Structural Analysis with Finite Elements

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Springer Science & Business Media, 2004 - Mathematics - 484 pages
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Structural Analysis with Finite Elements develops the foundations and applications of the finite element method in structural analysis in a language which is familiar to structural engineers. At the same time, it uncovers the structural mechanics behind the finite element method. This innovative text explores and explains issues such as: why finite element results are "wrong", why support reactions are relatively accurate, why stresses at midpoints are more reliable, why averaging the stresses sometimes may not help or why the equilibrium conditions are violated. An additional chapter treats the boundary element method and related software is available at www.winfem.de. Structural Analysis with Finite Elements provides a new foundation for the finite element method that enables structural engineers to address key questions that arise in computer modelling of structures with finite elements.
 

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Contents

1 What are finite elements?
1
13 Potential energy
5
14 Projection
8
15 The error of an FE solution
12
16 A beautiful idea that does not work
15
17 Set theory
16
18 Principle of virtual displacements
23
19 Taut rope
28
44 Plane elements
252
45 The patch test
258
46 Volume forces
260
47 Supports
261
48 Nodal stresses and element stresses
271
49 Truss models
277
410 Twobay wall
278
411 Multistory shear wall
280

110 Least squares
33
111 Distance inside distance outside
36
112 Scalar product and weak solution
39
113 Equivalent nodal forces
41
114 Concentrated forces
43
115 Greens functions
50
116 Practical consequences
52
117 Why finite element results are wrong
55
118 Proof
62
119 Influence functions
67
120 Accuracy
75
121 Why resultant stresses are more accurate
80
122 Why stresses at midpoints are more accurate
84
123 Why stresses jump
93
124 Why finite element support reactions are relatively accurate
94
125 Gauss points
99
126 Local errors and pollution
105
127 Adaptive methods
112
128 St Venants principle
127
129 Singularities
129
130 Actio reactio?
132
131 The output
135
132 Support conditions
137
133 Equilibrium
138
134 Changes in the temperature and displacement of supports
141
135 Stability problems
144
136 Interpolation
148
137 Polynomials
151
138 Infinite energy
158
139 Conforming and nonconforming shape functions
160
140 Partition of unity
161
141 Elements
163
142 Stiffness matrices
164
143 Coupling degrees of freedom
167
144 Numerical details
170
145 Warning
178
2 What are boundary elements?
181
21 Influence functions or Bettis theorem
182
22 Structural analysis with boundary elements
189
23 Comparison finite elementsboundary elements
204
3 Frames
211
32 The FE approach
212
33 Finite elements and the slope deflection method
227
34 Stiffness matrices
231
35 Approximations for stiffness matrices
237
4 Plane problems
241
42 Strains and stresses
248
43 Shape functions
251
412 Shear wall with suspended load
287
413 Shear wall and horizontal load
289
414 Equilibrium of resultant forces
292
415 Adaptive mesh refinement
296
416 Plane problems in soil mechanics
300
417 Incompressible material
306
418 Mixed methods
307
419 Influence functions
312
420 Error analysis
313
421 Nonlinear problems
314
5 Slabs
325
51 Kirchhoff plates
326
52 The displacement model
331
53 Elements
332
54 Hybrid elements
335
55 Singularities of a Kirchhoff plate
339
56 ReissnerMindlin plates
341
57 Singularities of a ReissnerMindlin plate
346
58 ReissnerMindlin elements
349
59 Supports
351
510 Columns
353
511 Shear forces
361
512 Variable thickness
362
513 Beam models
364
514 Wheel loads
369
516 T beams
372
517 Foundation slabs
378
518 Direct design method
384
519 Point supports
386
6 Shells
391
62 Shells of revolution
394
63 Volume elements and degenerate shell elements
396
64 Circular arches
397
65 Flat elements
399
66 Membranes
404
7 Theoretical details
409
72 Greens identities Integration by parts
414
73 Greens functions
419
74 Generalized Greens functions
421
75 Nonlinear problems
428
76 The derivation of influence functions
432
77 Shifted Greens functions
437
78 The dual space
447
79 Some concepts of error analysis Asymptotic error estimates
453
710 Important equations and inequalities
461
References
471
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