysis the asymmetric imperfection b u is assumed to be a linear combination of m buckling modes under consideration as b u ¼ xei ui

(14.75)

where i ¼ 1.m, x is the imperfection amplitude, ui are the buckling modes, and ei are considered as the direction of the imperfection vector b u in the unitary sphere defined as ei ei ¼ 1

(14.76)

In Eqs. (14.75) and (14.76), repeated indices indicate summation. When m > 1, there is an infinite number of ways for choosing a vector of unitary norm as defined by Eq. (14.76). Therefore, the minimum direction is sought out of the first few buckling modes [2] and the imperfection vector is considered in that direction. For the multimode analysis, respectively, two modes (m ¼ 2) and four modes (m ¼ 4) are considered. In case of m ¼ 2 in Fig. 14.2 the prebuckling mode q0, the first two buckling modes q1, q2, and the resulting three second-order modes q11, q12, q22 are shown. The skewness of the buckling modes is noticeable, which is due to anisotropy of the shell material. The shell is composed of a symmetric laminate [45/0/90]s. Therefore, bendingestretching coupling is absent, and also þ45 lamina is complemented by a 45 lamina and the other two types of lamina are 0 and 90; therefore, coupling between stretching-in-plane shear is absent too. However, the D16 and D26 terms in the D matrix of the constitutive relation are nonzero, which results in the bendingetwisting coupling effect and causes the observed skewness of the buckling modes. The worst-case imperfection vectors are reported in Table 14.10. The resulting limit-point buckling loads are reported in Table 14.11. Because of the modal interaction, somewhat lower buckling loads compared to the corresponding single-mode analysis case (Table 14.9) are noticed.

Computational aspects for stability and vibrations of thin-walled composite structures

715

Imperfection vectors for multimode analysis without axisymmetric imperfection   xaxi [ 0:0

Table 14.10

Number of modes (m)

Imperfection vector (e)

2

[0.70485, 0.70935]

4

[0.55980, 0.45716, 0.42798, 0.54262]

Table 14.11 Normalized buckling loads for multimode (two- and four mode) analysis without axisymmetric imperfection xaxi [ 0:0 Buckling load (two mode)

Buckling load (four mode)

x

Full model

Reduced model

Full model

Reduced model

0.0

0.32442

0.32442

0.32442

0.32442

7.371

0.27552

0.28053

0.27042

0.27686

14.742

0.24490

0.25337

0.23470

0.24669

29.484

0.19643

0.20812

0.18113

0.19595

14.6.1.2 With axisymmetric imperfection In this section the analysis case including an axisymmetric imperfection is carried out. The axisymmetric imperfection b u axi (positive outward) is defined as  x b u axi ¼ xaxi t cos ip L

(14.77)

where xaxi is the amplitude of the axisymmetric imperfection, x is the axial coordinate measured from one end of the cylindrical shell, and i is a positive integer. It is assumed that i ¼ 2, which is assumed to give the most likely axisymmetric imperfection of the steel mandrel used to lay up the type of composite cylindrical shell being studied [33]. In the present application example the axisymmetric imperfection amplitude xaxi is taken as 0.1. The axisymmetric part of the imperfection is defined in the DIANA input file as an initial displacement field. The axisymmetric imperfection lowers the first bifurcation buckling load and yields the first buckling mode with seven circumferential full waves, i.e., N ¼ 7. The bifurcation buckling loads are given in Table 14.12. In Fig. 14.3 the prebuckling, buckling, and second-order modes for N ¼ 7 are shown. First, a single-mode analysis using the first asymmetric mode (N ¼ 7) is carried out. The same imperfection amplitudes x Þ are used as before. The resulting limit-point buckling loads are reported in Table 14.13. For multimode analysis, respectively, three (m ¼ 3) and five (m ¼ 5) buckling modes are used. The minimum/worst imperfection vectors are reported in Table 14.14

716

Stability and Vibrations of Thin-Walled Composite Structures

Normalized bifurcation buckling loadsfor the  case with axisymmetric imperfection xaxi [ 0:1

Table 14.12

Bifurcation buckling load N

ANILISA

DIANA

7 (minimum)

0.32296

0.31221

9

0.33113

0.32737

10

0.32960

0.32606

11

0.32916

0.32600

12

0.33107

0.32859

(a)

(b)

(c)

  Figure 14.3 Deformation modes for the case with axisymmetric imperfection xaxi ¼ 0:1 : (a) q0, (b) q1 (N ¼ 7), and (c) q11.

Normalized limit-point buckling loads with axisymmetric imperfection  xaxi [ 0:1 and asymmetric mode (N [ 7)

Table 14.13

Buckling load x

Full model

Reduced model

0.0

0.31221

0.31221

7.371

0.25238

0.25835

14.742

0.21582

0.23207

29.484

0.16679

0.19679

for each case. The resulting limit-point buckling loads are reported in Table 14.15. It can be noticed that the single-mode case (for the first buckling mode with N ¼ 7) indeed shows the highest decrease in limit-point buckling load at the same level of imperfection amplitude (see Table 14.13). The underlying assumption behind the

Computational aspects for stability and vibrations of thin-walled composite structures

717

Imperfection vectors analysis with   for multimode axisymmetric imperfection xaxi [ 0:1

Table 14.14

Number of modes (m)

Imperfection vector (e)

3

[0.00008, 0.70813, 0.70607]

5

[0.10653, 0.69273, 0.51064, 0.31515, 0.38560]

Normalized limit-point buckling loads for multimode (three- and five-mode) analysis with   axisymmetric imperfection xaxi [ 0:1

Table 14.15

Buckling load (three mode)

Buckling load (five mode)

x

Full model

Reduced model

Full model

Reduced model

0.0

0.31221

0.31221

0.31221

0.31221

7.371

0.27297

0.27802

0.26531

0.26928

14.742

0.24235

0.24936

0.22960

0.23519

29.484

0.19643

0.20316

0.17858

0.18002

computation of the minimum direction [2] is that the buckling modes should be simultaneous. In this case, the first buckling mode is relatively away from the other clustered modes, which may be a possible reason why the imperfection in the computed minimum direction did not show the maximum imperfection sensitivity. In case of m ¼ 5 the effect of the asymmetric imperfection amplitude on buckling load has been studied. The result is shown in Fig. 14.4. A strong imperfection sensitivity is observed. It can also be noticed that for asymmetric imperfection amplitude up to 20% of the shell thickness, the prediction of limit-point buckling load made by perturbation approach compares well with that of fully nonlinear analysis.

14.7

Introduction to finite element dynamic analysis

This section presents the extension of the perturbation approach to dynamic buckling problems and nonlinear vibration problems. Dynamic buckling of a composite cylindrical shell under external pressure, in which the prebuckling state is linear, is presented. The results are compared with those obtained by Schokker et al. [34] who applied the same perturbation approach for dynamic buckling analysis of composite cylindrical shells under hydrostatic pressure using p-version of the finite element method. Additionally the dynamic buckling problem of a composite cylindrical shell (Booton’s shell [29]) under axial loading with the inclusion of prebuckling nonlinearity is presented and compared with full model dynamic analysis carried out using ABAQUS [35].

718

Stability and Vibrations of Thin-Walled Composite Structures

0.35

Normalized buckling load

0.3 0.25 Full model

0.2

Reduced order model

0.15 0.1 0.05 0

0

20 40 60 80 Asymmetric imperfection amplitude

100

Figure 14.4 Effect of asymmetric imperfection amplitude on buckling load:   five asymmetric imperfection modes and one axisymmetric imperfection mode xaxi ¼ 0:1 .

A perturbation approach for nonlinear or large-amplitude vibration problems was implemented earlier in a finite element framework within a MATLAB environment by Tiso [36], who applied the approach to beam structures as well as isotropic plates and cylindrical shells. As in case of static postbuckling, the implementation is available in the DIANA finite element code and using the layered curved shell element CQ40L [2], for nonlinear vibrations analysis of a composite cylindrical shell is illustrated in this chapter. The results of the analyses presented will be compared with semianalytical solutions [37,38].

14.8

Dynamic buckling

In this section the perturbation approach used for the dynamic buckling analysis will be discussed. It is basically an extension of the multimode formulation presented in section Perturbation method, with the inclusion of inertial effects.

14.8.1

Perturbation method

Because of the inertial forces, the variation equation (Eq. 14.6) takes the form € s$de  f$du þ MðuÞ$du ¼0

(14.78)

Computational aspects for stability and vibrations of thin-walled composite structures

719

€ which is linear where the dots represent differentiation with respect to time and MðuÞ, € represents the inertial loading. Here also, like Eq. (14.8), the reciprocal relation in u, MðuÞ$v ¼ MðvÞ$u

(14.79)

holds. The dynamic loading is assumed to take the form f ¼ lF(t)f0 where the time variation F(t) is normalized so that its maximum value is unity. Now one can write the dynamic counter part of Eq. (14.48) as u ¼ lFðtÞu0 þ ui xi ðtÞ þ uij xi ðtÞxj ðtÞ þ . e ¼ lFðtÞe0 þ ei xi ðtÞ þ eij xi ðtÞxj ðtÞ þ .

(14.80)

s ¼ lFðtÞs0 þ si xi ðtÞ þ sij xi ðtÞxj ðtÞ þ . By repeating the same procedure as for the static case and neglecting the inertial forces associated with the prebuckling displacements, one obtains !



1 € lFðtÞ lFðtÞ ðtÞ þ 1  ðtÞ þ a x ðtÞx ðtÞ þ b x ðtÞx ðtÞx ðtÞ ¼ xI x x Ijk j Ijkl j I I k k l lI lI u2I (14.81) where u2I is defined as u2I ¼

sI $eI MðuI Þ$uI

(14.82)

If uI happens to be a natural vibration mode then uI is its natural circular frequency, otherwise u2I has an interpretation as a Rayleigh quotient for circular frequency squared based on the buckling mode uI. In the present implementation, Eq. (14.81) is solved with the standard RungeeKutta scheme of the fourth order. To account for prebuckling nonlinearity, Eq. (14.82) is written as u2I ¼

bI lI D MðuI Þ$uI

(14.83)

b I in It can be noticed that the quantity sI $eI in Eq. (14.82) is replaced by lc D Eq. (14.83). Finally, Eq. (14.81) is modified to !

1 € lFðtÞ xI ðtÞ þ 1  xI ðtÞ þ aIjk xj ðtÞxk ðtÞ þ bIjkl xj ðtÞxk ðtÞxl ðtÞ lI u2I

lFðtÞ ¼ a I x I  bI 1  xI lI

(14.84)

720

Stability and Vibrations of Thin-Walled Composite Structures

where aI and bI are imperfection form factors [39], which are evaluated using Eqs. (14.65) and (14.66). In case of linear prebuckling state, both aI and bI are unity, and Eq. (14.84) becomes identical to Eq. (14.81). Furthermore, in case of nonlinear prebuckling state the buckling analysis is carried out at a state close to bifurcation buckling load in contrast to the undeformed state considered in case of linear prebuckling state. Now with the background established so far, a step-by-step procedure for the perturbation-type dynamic buckling analysis can be set up as in the following: computation of the prebuckling state u0, computation of the buckling loads lI and the corresponding buckling modes uI, computation of the second-order modes uij, computation of the bijkl coefficient (aijk ¼ 0 for the symmetric structures considered in the present application examples), 5. computation of the mode amplitude xI(t) corresponding to the applied dynamic load by solving Eq. (14.81) and identification of the dynamic buckling load level l ¼ ld at which xI(t) shows a sharp rise, 6. Recovering the displacement, stress, and strain by substituting the already computed terms in Eq. (14.80).

1. 2. 3. 4.

14.8.2

Finite element implementation

As far as finite element implementation is concerned, most of the ingredients are already present (see section Finite element implementation). To compute the additional Rayleigh quotient u2I of Eq. (14.83), computation of the denominator M(uI)$uI is necessary. In finite element context, it can be computed as MðuI Þ$uI ¼ uTI MuI

(14.85)

where M is the mass matrix that is already available in DIANA for the elements (CQ40S, CQ40L). The buckling problem can be cast into an eigenvalue problem as defined in Eq. (14.29). In case of follower loads such as fluid pressure, the loading direction can change to remain normal to the shell surface. In that case the follower load gives rise to an additional stiffness term known as load stiffness. In the present application examples the contribution of the load stiffness is accounted for according to Ref. [40]. The eigenvalue problem of Eq. (14.29) is then modified to, for instance, in case of linear prebuckling ½KM þ lI ðKG þ KLd ÞqI ¼ 0

(14.86)

where KLd is the load stiffness matrix.

14.9

Nonlinear vibrations

In this section the perturbation approach used for the nonlinear vibration problem will be discussed. It is an analogous extension of the initial static postbuckling analysis presented in section Single-mode initial postbuckling analysis, with the inclusion of inertial effects.

