An Introduction to Sandwich Structures (Student Edition) (Lecture Notes 41517)

455 Pages·2005·59.923 MB·English

by  Dan Zenkert

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ANI NTRODUCTTIOO N
SANDWICH
STRUCTURES
STUDENT EDITION
DANZENKERT
CONTENTS
PREFACE V
LIST OF SYMBOLS vii
CHAPTER 1. INTRODUCTION 1
CHAPTER 2. MATERIALS AND MATERIAL PROPERTIES 2.1
2.1F acMea terials 2.1
2.2E stimoafFt aicMoean t ePrrioaple rties 2.3
2.2.R1u le-of-mixtures 2.3
2.2."2P racrtuilcea-lo"f -mixtures 2.4
2.2.C3o nvewresiifrgoahnct t ion/frvaocltuimoen 2.5
2.2.T4h ickpnreesdsi ction 2.5
2.2.S5t ifpfnreospseo rftt hlieae msi na 2.6
2.2.S6t ifpfnreospseo rftt hlieae msi nate 2.7
2.72 .S treonfcg otmhp olsaimtien ates 2.10
2.3E xperiDmeetnetramloi fFn aacMteai toenPr rioaple rties 2.10
2.4C orMea terials 2.13
2.4.H1o neyccoormebs 2.14
2.4.B2a lwsoao d 2.17
2.4.C3e llfoualmasr 2.18
2.5F atiPgruoep eorfSt ainedswC iocrMhea terials 2.20
2.6E stimoafCt oirMoean tse Prrioaple rties 2.22
2.7E xperiDmeetnetramloi fCn oartMeia otneP rrioaple rties2 .24
2.8A dhes-iDveescsr ainpPdtr ioopne rties 2.25
2.8R.e1q uiroentm heaend thse sive 2.25
2.8A.d2h esaintvdhe epsir ro perties 2.28
2.9E xperiDmeetnetramloi ftn haAetd ihoenIs nitveePr rfaocpee rt2i.e3s0
2.1E0s timoafTt hieorInmn aslu lation 2.31
References 2.34
Exercises 2.37
CHAPTER 3. FUNDAMENTALS 3.1
3.1F lexRuirgaild ity 3.1
3.2A pproxiimtnah Fteli eoxnRusir gaild ity 3.2
3.3S treistnsh Seeas n dBweiacmh 3.3
3.4S heSatrr esses 3.3
3.5A pproxiimtna hStehi eoSantr r ess 3.4
3.6S ummoafAr pyp roximations 3.5
3.7" ThSea ndwEiffcecht " 3.5
3.8S andwwiiDtcihhs siFmaicleasr 3.6
3.9E quivWaildetnht 3.8
References 3.9
Exercises 3.9
AN INTRODUCTION TO SANDWICH CONSTRUCTION
CHAPTER 4. BEAM THEORY 4.1
4.1 Shear Deformations 4.2
4.2 Shear Stiffness 4.3
4.3 Equations in Terms of the Displacement Field 4.3
4.4 Governing Beam Equations 4.6
4.5 Effect of Thick Faces 4.7
4.6 Rigid Core 4.11
4.7 Energy Relations 4.12
4.8 General Solution of Beam Problems 4.14
4.9 Examples of Beam Calculations 4.15
4.9.1 Cantilever beam 4.15
4.9.2 Shear beam 4.18
4.9.3 Design example 4.19
4.9.4 Beam subjected to point load 4.21
4.9.5 Beam subjected to uniform pressure 4.26
4.9.6 Beam subjected to hydrostatic pressure 4.27
4.9.7 Hyperstatic beam example 4.29
4.10 Torsion 4.30
4.11 Testing of Sandwich Beams 4.31
4.11.1 The three-point bend (TPB) specimen 4.31
4.11.2 The four-point bend (FPB) specimen 4.32
References 4.33
Exercises 4.34
CHAPTER 5 BUCKLING AND FREE VIBRATION OF SANDWICH BEAMS 5.1
5.1 Governing equations 5.1
5.2 Boundary Conditions for Sandwich Beams 5.5
5.3 Buckling of Simply Supported Column -Simple Solution 5.8
5.4 Rigorous Solution to Beam Buckling 5.9
5.4.1 Clamped edges 5.9
5.4.2 Simply supported edges 5.14
5.4.3 One edge clamped, the other free (cantilever beam) 5.16
5.4.4 One edge clamped, the other simply supported 5.18
5.5 Examples of Sandwich Beam Buckling 5.21
5.6 Buckling of Sandwich Columns with Thick Faces 5.22
5.7 Buckling Stress Exceeding the Elastic Limit 5.23
5.8 Free Vibration of Sandwich Beams 5.24
5.8.1 Simply supported edges 5.26
5.8.2 Clamped edges 5.29
5.8.3 One edge clamped, the other simply supported 5.30
5.8.4 One edge clamped, the other free (c antilever beam) 5.31
