An Introduction to Sandwich Structures (Student Edition) (Lecture Notes 41517) [Second Edition] 9788770786751

Table of contents :
AN INTRODUCTION TO SANDWICH CONSTRUCTION
CONTENTS
PREFACE
PREFACE, 2ND EDITION
LIST OF SYMBOLS
CHAPTER 1 INTRODUCTION
References
CHAPTER 2 MATERIALS AND MATERIAL PROPERTIES
2.1 Face Materials
2.2 Estimation of Face Material Properties
2.2.1 Rule-of-mixtures
2.2.2 "Practical" rule-of-mixtures
2.2.3 Conversion weight fraction/volume fraction
2.2.4 Thickness prediction
2.2.5 Stiffness properties of the lamina
2.2.6 Stiffness properties of the laminate
2.2.7 Strength of composite laminates
2.3 Experimental Determination of Face Material Properties
2.4 Core Materials
2.4.1 Honeycomb cores
2.4.2 Balsa wood
2.4.3 Cellular foams
2.5 Fatigue Properties of Sandwich Core Materials
2.6 Estimation of Core Material Properties
2.7 Experimental Determination of Core Material Properties
2.8 Adhesives - Description and Properties
2.8.1 Requirements on the adhesive
2.8.2 Adhesives and their properties
2.9 Experimental Determination of the Adhesive Interface Properties
2.10 Estimation of Thermal Insulation
References
Exercises
CHAPTER 3 FUNDAMENTALS
3.1 Flexural Rigidity
3.2 Approximations in the Flexural Rigidity
3.3 Stresses in the Sandwich Beam
3.4 Shear Stresses
3.5 Approximation in the Shear Stress
3.6 Summary of Approximations
3.7 "The Sandwich Effect"
3.8 Sandwich with Dissimilar Faces
3.9 Equivalent Width
References
Exercises
CHAPTER 4 BENDING OF SANDWICH BEAMS
4.1 Shear Deformations
4.2 Shear Stiffness
4.3 Equations in Terms of the Displacement Field
4.4 Governing Beam Equations
4.5 Effect of Thick Faces
4.6 Rigid core
4.7 Energy Relations
4.8 General Solution of Beam Problems
4.9 Examples of Beam Calculations
4.9.1 Cantilever beam
4.9.2 "Shear beam"
4.9.3 Design example
4.9.4 Beam subjected to point load
4.9.5 Beam subjected to uniform pressure
4.9.6 Beam subjected to hydrostatic pressure
4.9.7 Hyperstatic beam example
4.10 Torsion
4.11 Testing of Sandwich Beams
4.11.1 The three-point bend (TPB) test
4.11.2 The four-point bend (FPB) test
References
Exercises
CHAPTER 5 BUCKLING AND FREE VIBRATION OF SANDWICH BEAMS
5.1 Governing Equations
5.2 Boundary Conditions for Sandwich Beams
5.3 Buckling of Simply Supported Column - Simple Solution
5.4 Rigorous Solution to Beam Buckling
5.4.1 Clamped edges
5.4.2 Simply supported edges
5.4.3 One edge clamped, the other free (cantilever beam)
5.4.4 One edge clamped, the other simply supported
5.5 Examples of Sandwich Beam Buckling
5.6 Buckling of Sandwich Columns with Thick Faces
5.7 Buckling Stress Exceeding the Elastic Limit
5.8 Free Vibration of Sandwich Beams
5.8.1 Simply supported edges
5.8.2 Clamped edges
5.8.3 One edge clamped, the other simply supported
5.8.4 One edge clamped, the other free (cantilever beam)
5.9 Examples of Sandwich Beam Free Vibration
5.10 Estimation of Elastic Properties on Free-Free Sandwich Beam
5.11 Approximate Solutions to Beam Buckling and Free Vibration Problems
5.11.1 Buckling of clamped sandwich beam
5.11.2 Free vibration of a cantilever sandwich beam
5.11.3 Free vibration of a clamped sandwich beam
References
Exercises
CHAPTER 6 FACE WRINKLING
6.1 Winkler Foundation Approach
6.2 Hoff's Method
6.3 Exponential Decay
6.4 Differential Equation Method
6.5 Wrinkling under Biaxial Load
6.6 Wrinkling under Multi-Axial Load
6.7 lntercellular Buckling
6.8 Imperfection Induced Wrinkling
6.9 Summary of Buckling Phenomena
References
CHAPTER 7 FAILURE MODES AND DESIGN CRITERIA
7.1 Formulae for Failure Loads
7.2 Failure Mode Maps
7.3 Design criteria
7.4 Determination of Thicknesses
(i) Core thickness
7.5 Single Parameter Optimum
(i) Flexural rigidity
(ii) Flexural strength
(iii) Face dimpling
7.6 Minimum Weight for Given Stiffness
(i) Core properties predetermined
(ii) Core properties varying with density
7.7 Minimum Weight for Given Strength
(i) Simultaneous face yield - core shear
(ii) Simultaneous face wrinkling - core shear
References
CHAPTER 8 SANDWICH PLATES - FUNDAMENTAL EQUATIONS
8.1 Governing Equations
8.2 Partial Deflections
8.3 Equation of Motion
8.4 Governing Buckling Equation
8.5 Isotropic Sandwich Plates
8.6 Isotropic Sandwich Plates with Thick Faces
8.7 Cross-section Properties
8.8 Energy Relations
8.9 Stresses and Strains
8.10 Thermal Stresses and Deformations