Computational aspects for stability and vibrations of thin-walled composite structures

721

14.9.1 Perturbation method Application of the perturbation method for nonlinear vibration problem (in a similar way as initial postbuckling analysis) was done by Rehfield [10] in a single-mode context using functional notation. Tiso [3] made a finite element implementation of this approach. In the following, only the essential equations following Rehfield and Tiso will be mentioned. The dynamics of a system under periodic motion with radial frequency u is governed by Hamilton’s principle, which can be written as Z

2p=u 

0

  

1 vu vu M $  s$de dt ¼ 0 2 vt vt

(14.87)

where t denotes time and the “dot” operation implies the inner multiplication of variables and the integration over the entire domain. The mass operator M and its properties are already defined in Eq. (14.79). Starting from Eq. (14.87) and introducing the new time variable s ¼ ut and considering the periodicity, one obtains Z

2p

€ ðs$de þ MðuÞ$duÞds ¼0

(14.88)

0

The vibration mode and the corresponding strain and stress are assumed as u ¼ xu1 e ¼ xe1

(14.89)

s ¼ xs1 where x is an amplitude parameter associated with mode u1. If the proposed form in Eq. (14.89) is substituted in Eq. (14.88) and only linear terms are retained, and by letting du ¼ u1, the natural frequency squared u20 is obtained. Z 0

2p 

 u20 Mðu€1 Þ$du þ s1 $de ds ¼ 0 Z

u20 ¼ Z

0 2p

2p

(14.90)

s1 $e1 ds (14.91)

Mðu_ 1 Þ$u_ 1 ds

0

It is assumed for now that only one mode u1 is associated with the frequency u0. To find how the structure behaves when the amplitude of vibration becomes finite, the solution is expanded as follows u ¼ xu1 þ x2 u2 þ x3 u3 þ /

(14.92)

e ¼ xe1 þ x2 e2 þ x3 e3 þ /

(14.93)

722

Stability and Vibrations of Thin-Walled Composite Structures

s ¼ xs1 þ x2 s2 þ x3 s3 þ /

(14.94)

To make the expansion unique, the second-order mode u2 is orthogonalized to u1 with respect to the mass operator Mðu_ 1 Þ$u_ k ¼ Mðu€1 Þ$uk ¼ 0;

ks1

(14.95)

By substituting the expansion Eq. (14.92) in the equilibrium Eq. (14.88) and letting du ¼ u1 and accordingly de ¼ e1 , as well as by introducing the expression for u20 in Eq. (14.91), one finds Z 2p   2   x u Mðu€1 Þ$du þ s1 $de þ x2 u2 Mðu€2 Þ$du þ s2 $de þ s1 $L11 ðu1 ; duÞÞ 0

 þ x3 u2 Mðu€3 Þ$du þ s3 $de þ s1 $L11 ðu2 ; duÞ   þ s2 $L11 ðu1 ; duÞ þ / ds ¼ 0

(14.96)

Finally, the relation between the frequency u and the amplitude x is found by making use of reciprocity relations u2 ¼ 1 þ aD x þ bD x 2 þ / u20

(14.97)

where Z aD ¼

2p 0

u20

Z

3 s1 $L2 ðu1 Þds 2 2p

(14.98)

Mðu_ 1 Þ$u_ 1 ds

0

and Z bD ¼

2p 0

ð2s1 $L11 ðu1 ; u2 Þ þ s2 $L2 ðu1 ÞÞds Z 2p 2 u0 Mðu_ 1 Þ$u_ 1 ds

(14.99)

0

Eq. (14.97) is a compact representation of the effect of the vibration amplitude on the frequency. The calculation of the second-order coefficient bD requires the calculation of the second-order field u2. By equating the term multiplying x2 in Eq. (14.96) to zero, the second-order state problem is obtained: u2 Mðu€2 Þ$du þ s2 $de þ s1 $L11 ðu1 ; duÞ ¼ 0

(14.100)

Computational aspects for stability and vibrations of thin-walled composite structures

723

The second-order field u2 is time dependent and it is actually constituted by two parts. To obtain the two contributions following the same line as considered by Tiso [3], the time dependence of the vibration mode u1 can be written explicitly as u1 ¼ b u 1 cos s e1 ¼ b e 1 cos s

(14.101)

b 1 cos s s1 ¼ s where the hatted quantities are the spatial shapes, which are multiplied by a harmonic time response. By substituting Eq. (14.101) into the second-order state problem (Eq. 14.100), one obtains   1 s 1 $L11 b u2 Mðu€2 Þ$du þ s2 $de ¼  ð1 þ cos 2sÞb u 1 ; du 2

(14.102)

It can be noticed that the right-hand side of Eq. (14.102) consists of a constant forcing term and a harmonic forcing term. The solution can therefore be split into two parts: u2 ¼ b u 21 þ b u 22 cos 2s

(14.103)

which are the solution to the two problems     1 b 22 $de b 21 $de ¼  s b 1 $L11 b s u 1 ; du  4u2 M b u 22 $du þ s 2   1 b 1 $L11 b ¼ s u 1 ; du 2

(14.104)

where the mass operator of the first problem has been dropped, as b u 21 does not depend on time. By accounting for the two different contributions of the second-order field u2 and carrying out the time integrations, the coefficients aD and bD assume the form aD ¼ 0

(14.105)

          u 21 þ s1 $L11 b u 22 þ H L1 b bD ¼ 2s1 $L11 b u1; b u1; b u 21 $L2 b u1 1      3      2   u 22 $L2 b u 1 þ H L2 b u 1 $L2 b u 1 u0 M b u1 u 1 $b þ H L1 b 2 8 (14.106) Unlike the coefficient in the perturbation expansion for the load parameter in the initial postbuckling analysis, the first coefficient of the perturbation expansion for the frequency in the dynamic analysis, aD, is always zero, also in the case of nonsymmetric structures. A positive coefficient bD represents a hardening behavior, i.e., the

724

Stability and Vibrations of Thin-Walled Composite Structures

frequency of vibration increases with increasing amplitude. Conversely, a negative coefficient bD indicates a softening behavior. An important case is constituted by multiple vibration modes associated with the same frequency. The vibration modes can interact and modify the frequencye amplitude curve. This situation can be treated by assuming the displacement field as a linear combination of the m modes ui (i ¼ 1, 2,., m) and the contribution of the second-order modes uij as u ¼ xi ui þ xi xj uij þ /

(14.107)

with the corresponding strain and stress fields e ¼ xi ei þ xi xj eij þ /

(14.108)

s ¼ xi si þ xi xj sij þ /

(14.109)

where the summation convention is used for repeated indices. The derivation follows the same line as the multimode analysis for initial postbuckling [30]. Only the main results are reported here. The nonlinear frequencyeamplitude relations are obtained in the following form: ! u2 xI 1  2 þ xi xj aijI þ xi xj xk bijkI ¼ 0; I ¼ 1; 2; /; m (14.110) u0 I The aD and bD coefficients are found as Z 2p 1 ½sI $L11 ðui ; uj Þ þ 2si $L11 ðuj ; uI Þds aDijI ¼ 2 u 0I D I 0 bDijkI

1 ¼ 2 u0 I D I

Z

2p 0

1h sIi $L11 ðuj ; uk Þ þ sij $L11 ðuk ; uI Þ þ sI $L11 ðui ; ujk Þ 2

i þ si $L11 ðuI ; ujk Þ þ 2si $L11 ðuj ; ukI Þ ds where DI ¼

Z 0

2p

(14.111)

Mðu_ I Þ$u_ I ds

(14.112)

(14.113)

The second-order fields uJK are the solutions of the second-order state problems   1 u2 M u€jk $du þ sjk $de ¼  ½sj $L11 ðuk ; duÞ þ sk $L11 ðuj ; duÞ 2

(14.114)

Computational aspects for stability and vibrations of thin-walled composite structures

725

14.9.2 Finite element implementation In finite element notation, the eigenvalue problem (Eq. 14.90) giving the natural frequency and the corresponding single vibration mode b u 1 can be written as   KM  u21 M b (14.115) u1 ¼ 0 where KM is the material stiffness matrix and M is the mass matrix. After the calculation of the natural frequency and the corresponding vibration mode, the initial curvature of the frequencyeamplitude relation can be computed with a modest additional computational cost, by solving for the corresponding second-order modes. The linear problem for the second-order modes b u 21 , b u 22 is   b1 ½KM b u 21 ¼ g u

(14.116)

    KM  4u21 M b u 22 ¼ g b u1

(14.117)

with the orthogonality conditions T b u 21 ¼ 0 u 1 Mb

(14.118)

T b u 22 ¼ 0 u 1 Mb

(14.119)

  b 1 is defined at the element level as The forcing term g u i       1h b 1 ¼  BTL HBNL b g u u1 u1 b u 1 þ 2BTNL b u 1 HBL b 2

(14.120)

  b 1 is formed through an assembly operation of contributions calcuThe global g u lated at the element level.

14.10

Numerical examples

In this section the following examples are considered: 1. Booton’s shell under dynamic external pressure, 2. Booton’s shell under dynamic axial loading, 3. nonlinear free vibration of Booton’s shell.

The first two examples are related to dynamic buckling problems. Composite cylindrical shells under external pressure and axial loading applied as a step load are considered. The last example is about nonlinear free vibration problems of composite plates and cylindrical shells. In the following numerical examples the buckling mode is normalized such that the maximum out-of-plane displacement is equal to the shell wall thickness.

726

Stability and Vibrations of Thin-Walled Composite Structures

14.10.1 Booton’s shell under dynamic external pressure Booton’s anisotropic cylindrical shell [29] is the first numerical example for composite shells. This shell was also used earlier in static stability investigation [15]. The cylindrical shell is loaded under external pressure. The pressure is applied as radial pressure where the direction of the loading remains radial throughout the deformation process as opposed to fluid pressure loading where the loading direction remains normal to the shell surface. Therefore, the load stiffness term is not included in the buckling analysis. The edges are subject to the “simply supported” boundary condition SS-3. Different types of simply supported boundary conditions applied to cylindrical shell edges are discussed in Ref. [28]. To apply the SS-3 boundary condition in the finite element model, both radial and circumferential displacements at one end of the cylinder are restrained and also one node in the axial direction is fixed to suppress the rigid body motion in that direction. At the other end, radial displacements are restrained and the relative circumferential displacements among all the nodes of that edge are also restrained by means of kinematic constraints. This shell has already been considered in the earlier sections, in which the material and geometric properties of the shell are given in Tables 14.1 and 14.2, respectively. For this example, L ¼ 3.776 in. is chosen. In Table 14.16, vibration frequencies and modes are compared between the DIANA analysis and a semianalytical method [38]. Next, in Table 14.17 the first bifurcation buckling loads and the b coefficients between the DIANA analysis and the semianalytical tool ANILISA [11,15] are compared. The buckling loads are normalized with respect to the classical buckling load, which in this case is 3.6  102 psi. The buckling load and b coefficient have already been reported in Table 14.3 in the context of singlemode static analysis, and they are mentioned here again for the sake of completeness. In Fig. 14.5 the first bifurcation buckling mode and the corresponding second-order mode is depicted by means of deformed mesh plots. Now with all the quantities at hand for the perfect structure, the calculation for the dynamic buckling load under step loading for the imperfect structure is made with the

Comparison of vibration frequency for Booton’s shell (SS-3 boundary condition)

Table 14.16

Cyclic frequency

a

Na

Ref. [38]

6

1.1595  10

DIANA 1.129  103

3

The number of circumferential full waves.

Comparison of normalized buckling load and b coefficients for Booton’s shell under external pressure

Table 14.17

b coefficient

Buckling load N

ANILISA

DIANA

ANILISA

DIANA

8

5.6569  102

5.7  102

6.3289  102

6.2664  102

Computational aspects for stability and vibrations of thin-walled composite structures

(a)

727

(b)

Figure 14.5 (a) Buckling mode and (b) second-order mode of Booton’s shell under external pressure.

reduced analysis. The imperfection is introduced as the shape of the first buckling mode and its maximum out-of-plane amplitude is taken as 10% of the shell thickness. With reduced analysis the dynamic response of the structure in terms of the mode amplitude x(t) is computed several times to identify the load level at which x(t) shows a sharp rise or becomes unbounded. This load level is considered as the dynamic buckling load ld. With reduced analysis, each run takes only a fraction of a second because it is solving only one ordinary differential equation, i.e., Eq. (14.80). In Fig. 14.6 the response of the structure is shown in terms of out-of-plane (radial) deflection of one of the nodes at the cylinder’s midlength for just below and above the dynamic buckling load level. At the load level l/lc ¼ 0.82 a bounded response is obtained from the reduced analysis. The same reduced analysis at load level l/lc ¼ 0.83 yields an unbounded response. Therefore, one can conclude that the dynamic buckling load is somewhere between l/lc ¼ 0.82 and l/lc ¼ 0.83. The same analysis is carried out with a full model explicit dynamic analysis using ABAQUS. With full model analysis the response takes a sharp rise at the same load level as the reduced analysis. Hence, the prediction of dynamic buckling load with reduced analysis is in good agreement with that of full model analysis. The response also matched reasonably in the asymptotic sense. In Table 14.18 the computational cost has been compared between the reduced order model and the full model analysis. The same number of elements (8624) is used in both models, but in the reduced order analysis an eight-node shell element (DIANA element CQ40L) is used, whereas in the full model analysis a four-node shell element (ABAQUS element S4R) is used. Therefore, the full model contains less number of nodes (8820) than the reduced order model (26264) and the buckling analysis takes less time. Buckling analysis is included in the full model analysis to generate the imperfection shape. The total time needed is close between full model analysis and the reduced model analysis. However, it should be noted that only a single run time of the transient analysis is reported. To find the dynamic buckling load a number of such run is required. Also a new set of run is needed for each imperfection shape and amplitude. In case of the reduced order model, the buckling and postbuckling analyses (computation of the second-order modes and the postbuckling coefficients) are done once for all and the subsequent transient analyses for each imperfection shape and amplitude take only a fraction of a second.

