Finite Element Methods for Engineers

Finite Element Methods for Engineers
اسم المؤلف
Roger T. Fenner
التاريخ
5 سبتمبر 2021
المشاهدات
التقييم
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Finite Element Methods for Engineers
Roger T. Fenner
Department of Mechanical Engineering
Imperial Colleg eof Science, Technology and Medicine
London
Contents
Preface
Noration
Some Program Variable Names
1 Introduction and Structural Analysis
1 . 1 Computer Programming
1.2 Structural Analysis
1.3 Case Study: Bending of a Tapered Beam
2 Continuum Mechanics Problems
2.1 Continuum Mechanics Equations
2.2 Some Physical Problems
Classification of Partial Differential Equations
Methods for Solving Harmonic and Biharmonic Equations
3 Finite Element Analysis of Harmonic Problems
Derivation of the Element Stiffness Matrix
Assembly of the Overall Stiffness Matrix
Comparison with the Finite Difference Approach
3.4 Variational Formulation
3.5 Boundary Conditions
Solution of the Linear Equations
Convergence of Finite Element Methods
A Computer Program for Harmonic Problems
4 Finite Element Meshes
4.1 Choice of Mesh
4.2 Mesh Data in Numerical Form
72vi Contents
4.3 Generation of Mesh Data
4.4 Mesh Modification
5 Some Harmonic Problems
Case Study: Downstream Viscous Flow in a Rectangular
Case Study: Torsion of Prismatic Bars
Channel
6 Finite Element Analysis of Biharmonic Problems
6.3 Variational Formulation
6.5 Boundary Conditions
Derivation of the Element Stiffness Matrix
Assembly of the Overall Stiffness Matrix
Solution of the Linear Equations
A Computer Program for Problems of the Biharmonic
Plane Strain or Plane Stress Type
7 Some Biharmonic Problems
Case Study: Plane Strain Compression
Case Study: Stresses in Concentric Cylinders
Case Study: Stress Concentration near a Hole in a
Flat Plate
8 Further Applications
Axi-symmetric Problems
Higher-order Elements
Three-dimensional Problems
Biharmonic Problems Involving Incompressible
Plate and Shell Problems
Isoparametric Elements
Nonlinear Problems
A Summary of the Finite Element Approach
Concluding Remarks
Materials
Appendix A Gaussian Elimination
Appendix B The Gauss-Seidel Method
Bibliogaphy
x
accuracy
Airy’s stress function 27, 28
algebraic equations see simultaneous
alphanumeric information 3, 66,
analytical solution 1, 2, 8-9, 36,
13, 48, 71, 90, 97-8,
102-3, 137, 146, 153, 154
algebraic equations
122
89-90, 96, 99, 101, 135, 136, 137,
138-9, 146
argument of subprogram
array, dimensioning of and order of
aspect ratio 144-6
assembly of overall stiffness 6-7,
axi-symmetric problems 135-8,
12, 69, 163
storage 11
12,43-5, 66-8, 110-1 1, 122
148-51, 154
back substitution 60, 159, I63
backing stores 60, 154
banded matrix
bandwidth 56, 59, 72, 154
beam 4, 6, 8- 14
bending 4-14, 30, 32, 155-6
biharmonic equations 32, 33-6, 155
biharmonic operator 28
biharmonic problems 71, 72, 76
binary arithmetic 3
bits 3
body forces
13, 55-6, 60, 154
104-47, 148, 149-51, 154-5
17, 20, 22, 27, 28, 29,
31,104, 108, 110,111, 112, 118,
121, 122, 124, 125, 127, 137
113
bound for a solution 63, 97, 103,
boundary conditions 9, 2 1, 22-3,
25, 26, 28, 29, 30-1, 33, 34-5,36,
48, 51, 53-5, 61, 89, 91, 93-5,96,
98, 101, 115-18, 122, 123,125-7,
129, 133, 136, 139, 144, 148, 155,
158
derivative 25, 26, 29, 30-1,
34-5, 51, 54-5,.93-5, 96,
158
boundary shape
boundary-value problems 31-2, 33
36, 37, 71-2, 74,
86, 103, 139, 140, 152, 156, 158
card 2
Cartesian co-ordinates 15, 36, 38,
centrifugal forces 124
coding 3, 97, 130