8.11 General Sandwich Theory for Anisotropic Plates
8.12 Boundary Conditions
(i) Free edge
(ii) Simply supported edge
(iii) Clamped edge
References
CHAPTER 9 SOLUTIONS TO PLATE PROBLEMS
9.1 Rotationally-Symmetric Plates
9.2 Bending of a Rectangular, Simply Supported, Isotropic Sandwich Plate
9.3 Rectangular, Simply Supported, Orthotropic Sandwich Plate
9.4 Solution by Energy Method - Ritz' Method
9.5 Approximate Solutions for Bending of Orthotropic Sandwich Plates
9.5.1 Simply supported plate
9.5.2 Clamped plate
9.5.3 Two sides clamped, the other two simply supported
9.6 Buckling of a Simply Supported, Isotropic Sandwich Plate
9.7 Buckling of a Simply Supported, Orthotropic Sandwich Plate with Thin Faces
9.8 Approximate Buckling Formulae for Orthotropic Sandwich Plates with Various Edge Conditions.
9.8.1 Simply supported plate
9.8.2 Loaded edges simply supported, the other edges clamped
9.8.3 Loaded edges simply supported, the other edges clamped
9.8.4 All edges clamped
9.9 Shear Buckling
9.10 Combined Buckling and Transverse Load
9.11 Free Vibration of a Simply Supported Sandwich Plate
9.12 Conclusions
References
Exercises and design examples
CHAPTER 10 SINGLE CURVED SANDWICH SHELLS
10.1 Fundamental Equations
10.2 Governing Buckling Equation
10.3 Buckling of a Simply Supported Isotropic Plate Subjected to Uniaxial Load
10.4 Buckling due to External Lateral Pressure
10.5 Local Buckling of Curved Sandwich Panels
10.6 Bending of Single Curved Sandwich Beams - a Practical Approach
References
CHAPTER 11 LOCALISED LOADS
11.1 Elastic Foundation Analogy
11.1.1 Classical Winkler foundation model
11.1.2 Two-parameter elastic foundation model
11.1.3 Specification of boundary conditions
11.1.4 Superposition with classical sandwich beam theory
11.1.5 Range of applicability - Winkler vs. two-parameter foundation model
11.2 Discussion: Application, Results and Parametric Effects
11.2.1 Application of the method for solving engineering design problems
11.2.2 Example
11.2.3 Parametric effects
11.3 Concluding Remarks
Acknowledgement
References
CHAPTER 12 SANDWICH AND FEM
12.1 General Remarks on FEM
12.2 Special Considerations for Sandwich Structures
12.3 Beam Analysis
12.4 Sandwich Beam Finite Element
12.4.1 Derivation of stiffness matrix by beam calculations
12.4.2 Governing equations
12.4.3 Weak form of DE using virtual work method
12.4.4 Displacement finite element formulation
12.4.5 Two-node shear deformable beam element
12.4.6 Shear Locking
12.4.7 Three-node Shear Deformable Beam
12.4.8 Stiffness matrix assembly
12.4.9 Boundary conditions
12.4.10 Solution of equation systems
12.4.11 Post-processing
12.4.12 Example of FEM-Calculation for a Cantilever Sandwich Beam
12.4.13 Example of FEM-Calculation for a Simply Supported Sandwich Beam
12.4.14 More Examples
12.5 Plate analysis
12.6 Shear Deformable Plate and Shell Elements
12.6.1 Governing equations
12.6.2 Loads on an arbitrary plane
12.6.3 Element construction
12.7 Boundary Conditions
12.8 Spurious zero energy modes
12.9 Effects of reduced integration
12.10 Point loads in plate formulations
12.11 Shell Elements
12.12 Alternative Modelling of Sandwich Structures
References
CHAPTER 13 JOINTS AND LOAD INTRODUCTIONS
13.1 Inserts
13.1.1 The elements involved
13.1.2 Face sheet/insert interface
13.1.3 Core/insert interface
13.1.4 Stress concentrations due to inserts
13.1.5 Inserts in panels subjected to shear
13.1.6 Summary
13.2 Insert Calculation Examples
13.2.1 Self-tapping screw or rivet
13.2.2 Partial insert
13.2.3 Through-thickness insert
13.2.4 Through-the-thickness insert with flared ends
13.3 Joints
13.3.1 Basic types
13.3.2 T-joints
13.3.3 L-joints
13.3.4 V-joints
13.3.5 Localised deflection
13.3.6 Calculation example, T-joint
13.3.7 Calculation example, L-joint
13.3.8 T-joints - tests and bbservations
Bibliography
CHAPTER 14 MANUFACTURING
14.1. Face Materials
14.2 Core Materials
14.3 Wet Lay-Up
14.3.1 Procedure
14.3.2 Characteristics
14.3.3 Applications
14.4 Prepreg Lay-Up
14.4.1 Procedure
14.4.2 Characteristics
14.4.3 Applications
14.5 Adhesive Bonding
14.5.1 Procedure
14.5.2 Characteristics
14.5.3 Applications
14.6 Liquid Moulding
14.6.1 Procedure
14.6.2 Characteristics
14.6.3 Applications
14.7 Continuous Lamination
14.8 Other Processes
14.9 Outlook
References
Back Cover

