Modern Trends in Structural and Solid Mechanics 1
Statics and Stability

This first volume is devoted to the statics and stability of solid and structural members. Modern Trends in Structural and Solid Mechanics 1 has broad scope, covering topics such as: buckling of discrete systems (elastic chains, lattices with short and long range interactions, and discrete arches), buckling of continuous structural elements including beams, arches and plates, static investigation of composite plates, exact solutions of plate problems, elastic and inelastic buckling, dynamic buckling under impulsive loading, buckling and post-buckling investigations, buckling of conservative and non-conservative systems and buckling of micro and macro-systems.

This book is intended for graduate students and researchers in the field of theoretical and applied mechanics.

Preface xi
Noël CHALLAMEL, Julius KAPLUNOV and Izuru TAKEWAKI

Chapter 1 Static Deformations of Fiber-Reinforced Composite Laminates by the Least-Squares Method 1
Devin BURNS and Romesh C BATRA

1.1 Introduction 2

1.2 Formulation of the problem 3

1.3 Results and discussion 7

1.3.1 Verification of the numerical algorithm 7

1.3.2 Simply supported sandwich plate 9

1.3.3 Laminate with arbitrary boundary conditions 10

1.4 Remarks 14

1.5 Conclusion 15

1.6 Acknowledgments 15

1.7 References 15

Chapter 2 Stability of Laterally Compressed Elastic Chains 17
Andrii IAKOVLIEV, Srinandan DASMAHAPATRA and Atul BHASKAR

2.1 Introduction 17

2.2 Compression of stacked elastic sheets 20

2.3 Stability of an elastically coupled cyclic chain 23

2.4 Elastic stability of two coupled rods with disorder 28

2.5 Spatial localization of lateral buckling in a disordered chain of elastically coupled rigid rods 31

2.6 Conclusion 39

2.7 References 40

Chapter 3 Analysis of a Beck’s Column over Fractional-Order Restraints via Extended Routh–Hurwitz Theorem 43
Emanuela BOLOGNA, Mario DI PAOLA, Massimiliano ZINGALES

3.1 Introduction 43

3.2 Material hereditariness 44

3.2.1 Linear hereditariness: fractional-order models 48

3.3 Dynamic equilibrium of an elastic cantilever over a fractional-order foundation 51

3.4 Stability analysis of Beck’s column over fractional-order hereditary foundation 54

3.4.1 The characteristic polynomial 55

3.4.2 State-space representation of the dynamic equilibrium equation 57

3.4.3 Stability analysis of fractional-order Beck’s column via the extended Routh–Hurwitz criterion 60

3.5 Numerical application 63

3.6 Conclusion 65

3.7 References 65

Chapter 4 Localization in the Static Response of Higher-Order Lattices with Long-Range Interactions 67
Noël CHALLAMEL and Vincent PICANDET

4.1 Introduction 68

4.2 Two-neighbor interaction – general formulation – homogeneous solution 70

4.3 Two-neighbor interaction – localization in a weakened problem 76

4.4 Conclusion 86

4.5 References 86

Chapter 5 New Analytic Solutions for Elastic Buckling of Isotropic Plates 91
Joseph TENENBAUM, Aharon DEUTSCH and Moshe EISENBERGER

5.1 Introduction 91

5.2 Equilibrium equation 93

5.3 Solution 94

5.4 Boundary condition 96

5.5 Numerical results 98

5.6 Conclusion 105

5.7 Appendix A: Deflection, slopes, bending moments and shears 109

5.8 Appendix B: Function transformation 116

5.9 References 119

Chapter 6 Buckling and Post-Buckling of Parabolic Arches with Local Damage 121
Uğurcan EROĞLU, Giuseppe RUTA, Achille PAOLONE and Ekrem TÜFEKCI