728

Stability and Vibrations of Thin-Walled Composite Structures

0.35 Full model (0.82) Full model (0.83) Reduced order model (0.82) Reduced order model (0.83)

0.3

Displacement (mm)

0.25 0.2 0.15 0.1 0.05 0

0

0.002

0.004 Time (s)

0.006

0.008

Figure 14.6 Comparison of response (radial displacement of a node at midlength ofthe shell)  between full model explicit dynamic and reduced order model analyses xaxi ¼ 0:1t . Table 14.18 Comparison of computational cost between full model and reduced order model found at about l/lc [ 0.718 Analysis type

Full model (s)

Reduced order model (s)

Buckling analysis

65.00

397.78

Postbuckling analysis

N/A

499.57

Transient analysis

857.00

0.50

Total

922.0

897.85

14.10.2 Booton’s shell under dynamic axial loading In this example the same cylindrical shell is considered as in the previous example but instead of external pressure, distributed axial load is applied. Therefore, the effect of prebuckling nonlinearity is considered. In this example, another variant of the simply supported boundary condition, namely, SS4, will be used. The difference between SS-4 and SS-3 is that in case of the former also the relative axial displacements of the nodes at an edge are restrained using kinematic constraints. In Table 14.19, vibration frequencies and modes are compared between the present analysis and a semianalytical method [38]. Next in Table 14.20 the first bifurcation

Computational aspects for stability and vibrations of thin-walled composite structures

729

Comparison of vibration frequency for Booton’s shell (SS-4 boundary condition)

Table 14.19

Cyclic frequency N

Ref. [38]

7

1.4224  10

DIANA 3

1.3936  103

Comparison of normalized buckling load and b coefficients for Booton’s shell under axial loading

Table 14.20

b coefficient

Buckling load N

ANILISA

DIANA

ANILISA

DIANA

8

0.4001

0.3942

0.3548

0.3658

(a)

(b)

(c)

Figure 14.7 (a) Prebuckling, (b) buckling, and (c) second-order modes of Booton’s shell under axial loading.

buckling loads and the b coefficients are compared. The buckling loads are normalized with respect to the classical buckling load given by Eq. (14.73), which in this case is 964.5108 lb/in. A reasonable agreement of results is observed in both the tables. In Fig. 14.7 the prebuckling mode, the first bifurcation buckling mode, and the corresponding second-order mode are shown using deformed mesh plots. Now the calculation for the dynamic buckling load under step loading for the imperfect structure will be made with reduced analysis. Imperfection shape of the first buckling mode with the amplitude of 10% of the shell thickness is used. With reduced analysis the dynamic buckling was detected around the load level l/lc ¼ 0.768, whereas with full model explicit dynamic analysis, it was found at about l/lc ¼ 0.718.

14.10.3 Nonlinear free vibration of Booton’s shell The same Booton’s shell that has already been considered in the context of dynamic buckling problem in an earlier example in this chapter is used in this example for the analysis of nonlinear vibration problem. In Table 14.21 the lowest vibration

730

Stability and Vibrations of Thin-Walled Composite Structures

Comparison of natural vibration frequency and bD coefficient (Booton’s composite shell) Table 14.21

Vibration frequency, f (Hz)

bD

N

Semianalytical [37]

DIANA

6

1.1595  10

1.129  10

3

(a)

3

Semianalytical [37]

DIANA

0.1415

0.1284

(b)

Figure 14.8 The linear vibration mode corresponding to the lowest natural frequency of Booton’s shell: (a) isometric view and (b) top view.

frequency (N ¼ 6) and the dynamic b coefficient bD are compared with semianalytical results. The vibration frequency has already been reported in Table 14.16, and they are mentioned here again for the sake of completeness. The dynamic b coefficient is negative, thus corresponding to a softening behavior, i.e., the frequency decreases with increasing vibration amplitude. A reasonably good agreement is observed considering the difference in kinematic relations used between the finite element approach and the semianalytical approach. The linear vibration mode corresponding to the lowest frequency is shown in Fig. 14.8. The vibration mode is scaled respect to the shell  with  b thickness. The constant part of the second-order mode u is shown in Fig. 14.9 21  and the time-dependent part b u 22 is shown in Fig. 14.10. All the second-order fields

(a)

(b)

  b 21 of Booton’s shell: (a) isometric view and Figure 14.9 The second-order mode u (b) top view.

Computational aspects for stability and vibrations of thin-walled composite structures

(a)

731

(b)

  b 21 of Booton’s shell: (a) isometric view and Figure 14.10 The second-order mode u (b) top view.

2 1.75

Mode amplitude

1.5 1.25 1 Semi analytical

0.75

Present study 0.5 0.25 0 0.5

0.6

0.7 0.8 0.9 Normalized frequency

1

1.1

Figure 14.11 Backbone curve of Booton’s shell (mode amplitude x vs. normalized frequency u/u0).

are constituted by a periodic contribution with 2N circumferential waves and an axisymmetric deformation, as predicted by the semianalytical results [37]. Finally, the backbone curve is plotted in Fig. 14.11 showing the softening behavior and compared with semianalytical result. The deviation between the curves is due to the difference in bD coefficients (see Table 14.21).

732

14.11

Stability and Vibrations of Thin-Walled Composite Structures

Concluding remarks

In this chapter, characteristics of the geometrically nonlinear behavior of composite shell-type structures have been illustrated through the treatment and application of finite element integrated reduced order models based on a perturbation approach. A finite element formulation of Koiter’s perturbation approach for single mode and multimode initial postbuckling analysis has been presented. The approach is available within a general purpose finite element code [41]. In this approach a perturbation expansion for both the load and the displacement field is made around the bifurcation buckling point. In the second part of this chapter an extension of Koiter’s perturbation approach was used to treat dynamic buckling problems and a similar perturbation-type approach was applied to analyze nonlinear vibration problems. Full model finite element analysis and reduced order model analysis results for specific cylindrical shells illustrated the characteristics of the static and dynamic geometrically nonlinear behavior of composite shell-type structures. Results based on semianalytical analysis were presented as reference results.

References [1] J. Arbocz, C. Bisagni, A. Calvi, E. Carrera, R. Cuntze, R. Degenhardt, N. Gualtieri, H. Haller, N. Impollonia, M. Jacquesson, E. Jansen, H.-R. Meyer-Piening, H. Oery, A. Rittweger, R. Rolfes, G. Schullerer, G. Turzo, T. Weller, J. Wijker, Space Engineering e Buckling of Structures ECSS-E-HB-32-24A, Technical report, ESA-ESTEC, Requirements and Standards Division, 2010. [2] T. Rahman, A Perturbation Approach for Geometrically Nonlinear Structural Analysis Using a General Purpose Finite Element Code (Ph.D. thesis), Delft University of Technology, 2009. [3] P. Tiso, Finite Element Based Reduction Methods for Static and Dynamic Analysis of Thin-Walled Structures (Ph.D. thesis), Delft University of Technology, 2006. [4] T. Rahman, E.L. Jansen, Finite element based coupled mode initial post-buckling analysis of a composite cylindrical shell, Thin-Walled Structures 48 (2010) 25e32. [5] T. Rahman, E.L. Jansen, Z. G€urdal, Dynamic buckling analysis of composite cylindrical shells using a finite element based perturbation method, Nonlinear Dynamics 66 (2011) 389e401. [6] T. Rahman, E.L. Jansen, P. Tiso, A finite element-based perturbation method for nonlinear free vibration analysis of composite cylindrical shells, International Journal of Structural Stability and Dynamics 11 (2011) 717e734. [7] C.M. Menken, W.J. Groot, G.A.J. Stallenberg, Interactive buckling of beams in bending, Thin-Walled Structures 12 (1991) 415e434. [8] B. Budiansky, Dynamic buckling of elastic structures: criteria and estimates, in: Dynamic Stability of Structures: Proceedings of International Conference, Northwestern University, Evanston, Illinois, Pergamon Press, 1965. [9] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1979. [10] L.W. Rehfield, Nonlinear vibration of elastic structures, International Journal of Solids Structures 9 (1973) 581e590.

Computational aspects for stability and vibrations of thin-walled composite structures

733

[11] J. Arbocz, J.M.A.M. Hol, ANILISA e Computational Module for Koiter’s Imperfection Sensitivity Theory, Technical Report LR-582, Delft University of Technology, 1989. [12] B. Budiansky, J.W. Hutchinson, Dynamic buckling of imperfection sensitive structures, in: Proceedings of the 11th IUTAM Congress, Springer-Verlag, Berlin/G€ ottingen/Heidelberg/ Newyork, 1964, pp. 636e651. [13] G.A. Cohen, Effect of a nonlinear prebuckling state on the postbuckling behavior and imperfection sensitivity of elastic structures, AIAA Journal 6 (1968) 1616e1619. [14] J.R. Fitch, The buckling and postbuckling behavior of spherical caps under concentrated loads, International Journal of Solids and Structures 4 (1968) 421e446. [15] J. Arbocz, J.M.A.M. Hol, Koiter’s stability theory in a computer-aided engineering (CAE) environment, International Journal of Solids Structures 26 (1990) 945e975. € [16] G. Kirchhoff, Uber das Gleichgewicht und die Bewegung einer elastichen Scheibe, Crelle’s Journal 40 (1850) 51e88. [17] S. Timoshenko, S. Woinowski-Krieger, Theory of Plates and Shells, McGraw-Hill, Newyork, 1969. [18] D.J. Allman, A simple cubic displacement element for plate bending, International Journal for Numerical Methods in Engineering 10 (1976) 263e281. [19] D.J. Allman, Evaluation of the constant strain triangle with drilling rotations, International Journal for Numerical Methods in Engineering 26 (1988) 2645e2655. [20] D.J. Allman, A basic facet finite element for the analysis of general shells, International Journal for Numerical Methods in Engineering 37 (1994) 19e35. [21] G. Garcea, R. Casciaro, G. Attanasio, F. Giordano, Perturbation approach to elastic post-buckling analysis, Computers and Structures 66 (1998) 585e595. [22] A.D. Lanzo, G. Garcea, Koiter’s analysis of thin-walled structures by a finite element approach, International Journal for Numerical Methods in Engineering 39 (1996) 3007e3031. [23] A.D. Lanzo, G. Garcea, R. Casciaro, Asymptotic post-buckling analysis of rectangular plates by HC finite elements, International Journal for Numerical Methods in Engineering 38 (1995) 2325e2345. [24] J.F. Olesen, E. Byskov, Accurate determination of asymptotic postbuckling stresses by the finite element method, Computer and Structures 15 (1982) 157e163. [25] P.N. Poulsen, L. Damkilde, Direct determination of asymptotic structural postbuckling behavior by the finite element method, International Journal for Numerical Methods in Engineering 42 (1998) 685e702. [26] R.D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics 18 (1951) 31e38. [27] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics 12 (1945) 69e77. [28] G. Zhang, Stability Analysis of Anisotropic Conical Shells (Ph.D. thesis), Delft University of Technology, 1993. [29] M. Booton, Buckling of Imperfect Anisotropic Cylinders Under Combined Loading, Technical Report 203, UTIAS, 1976. [30] E. Byskov, J.W. Hutchinson, Mode interaction in axially stiffened cylindrical shells, AIAA Journal 15 (1977) 941e948. [31] G.M. Van Erp, Advanced Buckling Analysis of Beams with Arbitrary Cross Sections (Ph.D. thesis), Eindhoven University of Technology, 1989. [32] L.T. Watson, S.C. Billups, A.P. Morgan, HOMPACK: a suite of codes for globally convergent homotopy algorithms, ACM Transactions on Mathematical Software 13 (1987) 281e310.