Citation preview

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An Introduction to Sandwich Structures Student edition

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Dan Zenkert

DTU Mechanical Engineering Department of Mechanical Engineering

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AN INTRODUCTION TO

SANDWICH STRUCTURES STUDENT EDITION

DANZENKERT

CONTENTS PREFACE

V

LIST OF SYMBOLS CHAPTER 1. CHAPTER

CHAPTER

2. 2.1 2.2

INTRODUCTION MATERIALS AND MATERIAL PROPERTIES

Face Materials Estimation of Face Material Properties 2.2.1 Rule-of-mixtures 2.2.2 "Practical" rule-of-mixtures 2.2.3 Conversion weight fraction/volume fraction 2.2.4 Thickness prediction 2.2.5 Stiffness properties of the lamina Stiffness properties of the laminate 2.2.6 2.2.7 Strength of composite laminates 2.3 Experimental Determination of Face Material Properties 2.4 Core Materials 2.4.1 Honeycomb cores 2.4.2 Balsa wood 2.4.3 Cellular foams 2.5 Fatigue Properties of Sandwich Core Materials 2.6 Estimations of Core Material Properties 2.7 Experimental Determination of Core Material Properties 2.8 Adhesives - Description and Properties 2.8.1 Requirements on the adhesive 2.8.2 Adhesives and their properties 2.9 Experimental Determination of the Adhesive Interface Properties 2.10 Estimation of Thermal Insulation References Exercises

3.

