Local Effects in the Analysis of Structures [1st Edition] 9781483295442

Table of contents :
Content:
Studies in Applied MechanicsPage II
Front MatterPage III
Copyright pagePage IV
ForewordPages V-VIP. LADEVEZE, R. OHAYON, M. PREDELEANU, E. SANCHEZ-PALENCIA, N.Q. SON
On Saint-Venant's Principle in ElasticityPages 3-34P. LADEVEZE
Edge Effects in the Stretching of plates*Pages 35-54R.D. GREGORY, F.Y.M. WAN
Stress Concentration for Defects Distributed Near a SurfacePages 55-74N. NGUETSENG, E. SANCHEZ-PALENCIA
On the Influence of Free Edges in Plates and ShellsPages 75-99A.M.A. VAN DER HEIJDEN
On a Method to Evaluate Edge Effects in Elastic PlatesPages 101-126F. PECASTAINGS
Energy Structure of LocalizationPages 127-158V. BERDICHEVSKII, L. TRUSKINOVSKII
Edge Effects in Rotationally Symmetric Composite ShellsPages 161-180MAHIR SAYIR
Some Theoretical Aspects in the Modelling of Delamination for Multilayered PlatesPages 181-197J.L. DAVET, Ph. DESTUYNDER, Th. NEVERS
Local Effects Calculations in Composite Plates by a Boundary Layer MethodPages 199-214D. ENGRAND
Boundary Layers Stresses in Elastic CompositesPages 215-232Hélène DUMONTET
High Stress Intensities in Focussing Zones of WavesPages 235-252J. BALLMANN, H.J. RAATSCHEN, M. STAAT
The Local Effects in the Linear Dynamic Analysis of Structures in the Medium Frequency RangePages 253-275C.H. SOIZE
Implementation of Local Effects into Conventional and Non Conventional Finite Element FormulationsPages 279-298J. JIROUSEK
Special Finite Elements for an Appropriate Treatment of Local EffectsPages 299-314R. Piltner
On the Consideration of Local Effects in the Finite Element Analysis of Large StructuresPages 315-324W. DIRSCHMID
Local Effects of Geometry Variation in the Analysis of StructuresPages 325-342E. SCHNACK

Citation preview

STUDIES IN APPLIED MECHANICS 1. Mechanics and Strength of Materials (Skalmierski) 2. Nonlinear Differential Equations (Fuclk and Kufner) 3. Mathematical Theory of Elastic and Elastico-Plastic Bodies An Introduction (Necas and Hlavacek) 4. Variational, Incremental and Energy Methods in Solid Mechanics and Shell Theory (Mason) 5. Mechanics of Structured Media, Parts A and B (Selvadurai, Editor) 6. Mechanics of Material Behavior (Dvorak and Shield, Editors) 7. Mechanics of Granular Materials: New Models and Constitutive Relations (Jenkins and Satake, Editors) 8. Probabilistic Approach to Mechanisms (Sandler) 9. Methods of Functional Analysis for Application in Solid Mechanics (Mason) 10. Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates (Kitahara) 11. Mechanics of Material Interfaces (Selvadurai and Voyiadjis, Editors) 12. Local Effects in the Analysis of Structures (Ladeveze, Ed itor)

STUDIES IN APPLIED MECHANICS 12

Local Effects in the Analysis of Structures

Edited by

Pierre Ladeveze Laboratoire de Mécanique et Technologie (E.N.S.E. T./Université Paris 6/C.N.R.S.), Cachan, France

ELSEVIER Amsterdam — Oxford — New York — Tokyo

1985

ELSEVIER SCIENCE PUBLISHERS B.V. 1 Molenwerf, P.O. Box 2 1 1 , 1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, NY 10017, U.S.A.

Library of Congress Cataloging-in-Publication Data Main e n t r y under t i t l e : Local effects in the analysis of structures. (Studies in applied mechanics ; 12) Selection of papers presented at the EUR0MECH Colloquium "Inclusion of Local Effects in the Analysis of Structures," held Sept. 11-14, 1984 at Laboratoire de mécanique et technologie, Cachan, France. Bibliography: p. 1. Structures, Theory of—Congresses. 2. Stress concentration—Congresses. I. Ladev^ze, Pierre, 1945. II. EUR0MECH Colloquium "Inclusion of Local Effects in the Analysis of Structures" (1984 : Laboratoire de mécanique et technologie, Cachan, France) III. Series. TA645.L63 1985 624.1f71 ISBN 0-444-42520-9 (U.S.)

