Fundamentals of Structural Mechanics

Front Cover
Springer Science & Business Media, Nov 12, 2004 - Technology & Engineering - 480 pages
The last few decades have witnessed a dramatic increase in the application of numerical computation to problems in solid and structural mechanics. The burgeoning of computational mechanics opened a pedagogical gap between traditional courses in elementary strength of materials and the finite element method that classical courses on advanced strength of materials and elasticity do not adequately fill. In the past, our ability to formulate theory exceeded our ability to compute. In those days, solid mechanics was for virtuosos. With the advent of the finite element method, our ability to compute has surpassed our ability to formulate theory. As a result, continuum mechanics is no longer the province of the specialist. What an engineer needs to know about mechanics has been forever changed by our capacity to compute. This book attempts to capitalize on the pedagogi cal opportunities implicit in this shift of perspective. It now seems more ap propriate to focus on fundamental principles and formulations than on classical solution techniques.
 

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Contents

Vectors and Tensors
1
The Geometry of Threedimensional Space
2
Vectors
3
Tensors
11
Vector and Tensor Calculus
33
Integral Theorems
45
Additional Reading
48
Problems
49
Problems
234
The Linear Theory of Beams
241
Equations of Equilibrium
243
The Kinematic Hypothesis
249
Constitutive Relations for Stress Resultants
252
Boundary Conditions
256
The Limitations of Beam Theory
257
The Principle of Virtual Work for Beams
262

The Geometry of Deformation
57
Uniaxial Stretch and Strain
58
The Deformation Map
62
The Stretch of a Curve
65
The Deformation Gradient
67
Strain in Threedimensional Bodies
68
Examples
69
Characterization of Shearing Deformation
74
The Physical Significance of the Components of C
77
Strain in Terms of Displacement
78
Principal Stretches of the Deformation
79
Change of Volume and Area
84
Timedependent motion
91
Additional Reading
93
Problems
94
The Transmission of Force
103
Normal and Shearing Components of the Traction
109
Principal Values of the Stress Tensor
110
Differential Equations of Equilibrium
112
Examples
115
Alternative Representations of Stress
118
Additional Reading
124
Problems
125
Elastic Constitutive Theory
131
Isotropy
138
Definitions of Elastic Moduli
141
Elastic Constitutive Equations for Large Strains
145
Limits to Elasticity
148
Additional Reading
150
Problems
151
Boundary Value Problems in Elasticity
159
Boundary Value Problems of Linear Elasticity
160
A Little Boundary Value Problem
165
Work and Virtual Work
167
The Principle of Virtual Work for the Little Boundary Value Problem
169
Essential and Natural Boundary Conditions
181
The Principle of Virtual Work for 3D Linear Solids
182
Finite Deformation Version of the Principle of Virtual WorkReference Configuration
186
Closure
188
Additional Reading
189
Problems
190
The Ritz Method of Approximation
193
The Ritz Approximation for the Little Boundary Value Problem
194
Orthogonal Ritz Functions
207
The Finite Element Approximation
216
The Ritz Method for Two and Threedimensional Problems
226
Additional Reading
233
The Planar Beam
266
The BernoulliEuler Beam
273
Structural Analysis
278
Additional Reading
282
Problems
283
The Linear Theory of Plates
293
Equations of Equilibrium
295
The Kinematic Hypothesis
300
Constitutive Equations for Resultants
304
Boundary Conditions
308
The Limitations of Plate Theory
310
The Principle of Virtual Work for Plates
311
The KirchhoffLove Plate Equations
314
Additional Reading
323
Problems
324
Energy Principles and Static Stability
327
Virtual Work and Energy Functionals
330
Energy Principles
341
Static Stability and the Energy Criterion
345
Additional Reading
352
Problems
353
Fundamental Concepts in Static Stability
359
Bifurcation of Geometrically Perfect Systems
361
The Effect of Imperfections
369
The Role of Linearized Buckling Analysis
375
Systems with Multiple Degrees of Freedom
378
Additional Reading
384
Problems
385
The Planar Buckling of Beams
389
Derivation of the Nonlinear Planar Beam Theory
390
Eulers Elastica
397
The General Linearized Buckling Theory
408
Ritz and the Linearized Eigenvalue Problem
415
Additional Reading
421
Problems
423
Numerical Computation for Nonlinear Problems
431
Newtons Method
433
Tracing the Equilibrium Path of a Discrete System
438
The Program NEWTON
444
Newtons Method and Virtual Work
446
The Program ELASTICA
452
The Fully Nonlinear Planar Beam
454
The Program NONLINEARBEAM
462
Summary
469
Problems
470
Index
473
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