FUNDAMENTALS

3.1 Flexural Rigidity 3.2 Approximations in the Flexural Rigidity 3.3 Stresses in the Sandwich Beam 3.4 Shear Stresses 3.5 Approximation in the Shear Stress 3.6 Summary of Approximations 3.7 "The Sandwich Effect" 3.8 Sandwich with Dissimilar Faces 3.9 Equivalent Width References Exercises

vii 1

2.1 2.1 2.3 2.3 2.4 2.5 2.5 2.6 2.7 2.10 2.10 2.13 2.14 2.17 2.18 2.20 2.22 2.24 2.25 2.25 2.28 2.30 2.31 2.34 2.37 3.1 3.1 3.2 3.3 3.3 3.4 3.5 3.5 3.6 3.8 3.9 3.9

AN INTRODUCTION TO SANDWICH CONSTRUCTION

CHAPTER

4. 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

BEAM THEORY

Shear Deformations Shear Stiffness Equations in Terms of the Displacement Field Governing Beam Equations Effect of Thick Faces Rigid Core Energy Relations General Solution of Beam Problems Examples of Beam Calculations Cantilever beam 4.9.1 Shear beam 4.9.2 4.9.3 Design example 4.9.4 Beam subjected to point load Beam subjected to uniform pressure 4.9.5 Beam subjected to hydrostatic pressure 4.9.6 4.9.7 Hyperstatic beam example 4.10 Torsion 4.11 Testing of Sandwich Beams 4.11.1 The three-point bend (TPB) specimen 4.11.2 The four-point bend (FPB) specimen References Exercises

CHAPTER 5

BUCKLING AND FREE VIBRATION OF SANDWICH BEAMS

4.1 4.2 4.3 4.3 4.6 4.7 4.11 4.12 4.14 4.15 4.15 4.18 4.19 4.21 4.26 4.27 4.29 4.30 4.31 4.31 4.32 4.33 4.34 5.1

5.1 Governing equations 5.1 5.2 Boundary Conditions for Sandwich Beams 5.5 5.8 5.3 Buckling of Simply Supported Column - Simple Solution 5.4 Rigorous Solution to Beam Buckling 5.9 5.9 5.4.1 Clamped edges 5.14 5.4.2 Simply supported edges One edge clamped, the other free (cantilever beam) 5.4.3 5.16 One edge clamped, the other simply supported 5.4.4 5.18 5.21 5.5 Examples of Sandwich Beam Buckling 5.6 Buckling of Sandwich Columns with Thick Faces 5.22 5.23 5.7 Buckling Stress Exceeding the Elastic Limit 5.8 Free Vibration of Sandwich Beams 5.24 5.26 5.8.1 Simply supported edges 5.8.2 Clamped edges 5.29 5.30 One edge clamped, the other simply supported 5.8.3 One edge clamped, the other free (cantilever beam) 5.31 5.8.4 5.9 Examples of Sandwich Beam Free Vibration 5.33 5.34 5.10 Estimation of Elastic Properties on Free-Free Sandwich Beam 5.11 Approximate Solutions to Beam Buckling and Free Vibration Problems 5.38 5.38 5.11.1 Buckling of clamped sandwich beam 5.40 5.11.2 Free vibration of a cantilever sandwich beam

5.11.3 References Exercises CHAPTER

Free vibration of a clamped sandwich beam

FACE WRINKLING 6 6.1 Winkler Foundation Approach 6.2 Hoff's Approach 6.3 Exponential Decay 6.4 Differential Equation Method 6.5 Wrinkling under Biaxial Load 6.6 Wrinkling under Multi-Axial Load 6.7 lntercellular Buckling 6.8 Imperfection Induced Wrinkling 6.9 Summary of Buckling Phenomena References