85-13150

ISBN 0-444-42520-9 (Vol. 12) ISBN 0-444^1758-3 (Series) © Elsevier Science Publishers B.V., 1985 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V./Science & Technology Division, P.O. Box 330, 1000 AH Amsterdam, The Netherlands. Special regulations for readers in the USA — This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. Printed in The Netherlands

V

F O R E W O R D

At the present time, the Inclusion of Local Effects in the Analysis of Structures is undoubtedly a question of prime importance for Engineering Design. The classical computational approaches are not readily adapted to take into account the local effects - appropriate treatments are necessary. This book attempts to provide an introduction to and a survey of the specific computational methods. It begins with the various theories which allow to separate and then to determine the local and global effects. Chapter 2 discusses edge effects for composite structures. Chapter 4 deals with general numerical methods, especially for effects due to large local variations of geometry. Chapter 3 concerns some dynamic problems - it is an opening towards non-conventional local effects in Structural Mechanics.

VI

The papers, for a part, have been presented at the EUROMECH Colloquium "Inclusion of Local Effects in the Analysis of Structures" which has been held on September 11-14, 1984 in CACHAN (France) - LABORATOIRE DE MECANIQUE ET TECHNOLOGIE . The Scientific Committee included : P. LADEVEZE R. OHAYON M. PREDELEANU E. SANCHEZ-PALENCIA N.Q. SON.

3

ON SAINT-VENANT'S PRINCIPLE IN ELASTICITY

P. LADEVEZE Laboratoire de Mécanique et Technologie, E.N.S.E.T./Université PARIS 6/C.N.R.S., 61, Avenue du Président Wilson - 94230 CACHAN (France)

The Saint-Venant's Principle is considered from the angle of its present practical interest in Structural Mechanics, especially for composite structures. The interior large wavelength effect must be separated from the edge or extremity effects with a small wavelength in order to be computed. In a more precise way, it is a theorem which expresses conditions ensuring localization of displacements and stresses. This point of view is built on certain characteristic properties of the solutions and not on the properties of zero resultant-moment loadings. Moreover, and this is an important point, the diameter 0 of the beam or the thickness 2h of the plate are not considered as small parameters. The approach has nothing to do with asymptotic methods. Beams and plates are studied as far as possible from a unitary point of view. The problem is restricted to the so-called

Saint-Venant Problem for

which the non-zero loadings and displacements are only prescribed on the edges of the structure. Additional loadings on the lateral surface of the beam or on the faces of the plate essentially only modify the interior effect. This effect is now well-known [6] [15] [16] [17] [18] [19] [20] [32]. This paper starts by proposing several major properties for the solutions which are localized or not. They are inferred from the particularities of the geometry. The localization concepts are stated precisely. The corresponding solutions decrease exponentially as a function of the distance to the edges such that the decaying length is 0(0) for beams and 0(h) for plates. In fact, the Saint-Venant Principle which characterizes such solutions expresses the orthogonality to the interior large wavelength solutions. Some auxiliary problems are used to write this condition in terms of the data on the edges for any boundary conditions. The splitting up of the solution into the interior effect and the edge effects is a central problem. Several results concerning the existence and the unicity of such a splitting up are presented.

4 1. GENERALITIES - HYPOTHESES 1.1. Beams Cylindrical beams are considered (figure 1). The domain Ω is defined by Ω = {M = X + t.N ; t e]0,L[;Xe S} where S denotes the c r o s s - s e c t i o n , t i s the N- c o o r d i n a t e . The boundary o f the c r o s s - s e c t i o n i s supposed as having the usual r e g u l a r i t y , namely corner p o i n t s can occur.