5.9 Examples of Sandwich Beam Free Vibration 5.33
5.10 Estimation of Elastic Properties on Free-Free Sandwich Beam 5.34
5.11 Approximate Solutions to Beam Buckling and Free Vibration Problems 5.38
5.11.1 Buckling of clamped sandwich beam 5.38
5.11.2 Free vibration of a cantilever sandwich beam 5.40
5.11.3 Free vibration of a clamped sandwich beam 5.41
References 5.42
Exercises 5.42
CHAPTER 6 FACE WRINKLING 6.1
6.1 Winkler Foundation Approach 6.2
6.2 Hoff's Approach 6.3
6.3 Exponential Decay 6.6
6.4 Differential Equation Method 6.7
6.5 Wrinkling under Biaxial Load 6.9
6.6 Wrinkling under Multi-Axial Load 6.11
6.7 lntercellular Buckling 6.15
6.8 Imperfection Induced Wrinkling 6.16
6.9 Summary of Buckling Phenomena 6.21
References 6.22
7 7.1
CHAPTER FAILURE MODES AND DESIGN CRITERIA
7.1 Formulae for Failure Loads 7.2
7.2 Failure Mode Maps 7.5
7.3 Design Criteria 7.7
7.4 Determination ofThicknesses 7.9
7.5 Single Parameter Optimum 7.10
7.6 Minimum Weight for Given Stiffness 7.12
7.7 Minimum Weight for Given Strength 7.14
References 7.16
CHAPTER 8 SANDWICH PLATES-FUNDAMENTAL EQUATIONS 8.1
8.1 Governing Equations 8.2
8.2 Partial Deflections 8.8
8.3 Equation of Motion 8.10
8.4 Governing Buckling Equation 8.12
8.5 Isotropic Sandwich Plates 8.13
8.6 Isotropic Sandwich Plate with Thick Faces 8.14
8.7 Cross-Section Properties 8.15
8.8 Energy Relations 8.17
8.9 Stresses and Strains 8.19
8.10 Thermal Stresses and Deformations 8.21
8.11 General Sandwich Theory for Anisotropic Plates 8.23
8.12 Boundary Conditions 8.29
References 8.32
CHAPTER 9 SOLUTIONS TO PLATE PROBLEMS 9.1
9.1 Rotationally-Symmetric Plates 9.1
9.2 Bending of a Rectangular, Simply Supported, Isotropic
Sandwich Plate 9.5
9.3 Rectangular, Simply Supported, Orthotropic Sandwich Plate 9.11
iii
AN INTRODUCTION TO SANDWICH CONSTRUCTION
9.4 Solution by Energy Method -Ritz's Method 9.21
9.5 Approximate Solutions for Bending ofOrthotropic Sandwich Plates 9.22
9.5.1 Simply supported plate 9.22
9.5.2 Clamped plate 9.25
9.5.3 Two sides clamped, the other two simply supported 9.30
9.6 Buckling of a Simply Supported, Isotropic Sandwich Plate 9.32
9.7 Buckling of a Simply Supported, Orthotropic Sandwich Plate
with Thin Faces 9.35
9.8 Approximate Buckling Formulae for Orthotropic Sandwich
Plates with Various Edge Conditions 9.38
9.8.1 Simply supported plate 9.38
9.8.2 Loaded edges simply supported, the other edges clamped 9.40
9.8.3 Loaded edges simply supported, the other edges clamped 9.41
9.8.4 All edges clamped 9.45
9.9 Shear Buckling 9.48
9.10 Combined Buckling and Transverse Load 9.49
9.11 Free Vibration of a Simply Supported Sandwich Plate 9.50
9.12 Conclusions 9.53
References 9.53
Exercises 9.55
CHAPTER 10 SINGLE CURVED SANDWICH SHELLS 10.1
10.1 Fundamental Equations 10.1
10.2 Governing Buckling Equation 10.3
10.3 Buckling of Simply Supported Isotropic Plate Subjected to
Uniaxial Load I 0.6
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
References I 0.15
CHAPTER 11 LOCALISED LOADS -by OLE T. THOMSEN 11.1
11.1 Elastic Foundation Analogy 11.2
11.1.1 Classical Winkler foundation model 11.2
11.1.2 Two-parameter elastic foundation model 11.6
I 1.1.3 Specification of boundary conditions 11.12
11.1.4 Superposition with classical sandwich beam theory 11 .14