CHAPTER 7

FAILURE MODES AND DESIGN CRITERIA

5.41 5.42 5.42 6.1

6.2 6.3 6.6 6.7 6.9 6.11 6.15 6.16 6.21 6.22 7.1

7.1 Formulae for Failure Loads 7.2 Failure Mode Maps 7.3 Design Criteria 7.4 Determination of Thicknesses 7.5 Single Parameter Optimum 7.6 Minimum Weight for Given Stiffness 7.7 Minimum Weight for Given Strength References

7.2 7.5 7.7 7.9 7.10 7.12 7.14 7.16

CHAPTER

SANDWICH PLATES-FUNDAMENTAL EQUATIONS 8 Governing Equations 8.1 Partial Deflections 8.2 Equation of Motion 8.3 Governing Buckling Equation 8.4 Isotropic Sandwich Plates 8.5 Isotropic Sandwich Plate with Thick Faces 8.6 Cross-Section Properties 8.7 Energy Relations 8.8 Stresses and Strains 8.9 8.10 Thermal Stresses and Deformations 8.11 General Sandwich Theory for Anisotropic Plates 8.12 Boundary Conditions References

8.1 8.2 8.8 8.10 8.12 8.13 8.14 8.15 8.17 8.19 8.21 8.23 8.29 8.32

CHAPTER

9 9.1 9.2 9.3

SOLUTIONS TO PLATE PROBLEMS

Rotationally-Symmetric Plates Bending of a Rectangular, Simply Supported, Isotropic Sandwich Plate Rectangular, Simply Supported, Orthotropic Sandwich Plate iii

9.1 9.1 9.5 9.11

AN INTRODUCTION TO SANDWICH CONSTRUCTION

Solution by Energy Method - Ritz's Method Approximate Solutions for Bending of Orthotropic Sandwich Plates 9.5.1 Simply supported plate 9.5.2 Clamped plate Two sides clamped, the other two simply supported 9.5.3 Buckling of a Simply Supported, Isotropic Sandwich Plate 9.6 Buckling of a Simply Supported, Orthotropic Sandwich Plate 9.7 with Thin Faces Approximate Buckling Formulae for Orthotropic Sandwich 9.8 Plates with Various Edge Conditions 9.8.1 Simply supported plate Loaded edges simply supported, the other edges clamped 9.8.2 Loaded edges simply supported, the other edges clamped 9.8.3 9.8.4 All edges clamped 9.9 Shear Buckling 9.10 Combined Buckling and Transverse Load 9.11 Free Vibration of a Simply Supported Sandwich Plate 9.12 Conclusions References Exercises 9.4 9.5

9.21 9.22 9.22 9.25 9.30 9.32 9.35 9.38 9.38 9.40 9.41 9.45 9.48 9.49 9.50 9.53 9.53 9.55

10.1 10.1 Fundamental Equations 10.3 Governing Buckling Equation Buckling of Simply Supported Isotropic Plate Subjected to I 0.6 Uniaxial Load 10.4 Buckling due to External Lateral Pressure 10.10 10.5 Local Buckling of Curved Sandwich Panels 10.11 10.6 Bending of Single Curved Sandwich Beams - a Practical Approach 10.12 I 0.15 References

CHAPTER 10

SINGLE CURVED SANDWICH SHELLS

CHAPTER 11

LOCALISED LOADS - by OLE T. THOMSEN

10.1 10.2 10.3

11.1

Elastic Foundation Analogy 11.1.1 Classical Winkler foundation model 11.1.2 Two-parameter elastic foundation model I 1.1 .3 Specification of boundary conditions 11.1.4 Superposition with classical sandwich beam theory 11.1.5 Range of applicability - Winkler vs. two-parameter foundation model 11.2 Discussion: Application, Results and Parametric Effects 1 1.2.1 Application of method for solving engineering design problems 11.2.2 Example 11.2.3 Parametric Effects 11.3 Concluding Remarks References iv

