Imperfect Bifurcation in Structures and Materials,9780387954097

Imperfect Bifurcation in Structures and Materials

by ;
ISBN13: 9780387954097
ISBN10: 0387954090
Format: Hardcover
Pub. Date: 2002-07-01
Publisher(s): Springer Verlag
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Summary

This book provides a modern investigation into the bifurcation phenomena of physical and structural problems. Systematic methods -- based on asymptotic, stochastic, and group-theoretic standpoints -- are used to examine experimental and computational data from numerous examples. Engineers will find this book, with its minimized mathematical formalism, to be a useful introduction to modern bifurcation theory. For mathematicians, static bifurcation theory for finite dimensional systems, as well as its implications for practical problems, is illuminated by the numerous examples.

Table of Contents

Preface v
List of Major Symbols
xv
Introduction to Bifurcation Behavior
1(32)
Introduction
1(1)
Simple Examples of Bifurcation Behavior
2(6)
One-Degree-of-Freedom System
2(3)
Two-Degree-of-Freedom System
5(3)
Overview of the Book
8(24)
Imperfection Sensitive Bifurcation Behavior
8(2)
Critical Initial Imperfection of Structural Systems
10(6)
Random Variation of Initial Imperfections
16(1)
Experimentally Observed Bifurcation Diagrams
17(2)
Bifurcation of Symmetric Systems
19(3)
Recursive Bifurcation and Mode Switching of Sands
22(4)
Echelon Modes on Uniform Materials
26(4)
Recursive Bifurcation of Steel Specimens
30(2)
Summary
32(1)
I Imperfect Behavior at Simple Critical Points 33(118)
Critical Points and Local Behavior
36(31)
Introduction
36(1)
General Framework
37(4)
Illustrative Example
41(4)
Liapunov-Schmidt Reduction
45(7)
Reduction Procedure
45(2)
Criticality Condition
47(2)
Direction of Bifurcated Paths
49(1)
Stability
50(1)
Expansion into Power Series
50(2)
Classification of Simple Critical Points
52(10)
Limit Point
54(2)
Transcritical Bifurcation Point
56(1)
Pitchfork Bifurcation Point
57(5)
Bifurcation Behavior of a Propped Cantilever
62(3)
Problems
65(1)
Summary
66(1)
Imperfection Sensitivity Laws
67(15)
Introduction
67(1)
Imperfection Sensitivity Laws
68(7)
Limit Point
69(1)
Transcritical Bifurcation Point
70(1)
Pitchfork Bifurcation Point
71(1)
Systematic Derivation
72(3)
Imperfection Sensitivity of Structures
75(6)
Propped Cantilever
75(1)
Truss Arches
76(5)
Problems
81(1)
Summary
81(1)
Critical Initial Imperfection (I)
82(19)
Introduction
82(1)
Illustrative Example
83(4)
Theory of Critical Initial Imperfection
87(4)
Formulation
87(2)
Derivation
89(2)
Imperfection with Multiple Categories
91(2)
Critical Initial Imperfection of Truss Structures
93(6)
Truss Arches
93(4)
Hexagonal Truss Dome
97(2)
Problems
99(1)
Summary
100(1)
Random Initial Imperfection (I)
101(21)
Introduction
101(1)
Probability Density Functions of Critical Loads
102(6)
Evaluation of Probability Density Functions
108(1)
Distribution of Minimum Values
108(3)
Scatter of Critical Loads of Structures
111(7)
Propped Cantilever
112(3)
Beam on a Nonlinear Foundation
115(3)
Appendix: Derivation of Scaling Factors
118(1)
Problems
119(2)
Summary
121(1)
Experimentally Observed Bifurcation Diagrams
122(29)
Introduction
122(3)
Illustrative Example
125(1)
Imperfection Sensitivity Laws
126(8)
The Koiter Two-Thirds Power Law
126(2)
Generalized Koiter Law
128(1)
Laws for Experimentally Observed Bifurcation Diagrams
129(5)
Recovering the Perfect System from Imperfect Systems
134(3)
Recovery from a Single Imperfect Path
135(1)
Recovery from a Series of Imperfect Paths
136(1)
Numerical and Experimental Examples
137(12)
Regular-Hexagonal Truss Dome
137(4)
Rectangular Plates
141(2)
Sand Specimens
143(6)
Problems
149(1)
Summary
149(2)
II Imperfect Bifurcation of Symmetric Systems 151(126)
Group-Theoretic Bifurcation Theory
155(27)
Introduction
155(1)
Bifurcation due to Reflection Symmetry
156(2)
Preliminaries on Group Representation
158(5)
Symmetry of Equations
163(12)
Group Equivariance of the Governing Equation
163(3)
Liapunov-Schmidt Reduction
166(6)