coefficient of thermal expansion 19,
123, 133, 135, 144
comment statement 3
COMMON block
118, 121, 124, 127
compatibility
103, 138
65, 66, 68, 76,
18, 22, 23, 27, 29, 30,
36,40, 52,61-2, 106, 151
inter-element 40, 52, 61-2,
106, 113, 152-3, 154-5,156
complementary stresses 16, 23
computer
conduction 25, 32, 124
conforming (compatible) elements
constant strain triangles (CST) 37,
1, 2, 36, 55, 56, 59, 60,
71, 73, 158
62,63, 106, 112, 150, 152-3, 155
40, 48, 52, 92, 104, 113, 116, 134,
151, 153Index
constitutive equations 19-20, 22,
28, 29, 42,45, 61, 108-9,
149-50
continuity
continuum mechanics 2, 14, 15,-36,
convergence, finite element methods
Gauss-Seidel method 57-9,
19, 24, 26, 29, 30, 154
49,61
48, 52,61-2, 113
61, 69, 70, 73, 93, 95-6, 102,
114, 131, 144, 164-5
co-ordinate systems 15, 36, 103,
core store 3, 60
136, 138, 158
amount of 3, 12, 55,56, 57,
59-60, 97, 98, 113, 147,
154, 158
cost of computing
couple 23, 31, 101
2,97, 154, 158
data
125
deformation
155, 156
density
dependent variable 30-2, 36, 39,
2, 12, 66,68, 71-88, 90, 121,
15, 21, 106, 131, 135,
17, 19, 133, 137, 142
40, 103
choice of 30, 35, 36, 103,
104-5, 154-5
determinant 114, 160
diagonal dominance 57-9,60.114,164
diagonal of a matrix
differential equations 17, 20, 23,
12, 54, 55, 57,
163
26, 30-6, 37, 49, 51, 53, 63, 89,
91, 101, 104, 112, 148
diffusion problems 25-6, 32
dimension matrix
67, 106, 108, 122, 151
DIMENSION statement 1 1
direct equilibrium formulation 37,
48, 53, 89, 104, 112, 150, 153,
157
39, 40, 42, 53,66,
discretisation 33
displacement 1, 4, 5, 6, 7, 8, 9, 11,
12, 15, 16, 33, 36, 103, 104, 105,
106, 109, 112, 113, 116, 150, 151,
153, 154, 157
displacement vector 6, 7, 106, 110,
111, 114
distributed external force 8, 112,
116, 125-6, 133, 144
domain of a solution 3 1, 33, 35, 36,
37, 38, 42,49, 50, 51, 61.65, 71,
72, 86, 90, 93, 97, 98, 99, 102,
104, 105, 112, 123, 124, 125, 132,
133, 134, 135, 137, 138, 139, 140,
144, 148, 156, 158
downstream viscous flow 20-1, 32,
37, 39, 46, 52-3, 54, 61, 63,
89- 100
drag flow 90, 98- 100
eigenvalue problems 148
elastic property matrix
elastic solid
149
109, 122, 150
19, 49, 104, 108, 148,
linear 19, 148
nonlinear 156
element, area of
73, 104, 153
39, 65, 66, 68, 72,
aspect ratio of 144-6
centroid of 42, 108, 124,
133, 134, 137, 144, 146, 150,
152
distribution of 71-2, 81, 86,
97, 99, 131, 139-47, 157,
158
numbering of
shape of
4, 13, 38, 72,
75, 77-8, 81, 84, 104, 142
58-9, 72, 76, 87,
100, 102, 114, 131, 140-7,
157, 158
size of 71-2, 81, 87, 97, 99,
102, 131, 140-7, 157, 158
element as subregion
104, 149, 157
element force vector
element stiffness matrix
1, 3, 36, 37,
6, 7, 42, 109
6, 7, 11, 12,
37-43,47, 58-9,66,67, 104-10,
111, 114, 121, 122, 155
elimination see gaussian elimination
elliptic differential equation 31-2
energy equation
equilibrium equations
17- 18, 25, 6 1
6, 17, 22, 23,
24, 27, 28, 29, 30, 36, 37, 43, 45,
47-8, 53,61, 110-11, 112
equilibrium problems 2, 3 1, 148
EQUIVALENCE statement 11
error, convergence 96, 144, 164
data 73, 74
relative 164 ~
roundoff 3, 160
truncation 34, 35, 48, 62, 96,
97,105 .-Index 169
error in analytical solution 139
error in axi-symmetric analysis 150
execution, control of
execution time
externally applied forces
68, 70, 123
termination of 12, 67, 122
3, 162, see also time,
computing
7, 11, 12,
42,43, 66,67-8, 104, 110-1 1,
112, 115, 116, 117, 118, 125, 129,
148, 156
factor of safety 1
finite difference methods 15, 33-6,
45-8, 53-4, 59, 62, 89, 90, 97,
103, 139, 158