11.1

11.2 11.2 11.6 11.12 11.1 4 11.16 11.19 11.1 9 11.20 11.23 11.27 11.28

CHAPTER

CHAPTER

12 12.1 12.2 12.3 12.4

SANDWICH AND FEM

General Remarks on FEM Special Considerations for Sandwich Structures Beam Analysis Sandwich Beam Finite Element 12.4.1 Derivation of stiffness matrix by beam calculations 12.4.2 Governing equations 12.4.3 Weak form of DE using virtual work method 12.4.4 Displacement finite element formulation 12.4.5 Two-node shear deformable beam element 12.4.6. Shear Locking 12.4. 7 Three-node Shear Deformable Beam 12.4.8 Stiffness matrix assembly 12.4.9 Boundary conditions 12.4.10 Solution of equation systems 12.4.11 Post-processing 12.4.12 Example of FEM-Calculation for a Cantilever Sandwich Beam 12.4.13 Example of FEM-Calculation for a Simply Supported Sandwich Beam 12.4.14 More Examples 12.4.15 Mixed Formulation for Two-Node Beam 12.5 Plate Analysis 12.6 Shear Deformable Plate and Shell Elements 12.6.1 Governing equations 12.6.2 Loads on an arbitrary plane 12.6.3 Element construction 12.7 Boundary Conditions 12.8 Spurious Zero Energy Modes 12.9 Effects of Reduced Integration 12.10 Point Loads in Plate Formulations 12.11 Shell Elements 12.12 Alternative Modelling of Sandwich Structures References 13 13.1

13.2

JOINTS AND LOAD INTRODUCTIONS

Inserts 13.1.1 The elements involved 13.1.2 Face sheet/insert interface 13.1.3 Core/insert interface 13.1.4 Stress concentrations due inserts 13.1.5 Inserts in panels subjected to shear 13 .1.6 Summary Insert Calculation Examples 13.2.1 Self-tapping screw or rivet V

12.1 12.1 12.2 12.3 12.4 12.5 12.7 12.9 12.10 12.12 12.13 12.14 12.17 12.18 12.21 12.23 12.25 12.27 12.30 12.30 12.31 12.32 12.32 12.37 12.40 12.46 12.48 12.51 12.52 12.53 12.53 12.55 13.l 13.1 13.3 13.4 13.5 13.5 13.6 13.6 13.7 13.9

AN INTRODUCTION TO SANDWICH CONSTRUCTION

13.2.2 13.2.3 13.2.4 13.3 Joints 13.3.1 13.3.2 13.3.3 13.3.4 13.3.5 13.3.6 13.3.7 13.3.8 Bibliography

Partial insert Through-thickness insert Through-the-thickness insert with flared ends Basic types T-joints L-joints V-joints Localised deflection Calculation example, T-joint Calculation example, L-joint T-joints - tests and bbservations

CHAPTER 14 14.1 14.2 14.3

MANUFACTURING- BY KARLSSON AND ASTROM Face Materials Core Materials Wet Lay-up 14.3.1 Procedure 14.3.2 Characteristics 14.3.3 Applications 14.4 Pre-preg Lay-up 14.4.1 Procedure 14.4.2 Characteristics 14.4.3 Applications 14.5 Adhesive Bonding 14.5.1 Procedure 14.5.2 Characteristics 14.5.3 Applications 14.6 Liquid Moulding 14.6.1 Procedure 14.6.2 Characteristics 14.6.3 Applications 14.7 Continuous Lamination 14.8 Other Processes 14.9 Outlook References

vi

13.11 13.14 13.15 13.16 13.16 13.17 13.18 13.20 13.21 13.23 13.25 13.26 13.28 14.1 14.2 14.2 14.3 14.3 14.6 14.7 14.8 14.8 14.10 14.11 14.11 14.12 14.13 14.13 14.13 14.14 14.16 14.17 14.19 14.20 14.23 14.24