734

Stability and Vibrations of Thin-Walled Composite Structures

[33] J. Arbocz, J.H. Starnes, M.P. Nemeth, On a high-fidelity hierarchical approach to buckling load calculations, in: 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Seattle, Washington, 2001, pp. 1e21. [34] A. Schokker, S. Sridharan, A. Kasagi, Dynamic buckling of composite shells, Computer and Structures 59 (1996) 43e53. [35] ABAQUS Analysis User’s Manual e Version 6.8, 2008. [36] P. Tiso, E.L. Jansen, A finite element based perturbation method for nonlinear free vibration of structures, in: 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu, Hawaii, AIAA, 2007, pp. 2007e2362. [37] E.L. Jansen, Nonlinear Vibrations of Anisotropic Cylindrical Shells (Ph.D. thesis), Faculty of Aerospace Engineering, Delft University of Technology, Delft, Netherlands, 2001. [38] E.L. Jansen, A perturbation method for nonlinear vibrations of imperfect structures: application to cylindrical shell vibrations, International Journal of Solids and Structures 45 (February 2008) 1124e1145. [39] T. Rahman, E.L. Jansen, Finite element based initial post-buckling analysis of shells of revolution, in: 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Schaumburg, Illinois, AIAA, 2008, pp. 2008e2120. [40] K. Schweizerhof, E. Ramm, Displacement dependent pressure loads in nonlinear finite element analyses, Computers and Structures 18 (6) (1984) 1099e1114. [41] DIANA FEA. DIANA User’s Manual - Release 10.1, Delft, The Netherlands, 2016.

Index ‘Note: Page numbers followed by “f” indicate figures, “t” indicate tables.’ A ABAQUS, 258e259, 373, 471e472, 473f ABAQUS-Python API, 260 FE model and analysis procedure in, 473f position of user subroutines within, 476f AC25.571C. See Advisory circular 25.571C (AC25.571C) Action Group (AG), 462e463 Ad hoc analytical approaches, 510 Adhesive American Cyanamid FM73 adhesive, 577e579 bonding, 634e636 material properties, 465e466, 465t Advanced sandwich-type constructions, 49 Advanced structural models, 53e56 class of refined plate models, 53e56 Advanced topometric sensor (ATOS), 310 Advisory circular 25. 571C (AC25.571C), 482 AFP. See Automated fiber placement (AFP) AG. See Action Group (AG) Aircraft flight loads, 505 primary structures, 497 Airframe structures, 482 Algebraic equation, 559 Aluminum shell edges casted in sand-resin rings, 634e636, 638f Ambartsumian multilayered theory (AMT), 107 American Cyanamid FM73 adhesive, 577e579 Amplitudeefrequency curves of anisotropic shell, 448e449, 449f AMT. See Ambartsumian multilayered theory (AMT) Angle-ply face sheets, sandwich plate with, 65te68t, 69

ANILISA, 453, 695e696, 705e706, 712e713, 713t, 716t, 726 Anisotropic shells, 545e546 ANSYS FE code, 659 Applied compressive force, 439 Applied loading, 433f, 437e441 Approximation functions, 305e309 in natural coordinate system, 330e332 ARAMIS, 409e410, 411f, 473e474 Arbitrary incrementing of pressure, 356e358, 357fe358f “As-built” structural model, 400 “As-designed” structural model, 400 “As-to-be-built” structure, 400, 413e414, 416e417 ASIs. See Axisymmetric imperfections (ASIs) Associated boundary conditions, 209e212, 229, 232, 439e440 Asymmetric modes, 441e442 Asymptotic approaches, 92e94 Asymptotic expansion, 699 ATL process. See Automated tape laying process (ATL process) ATOS. See Advanced topometric sensor (ATOS) Automated fiber placement (AFP), 413, 416 Automated tape laying process (ATL process), 414e415 Average (AVG) of IM7/8552 prepreg material, 487t “Average angle”, 621e623 Averaging method, 442e445 first-order approximation, 445 Axial loading Booton-type cylindrical and conical shells under, 704e706 Booton’s shell under dynamic, 728e729 waters’ shell under, 711e717

736

Axiomatic approaches, 92 Axisymmetric deformation, 440e441 Axisymmetric imperfections (ASIs), 260, 264, 264f effect, 447e448 mode, 447e448 normalized bifurcation buckling loads with, 715e717, 716t effect of asymmetric imperfection amplitude on buckling load, 718f deformation modes for case, 716f imperfection vectors for multimode analysis, 717t normalized limit-point buckling loads, 716t normalized limit-point buckling loads for multimode analysis, 717t normalized bifurcation buckling loads without, 712e714, 713t comparison of b coefficient corresponding to lowest buckling mode, 713t imperfection vectors for multimode analysis, 715t normalized buckling loads for multimode analysis, 715t normalized limit-point buckling loads, 714t Axisymmetric mode, 441e442 B B&H criterion. See Budiansky and Hutchinson criterion (B&H criterion) Barely visible impact damage (BVID), 500e501 Beam(s), 100, 559e563 buckling loads and natural frequencies, 562t buckling shape, 152 models, 91e92, 131e140 assessment on benchmark problems, 138e140 classical theories, 133e134 refined theories, 134e138 nondimensional circular frequency, 561f, 563t Benchmark problems, assessment on, 111e127, 112t, 138e140

Index

bending and vibration of composite plates, 112e121, 115te118t buckling of sandwich struts, 121e127, 124f Bending and vibration of composite plates, 112e121, 115te118t BenzeggagheKenane mixed-mode failure criterion (BeK mixed-mode failure criterion), 498e500 BernoullieEuler equation, 151 Bifurcation, 347e355, 348f, 349t arrangements for ply stacking, 348f effect of boundary conditions, 353t boundary conditions applied at equatorial plane, 353t buckling, 99e101, 351t, 352f CFRP deformed hemisphere, 351f cut-through 30-ply torisphere, 354f elastic constants, 349t ellipsoidal shells, 354f failed machined metal ellipsoids, 350f FE-computed eigenshape, 350f pressures in externally pressurized torispheres, 350t BeK mixed-mode failure criterion. See BenzeggagheKenane mixed-mode failure criterion (BeK mixed-mode failure criterion) Blade-stiffened cylinders, assembly for, 329e330 Booton’s anisotropic shell, 452e453 cylindrical shell, 726 under external pressure, 704 layer orientation effect on dynamic b-factor in, 455f shell length effect on dynamic b-factor in, 454f on lowest natural frequency, 455f Booton’s shell, 446, 446t, 545t, 546f, 704 conical shells, 704e706 cylindrical shells, 704e706 under dynamic axial loading, 728e729, 729f, 729t under dynamic external pressure, 726e727, 726t comparison of response between full model explicit dynamic, 728f

Index

computational cost between full model and reduced order model, 728t geometric properties, 704t material properties, 704t normalized buckling pressure and b coefficients, 705t Boundary conditions, 275, 370e375, 437e441, 488f, 489t, 544e546 benchmark case of isotropic plate, 371e373 finite element modeling considering VCT input, 373e375 Boundary conditions, 275, 370e375, 437e441, 488f, 489t, 544e546 benchmark case of isotropic plate, 371e373 finite element modeling considering VCT input, 373e375 Buckling, 224e226, 253, 400e403, 405, 408e409 analysis for buckling and vibrations of imperfect composite cylindrical shells, 540e543 of anisotropic shells, 452 behavior, 429 of columns CLT, 149e157 FSDT, 158e163 criteria, 226f equation for columns, 552 experiments, 409t loads, 60, 452, 452t mode, 569e573, 659e667, 693e694, 726, 727f of laminated cone, 410f mode of laminated cone, 410f of plates, 558f CLPT, 163e179 FSDPT, 179e182 of rectangular plates, 556fe557f of sandwich struts, 121e127, 124f tests, 416 of thin-walled structures, 558f VCT approach, 381 vibration buckling correlation, 559e567 Budiansky and Hutchinson criterion (B&H criterion), 225f, 227

737

BudianskyeRoth criterion, 694e695 BVID. See Barely visible impact damage (BVID) C Capeknuckle junction, 362e363 Carbon fiberereinforced polymer (CFRP), 600e601, 626e627 collapsed CFRP domes, 361 composites, 497 material properties for CFRP prepreg IM7/8552 UD, 465e466, 465t plastic cones, 401 plastic dome, 346e347 plastic panels, 463 Carrera’s unified formulation (CUF), 110e111, 135e138 Cartesian reference frame, 95e96 Casted sand-epoxy boundary box, 631 CeC column. See Clampedeclamped column (CeC column) CDF. See Cumulative distribution function (CDF) Cerrobend, 601 CFRP. See Carbon fiberereinforced polymer (CFRP) Circumferential periodicity condition, 530e532 CL. See Collapse load (CL) Clampedeclamped column (CeC column), 561e562, 570, 572 Classical lamination plate theory (CLPT), 101e103, 141, 191e194, 215e217, 286. See also First-order shear deformation plate theory (FSDPT) buckling of plates, 163e179 simply supported on two opposite edges of special orthotropic plates, 170e172 simply supported special orthotropic plates, 163e169 unsymmetric orthotropic plates, 172e179 force and moment resultants, 142, 143f simply supported on two opposite edges of special orthotropic plates, 216e217 simply supported special orthotropic plates, 215e216

738

Classical lamination plate theory (CLPT) (Continued) strainedisplacement equations for, 289e292 stress resultants, 142e145 Classical lamination theory (CLT), 12e14, 149e150, 191, 196e205, 434 buckling of columns, 149e157 buckling loads and modes of laminated composite columns, 159t coupled equations, 150 laminated composite column model, 150f nonsymmetric laminate, 156e157 symmetric laminate, 151e155 dynamic buckling of columns using, 227e230 energy terms for plates using, 187e189 matrix notation for equilibrium equations, 189e190, 220e221 nonsymmetric laminate, 200e205 nonsymmetric laminated composite beam, 184e187 solution for nonsymmetric beam resting, 218e220, 220t symmetric laminate, 197e200, 201te203t Classical sandwich structures, 50f, 61e71 free vibration analysis, 61e69 thermal stability analysis, 69e71 Classical theories, 101e104 CLPT, 101e103 FSDT, 103e104 Closure, 367e368 CLPT. See Classical lamination plate theory (CLPT) CLT. See Classical lamination theory (CLT) CNC. See Computer numerical control (CNC) COCOMAT program, 596 campaign, 594, 597te598t COCOMAT10 panel, 595f, 596, 599f European Commission project, 497 experimental first buckling load and collapse load, 597te598t four-year follow-up project, 462e463, 467f design process within, 470e471 load-shortening curve, 472f Collapse load (CL), 481e483, 581 histogram, PDF, and CDF of, 494f POSICOSS campaign, 593t Columns, 559e563 buckling loads and natural frequencies, 562t

Index

dynamic buckling using CLT, 227e230 using FSDT, 230e232, 233f nondimensional circular frequency, 561f, 563t Companion mode, 441e442 Compatibility enforcement using penalty approach, 333e334 Completeness, 305 Composite columns applications, 565e567 determination of first buckling load, 567f experimental definition of buckling load, 564f, 566f columns, 559e563 buckling loads and natural frequencies, 562t nondimensional circular frequency, 561f, 563t setups for columns, 549e554 hinged boundary condition, 550f hingedehinged column, 553f modeling buckling of thin-walled structure, 553f Southwell’s plot, 554f test columns under compressive load, 550fe552f vibration, 191 CLT, 196e205 FSDT, 205e214 Composite cylindrical shells imperfections effect on stability and vibrations, 522e540 first-order state, 527e528 governing equations, 522e526, 523f perturbation expansion, 526e527 second-order states, 528e530 standard boundary conditions, 525t R07, R08, and R09, with R/t¼399, 379e381 compressive loading on cylindrical shells, 380f geometric parameters, 379t material properties of UD, 380t second-order fit based on all experimental data, 381f R15 and R16, with R/t¼478, 381e384 compressive loading in R15 and R16 cylindrical shells, 383f

Index

geometric parameters, 382t measured material properties of UD, 382t second-order fit based on all experimental data, 383f simplified analysis for buckling and vibrations, 540e543 Z37, with R/t¼510, 384e385 compressive loading in Z37 cylindrical shell, 384f geometric parameters for Z37 cylinder, 384t second-order fit, 385f Composite materials, 1, 1fe2f in aerospace structures, 5f experimental tension and compression strengths for, 16t properties of single ply, 5e7 and resins, 16te18t transformation of stresses and strains, 7e15 unidirectional composites, 2e5, 6t Composite plates bending and vibration, 112e121, 115te118t plates, 563e564 nondimensional circular frequency, 561f setups for plates, 554e559 buckling of plates, 558f buckling of thin-walled structures, 558f Technion test setup, 555fe556f test setup for buckling of rectangular plates, 556fe557f Composite shell-type structure vibrations, 429. See also Shells vibrations application of Level 3 analysis, 432 boundary conditions, 450e453 effect of layer orientation, 455f results and discussion, 451e453 effect of shell length, 455f two-point boundary value problem and perturbation approach, 450e451 boundary conditions and applied loading, 437e441 buckling behavior, 429 buckling problems, 432 cylindrical shells, 430 dynamic stability, 430 finite element method, 432 governing equations, 435e437 nonlinearity of flexural vibrations, 430