flexural rigidity 30
flow rate
fluid flow
148
fluid mechanics
forces on an element
FORMAT statements 3, 68
functional 49-53, 61, 63, 112
21, 25, 30, 31, 89, 90,
92, 95,97,98-9
1, 15, 17, 18, 30, 36, 37,
1, 2, 17, 18, 37
4, 6, 7, 40-2,
43, 107-8, 110
FORTRAN 2-3,9,11
Gauss-Seidel method 55, 56-61,
63, 67, 68-70, 72, 73, 98, 102,
113-14, 116, 118, 122, 131, 144,
146, 147, 154, 156, 164-5
subprograms for 63, 68-70,
gaussian elimination method 9, 12,
118, 127-8
55-6, 59-61, 72,98, 113, 118,
147, 154, 156, 159-63
subprogram for 9, 12, 161-3
generation of meshes
global co-ordinates
7 1, 74-86,
6, 38, 65, 68,
73, 74, 77, 81, 83, 87, 115, 131,
138, 141
90, 139, 140-2
graph plotting 73, 130
Green’s theorem 24, 50
grid 33, 36
hZ extrapolation 97-8, 102
harmonic equations 32-6, 46, 49
harmonic operator 20
harmonic problems
148, 149, 154
37-70, 7 1, 72,
89-103, 104, 111, 114, 116, 121,
heat transfer
heat transfer coefficient 25
higher-order elements 151-3, 156,
homogeneous materials 19, 24, 123,
hydrodynamic lubrication 32
hyperbolic differential equation 3 1
ideal fluid 24-5, 26, 32
ill-conditioned equations 160, 162
incompressibility 19, 24, 30, 104,
independent variables 30-2
inertia forces 17, 24
infinite series 33, 90
integration
1, 2, 17-18, 25, 148,
157
157
138, 158
109, 150, 154-5
9, 21, 24, 31, 49, 50, 52,
61, 92, 98, 102, 112, 113, 150,
153, 158
irrotational flow 25
isoparametric elements 156
isotropic materials 19, 24
Lam6 equations 136
laminarflow 20
Laplace’s equation 32
line printer 3, 68
linear algebraic equations see
simultaneous algebraic equations
linearity of algebraic equations 156,
159
linearity of differential equations
linearity of material properties 19,
local co-ordinates
local variable 69, 127
magnetic tape 2, 60
main program
matrix notation
matrix transposition
mesh
31,32
20,45, 156-7
4, 38, 45, 62, 72,
75, 78, 81, 84, 105, 155
9, 11, 12, 63-8, 86,
6, 7, 53, 159, 160
42, 108, 113
92, 118-23
37, 43, 44, 46, 47, 55, 56, 59,
63, 66, 67, 68, 71-88, 90, 97,
99-100, 102, 103, 104, 140-2,
156, 158
Circular 81-6, 102, 135, 137
irregular 47
square 74-80, 90, 99- 100,
133, 140
triangular 80-1, 103, 137Index
mesh data
mesh plotting 73, 130
modification of meshes
63, 66, 68, 7 1-88, 90,
118, 122, 125, 133, 139, 140-2
7 1, 74, 76,
86-8, 90, 99-100, 102, 103, 133,
139, 140-2
molecule, computing 3 5
moment 4, 5, 7, 40, 107
Navier-Stokes equations 17
newtonian fluid
node (nodal point)
19-20, 25, 42, 148
4, 5, 7, 11, 36,
37, 38, 54, 55, 59, 71, 72, 104,
152-3
numbering of 4, 13, 38, 71,
72, 74-5, 76-8, 80-1,
82-5, 104, 141-2
nonlinear problems 60, 148, 156-7,
non-newtonian fluid 32, 156
numerical methods
158
1, 33, 139, 158
one-dimensional problems 2, 4, 37,
overall stiffness matrix
155, 157
7, 11, 12,
43-5, 54, 55, 56, 57, 58, 59, 60,
66, 67, 72, 110-11, 114, 116, 118,
122, 154, 155, 156
144, 165
over-relaxation 69, 93, 95-6, 133,
paper tape 2
parabolic differential equation 3 1
permeability 26
photoelasticity 1
pin-jointed structures 3
pivotal coefficient 162
plane strain
plane stress
Poisson’s equation 32
Poisson’sratio 19, 123, 133, 135,
142, 154
polynomial 5, 51, 151, 153, 155,
157
porous medium 25
potential energy
precision of a stored number
pressure
pressure flow 90- 100
26-8, 29, 30, 32,
26, 28-9, 32, 104-30
104-30, 131-8, 149, 150, 154
131, 138-47, 149
49, 112, 113, 150
3
1, 17, 20, 24, 25, 26, 32,
42,43
pressure forces
pressure gradient
program
17, 24, 42, 43, 45,
20, 25, 29, 42,
4 7 , 4 8
45, 63, 67, 90, 98
71, 73, 89, 90, 118, 158
