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

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http://www.mek.dtu.dk, ISBN: 978-87-70786-75-1 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 toemphasise 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. 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. 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. 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