Jk=L·

Figure 1

The end cross-sections are denoted by S 0 and S, . The Hooke tensor IK is taken to be constant on lines parallel to the center line. 1.2. Plates The thickness is constant. It is supposed that the material is homogeneous on planes parallel to the middle surface

Σ. The domain is defined by

Ω = {M = m + Nz , m 6 Σ , ze ]-h,h[} and the boundary of the middle surface is supposed as having the usual regularity, namely corner points can occur.

i N3=N

>

Figure 2

1.3. Basic problem Our aim is to study end effects, so we shall restrict the paper to this specific problem which, by analogy with the beam theory, will be designated as

5 the Saint-Venant Problem. It is characterized by the following assumptions : - the body-forces are zero - the lateral surfaces of the beam and the upper and lower surfaces of the plate are free. In other words, the non-zero forces and displacements are only prescribed on the end surfaces S 0

and S..

Moreover, we will consider only physically admissible solutions namely those with finite energy. Remarks - This framework contains two important particular cases : . composite beams with homogeneous layers parallel to the center line . composite plates with homogeneous layers parallel to the middle surface. - The boundary conditions on the end cross-section can be of any type. Notations The scalar product of the vectors V,W w i l l be written VW. π denotes both the orthogonal projection on the cross-section for the beams and the orthogonal projection on the middle surface for the plates. 1.4. Saint-Venant's solutions They are large wavelength "solutions" of the Saint-Venant Problem. They verify a l l the equations except for the boundary conditions on S0

and S, .

- Beams The Saint-Venant solutions

can be written as follows :

U* = A T* + B M* + V t + Qt X c* NL = A0 T* + B°M* where - T*,M* : resultant-moment of normal stress vector - V, Ω : vectors which are constant related to X-coordinates - A , B , A ° B 0 : linear operators which are constant related to t. - Plates For the sake of simplicity, the Saint-Venant solution is written for homogeneous isotropic materials. It is the classical Kirchhoff-Love solution : TTU* = grad Γ-ζ W + 3λ + 4 μ z 3 Ají] + u + — - — z 2 grad m L 6(λ + 2μ) -I 8(λ + μ)

m

ω

6 N U*

^ z 2 - 2η 2 (λ+μ)

W +

Δ W m

λ+2μ with

Δ Δ

mm

W(m), u(m), w(m)

W= 0 - ^ -

λ+2μ

4(λ+μ)

such as :

grad f d i vm u) + 2μ divm ( Ï Ï E ( U ) Ï Ï ) m

di v u =-Aí2lL· m

λζ

= 0

ω

4(λ+μ)

1.5. Interior and exterior problems The beam and the plate described by the figures (1) and (2) have in fact two edges. Therefore, it is natural to introduce : - the interior problem related to an end section S 0

So

Figure 3

So the exterior problem related to an end section S0

XSo

S^ Figure U

7 These problems are p a r t i c u l a r Saint-Venant Problems. I t should be r e c a l l e d t h a t the s o l u t i o n s have to lead to f i n i t e energy. " + " s h a l l denote q u a n t i t i e s connected w i t h the i n t e r i o r problem, " - " those connected w i t h the e x t e r i o r one. For p l a t e s , the f o l l o w i n g coordinates system i s introduced : C. , t €]-OO,-HX>[ i s a f a m i l y o f closed curves such t h a t :

"Ct

= Cn

t=0

- the domain i n t e r i o r t o C. decreases w i t h t and tends to zero w i t h - the domain e x t e r i o r t o C.i decreases w i t h ( - f ) . tends to i n f i n i t y w i t h

t -> °°

The i n t e r i o r diameter of C.,

(-f).

JHîLJ"^ QN3

Figure 5

For starshaped domains, it is possible to use I Onu i t = -log ' tl |0m0| where nu, m are homothetic points. Moreover, the t-section is defined by S t = C t x]-h,h[. So, a plate is mapped on an abstract beamwith a variable "cross-section", t is the coordinate generating this different "cross-sections". With these notations, beams and plates can be studied using the same terminology. 2. INTERIOR AND EXTERIOR PROBLEMS - BASIC PROPERTIES 2.1. Semi-groups IR ,IR~ - Beams Let D be the space of displacement values at the cross-section S (D = [H 2 ( S ) ] 3 ) . S 0 is the chosen reference cross-section and U 0 a given displacement belonging to D .