11.1.5 Range of applicability -Winkler vs. two-parameter
foundation model 11.16
11.2 Discussion: Application, Results and Parametric Effects 11.19
1 1.2.1 Application of method for solving engineering
design problems 11.19
11.2.2 Example 11.20
11.2.3 Parametric Effects 11.23
11.3 Concluding Remarks 11.27
References 11.28
iv
CHAPTER 12 SANDWICH AND FEM 12.1
12.1 General Remarks on FEM 12.1
12.2 Special Considerations for Sandwich Structures 12.2
12.3 Beam Analysis 12.3
12.4 Sandwich Beam Finite Element 12.4
12.4.1 Derivation of stiffness matrix by beam calculations 12.5
12.4.2 Governing equations 12.7
12.4.3 Weak form of DE using virtual work method 12.9
12.4.4 Displacement finite element formulation 12.10
12.4.5 Two-node shear deformable beam element 12.12
12.4.6. Shear Locking 12.13
12.4. 7 Three-node Shear Deformable Beam 12.14
12.4.8 Stiffness matrix assembly 12.17
12.4.9 Boundary conditions 12.18
12.4.10 Solution of equation systems 12.21
12.4.11 Post-processing 12.23
12.4.12 Example of FEM-Calculation for a Cantilever
Sandwich Beam 12.25
12.4.13 Example of FEM-Calculation for a Simply
Supported Sandwich Beam 12.27
12.4.14 More Examples 12.30
12.4.15 Mixed Formulation for Two-Node Beam 12.30
12.5 Plate Analysis 12.31
12.6 Shear Deformable Plate and Shell Elements 12.32
12.6.1 Governing equations 12.32
12.6.2 Loads on an arbitrary plane 12.37
12.6.3 Element construction 12.40
12.7 Boundary Conditions 12.46
12.8 Spurious Zero Energy Modes 12.48
12.9 Effects of Reduced Integration 12.51
12.10 Point Loads in Plate Formulations 12.52
12.11 Shell Elements 12.53
12.12 Alternative Modelling of Sandwich Structures 12.53
References 12.55
CHAPTER 13 JOINTS AND LOAD INTRODUCTIONS 13.l
13.1 Inserts 13.1
13.1.1 The elements involved 13.3
13.1.2 Face sheet/insert interface 13.4
13. 1.3 Core/insert interface 13.5
13.1.4 Stress concentrations due inserts 13.5
13.1.5 Inserts in panels subjected to shear 13.6
13 .1.6 Summary 13.6
13.2 Insert Calculation Examples 13.7
13.2.1 Self-tapping screw or rivet 13.9
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AN INTRODUCTION TO SANDWICH CONSTRUCTION
13.2.2 Partial insert 13.11
13.2.3 Through-thickness insert 13.14
13.2.4 Through-the-thickness insert with flared ends 13.15
13.3 Joints 13.16
13.3.1 Basic types 13.16
13.3.2 T-joints 13.17
13.3.3 L-joints 13.18
13.3.4 V-joints 13.20
13.3.5 Localised deflection 13.21
13.3.6 Calculation example, T-joint 13.23
13.3.7 Calculation example, L-joint 13.25
13.3.8 T-joints -tests and bbservations 13.26
Bibliography 13.28
CHAPTER 14 MANUFACTURING-BY KARLSSON AND ASTROM 14.1
14.1 Face Materials 14.2
14.2 Core Materials 14.2
14.3 Wet Lay-up 14.3
14.3.1 Procedure 14.3
14.3.2 Characteristics 14.6
14.3.3 Applications 14.7
14.4 Pre-preg Lay-up 14.8
14.4.1 Procedure 14.8
14.4.2 Characteristics 14.10
14.4.3 Applications 14.11
14.5 Adhesive Bonding 14.11
14.5.1 Procedure 14.12
14.5.2 Characteristics 14.13
14.5.3 Applications 14.13
14.6 Liquid Moulding 14.13
14.6.1 Procedure 14.14
14.6.2 Characteristics 14.16
14.6.3 Applications 14.17
14.7 Continuous Lamination 14.19
14.8 Other Processes 14.20
14.9 Outlook 14.23
References 14.24
v
i
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 1 1, which is fol lowed 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