Bifurcation and Buckling in Structures

✍ Scribed by Kiyohiro Ikeda, Kazuo Murota

• Publisher: CRC Press
• Year: 2021
• Tongue: English
• Leaves: 278
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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