739

problem definition, 432e435 simplified analysis for, 441e450 amplitudeefrequency curves of anisotropic shell, 449f Galerkin’s method and averaging method, 442e445 geometric imperfections, 441 influence of imperfection amplitude, 447f effect of layer orientation, 447f nonlinear behavior, 441e442 results and discussion, 446e450 static state response, 442 single-and driven-and coupled-mode response, 431f sonic fatigue, 430e431 stress and moment resultants, 435f Composite shells empirical KDFs and geometric properties, 393 stability of composite shelletype structures design and manufacture of composite cones, 254 VCT, 253e254 stability of shells, 619e678 vibrations of shells, 678e690 Composite spheroidal shells closure, 367e368 geometry, 343f imperfect composite spheroids, 361e367 metallic and composite spheroids, 344e361 Composite stringer-stiffened panel stability, 461 design and analysis of stiffened composite panels, 462e463 postbuckling composite aerospace structures, 497 stiffened structures, 461 structural robustnessebased design strategy, 481e483 Compressive crippling, 615e616 Compressive failure, 355 Compressive load, 379, 382, 384, 482e483 Compressive strain, 232 Computer numerical control (CNC), 625e626 Cones. See also Laminated cone(s) boundary conditions for, 404f buckling experiments, 409te410t

740

Cones (Continued) conical shell stability, 401e405 converted to equivalent cylinder, 407f experiments of conical shell structures, 408e411 linear buckling analysis of unstiffened, 316e318 load components in, 405f lower-bound design for truncated cones, 405e408 strainedisplacement equations for laminates, 286e289 strainedisplacement relations for, 281e285 theoretical buckling load and compressive stress, 402f thin-walled conical structures, 400e401 Conical mandrels, 620e621, 622f Conical shells, 706, 706t governing equations for, 281e292 strainedisplacement equations for laminates, 286e289 strainedisplacement relations for cones and cylinders, 281e285 stability, 401e405 boundary conditions for cylinders and cones, 404f geometry of truncated cone, 402f load components in vone, 405f theoretical buckling load, 403f structure experiments, 408e411 buckling mode of laminated cone, 410f load-shortening curve of cone K06, 411f Constant forcing term, 723 “Continuous fiber”, 418 composites, 1e2 Convergence, 300e301, 305 Correlation between bending curvature and measured strains, 573 Coupled equations, 150 Coupled-mode response, 441e443 Critical buckling load, 382e384 of column, 152 per unit width, 176 Critical temperatures, 83te87t parameter, 75te79t Cross-ply face sheets, sandwich plate with, 61e62, 62te64t Cross-ply laminated beams, first natural frequency of simply-supported, 135t

Index

Cross-ply laminates, first natural frequency of, 114t CUF. See Carrera’s unified formulation (CUF) Cumulative distribution function (CDF), 493 Curve-fitting process, 243 “Cylinder Z15”, 376e377 Cylinders, 403 boundary conditions for, 404f theoretical buckling load and compressive stress of, 402f Cylindrical coordinate system, 488e489 Cylindrical mandrels, 620e621, 622f Cylindrical shells, 261f, 264e265, 408, 430, 705e706, 711e712 buckling load and b coefficients of Booton’s anisotropic shell, 705t comparing imperfection patterns and lowerbound methods, 266e272 considerations regarding test loading strategy and boundary conditions, 275 deformation modes of Booton’s anisotropic shell, 705f governing equations for, 281e292 strainedisplacement equations for laminates, 286e289 strainedisplacement relations for cones and cylinders, 281e285 imperfection types in literature, 259e266 linear buckling analysis of unstiffened, 316e318 lower-bound methods to calculate KDF, 256e259 new developments regarding perturbation load approaches, 272e275 D D300 L440 ISS+JAVE shell specimen, 629 D500 ISS+JAVE shells, 629, 631 DAEDALOS. See Dynamics in Aircraft Engineering Design and Analysis for Light Optimized Structures (DAEDALOS) DAEDALOS program, 600 Damage propagation, 502 Damping, 437 DCB. See Double cantilever beam (DCB) Deflection function, 441e442

Index

Deformations, 433 modes of Booton’s anisotropic shell, 705f “Degenerated” cylinders, 401e402 Degrees of freedom (DOF), 98e99, 255e256, 280 Dependence of frequency, 450e451 DESICOS program, 620, 621t DESICOS project, test campaign during, 638e678 ANSYS FE results of cone C-1, 654f buckling pattern for nonlinear buckling analyses, 678f circumferential thickness distributions, 642f Cone C-1, 654fe657f Cone C-2, 658fe661f cones, 642f end shortening vs. axial compression load, 645f experimental application of SPLA at A and B positions, 673f of SPLA at C and D positions, 673f experimental curves for first buckling at perturbation load, 671f KDF curve using SPLA, 670f along nonlinear imperfect FE prediction, 668f experimental results of ISS+JAVE D500 L700 specimen, 665f experimental setup used to test ISS+JAVE D300 L440 composite shell, 662f load vs. end-shortening curves, 663f for ISS+JAVE D500 L700 specimen, 665f locations of bonded strain gages, 643f nominal dimensions for four tested specimens by technion, 639t numerical application of SPLA at A and B positions, 674f of SPLA at C and D positions, 674f photogrammetry-based buckling pattern vs. experimental buckling mode, 668f properties of HexPly IM7/8552 UD carbon prepreg, 640te641t realization of clamped boundary conditions in shell, 643f sandwich cylinders and cones, 662t Shell R15, 675f, 675t, 676f

741

Shell R16, 676t, 677f Shell SH-1, 643f, 645f Shell SH-2, 649fe653f SPLA experimental curves, 670f spring back phenomena, 668f strain gage arrangement for ISS+JAVE D300 L440 specimen, 662f for ISS+JAVE D500 L700 specimen, 663f of bonded strain gages, 670f strain gage readings for bending vs. compression strains, 648f for load vs. bending strains, 647f for load vs. compression strains, 646f test setup for D500 ISS+JAVE specimens, 664f for SST_1 specimen, 667f for SST_2 shell, 669f Design of experiment (DOE), 485 Design variables, 481 Deterministic framework, 485, 485f. See also Probabilistic framework application, 489e491 alternative design, load-shortening curves for, 490f baseline model, load-shortening curves for, 490f failure load, structural robustness, and weight of all panels, 491t DIANA, 695e696, 705e706, 712, 713t, 716t Differential equation, 197, 216e217, 444, 559 Donnell-type, 451 governing mechanical behavior of conical shells, 279e280 inhomogeneous linear partial, 443 three-dimensional, 92e94 Discrepancy, 509 Displacement, 436e437 displacement-based axiomatic approach, 97 field, 141e142 models, 94 Diverse boundary conditions, achieving, 302e305, 304t DLF. See Dynamic load factor (DLF)

742

DLR, 469 DLR-manufactured shell Z36, 680f test, 384 DOE. See Design of experiment (DOE) DOF. See Degrees of freedom (DOF) Domains assembly, semianalytical approach based on, 326e339 approximation functions and kinematic equations in natural coordinate system, 330e332 assembly for blade-stiffened cylinders, 329e330 assembly for unstiffened cylinders, 328e329 enforcing compatibility using penalty approach, 333e334 linear buckling analysis of stiffened cylinders using multidomain approach, 336e339 linear buckling analysis of unstiffened cylinders, 334e336 Donnell-type differential equations, 451 equations, 526 strainedisplacement relations, 524e525 thin shell theory, 433 Donnell’s equations, 298, 332, 434, 436e437 Double cantilever beam (DCB), 498e500 Draped domes, numerical results for, 359, 360f Draping algorithm, 353e355 Driven mode, 441e442 Dry fibers, 413 Dual-phase technology, 414e415 ATL “dual-phase” head of Forest-line ATLAS, 415f Dynamic “a-factor”, 517e519 Dynamic “b-factors”, 517e519 Dynamic analysis, 511e513 dynamic state, 513 nontrivial static state, 513 static fundamental state, 512e513 Dynamic approach, 566 Dynamic behavior of shell, 430 Dynamic buckling, 223, 225e226, 437e439, 717e720. See also Linear buckling buckling criteria, 226f buckling under parametric resonance, 223f, 224

Index

calculation of uniaxial loaded plate critical buckling load, 251e252 of columns using CLT, 227e230 using FSDT, 230e232, 233f finite element implementation, 720 load, 694e695 nonlinear model, 225f perturbation method, 718e720 of plates, 233e239 thin orthotropic plate uniaxially loading, 233f thin-walled structures, 239e247 Dynamic equilibrium equation, 517e519 Dynamic load factor (DLF), 224, 227, 240 Dynamic radial loading, 436e437 Dynamic stability, 430 Dynamic state, 442, 513e514, 516e520 “Dynamic” b-factor (bd), 453, 454f Dynamics in Aircraft Engineering Design and Analysis for Light Optimized Structures (DAEDALOS), 685e687 E EBBT. See EulereBernoulli beam theory (EBBT) EBL. See Experimental buckling load (EBL) EC project. See European Commission project (EC project) ECSS. See European Convention for Constructional Steelwork (ECSS) ECSS Buckling Handbook, 693 Effective width approach, 609e617 buckling and postbuckling behaviors, 611f flat plate, 611f laminated curved-stringer-stiffened panel, 616fe617f stringer-stiffened plate, 612f Eigenmode affine imperfections imperfect ellipsoidal shells, 366 imperfect hemispheres, 363 imperfect torispheres, 364 Eigenvalue, 560e562 Elastic edge restraint, 441 Elastic microbuckling, 619 Elastic restraint, 374 Elastic stiffness parameters, 441

Index

Elephant-foot-type deformation, 408e409 Ellipsoidal shells, imperfect, 365e367, 365f eigenmode affine imperfections, 366, 366f force-induced inward dimple, 367 Empirical KDFs and geometric properties for composite shells, 393 Energy methods, 229, 232 Energy reserve, 484 Energy terms for plates, 187e189 Energy-based structural robustness measures, 483e485 collapse, 483e484 energy reserve, 484 GB, 484 inherent variations, 485 load-shortening curve, 484 local failure, 483 relative measure, 484 Equilibrium equation, 448 matrix notation for, 220e221 Equivalent cylinder philosophy, 408 Equivalent single layer (ESL), 51e52, 94e95, 285e286 displacement field, 54 kinematics, 55f ESA. See European Space Agency (ESA) ESL. See Equivalent single layer (ESL) Euler-Bernoulli’s theory, 132 Euler’s method, 59e60 EulereBernoulli beam theory (EBBT), 133 EulereLagrange equations, 279e280 of Hamilton’s principle, 98 European aircraft industry, 462e463 European Commission project (EC project), 462e463 European Convention for Constructional Steelwork (ECSS), 408 European Space Agency (ESA), 376e377, 492 Exascan external laser scan, 633e634 Experimental buckling load (EBL), 620 Extended Analysis, 522, 530, 546 Extensions of theory, 520e521 forced vibrations, 520 higher order analysis, 520e521 multimode analysis, 521 F Face asymmetry, 49 Failure index (FI), 353e355, 354f

743

Failure load. See Collapse load (CL) FBG sensors. See Fiber Bragg grating sensors (FBG sensors) FBL. See First buckling load (FBL) FE. See Finite element (FE) FEM. See Finite element model (FEM) FG sandwich plates. See Functionally graded sandwich plates (FG sandwich plates) FGMs. See Functionally graded materials (FGMs) FI. See Failure index (FI) Fiber, 1 Fiber Bragg grating sensors (FBG sensors), 604e606 Fiber-reinforced composites, 401 FID. See Force-induced dimple (FID) Filament winding, 412e413, 415 manufacturing technology, 621e623 Finite element (FE), 575, 619 analysis tool, 497 buckling, 693e695 dynamic analysis, 717e718 implementation, 710e711, 720, 725 nonlinear buckling analysis, 700e701 postbuckling analysis, 702e703 matrices, 702 Finite element method. See Finite element model (FEM) Finite element model (FEM), 51e52, 95, 280, 346e347, 373e375, 432, 465e466, 473f, 485, 620 comparison of vibration frequencies, 374t variation of natural frequencies of vibration, 375f First buckling load (FBL), 586 COCOMAT campaign, 597te598t POSICOSS campaign, 593t First global buckling, 464 First-order mode, 516e517 First-order shear deformation plate theory (FSDPT), 141, 145e149, 195e196. See also Classical lamination plate theory (CLPT) buckling of plates, 179e182 simply supported symmetric plates, 179e182 constants, 148 equations of motion, 148e149

