Deformation and Vibration by FEA
1 Introduction 1
Page References 3
How to use this Book 3
Preliminaries
2 Graphical Facilities 5
Demonstration, Student, or Professional Edition? 5
Downloading FlexPDE 6
The FlexPDE Editor 6
Plotting Functions of One Variable 7
Running the Problem Descriptor File 8
Plotting Functions of Two Variables 10
Running FlexPDE from the Explorer 14
Help and Manual 15
3 "Curly" Velocity Fields 17
Liquid, Rotating as a Disk 17
Non-Constant w 19
More General Velocity Fields 24
4 Fields of Gravitation 26
Earth and Moon as Point Masses 26
Planets of Finite Size 28
Divergence of g 33
5 Laplace and Poisson Equations 35
Known Values on Boundaries 35
Values and Derivatives on Boundaries 37
Derivative Boundary Conditions Only 38
Solution Over a Quarter-Circle 39
The Poisson Equation 40
Elastic Deformation
6 Elementary Elasticity in (x,y) Space 43
Plane Stress 44
Boundary Conditions 46
Steel Plate Under Compressive Strain 48
Strained Plate, No Sliding 52
Plate Subjected to Pressure at One End 54
Plate with Tension Applied at both Ends 55
Plate Subjected to Shear Forces 57
7 Straight Beams 59
Preparing for the Student Edition 59
Beam with Uniform Load 59
Comparison with an Analytic Solution 62
Further Analysis of Stress 63
Gravity Bending 67
Forces and Moments 68
Beam with Bending Moments at Ends 70
Elastic Energy 73
Plane Strain 75
8 Hooks, Rings and Tubes 79
Semi-Circular Beam 79
Compound Semi-Circular Beam 81
Tube with Internal and External Pressure 83
Comparison with Exact Solution 86
Tube with Tangential Load 87
Ring Strained between Two Edges 89
9 Elasticity in (r ,z) Space 94
Tube with Internal Pressure 96
Tube Pressurized over Part of its Length 99
Stress Concentration in a Shaft 101
10 Torsion 106
Twisted Circular Steel Shaft 107
Twisted Square Bar 109
General PDE for Torsion 111
Compound Bar 112
Hollow Bar 113
PDE for Warping 114
Warping of an Elliptical Shaft 117
11 Thermo-Elasticity 120
Uniformly Heated Semi-Circular Rod 121
Thermo-Elasticity in (x,y) Space 123
Steel Block, Strained by a Temperature Gradient 124
Bi-Metallic Block 126
Simultaneous PDEs for (u, v, dtemp) 129
PDEs for Axially Symmetric Problems 131
Axially Symmetric heating of a Cylinder 133
Compound Cylinder 136
12 Deformation in Three Dimensions 141
Two Point Charges of Different Signs 141
Non-Linear Set of Point Charges 144
Laplace Equation in 3D 146
Elasticity in Three Dimensions 148
Include-File for 3D Elasticity 149
Capturing Descriptor Errors 150
Plate under Pressure from Top 150
Bending of a Beam under Gravity 152
Beam under Asymmetric Load 154
Tube under Asymmetric Load 156
Torsion and Warping of a Hollow Bar 158
Vibration
13 Waves Along a Beam 162
Complex Solution 163
Forced Shear Vibration 164
Animated Shear Vibration 167
Driving Force 168
Wavelength 169
Longitudinal Vibration 170
Bending Vibration 171
14 Eigenstates in Two Dimensions 175
Frequency Scan 175
Finding Eigenstates 177
Plate with Free Boundaries 180
Animation of a Vibrating Plate 182
15 Eigenstates of a Membrane 184
Circular Membrane 185
Triangular Drumhead 187
Fancy Drumhead 189
16 Eigenstates in Three Dimensions 191
Vibration of a Rectangular Block 192
Vibration of a Tube 194
Vibrating Circular Plate 196
Plate Resting on Rim 198
Vibrating Sledgehammer 199
17 Waves in Fluids 203
Complex Solution 204
Plane Acoustic Wave 206
Resonance 208
Constricted Cavity 209
Cylindrical Resonator 211
Eigenstates 214
Appendix
Principles of Finite Element Analysis 217
Interpolation and Differentiation in 2D 217
Interpolating to Obtain a Solution 219
Solving for Node Values 220
Natural Boundary Conditions 223
Conclusion 224
References 225
Vocabulary