Bifurcation and Buckling in Structures

Description

Bifurcation and Buckling in Structures describes the theory and analysis of bifurcation and buckling in structures. Emphasis is placed on a general procedure for solving nonlinear governing equations and an analysis procedure related to the finite-element method. Simple structural examples using trusses, columns, and frames illustrate the principles. Part I presents fundamentals issues such as the general mathematical framework for bifurcation and buckling, procedures for the buckling load/mode analyses, and numerical analysis procedures to trace the solution curves and switch to bifurcation solutions. Advanced topics include asymptotic theory of bifurcation and bifurcation theory of symmetric systems. Part II deals with buckling of perfect and imperfect structures. An overview of the member buckling of columns and beams is provided, followed by the buckling analysis of truss and frame structures. The worst and random imperfections are studied as advanced topics. An extensive review of the history of buckling is presented. This text is ideal for advanced undergraduate and graduate students in engineering and applied mathematics. To assist readers, problems are listed at the end of each chapter, with their answers at the end of the book. Cover Half Title Title Page Copyright Page Contents Preface The Authors Part I: Bifurcation in Structures Chapter 1: Introduction to Buckling and Bifurcation 1.1. Summary 1.2. What are buckling and bifurcation? 1.3. Mathematical framework 1.4. Snap buckling of a structure 1.5. Bifurcation buckling of a structure 1.6. Problems Chapter 2: Analysis of Buckling Load and Mode 2.1. Summary 2.2. Mathematical framework 2.3. Procedure of buckling analysis 2.4. Jacobian matrix and critical point 2.5. Total potential energy 2.5.1. Derivation of governing equation 2.5.2. Existence of a potential 2.6. Stability 2.7. Example of buckling load analysis 2.8. Examples of buckling mode ana 2.8.1. Bar-spring system with two degrees of freedom 2.8.2. Bar-spring system with many degrees of freedom 2.9. Problems Chapter 3: Numerical Analysis I: Path Tracing 3.1. Summary 3.2. Theoretical foundation 3.3. Path tracing methods 3.3.1. Load control method 3.3.2. Displacement control method 3.3.3. Arc-length method 3.4. Structural example 3.5. Problems Chapter 4: Numerical Analysis II: Branch Switching 4.1. Summary 4.2. Theoretical foundation 4.2.1. Direction of bifurcating path 4.2.2. Bifurcation analysis pro 4.3. Bar-spring system 4.3.1. Linear-symmetric spring 4.3.2. Nonlinear-asymmetric spring 4.4. Appendix: Linear simultaneous equations 4.5. Problems Chapter 5: Bifurcation Theory I: Basics 5.1. Summary 5.2. Simple example of bifurcation equation 5.3. Derivation of bifurcation equation 5.4. Classification of simple critical points 5.4.1. Maximal and minimal points of load 5.4.2. Asymmetric bifurcation point 5.4.3. Symmetric bifurcation point 5.5. Direction of bifurcating paths 5.6. Structural examples of three kinds of bifurcations 5.6.1. Unstable–symmetric bifurcation 5.6.2. Stable–symmetric bifurcation 5.6.3. Asymmetric bifurcation 5.7. Problems Chapter 6: Bifurcation Theory II: Symmetric Structures 6.1. Summary 6.2. Bifurcation due to reflection symmetry 6.2.1. Propped cantilever 6.2.2. Two-bar truss arch with bilateral symmetry 6.3. Basics of groups 6.3.1. Groups and subgroups 6.3.2. Dihedral and cyclic groups 6.3.3. Truss dome with regular-triangular symmetry 6.4. Group-theoretic bifurcation theory 6.4.1. General theory 6.4.2. Bifurcation of a regular-triangular system 6.4.3. Truss dome with regular-hexagonal symmetry 6.5. Studies of symmetry and bifurcation of structures 6.5.1. Symmetry in structural mechanics 6.5.2. Development of catastrophe theory 6.5.3. Mathematics on symmetry and bifurcation 6.6. Problems Part II: Buckling of Structures Chapter 7: Member Buckling of Columns and Beams 7.1. Summary 7.2. Beam-column equation 7.2.1. Total potential energy 7.2.2. Derivation of beam-column equation 7.3. Beam-column subjected to axial compression 7.3.1. Procedure of buckling analysis 7.3.2. Buckling under typical boundary conditions 7.3.3. Effective buckling length 7.3.4. Cross-sectional shape and buckling stress 7.4. Beam-column on elastic foundation 7.5. Beam-column subjected to axial force and distributedload 7.6. Initial deflection 7.7. Inelastic buckling 7.7.1. Elastic-perfectly plastic body 7.7.2. General inelastic body 7.8. Appendix: Linear ordinary differential equations 7.9 Problems Chapter 8: Structural Buckling I: Truss 8.1. Summary 8.2. Finite displacement analysis 8.2.1. Member stiffness equation 8.2.2. Structural equilibrium equation 8.2.3. Structural example 8.3. Small displacement analysis 8.3.1. Member stiffness equation 8.3.2. Structural stiffness equation 8.3.3. Structural example 8.4. Linear buckling analysis 8.4.1. Formulation 8.4.2. Structural example 8.5. Buckling of truss members 8.6. Problems Chapter 9: Structural Buckling II: Frame 9.1. Summary 9.2. Stiffness equations of beam-column 9.2.1. Member stiffness equation 9.2.2. Structural stiffness matrix 9.2.3. Structural example of small-displacementanalysis 9.3. Linear buckling analysis: Introductory example 9.4. Formulation implementing axial deformation 9.4.1. Member stiffness matrix 9.4.2. Procedure to obtain axial forces 9.5. Structural example of linear buckling analysis 9.5.1. Definition of variables 9.5.2. Small displacement analysis 9.5.3. Linear buckling analysis 9.6. Linear buckling analysis in the global coordinates 9.6.1. Global coordinate system 9.6.2. Structural example 9.7. Problems Chapter 10: Advanced Topics on Imperfect Systems 10.1. Summary 10.2. Structural example with imperfection 10.3. Formulation of imperfection sensitivity 10.3.1. Maximal/minimal point of load 10.3.2. Symmetric bifurcation point 10.3.3. Structural example 10.4. Worst imperfection pattern 10.4.1. Formulation 10.4.2. Structural example 10.5. Buckling loads for random imperfections 10.5.1. Formulation 10.5.2. Structural examples 10.6. Imperfection sensitivity of elastic–plastic plates 10.7. Problems Chapter 11: History of Imperfect Buckling 11.1. Summary 11.2. Initial post-buckling behaviors 11.3. Search for prototype initial imperfections 11.4. Probabilistic scatter of buckling loads 11.5. Asymptotic method and plastic bifurcation of materials 11.6. Hilltop branching for materials and structures Appendix A: Answers to Problems Index