8 U. designates the solution of the interior problem for the displacement boundary conditions : Ό

μ=ο

U", i s defined i n the same way f o r t ' < 0. I t i s c l e a r t h a t :

uj = RJ.U 0

t*o

U~, = TR~V U0

t'< 0

where IR.

,IR",

are operators on D which v e r i f y :

1R+ = Id o

IR!

. IRT = IR! ^

tl t2 IR" = Id o IR" .IR'

= J R l ,+. f ,

1*2

tl

Property

v t! >, o v t 2 >, o

t l + C2

*'l

2-1 {IR.

«'2

V tj < 0

t > 0}

V t2 < 0

and {IR",

t ' < 0} are one parameter semi-groups.

- Plates

For plates, the same kind of properties exist but the semi-groups involve two parameters, namely U+ = IR+(t,t')U*,

t> V

U", = ]R"(t',t)U~

t' < t

2.2.

Scalar products on D Let U,V be two displacements o f D. The f o l l o w i n g energy s c a l a r product

can be defined on D ( i n f a c t on the q u o t i e n t of D by the space of r i g i d body displacements) : < U,V > , Q < U,V > i s I o

=

=

/

Τ ^ Κ . ε ( υ ! ) ε ( ν ! ) 1 dfi

/_ Τ Γ Γ κ . ε ( υ ^ , ) ΐ ( ν ^ ) 1 dfi ^o

where U"t, V t > U t ,, V~, are prolongations of U,V. Ω 0

is the interior domain

to the cross-section S 0 , Ω^ is the complementary part. From [25] [27], it can be seen that :

9

+|So=

+

/

V0(U>| t=o dS 0

_ls

= - / o

_ Vc(U-,)n|t,=0dS

b0

where n i s the unit outward normal to SQ . and + | S

and < U,V > , ς are independent of the reference

cross-section. For plates, however the scalar products

related to the reference cross-section S. have to be introduced. The following property expresses a relation between IR and IR~ , that i s IR~ can be defined through IR . Property 2-2

V t,t'

t >, f

V U,V € D < U,IR" ( f , t ) V >*,

+ < U,1R~ ( t ' , t ) V >~, =

= < ν,Κ+(ΐ,ί')υ>*

+ < ν , Κ + ( ΐ , ΐ ' ) υ " >~

The proof is obtained by applying Stokes' formula to the quantity A. . , = /

Tr Γκ. l i K ^ t . t ' J U J c i D M t ' . t O V ) ! dfi

Remarks Supplementary properties have been obtained in the case of beams where the cross-section is a symmetric plane for the material [26] [27]. - Let us introduce the scalar product - « U,V »

= < U,V >^ + < U,V >^

The previous property can be expressed in the form «

U,]R~(t\t)V » , = « V,JR+(t,t')U »

t

Let U t , V be solutions of the i n t e r i o r and exterior problems related to d i f ferent end cross-section S t , S t . I t can be deduced from (2.2) that the quantity

« u;,v; » t is a constant γ independent of t on the common domain, i.e.,

10 t

< t

-

w e D _

Proof : for u; e D ,one has [.

Remarks - The previous result can be written E+ < exp(-2t/J¿+) E*

(2)

where E. denotes the strain energy contained in the part of beam situated beyond the cross-section S.. - The norm L 2 (S) of the displacement follows a similar decaying property. Using the norm associated to the scalar product « , » , the same kind of results for U ,U~ are obtained. The decaying length is the same : I - inf I V S D

« A + W , W » - - inf «A~W,W»

' w e o «w>w»

(3) ()

- The above improves Toupin's famous results [ 2 ] on two points. The decaying is purely exponential near the end cross-section, a property which has already been proved by another method in [ 5 ] . Moreover, the lengths I , € which characterize the cross-section geometry and the material are optimal.

- Plates The decaying properties are the same for the interior and the exterior problem.