744

First-order shear deformation plate theory (FSDPT) (Continued) matrix notation for equilibrium equations, 221e222 stress resultants and assumed displacements, 147e148 transverse force resultants, 146 First-order shear deformation theory (FSDT), 101, 103e104, 158, 191, 205e214, 286, 288 buckling of columns, 158e163 boundary conditions of problem, 161 buckling loads and modes, 163t coupled equations of motion, 160 critical buckling load per unit width, 162 decoupling equations, 160 simply supported laminated composite beam, 162 symmetric case, 163 dynamic buckling of columns using, 230e232, 233f nonsymmetric laminate, 209e214, 210te211t boundary conditions for, 214t symmetric laminate, 206e208, 208te209t First-order state, 527e528 equation, 517 problem, 451, 533e535 First-ply failure (FPF), 346 analyses, 347e355 arrangements for ply stacking, 348f bifurcation buckling, 351t, 352f CFRP deformed hemisphere, 351f elastic constants, 349t failed machined metal ellipsoids, 350f FE-computed eigenshape, 350f pressures in externally pressurized torispheres, 350t Flat-stringer-stiffened composite panel, 577f, 579f Fl€ ugge’s and Donnell’s theories, 255 Fluid pressure, 720 Follow-up study, 579 Force resultants, 142, 143f Force-induced dimple (FID), 345 Force-induced inward dimple imperfect ellipsoidal shells, 367 imperfect hemispheres, 363 imperfect torispheres, 364e365

Index

Forced vibrations, 520 Forest-Line ATLAS machine, 414e415, 415f FORTRAN program, 446 Fourier Series, 305e307 Fourier transform method, 567 Fourth-order linear differential operators, 435e436 FPF. See First-ply failure (FPF) FPF-controlled incrementing of pressure, 358e359 Free harmonic vibration, 569e570 Free vibration analysis, 61e69, 73e77 sandwich plate with angle-ply face sheets, 65te68t, 69, 71t configurations, 69f with cross-ply face sheets, 61e62, 62te64t FSDPT. See First-order shear deformation plate theory (FSDPT) FSDT. See First-order shear deformation theory (FSDT) Full NewtoneRaphson method, 299 Full NewtoneRaphson nonlinear algorithm, 319 Functional notation, 696 Functionally graded materials (FGMs), 50e53 Functionally graded sandwich plates (FG sandwich plates), 52e53, 72e85 free vibration analysis, 73e77 thermal stability analysis, 83e85 Fundamental dimensionless circular frequency parameter, 72te74t Fuselage structures, 461 Fuselage-representative stiffened panel designs, 500e501, 501f G Galerkin’s method, 229, 232, 442e445 weighting functions, 443 Gap elements, 498e500 GARTEUR Structures and Materials. See Group for Aeronautical Research and Technology in EURope Structures and Materials (GARTEUR Structures and Materials)

Index

Gauss points (GPs), 355 GB. See Global buckling (GB) GDIs. See Geometric dimple imperfections (GDIs) GDQ. See Generalized differential quadrature (GDQ) General nonlinear formulation, 292e298 geometric stiffness matrix (KG), 297e298 “General Theory of Elastic Stability”, 255 Generalized differential quadrature (GDQ), 52e53 Generalized unified formulation (GUF), 111 Geometric dimple imperfections (GDIs), 260e262 dimple depths, 262t imperfection pattern for, 261f shapes, 262f Geometric imperfections, 253, 255, 264e266, 441 Geometric stiffness matrix (KG), 297e298 Geometrical imperfection, 492 Glass-fiber-reinforced plastic (GFRP), 415 Glass-reinforced plastic (GRP), 344e345 Global asymmetry, 49 Global buckling (GB), 124, 124f, 125te126t, 482e483, 671e672 histogram, PDF, and CDF of, 494f reliability for, 495t “Global” levels, 473e474 Glue, 1 Governing equations, 56e61, 435e437, 510e511, 522e526, 523f linear temperature distributions, 60e61 nonlinear temperature distributions, 60e61 Ritz method, 56e60 uniform temperature distributions, 60e61 GPs. See Gauss points (GPs) Gr-Ep/foam sandwich strut, 125, 126t Gr-Ep/honeycomb sandwich, 125 Graphiteeepoxy lamina, properties of, 578t GRIPHUS, 620, 621t Group for Aeronautical Research and Technology in EURope Structures and Materials (GARTEUR Structures and Materials), 462e463 GARTEUR AG 25/POSICOSS, 467f GRP. See Glass-reinforced plastic (GRP) GUF. See Generalized unified formulation (GUF)

745

H Hamilton’s principle, 434e435, 721 Hand layup process, 413e414 Hardening nonlinearity, 430 Harmonic forcing term, 723 Harmonic lateral excitation, 441 Harmonic vibrations, 559, 570 Hat-stringer-stiffened curved composite panel, 576f Hierarchical trigonometric Ritz formulation (HTRF), 52e53 Higher order analysis, 520e521 Higher-order effect of asymmetric imperfections, 448e449 Higher-order shear deformation theories (HSDTs), 51e52, 105e106, 289 kinematics, 134 HSDTs. See Higher-order shear deformation theories (HSDTs) HTRF. See Hierarchical trigonometric Ritz formulation (HTRF) I IAI. See Israel Aircraft Industries (IAI) Imperfect composite spheroids, 361e367 imperfect ellipsoidal shells, 365e367 imperfect hemispheres, 362e363 imperfect torispheres, 363e365 “Imperfect dynamic” second-order modes, 546 Imperfect ellipsoidal shells, 365e367 eigenmode affine imperfections, 366, 366f force-induced inward dimple, 367 Imperfect hemispheres Eigenmode affine imperfections, 363 force-induced inward dimple, 363 localized flattening in hemispheres, 362e363, 362f Imperfect structure, 699e700 Imperfect torispheres, 363e365 eigenmode affine imperfections, 364 force-induced inward dimple, 364e365 increased-radius flattening, 364, 364f Imperfection, 273, 442 comparing imperfection patterns and lowerbound methods, 266e272 KDFs using REM, 267f

746

Imperfection (Continued) knockdown curves for ASIs and LBMIs, 271f knockdown curves for MSI, SPLI, and GDIs, 270f knockdown curves for MSI and SPLI, 271f knockdown curves for SPLI and LBMI, 268fe269f nonlinear analysis using initial, 322e325 cylinder Z23 with geometric imperfection, 326f sensitivity theory, 451 types in literature, 259e266 ASIs, 264 GDIs, 260e262 geometric imperfections, 264e266 LBMIs, 262e263 SPLI, 259e260 In-plane displacement, 107e108, 156 inertia of radial modes, 434 loading, 438e441 Incremental algorithms, 298e299 Industrial structures, 463e464 Inherent uncertainties, 481 Initial geometric imperfections, 509 analysis for buckling and vibrations, 540e543 effect on stability and vibrations of composite cylindrical shell, 522e540 extensions of theory, 520e521 governing equations, 510e511 perturbation expansion, 513e520 static and dynamic analysis, 511e513 effect of imperfections and loading on vibrations of shells, 544e546 Initial imperfection, 513 Initial postbuckling, 453 analysis, 693e695 theory, 451 Initial stress nucleus, 59e60 Initial value techniques, 450 initialInc parameter, 298e299 INSTRON 1195, 557 INSTRON 8002 test frame, 631 INSTRON 8802 test frame, 655 Interlaminar damage growth, 498e500 initiation, 498

Index

Inverse-weighted interpolation (IW interpolation), 323e325 Isotropic plate, benchmark case of, 371e373 characteristics of first four vibration, 373f evolution of four vibration modes, 372f experimental fixture frame for aluminum plate, 372f geometry and boundary conditions of aluminum plate, 371f Israel Aircraft Industries (IAI), 584e585 ISS+JAVE D300 L440 specimen, 629e631, 630f, 655 ISS+JAVE D500 L1000 specimen, 631, 632f, 635f, 659, 666f ISS+JAVE D500 L700 specimen, 631, 632f, 634f, 655, 664f Iterative algorithms, 298e299 IW interpolation. See Inverse-weighted interpolation (IW interpolation) K KDFs. See Knockdown factors (KDFs) Khot’s parameter, 446e447 Kinematic assumption CLPT, 101 EBBT, 133 FSDT, 103 TBT, 133 Kirchhoff assumptions, 433 Kirchhoff-type shell elements, 695 KirchhoffeLove classical plate theory, 141e142, 191e192 Knockdown factors (KDFs), 256, 388, 650 comparison of SBPA thresholds, 391f conversion factors for SI units empirical KDFs and geometric properties for composite shells, 393 experimental data distribution of axial compressed composite cylindrical shells, 388f lower-bound buckling load, 391 lower-bound KDF, 389f, 390 lower-bound methods to calculate, 256e259 Koiter’ theory, 255, 695 Koiter’s imperfection sensitivity theory ANILISA, 453

Index

Koiter’s perturbation approach, 693 KolmogoroveSmirnov test, 493e495 L Lagrange hierarchical polynomials for semianalytical models, 280 Lamé coefficients, 281e282 Lamina notations within laminate, 13f Lamina strainestress relationship, 8e9 Laminated cone(s), 424t. See also Cones buckling mode, 410f examples of manufactured, 415e418 AFP, 416 filament winding, 415 tape laying, 416e418 manufacturing, 412e425 AFP, 413, 416 filament winding, 412e413, 415 offline tape cutting system, 414f tape laying, 413e418 manufacturing method effects in analysis, 419e425 AFP, 419e421 filament winding, 419 number of layers of overlapped laminate, 420f ply piece, 423f tape laying, 421e425 Laminates laminated composite column model, 149e150, 150f laminated curved-stringer-stiffened panel, 616fe617f laminated plates, 113 laminated structures, 91 strainedisplacement equations for, 286e289 Large postbuckling region, 469e471, 474e475, 486e487 Laser scanner mechanical assembly, 626e627 Last-ply failure (LPF), 346, 355e359 arbitrary incrementing of pressure, 356e358, 357fe358f comparison of numerical and experimental results, 359e361, 361f FPF-controlled incrementing of pressure, 358e359 numerical results for draped domes, 359, 360f stiffness matrix degradation, 356

747

Lateral vibration, 563 Latvia, 557 Layered anisotropic shell, 434e435 Layerwise (LW), 94e95 displacement field, 55 kinematics, 55f models, 62, 109e110 theories, 51e52, 286 LB. See Local buckling (LB) LBMIs. See Linear buckling modeeshaped imperfections (LBMIs) Least squares method of NumPy, 310 Level-2 analysis, 522 Lévy method, 170, 216 Limit load (LL), 482e483 Line search algorithms, 299e300 Linear buckling. See also Dynamic buckling analysis of stiffened cylinders using multidomain approach, 336e339, 338f of unstiffened cones and cylinders, 316e318 of unstiffened cylinders using multidomain approach, 334e336, 337f formulation, 301e302 axial compression for cylinder Z33, 318te319t, 320f cone C02 axial compression for, 321te322t, 323f eigenvalue analysis for, 324f knockdown curve, 324f Linear buckling modeeshaped imperfections (LBMIs), 262e263 imperfection pattern for, 263f Linear temperature rise, 60 Linear variable differential transducers (LVDTs), 594, 626, 655 Linear vibration of anisotropic shells, 452, 454f Linearized buckling problem, 535 Linearized flutter problem, 535 Linearized vibration problem, 535 Lipowitz’s alloy, 601 Lissajous curve, 566e567 Lissajous figures, 549e550, 566e567 Liu’s equations, 444 LL. See Limit load (LL) Load stiffness, 720

748

Load-controlled iterative methods, 293 Load-shortening curve, 482e485, 483f alternative design, 490f baseline model, 490f of each model from sample, 492f Loadedisplacement curves, 472 Local buckling (LB), 482e483, 615e616 Local failure, 482e483 “Local” levels of validation, 473e474 Lower-bound methods, 253, 255 to calculate KDFs NASA SP-8007, 256e257 RSM, 257e259 comparing different imperfection patterns and, 266e272 considerations regarding test loading strategy and boundary conditions, 275 design for truncated cones, 405e408 new developments regarding perturbation load approaches, 272e275 LPF. See Last-ply failure (LPF) LVDTs. See Linear variable differential transducers (LVDTs) LW. See Layerwise (LW) M Marc. See MSC. Marc v2005r3 (Marc) Mass nucleus, 59 Matrix, 1e2 Matrix notation for equilibrium equations using CLT, 220e221 CLT approach, 189 FSDPT, 221e222 FSDT approach, 189 Metallic shells, 361e362 Metallic spheroids, 344e361 bifurcation and FPF analyses, 347e355, 348f, 349t composite domed closures, 346fe347f, 346e347 magnitude of buckling pressure, 344e345 progressive failure, 355e359 “single eigen-dimple”, 345 Midsurface imperfections (MSIs), 264e265, 270e272 Mindlin-type shell elements, 695 MindlineReissner plate theory, 195