Equivariance of the Jacobian Matrix
172(3)
Symmetry of Solutions
175(3)
Example of a Symmetric System
178(3)
Problems
181(1)
Summary
181(1)
Bifurcation Behavior of Dn-Equivariant Systems
182(51)
Introduction
182(1)
Dihedral and Cyclic Groups
183(5)
Definition of Groups
183(3)
Irreducible Representations
186(2)
Perfect Bifurcation Behavior
188(5)
Symmetry of Solutions
188(4)
Recursive Bifurcation
192(1)
Bifurcation of Dome Structures
193(5)
D3-Symmetric Regular-Triangular Dome
194(3)
C6-Symmetric Schwedler Dome
197(1)
Bifurcation Equations for a Double Critical Point
198(5)
Local Analysis near a Double Critical Point: Perfect System
203(9)
Bifurcated Branches
203(6)
Stability
209(3)
Local Analysis near a Double Critical Point: Imperfect, System
212(12)
Solution Curves
212(7)
Imperfection Sensitivity Laws
219(5)
Experimentally Observed Bifurcation Diagrams
224(4)
Simple Bifurcation Point
224(1)
Double Bifurcation Point
225(3)
Appendix: Alternative Stability Analysis for Bifurcated Branches
228(3)
Problems
231(1)
Summary
232(1)
Critical Initial Imperfection (II)
233(17)
Introduction
233(1)
Theory of Critical Initial Imperfection
234(8)
Formulation
234(3)
Exploiting Group Equivariance
237(3)
Simple Critical Points
240(1)
Double Critical Points
241(1)
Resonance of Symmetry
242(1)
Critical Imperfection of Symmetric Truss Structures
243(6)
Truss Tents
243(3)
Regular-Hexagonal Truss Dome
246(3)
Problems
249(1)
Summary
249(1)
Random Initial Imperfection (II)
250(16)
Introduction
250(1)
Probability Density Functions of Critical Loads
251(7)
Formulation
251(1)
Derivation of Probability Density Functions
252(5)
Semiempirical Evaluation
257(1)
Distribution of Minimum Values
258(1)
Scatter of Critical Loads of Structures and Materials
259(6)
Regular-Polygonal Truss Structures
259(3)
Cylindrical Material Specimens
262(3)
Problems
265(1)
Summary
265(1)
Description of Bifurcation Behaviors
266(11)
Introduction
266(1)
Perfect Bifurcation Behavior of Truss Domes
267(1)
Imperfect Behavior of Structures and Materials
268(7)
Regular-Pentagonal Truss Dome
269(3)
Sand Specimens
272(3)
Problems
275(1)
Summary
275(2)
III Modeling of Bifurcation Phenomena 277(115)
Bifurcation of Cylindrical Sand Specimens
281(27)
Introduction
281(4)
Groups for Spatial Symmetry
285(4)
Experiments on Sand Specimens
289(9)
Recursive Bifurcation Behavior
289(8)
Mode Switching Behavior
297(1)
Appendix: Derivation of Bifurcation Rules
298(9)
Bifurcation of D∞h- and Dnh-Equivariant Systems
299(6)
Bifurcation of a Dnd-Equivariant System
305(2)
Problems
307(1)
Summary
307(1)
Echelon-Mode Formation
308(50)
Introduction
308(5)
Underlying Symmetries
313(2)
Subgroups Representing Patterns of Interest
315(5)
Recursive Bifurcation Leading to Echelon Modes
320(2)
Experiment on a Soil Specimen
322(3)
Rectangular Plate with Periodic Boundaries
325(6)
Underlying Symmetry
325(2)
Numerical Analysis
327(4)
Image Simulations
331(9)
Image Simulation Procedure
331(3)
Kaolin
334(3)
Steel
337(3)
Appendix: Derivation of Bifurcation Rules
340(16)
Bifurcation of a C∞v x C∞v-Equivariant System
340(8)
Bifurcation of an OB±nn-Equivariant System
348(5)
Bifurcation of a D∞∞-Equivariant System
353(2)
Symmetry of Fourier Terms
355(1)
Problems
356(1)
Summary
357(1)
Bifurcation of Steel Specimens
358(20)
Introduction
358(1)
Symmetry of a Rectangular Parallelepiped Domain
359(2)
Recursive Bifurcation Rule
361(1)
Experimental Study
362(10)
Effect of Cross-Sectional Shape
364(5)
Recursive Bifurcation
369(3)
Computational Study
372(3)
Analysis Conditions
372(1)
Bifurcation Analyses
372(3)
Appendix: Derivation of Bifurcation Rules
375(1)
Problems
376(1)
Summary
376(2)
Miscellaneous Aspects of Bifurcation Phenomena
378(14)
Introduction
378(1)
Clustered Bifurcation Point
379(1)
Size Effect
380(9)
Viewpoint of Fracture Mechanics
380(1)
Viewpoint of Imperfection Sensitivity
381(2)
Viewpoint of Hilltop Bifurcation
383(2)
Viewpoint of Mode Switching
385(4)
Explosive Bifurcation
389(2)
Problems
391(1)
Summary
391(1)
References 392(13)
Index 405

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