The volume showcases how basic formulation and solution methods can be used for an improved understanding of stability-affected structures. It also highlights the numerical treatment of the resulting equations, demonstrating and quantifying many aspects of structural stability not available to engineers by the traditional methods. The elaborated theoretical background is also used to show how methods included in commercial software can give misleading conclusions. The book will be of interest to researchers and professionals working in computational modelling of structures. The book includes exercises and projects, appended to each chapter, which further extends its usability as a text book in classroom teaching and for professional courses. Preface Contents About the Authors 1 Engineering Stability Evaluations 1.1 Stability and Instability 1.2 Spring-Supported Rigid Link 1.3 Plane 2-Bar Truss 1.4 Compressed Beam 1.5 System Stability 1.6 Plate Buckling 1.7 Thin-Walled Column 1.8 Pressurized Sphere 1.9 Spinning Rigid Object References 2 Underlying Theories 2.1 Reference Frame 2.2 Kinematics 2.3 Deformation and Strain 2.4 Strain Energy 2.4.1 Stress 2.4.2 Hyper-Elastic Materials 2.5 Continuum Mechanics Problems 2.5.1 Interior Forces 2.5.2 Boundary Traction 2.5.3 Boundary Conditions 2.5.4 Displacement Basis 2.5.5 Structural Regions 2.5.6 Dimensional Reduction 2.6 Energy, Work and Power 2.6.1 Kinetic Energy 2.6.2 Strain Energy 2.6.3 Exterior Forcing 2.6.4 Power and Work 2.6.5 Conservative Systems 2.7 Physical and Virtual Displacement Increments 2.8 Fundamental Relations 2.8.1 General Equation 2.8.2 The Time Dimension 2.8.3 Time-Dependent Motion 2.8.4 Time-Independent Equilibrium 2.8.5 Parametric Equilibrium 2.9 Stability 2.9.1 Stability of Equilibrium 2.9.2 Stability of Motion 2.10 Modelling of Reality References 3 Discretization of Structural Models 3.1 General Discretization 3.1.1 Interpolation and Curve-Fitting 3.1.2 Differential Equation 3.2 Discretized Balance 3.2.1 Continuum Displacement 3.2.2 Derivatives and Differentials 3.2.3 Virtual Work Terms 3.2.4 Discretized Mechanics Equation 3.2.5 Discretized Equilibrium 3.2.6 Differential of Equilibrium 3.2.7 Discretized Motion 3.2.8 Conclusions from Sect.3.2 3.3 Approaches to Discretization 3.3.1 Global Approximation 3.3.2 Local Approximation 3.3.3 Finite Elements for 3D Continuum 3.3.4 Isogeometric Analysis 3.3.5 Mesh-Free Methods 3.4 Dimensionally Reduced Models 3.4.1 Membranes and Trusses 3.4.2 Shells and Frames 3.5 Displacement Boundary Conditions 3.5.1 Symmetry Modelling 3.5.2 Coupling of Subregions 3.6 Mechanical Constraints 3.6.1 Energy Form 3.6.2 Supports 3.6.3 Symmetry Conditions 3.6.4 Hard Contacts 3.7 Discretized Stability 3.7.1 Constrained Stability 3.7.2 Comparison of Criteria 3.7.3 Unstable Equilibrium 3.7.4 Critical Equilibrium 3.7.5 Classification of Critical States 3.8 Model Creation References 4 Solution Algorithms 4.1 Discretized Equilibrium Analyses 4.1.1 Linear Stiffness 4.1.2 Secant Stiffness 4.1.3 Geometric Stiffness 4.1.4 Non-linear Equilibrium 4.1.5 Basic Solution Method for Non-linear Equilibrium 4.1.6 One-Parametric Forcing 4.2 Discretized Motion Analyses 4.2.1 Dynamic Solution Methods 4.2.2 Stability and Accuracy 4.2.3 Explicit Algorithms 4.2.4 Implicit Algorithms 4.3 Parametric Equilibrium Analyses 4.3.1 Variations to the Newton Iteration 4.3.2 Stepping Procedures 4.4 Multi-parametric Equilibrium 4.4.1 Selectors and Parameters 4.4.2 Sequence Definition 4.4.3 Increment Selection 4.4.4 Equilibrium Surfaces 4.5 Equilibrium Properties 4.5.1 Operative Stiffness 4.5.2 Stability Properties 4.5.3 Tangent Space 4.5.4 Tangent Vector 4.5.5 Contact Status 4.5.6 Transition Equilibrium States 4.6 Critical States 4.6.1 Identification 4.6.2 Secondary Branches 4.6.3 Critical Parameter Traces 4.6.4 Grazing Contacts 4.7 Algorithmic Implementation References 5 A Wider View 5.1 Special Situations 5.2 Critical Mode Interaction 5.2.1 Model Setup 5.2.2 Basic Results 5.2.3 Trigger Force 5.2.4 Comments to the Problem 5.3 Symmetry in Modeling 5.3.1 Group-Theoretical Background 5.3.2 Numerical Tests 5.3.3 Secondary Branches 5.3.4 Comments to the Problem 5.4 Stability Under Contact 5.4.1 Primary Branch 5.4.2 Contact Evolution 5.4.3 Secondary Branches 5.4.4 Parametric Investigations 5.4.5 Soft Contact Modelling 5.4.6 Comments to the Problem 5.5 Final Comments References Appendix Notation General Symbols Mechanical Quantities Engineering Parameters Consolidated List of References Index