17 So, we will only deal with the interior problem. The index + will be omitted. • A_gr^lijiiin^r^_ property I f the c o n d i t i o n s (3-3) are s a t i s f i e d on the end c r o s s - s e c t i o n S , we have seen t h a t they are e q u a l l y v e r i f i e d on the whole middle surface account

Σ. Taking i n t o

the l o c a l equations v e r i f i e d by any s o l u t i o n , i t i s easy t o show the

f o l l o w i n g r e l a t i o n s which hold on the i n t e r i o r of Σ: (3λ+4μ) Jhh

z 3 a dz - 12μ(λ+μ) _ £ \ z 2 - h 2 ) ÏÏ U dz = 0

λ _ ¿ h ( z 2 - h 2 ) grad a + 8μ(λ+μ) _£ h ττυ dz = 0 (4) /

h

-h 1

H d* i v

'}

ΝΛττυ

πΙΙζ dz +

dz = 0 . πϋΝ.

dz + 2μ

— _^ 4μ(λ+μ)

z.grad (ΝατΝ) dz = 0

These s i x scalar conditions l i n k c e r t a i n mean values and moments of the d i s p l a cement t o the s t r e s s e s .

• lDfiDiï§-Bl§î§_§y^0?lîï§^_Ï2.2§!2i2^Î9_^tÈ_2D_î!]Ë_Ë^2e

^

\ L: period

Figure 6 Let us consider the domain r e s t r i c t e d t o a p e r i o d . I t can be classed as a beam w i t h a r e c t a n g u l a r c r o s s - s e c t i o n , and thus the semi-group IR involves only one parameter. Property 3-5 The localized solutions verify - ||Ut|| < exp(-t/kh) ||Uo||

18

lu,t l l L 2 ( s ) where k,k'

< k' k 1 ' 2

exp(-Vkh)

||U0

are dimensionless constants 0 ( 1 ) . The optimal value of k i s defined

by

1=

,

+

Ψ>!ΐ 0

=

on

°

{X

i»X3'

Χχ > 0 - h

< X3 < h }

for

Plane_groblem Displacement : V = V X N X + V 3 N Normal stress vector

σ

σ1χ

div (Hc(V)) = 0 «Í(V).N

h

= 0

for

on

{X 1 ,X 3 ; Χχ > 0

X3 = ±h

: Hooke tensor for plane strain

-h < X 3 < h}

26

f N3 L

2h

i

*N1

®N2 Figure 12 The edge effect is described by the superposition of the both solutions. This is to point out that the localization conditions are related to the plane problem : / σ -h

z dz = 0

π

(3λ+4μ)

fh a ll -h

on S 0 z 3 - 12μ(λ+μ)

fh -h

V (z 2 -h 2 )dz = 0 3

(7)

on S 0

Furthemore, all the values at the edge namely at X1 = 0, will be underlined and let (U, ,\}¿,U_3 ) and (f_ »F^.fj) be displacements and forces at the edge :

"l

=*h 2

U,

=

+

fi = - 2μ

1

l,l2

+ £u

L ^ [ ΐ ' , χ -Ι»2

Ψ,,

+ μϊ,,

F3 = - μ Ψ , 2 3 + σ 1 3

Ü3 = I ,

We have to add t o the l o c a l i z a t i o n c o n d i t i o n s those conditions making

it

possible to b u i l d the s o l u t i o n s of the plane and a n t i - p l a n e problems. Ψ i s an even z - f u n c t i o n and s o , the Ψ -problem does not

i n v o l v e any c o n d i t i o n . At the

contrary , f o r the plane problem,conditions are needed. Two of them are (7) and the others are the f o l l o w i n g : / h σ 13 dz = 0 3 -h 3λ fh -h 24μ(λ+μ)

z2

dz

+

/h -h

z

v

dz

= 0

A l l these c o n d i t i o n s come from the Saint-Venant P r i n c i p l e f o r the s e m i - i n f i n i t e s t r i p which i s a p a r t i c u l a r beam. Property 5-1 The second order approximation of the Saint-Venant P r i n c i p l e i s defined through the f o l l o w i n g c o n d i t i o n s :

27 /h -h

(F

+ 2μΨ.