Index

Minor modulus, 4 Mixed models, 94 MMB. See Mode IeII mixed-mode bending (MMB) Mode II end-notched flexure, 498e500 Mode III edge crack torsion, 498e500 Mode IeII mixed-mode bending (MMB), 498e500 Mode shapes, 203t Modified Donnell’s approach, 243 Modified NewtoneRaphson method, 299 Moiré method, 585e586 Moment resultants, 142, 143f Monte Carlo simulation, 491, 493 Mosquito aircraft, 51 “Movability” conditions of edge planes, 439e440 MPCs. See Multipoint constraints (MPCs) MSC. Marc v2005r3 (Marc), 497 MSIs. See Midsurface imperfections (MSIs) Multidomain approach linear buckling analysis of stiffened cylinders using, 336e339 unstiffened cylinders linear buckling analysis using, 334e336 Multimode analysis, 521 Multimode initial postbuckling analysis, 707e711 contribution of prebuckling nonlinearity, 707 finite element implementation, 710e711 perturbation method, 707e710 Multipoint constraints (MPCs), 498e500 Multiskin and multilayered core constructions, 51e52 Multistiffener curved panels, 500e501 Multivariate normal distribution function, 492 Murakami’s function, 71 Murakami’s zigzag function (MZZF), 53e54, 107e108, 134e135 N NASA SP-8007, 256e257, 388, 405e406 for cylinders, 401 guideline, 253 NASA SP-8019, 401, 405 NASA TP-2009e215778, 620 Natural coordinate system, 330e332

Index

Natural frequency, 197, 205 Navier approach, 179 Navier problem, 179e180 NDT inspection. See Nondestructive testing inspection (NDT inspection) Newton’s second law, 279e280 NewtoneRaphson method, 294, 472 914 epoxy resin, properties of, 5, 6t NLA. See Nonlinear finite element analysis (NLA) Nondestructive testing inspection (NDT inspection), 628f Nondimensional global buckling, 126e127, 126f, 127t Nondimensional variable (p), 680e683 Nonlinear algebraic equation, 693e694 Nonlinear algorithms, 298e301 convergence criteria and other nonlinear parameters, 300e301 full NewtoneRaphson method, 299 line search algorithms, 299e300 modified NewtoneRaphson method, 299 Nonlinear analysis, 377 using initial imperfections, 322e325 using SPLA, 319e322 Nonlinear behavior, 441e442 Nonlinear buckling analysis, 693, 700e701 Nonlinear Donnell-type equations, 451, 522 governing equations, 442 Nonlinear equations, 433 Nonlinear FE analyses, 671e672 Nonlinear finite element analysis (NLA), 485 without degradation, 472e476 with degradation, 475e476 Nonlinear free vibration of Booton’s shell, 729e731, 730t backbone curve, 731f linear vibration mode, 730f Nonlinear frequencyeamplitude relations, 724 Nonlinear model, 225f Nonlinear strainedisplacement relation, 696 Nonlinear temperature rise, 61 Nonlinear vibrations, 430, 437, 442, 720e725 of anisotropic shells, 453, 453t finite element implementation, 725 perturbation method, 721e724

749

Nonlinearity of flexural vibrations, 430 Nonstationary loads, 436e437 Nonsymmetric laminate, 156e157. See also Symmetric laminate buckling of, 157te158t CLT, 200e205 FSDT, 209e214, 209te211t, 214t in-plane displacement, 156 pinnedepinned column, 156e157 Nonsymmetric laminated composite beam, 184e187 Nontrivial static state, 513e516 Normalized frequency parameter, 445 Numerical examples, 703e706, 725e731 Booton-type cylindrical and conical shells under axial loading, 704e706 Booton’s anisotropic shell under external pressure, 704 Booton’s shell under dynamic axial loading, 728e729 Booton’s shell under dynamic external pressure, 726e727 nonlinear free vibration of Booton’s shell, 729e731 waters’ shell under axial loading, 711e717 Numerical methods and errors, 95 Numerical models, 492 VCT applied to cylindrical shells, 376e377 buckling load predicted using proposed VCT, 378t geometric parameters of Z15 cylinder, 377t material properties of Z15 cylinder, 377t VCT methodology, 378f Numerical shear correction factors, 104 O Oblate spheroid, 343e344 Offline tape cutting system, 414f One-dimensional element, 149e150 One-term Ritz solution, 61e62 Optical lock-in thermography, 475e476 Out-of-plane deflection, 165e166, 551e552 P Panel Analysis and Sizing Code (PASCO), 579 Parallel shooting method, 450

750

Parametric excitation, 437e439 Parametric resonance, 223e224, 223f PASCO. See Panel Analysis and Sizing Code (PASCO) Patran functions, 500 PDF. See Probability density function (PDF) Penalty approach, enforcing compatibility using, 333e334 Perfect structure, 696e699 Perturbation approach, 450e451 expansion, 513e520, 526e527 dynamic state, 516e520 nontrivial static state, 515e516 load approaches, 272e275 method, 707e710, 718e724 theory, 451 PFA scheme. See Ply progressive failure scheme (PFA scheme) Photogrammetry method, 644 photogrammetry-based buckling pattern, 672e676, 676f, 678f Plane stress constitutive relation, 101e102 Plastic microbuckling, 619 Plates, 100, 563e564 CLPT, 215e217 simply supported on two opposite edges of special orthotropic plates, 216e217 simply supported special orthotropic plates, 215e216 dynamic buckling, 233e239 thin orthotropic plate uniaxially loading, 233f models, 91e92 nondimensional circular frequency, 561f vibration, 191 Ply damage, 498 number, 347 properties of single, 5e7 Ply progressive failure scheme (PFA scheme), 486 Poisson locking, 71, 109 Poisson ratio, 403 Poisson’s coefficient, 4, 554e555 Polyurethane (PU), 629 Polyvinyl chloride (PVC), 638e639 POSICOSS project, 584e585 campaign, 591e593, 593t

Index

EC project, 462e463, 470f design panel P12, 470f design process within, 469e470 Postbuckling analysis, 702e703 Postbuckling composite aerospace structures, 497 analysis tool, 497e500 damage definition, 500f in-plane failure criteria and property reduction, 499t interlaminar damage growth, 498e500 interlaminar damage initiation, 498 ply damage, 498 user interface, 500 design and analysis, 500e503 comparison with experiment, 503, 504f D2 panel and FE mesh, 502f design studies, 501e502 fuselage-representative stiffened panel designs, 501f in-plane damage and applied loadedisplacement curve, 502f PPLA. See Probabilistic perturbation load approach (PPLA) Prebuckling equilibrium state, 697 Pressure loading, 437e438 in-plane loading and boundary conditions, 438e441 summary of loads, 438t Primary mode. See First-order mode Principle of minimum of potential energy. See Principle of virtual displacements (PVD) Principle of virtual displacements (PVD), 56, 94 Probabilistic framework, 485e486, 486f. See also Deterministic framework application, 491e495 histogram, PDF, and CDF of collapse load, 494f histogram, PDF, and CDF of global buckling load, 494f histogram, PDF, and CDF of structural robustness index, 494f load-shortening curve of each model from sample, 492f reliability for global buckling load and structural robustness index, 495t statistical parameters, 493t

Index

Probabilistic perturbation load approach (PPLA), 273, 274t Probability density function (PDF), 481 Prolate ellipsoids, 347e349 Prolate spheroid, 343e344 PU. See Polyurethane (PU) Pulse-type buckling, 223f, 224 PVC. See Polyvinyl chloride (PVC) PVD. See Principle of virtual displacements (PVD) R R29AL shell, 634e636, 638f, 671e672, 684 R30AL shell, 634e636, 671e672 Radial displacement, 443 Rayleigh’s principle, 571 Reddy’s model, 110 REDUCE symbolic manipulation program, 446 Reduced boundary conditions, 440 Reduced energy method (REM), 256 Reduced stiffness method (RSM), 255, 257e259 Redux 312, 465t Refined equivalent single-layer theories, 105e109 HSDT, 105e106 theories including transverse normal stress, 109 ZZTs, 107e108 Refined plate models, 53e56 Refined theories, 134e138 Reinforced continuous fibers, properties of, 3te4t Reissner multilayered theory (RMT), 107 Reissner’s mixed variational theorem (RMVT), 107 REM. See Reduced energy method (REM) Rhumb line, 621e623, 623f Riga Technical University (RTU), 620, 679t, 680 OD500 RTU mandrel, 625e626, 625f Ritz method, 56e60, 280, 294, 305, 405 initial stress nucleus, 59e60 mass nucleus, 59 stiffness nucleus, 58e59 RMT. See Reissner multilayered theory (RMT)

751

RMVT. See Reissner’s mixed variational theorem (RMVT) Robust knockdown factors, 254 Robustness, 481 Rodrigues polynomials, 331 Rods, 559e563 buckling loads and natural frequencies, 562t nondimensional circular frequency, 561f, 563t Rotatory inertia, 434 RSM. See Reduced stiffness method (RSM) RTU. See Riga Technical University (RTU) Rule of mixtures, 2e4 RungeeKutta methods, 237 S S-GUF. See Sublaminate-GUF (S-GUF) Safety factors, 481 Saint-Venant’s theory, 132, 135 Sanders’ nonlinear theory, 298 Sandwich buckling problems, acronyms of three-layer models for, 124t Sandwich plate with angle-ply face sheets, 65te68t, 69 bending and vibration, 120 configurations, 69f with cross-ply face sheets, 61e62, 62te64t first natural frequency, 121t with laminated angle-ply face sheets, 71t Sandwich shells, 650 Sandwich square plate under bisinusoidal load bending, 122te123t Sandwich structures, 49, 91, 120 advanced structural models, 53e56 classical sandwich structures, 50f, 61e71 functionally graded sandwich structures, 72e85 governing equations, 56e61 historical notes and literature, 51e53 Sandwich struts, buckling of, 121e127, 124f SBPA. See Single boundary perturbation approach (SBPA) Scaling effect, 448e449 SD. See Standard deviation (SD) Second-order modes, 693e694, 729e731, 730fe731f Second-order polynomial, 687e689

752

Second-order shear deformation theory, 288 Second-order states, 528e530 circumferential periodicity condition, 530e532 equations, 528 first-order state problem, 533e535 problem, 535e540 second-order state problem, 535e540 Semianalytical approaches, 253, 405 based on assembly of domains, 326e339 cone/cylinder model, 279f force and moment resultants on a plate element, 280f general nonlinear formulation, 292e298 governing equations for cylindrical and conical shells, 281e292 linear buckling formulation, 301e302 nonlinear algorithms, 298e301 using single domain for approximation functions achieving diverse boundary conditions, 302e305, 304t approximation functions, 305e309 fitting measured imperfection data into continuous function, 309e316, 311fe313f, 316f linear buckling analysis of unstiffened cones and cylinders, 316e318 nonlinear analysis using initial imperfections, 322e325 nonlinear analysis using SPLA, 319e322 SH-1, 641 SH-2, 641 Shear deformation laminated plate theories, 286 Shear modulus, 4 Shell(s), 100 geometry, 432e433, 433f imperfections and loading effect on shells vibrations, 544e546 on linearized vibrations of shells, 545e546, 545f models, 91e92 stability, 619e678 vibrations, 678e690 Shells vibrations, 678e690. See also Composite shell-type structure vibrations

Index

calculated and experimental buckling and collapse loads, 689t DLR-manufactured shell Z36 and RTU VCT system, 680f experimental results using VCT on DLR shell Z36, 680f geometric and material properties of shell SH-1, 688f Shell R15, 687f Shell R16, 688f Shell R29AL, 685f Shell R30Al, 686f Shell SH-1, 689f Shell SST_1, 683f Shell SST_2, 684f Shell Z36, 681fe682f Shooting methods, 450 Simply supported special orthotropic plates, 163e169 biaxial loads, 168 boundary conditions, 164e165 coefficients for buckling parameters, 170t compressive loads, 164 critical value, 169 deflection, 168e169 nondimensional term, 166 out-of-plane deflection, 165e166 schematic plate under bidirectional compression, 167f shear, 168f unidirectional compression, 164f special orthotropic plates, 164 uniaxial buckling of rectangular plate, 166f yields, 167 Simply supported square sandwich plates, 14, 69e71 dimensionless circular frequency parameters, 63te64t on two opposite edges of special orthotropic plates, 170e172 Simply supported symmetric plates, 179e182 Simply supported-3 boundary conditions (SS-3 boundary conditions), 453, 726, 726t Simply supportedesimply supported boundaries (SSeSS boundaries), 572e573 Simply-supported sandwich beam, 138, 138t

