One-Dimensional Finite Elements

by Ochsner, Andreas; Merkel, Markus

• ISBN13: 9783642317965
• ISBN10: 3642317960
• Format: Hardcover
• Copyright: 2012-10-06
• Publisher: Springer Verlag

Summary

This textbook presents finite element methods using exclusively one-dimensional elements. The aim is to present the complex methodology in an easily understandable but mathematically correct fashion. The approach of one-dimensional elements enables the reader to focus on the understanding of the principles of basic and advanced mechanical problems. The reader easily understands the assumptions and limitations of mechanical modeling as well as the underlying physics without struggling with complex mathematics. But although the description is easy it remains scientifically correct. The approach using only one-dimensional elements covers not only standard problems but allows also for advanced topics like plasticity or the mechanics of composite materials. Many examples illustrate the concepts and problems at the end of every chapter help to familiarize with the topics.

Table of Contents

Introduction	p. 1
The Finite Element Method at a Glance	p. 1
Foundations of Modeling	p. 2
Reference	p. 4
Motivation for the Finite Element Method	p. 5
From the Engineering Perspective Derived Methods	p. 5
The Matrix Stiffness Method	p. 6
Transition to the Continuum	p. 11
Integral Principles	p. 16
Weighted Residual Method	p. 18
Procedure on Basis of the Inner Product	p. 19
Procedure on Basis of the Weak Formulation	p. 23
Procedure on Basis of the Inverse Formulation	p. 25
Sample Problems	p. 25
References	p. 32
Bar Element	p. 33
Basic Description of the Bar Element	p. 33
The Finite Element Tension Bar	p. 35
Derivation Through Potential	p. 39
Derivation Through Castigliano's Theorem	p. 40
Derivation Through the Weighted Residual Method	p. 41
Sample Problems and Supplementary Problems	p. 45
Sample Problems	p. 45
Supplementary Problems	p. 49
References	p. 50
Torsion Bar	p. 51
Basic Description of the Torsion Bar	p. 51
The Finite Element Torsion Bar	p. 54
References	p. 56
Bending Element	p. 57
Introductory Remarks	p. 57
Basic Description of the Beam	p. 59
Kinematics	p. 59
Equilibrium	p. 64
Constitutive Equation	p. 65
Differential Equation of the Bending Line	p. 69
Analytical Solutions	p. 69
The Finite Element Method of Plane Bending Beams	p. 74
Derivation Through Potential	p. 79
Weighted Residual Method	p. 82
Comments on the Derivation of the Shape Functions	p. 85
The Finite Element Bending Beam with two Deformation Planes	p. 87
Transformation Within the Plane	p. 89
Transformation Within the Space	p. 92
Determination of Equivalent Nodal Loads	p. 95
Sample Problems and Supplementary Problems	p. 100
Sample Problems	p. 100
Supplementary Problems	p. 108
References	p. 110
General ID Element	p. 111
Superposition to a General ID Element	p. 111
Sample 1: Bar Under Tension and Torsion	p. 113
Sample 2: Beam in the Plane with Tension Part	p. 114
Coordinate Transformation	p. 115
Plane Structures	p. 117
General Three-Dimensional Structures	p. 118
Numerical Integration of a Finite Element	p. 121
Shape Function	p. 123
Unit Domain	p. 126
Supplementary Problems	p. 127
References	p. 127
Plane and Spatial Frame Structures	p. 129
Assembly of the Total Stiffness Relation	p. 129
Solving of the System Equation	p. 133
Solution Evaluation	p. 134
Examples in the Plane	p. 135
Plane Structure with Two Bars	p. 135
Plane Structure: Beam and Bar	p. 139
Examples in the Three-Dimensional Space	p. 145
Supplementary Problems	p. 152
References	p. 153
Beam with Shear Contribution	p. 155
Introductory Remarks	p. 155
Basic Description of the Beam with Shear Contribution	p. 160
Kinematics	p. 160
Equilibrium	p. 162
Constitutive Equation	p. 162
Differential Equation of the Bending Line	p. 163
Analytical Solutions	p. 164
The Finite Element of Plane Bending Beams with Shear Contribution	p. 168
Derivation Through Potential	p. 169
Derivation Through the Castigliano's Theorem	p. 174
Derivation Through the Weighted Residual Method	p. 175
Linear Shape Functions for the Deflection and Displacement Field	p. 179
Higher-Order Shape Functions for the Beam with Shear Contribution	p. 191
Sample Problems and Supplementary Problems	p. 196
Sample Problems	p. 196
Supplementary Problems	p. 205
References	p. 207
Beams of Composite Materials	p. 209
Composite Materials	p. 209
Anisotropic Material Behavior	p. 210
Special Symmetries	p. 212
Engineering Constants	p. 214
Transformation Behavior	p. 216
Plane Stress States	p. 218
Introduction to the Micromechanics of the Fiber Composite Materials	p. 222
Multilayer Composite	p. 224
One Layer in the Composite	p. 224
The Multilayer Composite	p. 226
A Finite Element Formulation	p. 227
The Composite Bar	p. 228
The Composite Beam	p. 229
Sample Problems and Supplementary Problems	p. 230
References	p. 231
Nonlinear Elasticity	p. 233
Introductory Remarks	p. 233
Element Stiffness Matrix for Strain Dependent Elasticity	p. 235
Solving of the Nonlinear System of Equations	p. 241
Direct Iteration	p. 241
Complete Newton-Raphson Method	p. 244
Modified Newton-Raphson Method	p. 257
Convergence Criteria	p. 260
Sample Problems and Supplementary Problems	p. 261
Sample Problems	p. 261
Supplementary Problems	p. 269
References	p. 271
Plasticity	p. 273
Continuum Mechanics Basics	p. 273
Yield Condition	p. 274
Flow Rule	p. 276
Hardening Law	p. 277
Elasto-Plastic Material Modulus	p. 278
Integration of the Material Equations	p. 279
Derivation of the Fully Implicit Backward-Euler Algorithm	p. 285
Mathematical Derivation	p. 285
Interpretation as Convex Optimization Problem	p. 290
Derivation of the Semi-Implicit Backward-Euler Algorithm	p. 294
Sample Problems and Supplementary Problems	p. 296
Sample Problems	p. 296
Supplementary Problems	p. 309
References	p. 310
Stability (Buckling)	p. 313
Stability for Bar/Beam	p. 313
Large Deformations	p. 315
Stiffness Matrices in Large Deformations	p. 317
Bar with Large Deformations	p. 318
Beams with Large Deformations	p. 319
Examples of Buckling: The Four Euler's Buckling Loads	p. 321
Analytical Solutions for Euler's Buckling Loads	p. 322
The Finite Element Method	p. 323
Supplementary Problems	p. 324
References	p. 324
Dynamics	p. 327
Principles of Linear Dynamics	p. 327
The Mass Matrices	p. 330
Modal Analysis	p. 330
Forced Oscillation, Periodic Load	p. 332
Direct Methods of Integration, Transient Analysis	p. 333
Integration According to Newmark	p. 334
Central Difference Method	p. 335
Examples	p. 337
Provision of Mass and Stiffness Matrices	p. 337
Axial Vibration of a Bar	p. 341
Supplementary Problems	p. 357
References	p. 358
Appendix A	p. 359
Short Solutions of the Exercises	p. 375
Index	p. 395
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