PREFACE

This text is intended to cover some of the most important aspects of the theory on load carrying sandwich panels. In that respect it is very similar to the now almost classical texts from the 1960s by Allen and Plantema. I have tried, however, to be more practical and to emphasise problem solving. Thus, this text covers less of the purely theoretical side of the topic, fewer special cases, but on the other hand contains more examples and_ solved problems. The problems are there not only to provide a useful engineering tool, but are also sometimes included for the purpose that they clarify a physical behaviour. In that respect it is important not only to study the theory but to do so along with solving the problems. This book has fourteen separate chapters in which the theory of structural sandwich construction is evolved. The chapters should preferably be read in the sequence they appear and although some chapters are more or less stand-alone much of the theory in each new chapter is based on the preceding chapters. The first chapter is only a brief introduction to the concept; it gives some of the historical background and then mentions some of the pioneer work and applications in the field. The materials used and their characteristics are covered in chapter 2 along with some practical aspects of bonding face sheets to different kinds of core materials. Special emphasis is given to core materials, their special features and how to predict their characteristics. For each material category, some information is given on how to extract relevant properties, either by theoretical estimations or from testing. Basic definitions, stress and strain calculations and important approximations are given in chapter 3. This is followed by a comprehensive beam theory development in chapter 4. A cross-section shear stiffness is introduced, governing differential equations including transverse shear and rotary inertia effects are derived and expressions for the strain energy given. Some special features like the effect of a rigid core and thick faces are also included. The chapter also contains quite a large number of examples and solutions to standard beam problems. Buckling of sandwich columns with thin and thick faces are treated in chapter 5 and the so typical instability mode for sandwich construction called wrinkling or local buckling is treated in chapter 6. A summary of failure modes and how to predict their respective failure loads are given in chapter 7. Design procedures including the stiffness and the failure modes of the preceding chapter are outlined in chapter 8. The evolution of design criteria leads to some simple formulae for minimum weight design which can easily be included in an engineering design process. In chapter 9 the theory for sandwich plates is derived, in which plates with generally anisotropic, orthotropic or isotropic behaviour as well as thin and thick faces are included. Energy expressions, boundary conditions and cross-section properties are also presented. Chapter I O contains a large number of solutions to bending, buckling and free vibration of sandwich plate problems. Solutions are derived using direct integration of the governing equation and approximate solutions based on the Ritz energy approach. The theory of flat vii

AN INTRODUCTION TO SANDWICH CONSTRUCTION

plates is extended to single curved sandwich shells in chapter 11, which is followed by solutions to some fundamental problems. A more practical approach to the design of curved sandwich shells is also presented. The last three chapters in the book are slightly different. Chapter 12 treats the very important problem of localised loads. The theory given is based on two foundation models and the chapter is concluded with its application to engineering design problems. In chapter 13 the finite element method (FEM) is discussed in conjunction with sandwich construction. The basis of FEM and the formulation for sandwich elements is not given as that lies outside the scope of this text, but some important considerations when using FEM for sandwich structures are given. The final chapter has no theoretical background but simply gives some practical aspects on the joining and load introduction problems encountered when designing sandwich structures. The chapter is included merely for the sake of completeness rather than being any basis for the foundations of the concept. This book started out as an attempt to tidy up some old lecture notes, add to them and prepare material for an extended course on the topic of structural sandwich constructions. 1 guess that is how most books start out. I had the opportunity to read calmly through a lot of collected material and properly learn the topic myself. There is no teacher like having to prepare material for teaching a subject. Most parts of this text have been written and compiled during my stay at the Department of Mechanical Engineering, The University of Auckland, New Zealand, between May 1991 and July 1992. I wish to acknowledge Professor Jan Backlund for his encouragement and interest in the progress of my writing. I also want to take the opportunity to express my appreciation to all the staff at the Department of Mechanical Engineering, The University of Auckland, and especially Professor John Duncan for giving me opportunity to spend a wonderful year in New Zealand. I am also greatly in debt to Mr. Lars Falk for the many and long hours of editing and proof-reading. Thanks also to Mr. Mats Roslund for help with some of the drawings. Chapter 12 � Localised Loads, has been written and compiled by Dr. Ole Thybo Thomsen, Department of Mechanical Engineering, Aalborg University Centre, Aalborg, Denmark. Chapter 13 - Sandwich and FEM is co-authored by Prof. Jan Backlund, Department of Lightweight Structures, Royal Institute of Technology, Stockholm, Sweden. Since this is the first issue I expect there are still several mistakes, misprints or even errors present. Certainly some parts need further explanation while others may be redundant. I am therefore more than happy to receive comments, corrections and suggestions for improvements.