6.1 Introduction 121

6.2 A one-dimensional model for arches 122

6.2.1 Finite kinematics and balance, linear elastic law 124

6.2.2 Non-trivial fundamental equilibrium path 126

6.2.3 Bifurcated path 127

6.2.4 Special benchmark examples 128

6.3 Parabolic arches 130

6.4 Crack models for one-dimensional elements 133

6.5 An application 135

6.5.1 A comparison 138

6.6 Final remarks 138

6.7 Acknowledgments 139

6.8 References 139

Chapter 7 Inelastic Microbuckling of Composites by Wave-Buckling Analogy 145
Rivka GILAT and Jacob ABOUDI

7.1 Introduction 145

7.2 Buckling-wave propagation analogy 146

7.3 Microbuckling in elastic orthotropic composites 148

7.4 Inelastic microbuckling 150

7.5 Results and discussion 152

7.6 References 156

Chapter 8 Quasi-Bifurcation of Discrete Systems with Unstable Post-Critical Behavior under Impulsive Loads 159
Mariano P AMEIJEIRAS and Luis A GODOY

8.1 Introduction 160

8.2 Case study of a two DOF system with unstable static behavior 161

8.3 Exploring the static and dynamic behavior of the two DOF system 164

8.4 The dynamic stability criterion due to Lee 167

8.5 New stability bounds following Lee’s approach 170

8.6 Conclusion 174

8.7 Acknowledgments 175

8.8 References 175

Chapter 9 Singularly Perturbed Problems of Drill String Buckling in Deep Curvilinear Borehole Channels 177
Valery I GULYAYEV and Natalya V SHLYUN

9.1 Introduction 177

9.2 Singular perturbation theory: elements and history 179

9.3 Posing the problem of a drill string buckling in the curvilinear borehole 184

9.4 Modeling the drill string buckling in lowering operation 195

9.5 References 199

Chapter 10 Shape-optimized Cantilevered Columns under a Rocket-based Follower Force 201
Yoshihiko SUGIYAMA, Mikael A LANGTHJEM and Kazuo KATAYAMA

10.1 Background 201

10.2 Aims 204

10.3 Numerical analysis 206

10.3.1 Stability analysis 206

10.3.2 Optimum design 209

10.4 Experiment 213

10.4.1 General description 213

10.4.2 Rocket motor 214

10.4.3 Columns 216

10.4.4 Free vibration test 219

10.5 Flutter test 220

10.6 Concluding remarks 223

10.7 Acknowledgments 224

10.8 Appendix 224

10.9 References 225

Chapter 11 Hencky Bar-Chain Model for Buckling Analysis and Optimal Design of Trapezoidal Arches 229
Chien Ming WANG, Wen Hao PAN and Hanzhe ZHANG

11.1 Introduction 230

11.2 Buckling analysis of trapezoidal arches based on the HBM 231

11.2.1 Description of the HBM 232

11.2.2 HBM stiffness matrix formulation 234

11.2.3 Governing equation considering compatibility conditions 235

11.2.4 Verification of the HBM 236

11.3 Optimal design of symmetric trapezoidal arches 239

11.3.1 Problem definition 239

11.3.2 Optimization procedure 240

11.3.3 Optimal solutions 240

11.3.4 Sensitivity analysis of optimal solutions 243

11.3.5 Comparison with the buckling load of optimal fully stressed trapezoidal arches 245

11.4 Concluding remarks 245

11.5 References 246

List of Authors 249

Index 251

Summaries of Volumes 2 and 3 255

Noel Challamel is Professor at the University of Southern Brittany, France. He is the co-author of several books and over a hundred journal papers in the field of mechanics and civil engineering and is on the editorial board of numerous international journals. He is also Editor and Head of the Solid Mechanics and Mechanical Engineering series published by ISTE-Wiley.

Julius Kaplunov is Professor at Keele University, UK. He is the co-author of over a hundred publications in mechanics, including three books. He is a member of the European Academy of Sciences and sits on the editorial boards of more than ten journals.

Izuru Takewaki is Professor of building structures at Kyoto University, Japan, and is the 56th President of the Architectural Institute of Japan. He is the Field Chief Editor of Frontiers in Built Environment and has published over 200 international journal papers.