) ζ dz = O

_1

/ h ( £ x 4-2μ Ψ , 2 ΐ ) -h

(3λ+4μ)

z 3 dz - 12μ(λ+μ)

/ h ^ 3 ( z 2 - h 2 ) dz = O -h

/ h (F 3 + μΨ, 2 3 ) dz = O -h ~ 23 ;h

3λ

-h

{F

24μ(λ+μ)

~

Ψ = / Z dn / % ' O O

üx

*

+ μ 3

z2dz

" QJ

+

;h

-h

z

+ ψ}2)

dz

=

0

l

- —) μ

"Ι>2 + Ιι

=

Ü2 =

I M

Proof : From the r e l a t i o n s

I2 = u [ l . n Ψ,

+ Ψ,

11

33

- I . 2 2 ] +μ v l j 2 = 0

Mi = - i ' 2 + vx it is easy to determine that the value of Ψ at the edge is equal to : i t i s easy to determine t h a t the value of Ψ a t the edge is equal t o :

Ψ = fZ dn / η dn' Γϋ

- F /μ]

These conditions can be easily written in terms of the data for any boundary conditions. To illustrate this point, we will consider two cases : given loadings and prescribed displacements. • QlyËD-l2§dings The only conditions to take into account are /h σ l l z dz = 0 ^ — h

(8) (8)

r σ13 dz = o -h One introduces the f u n c t i o n G ( X i , X 2 ) , s o l u t i o n o f the a n t i - p l a n e problem f o r G|Xi=

0

=

( z / h ) . Then,

28 O =

/h σ ζ dz = / h F1 z dz + 2μ -h - 1 1 -h

fh -h

û =

/ h F z dz + 2μ 1 -h

dz

/ h h G, 2 Ψ, 2 -h

z

Ψ, 2 1 dz

Let us consider the value of Ψ at the edge

ψ = / z dn / η dn' Γυ1 0

2

L- '2

-h

- ί,/μΐ = fz dn / η d n ' k J

o

-h

L-

29

- F2/ -

Vi

- ψ,22 1 -

J

So, w i t h an e r r o r of ( h 2 / L 2 ) , Ψ can be approximated by :

= ;z dn /η άη' li,2 0 0

Ψ

-f2^

i

and we obtain : 0 = /h F, z dz + 2y /h dz G91 h/Z Λΐηαη'ίν, 2 - F^/y)^ -h -h " 0 -h As σ = 0(χχ J and then V_x 22 h = 0((h2/L2) σ ^ ) , it follows that the previous condition can be written : (/ Fi z dz / 0) -h 0 = /h F z dz - 2 /h dz {G91 fZ /η dndn' F, 2 }h -h _ 1 -h " 0 -b The other condition is 0 = /h σ13 dz = /h(F3 + μ l,23)dz = /h{F3 +y/Zdn(V1 22 - F , , » } dz -h -h -h 0 ' 0 = / h F3 dz - / h z dz {μνχ 22 - F2j2} -h -h - 1 ' 22 ' From

/h

z

v dz - — 3 L .

-h

f" αΐ3 z>dz 24μ(λ+μ) -h

it can be determined that _/" z y V 1>22 dz = h 0(£l3(h2/L2)) and then / (F, + z F2 ) dz = 0 -h " 9 ~2'2

29 Property 5-2 The second order approximation of the Saint-Venant P r i n c i p l e i s defined f o r

given loadings by /h F z dz - 2 /h h { G M fz / n dnd n ' Fo J dz = 0 -h _ 1 -h 0 -h ~2'2

.

• {h da + Z I 2 , 2 ) dz = û -h • ?!T˧£!2l!^_di§ElÊÇËmËDÎ:5 The conditions to take into account are (3λ+4μ)

/" σ „ ζ3 - 12μ(λ+μ) /" V, (z 2 -h 2 ) dz = 0 -h - 11 -h

(9) ¿

/ σ z dz + /" z _V1 dz = 0 8μ(λ+μ) -h ~~13 -h One introduces two auxiliary plane problems defined by the following boundary displacement conditions :

ù

k=o =Z X

ö|Xl-o ■z2 N 3

The corresponding stresses are l o c a l i z e d . Let us denote by Ç7 l l 9 â 1 3 and