2, 36, 48, 53, 55, 56, 63,
biharmonic problems 71, 86,
harmonic problems 59, 63-70,
rigid-jointed structures 9- 13
programming 2-3, 14, 36, 61, 63,
propagation problems 31, 32, 148
quasi-harmonic equations 32
READ statement 2
recirculating viscous flow 29-30, 32,
35, 104, 154, 156
rectangularised matrix 56, 57, 59,
60, 66, 67, 98, 114, 118, 122
restraint conditions 7, 11, 12, 13,
54, 116, 117, 127, 144, 158
rigid-jointed structures
155
roundoff error 3. 160
104, 118-30, 131, 133, 151
71, 86, 89, 110
110, 141, 165
3, 4- 14, 54,
St Venant’s theory of torsion
scale factor 141-7
second moment of area
12, 13
self-flexibility
122, 127
self-stiffness
127
shape factor, channel flow
shape function
shear 5, 16, 17, 40, 54, 149
shear modulus 19, 103
simultaneous algebraic equations 2,
21
4, 8, 9, 11,
1 14, 1 16, 1 17, 1 18,
55, 66, 67, 114, 122,
90- 100
39, 42, 62, 63, 92,
105, 150, 151-5, 156, 157
9, 13, 33, 36, 53, 55-61, 63, 66,
68, 72, 90, 97, 110, 113-14, 118,
146-7, 156, 158, 159-65
162
singular set of equations
slip 21, 29, 132, 134-5
slow flows 17
solid medium
sparse matrix
1 18, 160,
1, 15, 17, 19, 30, 36,
7, 12, 44, 55, 56, 59,
37, 49, 104, 108, 148, 149
63Index 171
specific heat 18
statement numbering 2-3
storage requirement
strain
12, 55, 56, 57,
59-60, 97, 98, 113, 147, 154, 158
4, 15-17, 18, 19, 20, 23,
105, 107, 129, 149
element 129
volumetric 19
strain rate
156
stream function
154, 157
stress
149
15- 17, 18, 19, 20, 42,
24-5, 26, 29, 30,
1, 15-16, 19, 33, 36, 54, 107,
element 129, 134, 146, 158
nodal point 130, 146, 158
stress analysis 1 , 17, 18, 148
stress concentration 131, 138-47
stress function 23, 24, 27, 30, 49,
structural analysis 1, 2, 3-14, 37,
submatrix 110, 111, 114, 162
subprogram
101, 102, 103, 112, 157
49, 54, 55, 60, 72, 104
3, 9, 12, 63, 7 1, 76
boundary conditions 9 1, 123,
Gauss -Seidel method 63,
gaussian elimination 9, 12,
mesh data 72-3, 76-7,
mesh data output 68, 125
mesh modification 9 1,
output of results 92-3,
125-7
68-70, 123, 127-8
161-3
78-80, 81-2, 85-6
142-4
129-30
subregions of continua
subscripts 11, 15-16
symmetry of a matrix
54, 56, 110, 111
symmetry of problems
1, 36, 37,
51, 104, 157
6, 43, 45,
90, 93, 98,
116, 131, 136, 137, 139, 148-51
Taylor series 34, 62
temperature 18, 19, 25, 27, 28, 61,
104, 108-10, 118, 121, 122, 124,
125, 131, 133, 135, 157
testing of programs
thermal conductivity 18
thermal convection
thermalforces 110, 111, 118, 121,
122, 127
thermal strain
121, 122, 135, 149
three-dimensional problems 2, 15,
time, computing
146, 154, 162
time-dependent problems 17, 148
tolerance, convergence 69, 93,
torsion
triangular elements
3, 89, 131, 134,
158
1, 17- 18, 25
19, 31, 104, 109,
30, 37,49, 148, 150, 153-4, 157
3, 57, 59-60, 97,
95-6, 133, 144, 164, 165
3, 4, 21-4, 31, 32, 100-3,
37, 47, 71, 74,
148, 151
76, 80, 90, 99, 104, 132, 140, 149,
152, 153, 154, 155, 156
tridiagonal matrix 55-6
truncation error
two-dimensional problems 2, 15,
34, 35, 48, 62, 96,
97, 105
24, 26, 29, 33, 37,49, 63, 71, 104,
108, 112, 118, 148, 151, 154, 155,
157
variable names 3, 11, 65-6, 69-70,
76, 78, 85, 118-22, 123, 124,
125-6, 127, 129-30, 142
variational formulation 37, 40
48-53, 54,61-2,63, 112-13,
150-1, 153, 157
velocity 1 , 15, 16, 18, 20, 21, 25,
33, 39, 40, 52, 89, 103, 151,
154-5, 157
viscoelastic materials 19
viscosity
viscous forces
vorticity 36
20, 32, 45, 63, 156, 157
17, 24, 25, 40, 43,
46-7
word 3
WRITE statement 2
Young’s modulus

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