Stockholm, March 1995 © Dan Zenkert viii

PREFACE, 2:ND EDITION

The main reason for compiling this second edition is that this version is updated to comply with current educational activities at KTH and is therefore to be used in only that context. It is merely an update of the original text. However, there are some major changes. The first chapters, 1-4, are basically unaltered albeit with some minor amendments here and there. Chapter 5 has been revised substantially and is now much more complete with stricter derivations, complete solutions to some fundamental beam cases now also including comparisons with FE-calculations. Chapter 6 has been going through some updates, though not very extensive ones apart from the inclusion of some more recent theories on wrinkling that has appeared from research in the last few years. Chapters 7 and 8 have been merged but are otherwise more or less unchanged. Chapters 8-11 and 14 are basically unchanged except for some minor updates. Chapter 12 on the FEM has gone through quite a revision now including the basic theory of finite element construction for shear deformable beams and plates. I have added a chapter on manufacturing which is based on a paper by Karlsson and Astrom. This is added for completeness of the book contents rather for inclusion in the present course syllabus. Another major change is that some chapters are followed by a section of examples. These are examples coming from exams in the undergraduate courses taught at KTH since around 1993. The examples have more or less worked through solutions although these are rather brief in most cases. In its present form, this text is only to be used for teaching activities at KTH.

Stockholm, December 2005 © Dan Zenkert The Department of Mechanical Engineering would like to acknowledge Professor Dan Zenkert for permission to re-print and use this compendium in the course 41517 Stiffened Plates and Sandwich Composites. The courtesy is highly appreciated. DTU, Kgs. Lyngby 2016 Christian Berggreen ([email protected])

ix

AN INTRODUCTION TO SANDWICH CONSTRUCTION

LIST OF SYMBOLS

The following is a list of the notation and symbols used throughout the text. Some of these may have duplicate definitions in parts of the text and are in those parts locally defined. Other symbols not mentioned are defined in the text when appearing. Mostly millimetres (mm) are used in the problems used, but metres (m) could equally well be used. Latin symbol A B C CE, Ca, C,

D Do, DP De E G K L M

N

p

Q

R R

s

T T1 , T2 Tm '

u w z

A, B, D N B

F K Ka M a, b d

d e k

kh> ks

Description Extensional stiffness First moment of area (function of z) Compliance Constants defining core material properties Flexural rigidity (bending stiffness) Flexural rigidity of components in a sandwich Young's modulus of elasticity Shear modulus Buckling coefficient Length of beam Bending moment Normal force Load Point load Reaction force or initial radius of shell (chapter 11) Rotary inertia in chapters 4, 9 and I 0 Shear stiffness Transverse force Temperatures Strain energy, potential energy Weight Curvature parameter Extension, coupling and bending stiffness matrices Vector or matrix of element shape functions Vector or matrix of element shape function derivatives Load vector Stiffness matrix Geometric stiffness matrix Mass matrix Sides of rectangular panel (width of beam) Distance between centroids of the sandwich Vector of nodal degrees-of-freedom Distance defining position of neutral axis (see Fig.2.7) Thermal transmittance Deformation coefficients for beams and panels

X

Unit N/mm N mm/N see eq.(8.3) Nmm Nmm N/mm2 (MPa) N/mm2 mm Nmm N/mm N/mm N N/mm or mm kg N/mm N/mm K Nmm kg N/mm, N, Nmm N N/mm N/mm kg mm mm mm mm W/m2 K

kr, kM m, n

Px• Py q, q T t

U, V, W

x, y, z

Greek symbol

'¥

r

a

Ii

qi

K If/

r

Yo 1 1 V

OJ

0 p p' a­ T

Subscript x, y, z r,