Index

under sinusoidal load, 139t Single boundary perturbation approach (SBPA), 388e389 “Single eigen-dimple”, 345 Single perturbation load (SPL), 319e322 Single perturbation load approach (SPLA), 255e256, 259, 274t, 319, 626, 659 nonlinear analysis using, 319e322 Single perturbation load imperfection (SPLI), 259e260, 261f Single-domain approach, 334 Single-mode initial postbuckling analysis, 695e700 functional notation, 696 imperfect structure, 699e700 perfect structure, 696e699 Single-phase technology, 414e415 Single-stiffener flat panels, 500e501 Sinus theory, 105e106 Sinusoidal load, simply-supported beam bending under, 136te137t, 139t Skewedness parameter of imperfection, 442 Skinestiffener interface, 498e500 Skinestringer debonding, 593e594 Softening nonlinearity, 430 Solution methods for two-dimensional model, 98e99 Sonic fatigue, 430e431 Southwell plot, 551e552, 554, 565 Special orthotropic plates, 164, 215e216 simply supported on two opposite edges, 216e217 Spheroid, 343e344 SPL. See Single perturbation load (SPL) SPLA. See Single perturbation load approach (SPLA) SPLI. See Single perturbation load imperfection (SPLI) Spring elements, 472 SS-3 boundary conditions. See Simply supported-3 boundary conditions (SS-3 boundary conditions) SS-4 boundary condition, 728, 729t SSeSS boundaries. See Simply supportedesimply supported boundaries (SSeSS boundaries) SST_2 shell, 669 Stability and vibrations

753

of composite cylindrical shell, imperfections effect on, 522e540 first-order state, 527e528 governing equations, 522e526, 523f perturbation expansion, 526e527 second-order states, 528e530 standard boundary conditions, 525t extensions of theory, 520e521 forced vibrations, 520 higher order analysis, 520e521 multimode analysis, 521 initial geometric imperfections on governing equations, 510e511 perturbation expansion, 513e520 static and dynamic analysis, 511e513 Stability of composite columns and plates, 141e149 buckling of columns, 149e163 buckling of plates, 163e179 CLPT, 141e145 FSDPT, 145e149 Stability of shells, 619e678, 636f additionally laminated composite cylindrical shells, 639t aluminum shell edges casted in sand-resin rings, 638f CNC-machined end plates, 627f deadweight and view on SPLA device, 628f flattened final configuration, 625f geometry of ISS+JAVE scaled-down specimen, 630f initial geometric imperfection scans, 636f internal content of geometry scanner based on laser distance sensor, 628f ISS+JAVE D500 L1000 specimen, 628f mandrel revolving machine with D500 mandrel, 627f OD500 RTU mandrel, 625f original designed angle of 60 degrees, 623f realized ISS+AVE scaled-down specimen, 630f SST_1 shell, 637f SST_2 shell, 637f support frame for ISS+JAVE specimens, 633f test campaign during DESICOS project, 638e678 various trials with layups, 624f Stabilizing curvature effect, 447e448

754

Stabilizing membrane stress, 447e448 STAGSC-1. See Structural Analysis of General Shells computer program (STAGSC-1) Standard boundary conditions, 440, 440t, 525t Standard deviation (SD), 487t Standard Hilgus USPC 3010 HF, 627 Start design, 466e469 Static analysis, 511e513 dynamic state, 513 nontrivial static state, 513 static fundamental state, 512e513 Static assumption CLPT, 101 FSDT, 103 Static fundamental state, 512e513 Static loading, 243, 433 Static radial loading, 436e437 Static state, 437e438, 442 response, 442 “Static” b-factors (bs), 453 formula, 539 Stationary loads, 436e437 Steady-state nonlinear flexural vibration behavior, 441 Stiffened composite panels, 462e463. See also Unstiffened cylindrical shells analysis, 471e472 comparison of experiment and different simulation tools, 476f FE model and analysis procedure in ABAQUS, 473f nonlinear finite element analysis, 472e476 numerical simulation of failure propagation of adhesive layer, 477f out-of-plane deformations, 474f position of user subroutines within ABAQUS calculation process, 476f ultrasonic flaw echo and thermographic investigation, 478f CFRP panels, 463 design, 463e465 boundary conditions, 468f design process within COCOMAT, 470e471 design process within POSICOSS, 469e470

Index

finite element analyses of undamaged DLR benchmark, 468f first local and global buckling load and collapse load, 464f geometrical and material data of panel designs, 465e466 material properties for CFRP prepreg IM7/8552 UD, 465e466, 465t material properties of adhesive, 465t nominal geometrical data and layup for panel designs, 466t process of DLR, 464e465 start design, 466e469 kinds of tools, 463 timetable of European Union projects, 462f Stiffened panels, 594 Stiffened structures, 461 Stiffness matrix degradation, 356 nucleus, 58e59 parameters, 435e436 “Straightforward” polar winding, 346e347 Strain stress, and constitutive relation, 96e97 transformation of, 7e15 Strainedisplacement equations for CLPT, 289e292 for laminates, 286e289 relation, 281e285, 696 Strength-based “degenerated Tsai” criterion, 498 Stress, 436e437 resultants, 142e145, 439 thermal, 15t transformation, 7e15 Stringer-stiffened composite panels, 575 curved stiffened composite panel, 589f effective width approach, 609e617 flat-stringer-stiffened composite panel, 577f, 579f curved composite panel, 576f stability, 575e601 aluminum panels, 602f composite panels, 603f experimental and analytical results, 583t, 585t experimental results, 580t FE numerical predictions, 601t

Index

first buckling and collapse loads of panels AXIAL 1eAXIAL 4, 592t geometry, dimensions, and stringer layup stiffening, 594f geometry and layups for stringers, 586f geometry of Z stiffened flat panels under axial compression, 584f panel COCOMAT10, 595f, 599f panel COCOMAT7, 595f panel geometry of curved-stringerstiffened composite panel, 581f panels’ geometry and initial geometric imperfections, 582t properties of graphiteeepoxy lamina, 578t size of stringer, 604f skin and stiffener laminates, 578t torsion box, 587t, 588f vibrations, 601e607 postcollapse modes of failure, 605f Technion excitation system, 605f vibrational test setup, 607f Stringer-stiffened plate, 612f Structural Analysis of General Shells computer program (STAGSC-1), 579 Structural models, 91e92 Structural robustnessebased design strategy, 481e483, 485e486 airframe structures, 482 alternative design methodologies, 482 application of deterministic framework, 489e491 of probabilistic framework, 491e495 compressive loading, 482e483 design variables, 481 deterministic framework, 485, 485f energy-based structural robustness measures, 483e485 failure load, 481e482 histogram, PDF, and CDF of, 494f inherent uncertainties, 481 local failure, 482 probabilistic framework, 485e486, 486f reliability for, 495t thin-walled composite structures in postbuckling, 482 of thin-walled stiffened composite shells, 486e489

755

typical factors, 481 Sublaminate, 110 Sublaminate-GUF (S-GUF), 111 Submenus, 500 Symmetric laminate, 151e155. See also Nonsymmetric laminate buckling loads and modes of laminated composite columns, 154t buckling of laminated composite columns, 153t CLT, 197e200, 201te203t FSDT, 206e208, 208t laminated composite column under axial compression, 155f out-of-plane boundary conditions, 155f Syntactic foams, 343e344 T T300 carbon fibers, properties of, 6t Tape laying, 413e418, 416f sandwich cone, 417f tapes in radial direction, 417f TBT. See Timoshenko beam theory (TBT) Technion, 554e555, 555fe556f, 638e639, 640f excitation system, 605f Tensile failure, 355 Test loading strategy, 275 Thermal stability analysis, 69e71, 83e85 Thermal stresses, 15t Thermoplastic materials, 413 Thermoplastics, 1e2 Thermoset, 1e2 prepreg materials, 413 Thickness locking, 71, 109 Thin-walled composite shells, 678 Thin-walled composite structures dynamic buckling, 718e720 finite element buckling, 693e695 finite element dynamic analysis, 717e718 finite element implementation, 700e703 initial postbuckling analysis, 693e695 multimode initial postbuckling analysis, 707e711 nonlinear vibrations, 720e725 numerical examples, 703e706, 711e717, 725e731 single-mode initial postbuckling analysis, 695e700

756

Thin-walled conical structures, 400e401 Thin-walled stiffened composite shells, structural robustness assessment of, 486e489 AVG and SD, IM7/8552, 487t boundary conditions, 488f, 489t stiffened panel design parameters, 487f stringer mesh, 488f Thin-walled structures, dynamic buckling of, 239e247 clamped shallow spherical cap, 239f DLF vs. nondimensional impulse period, 248f isotropic flat plate, 240f tested cylindrical shell, 247f tested plates experimental results, 242t material properties, 242t properties, 242t variation of DLF, 241f, 244fe245f variation of dynamic buckling mode shapes, 246f Third-order shear deformation theory, 288 Third-order theory, 106 Three-dimension (3D) brick model, 498 elasticity analysis, 51e52 elasticity theory, 281e282, 286 Timoshenko beam theory (TBT), 133, 195 kinematics, 134 Timoshenko’s theory, 132 Torisphere, 353e355. See also Imperfect torispheres Torque, 439 Traditional 3D elasticity formulations, 286 Transverse force resultants, 146 modulus, 4 normal stress, 109 stresses, 94 Trigonometry-based approximation for semianalytical models, 280 Truncated cone, 401, 402f lower-bound design for, 405e408 TsaieWu criterion, 347e349 interactive failure criterion, 355 Two coordinate systems, 7e8, 8f Two-dimension (2D) construction of 2D model, 97e98

Index

higher-order deformation theory, 52e53 solution methods for 2D model, 98e99 Two-dimensional plate model, 100 Two-mode imperfection, 442 Two-phase technology, 414e415 Two-point boundary value problem, 450e451 U Ultimate load (UL), 482e483 UMAT explicit, 475 UMAT implicit, 475 Uniaxial loaded plate critical buckling load calculation, 251e252 Unidirectional composites, 2e5, 6t Unified formulation, 110e111. See also Carrera’s unified formulation (CUF); Generalized unified formulation (GUF) Uniform temperature rise, 60 Unstiffened cylinders assembly for, 328e329 linear buckling analysis of, 316e318 Unstiffened cylindrical shells, 376e385. See also Stiffened composite panels VCT applied to cylindrical shells, experimental verification, 379e385 composite cylindrical shell Z37, with R/t¼510, 384e385 composite cylindrical shells R07, R08, and R09, with R/t¼399, 379e381 composite cylindrical shells R15 and R16, with R/t¼478, 381e384 VCT applied to cylindrical shells, numerical models, 376e377 Unsymmetric orthotropic plates, 172e179 biaxial compression buckling loads, 177f boundary conditions, 179 critical buckling load per unit width, 176 displacements, 173, 175 in-plane boundary conditions, 172e173 in-plane resultants, 173, 176 loading parameter, 174 relative uniaxial buckling loads, 175f symmetric, antisymmetric, and unsymmetric cross-ply laminates, 172f uniaxial and biaxial buckling loads, 178t

Index

uniaxial buckling loads for square antisymmetric angle-ply laminated plates, 177f uniaxial nondimensional buckling loads, 174f User Defined Field (USDFLD), 475 User interface, 500 User subroutines, 471e472 User-defined MPCs, 498e500, 499f V V groove, 549e551, 550f Validation structures, 463e464 VCCT. See Virtual crack closure technique (VCCT) VCT. See Vibration correlation technique (VCT) Vibration. See also Composite shell-type structure vibrations; Shells vibrations behavior, 442 of composite columns, 191 CLT, 196e205 FSDT, 205e214 frequencies, 377e379 modes, 724 of plates, 191 CLPT, 215e217 Vibration buckling, 223e224 correlation, 569e573 columns, beams, and rods, 559e563 nondimensional circular frequency, 572f plates, 563e564 Vibration correlation technique (VCT), 253e254, 370, 429, 565, 604, 659e667

757

applied for determination of boundary conditions, 370e375 applied to unstiffened cylindrical shells, 376e385 Virtual crack closure technique (VCCT), 498e500 Viscous damping, effect of, 451 W Warping function f (z), 105e106 Waters’ shell under axial loading, 711e717, 711t, 712f without axisymmetric imperfection, 712e714, 713t with axisymmetric imperfection, 715e717 Worst multiple perturbation load approach (WMPLA), 273e275 Wrinkling, 125e126 loads, 126e127, 127t modes of sandwich strut, 124f Y Young’s moduli ratio, 174 Young’s modulus of column, 559 Z Zigzag (ZZ) displacement field, 54 effect, 120 kinematics, 55f models, 94e95 Zigzag theories (ZZTs), 51e52, 107e108, 134e135