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 |  موضوع: كتاب Essentials of the Finite Element Method - For Mechanical and Structural Engineers السبت 30 أبريل 2022, 5:43 am | |
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أحضرت لكم كتاب
Essentials of the Finite Element Method - For Mechanical and Structural Engineers
Dimitrios G. Pavlou, PhD
Department of Mechanical and Structural Engineering and Materials Science,
University of Stavanger, Norway
و المحتوى كما يلي :
Contents
Preface xiii
Acknowledgments . xv
CHAPTER 1 An Overview of the Finite Element Method 1
1.1 What Are Finite Elements? 1
1.2 Why Finite Element Method Is Very Popular? . 1
1.3 Main Advantages of Finite Element Method 1
1.4 Main Disadvantages of Finite Element Method 1
1.5 What Is Structural Matrix? . 2
1.5.1 Stiffness Matrix 2
1.5.2 Transfer Matrix . 3
1.6 What Are the Steps to be Followed for Finite Element Method Analysis
of Structure? . 3
1.6.1 Step 1. Discretize or Model the Structure . 4
1.6.2 Step 2. Define the Element Properties . 4
1.6.3 Step 3. Assemble the Element Structural Matrices . 4
1.6.4 Step 4. Apply the Loads . 4
1.6.5 Step 5. Define Boundary Conditions . 4
1.6.6 Step 6. Solve the System of Linear Algebraic Equations . 4
1.6.7 Step 7. Calculate Stresses . 4
1.7 What About the Available Software Packages? 4
1.8 Physical Principles in the Finite Element Method 5
1.9 From the Element Equation to the Structure Equation . 7
1.10 Computer-Aided Learning of the Finite Element Method 7
1.10.1 Introduction to CALFEM . 7
1.10.2 Spring elements 10
1.10.3 Bar Elements for Two-Dimensional Analysis . 11
1.10.4 Bar Elements for Three-Dimensional Analysis . 12
1.10.5 Beam Elements for Two-Dimensional Analysis 13
1.10.6 Beam Elements for Three-Dimensional Analysis 14
1.10.7 System Functions 15
1.10.8 Statement Functions . 15
1.10.9 Graphic Functions . 16
1.10.10 Working Environment in ANSYS 16
References 17
viiCHAPTER 2 Mathematical Background . 19
2.1 Vectors 19
2.1.1 Definition of Vector . 19
2.1.2 Scalar Product . 19
2.1.3 Vector Product 21
2.1.4 Rotation of Coordinate System 22
2.1.5 The Vector Differential Operator (Gradient) . 23
2.1.6 Green’s Theorem 23
2.2 Coordinate Systems 24
2.2.1 Rectangular (or Cartesian) Coordinate System 24
2.2.2 Cylindrical Coordinate System 25
2.2.3 Spherical Coordinate System . 25
2.2.4 Component Transformation 26
2.2.5 The Vector Differential Operator (Gradient) in Cylindrical and
Spherical Coordinates . 28
2.3 Elements of Matrix Algebra . 28
2.3.1 Basic Definitions 28
2.3.2 Basic Operations . 29
2.4 Variational Formulation of Elasticity Problems 34
2.4.1 Definition of the Variation of a Function 34
2.4.2 Properties of Variations 35
2.4.3 Derivation of the Functional from the Boundary Value Problem . 35
References 40
CHAPTER 3 Linear Spring Elements . 41
3.1 The Element Equation 41
3.1.1 The Mechanical Behavior of the Material . 41
3.1.2 The Principle of Direct Equilibrium 42
3.2 The Stiffness Matrix of a System of Springs 43
3.2.1 Derivation of Element Matrices . 43
3.2.2 Expansion of Element Equations to the Degrees of Freedom of the
Structure 44
3.2.3 Assembly of Element Equations 44
3.2.4 Derivation of the Field Values . 44
References 55
CHAPTER 4 Bar Elements and Hydraulic Networks . 57
4.1 Displacement Interpolation Functions . 57
4.1.1 Functional Form of Displacement Distribution . 57
4.1.2 Derivation of the Element Equation 59
viii Contents4.2 Alternative Procedure Based On the Principle of Direct Equilibrium 60
4.2.1 The Mechanical Behavior of the Material . 60
4.2.2 The Principle of Direct Equilibrium 61
4.3 Finite Element Method Modeling of a System of Bars 61
4.3.1 Derivation of Element Matrices . 62
4.3.2 Expansion of Element Equations to the Degrees of Freedom of the
Structure 62
4.3.3 Assembly of Element Equations 63
4.3.4 Derivation of the Field Values . 63
4.4 Finite Elements Method Modeling of a Piping Network 67
References 79
CHAPTER 5 Trusses 81
5.1 The Element Equation for Plane Truss Members . 81
5.2 The Element Equation for 3D Trusses 83
5.3 Calculation of the Bar’s Axial Forces (Internal Forces) . 85
References 133
CHAPTER 6 Beams 135
6.1 Element Equation of a Two-Dimensional Beam Subjected to Nodal Forces 135
6.1.1 The Displacement Function 135
6.1.2 The Element Stiffness Matrix 137
6.2 Two-Dimensional Element Equation of a Beam Subjected to a
Uniform Loading 150
6.3 Two-Dimensional Element Equation of a Beam Subjected to an
Arbitrary Varying Loading 153
6.4 Two-Dimensional Element Equation of a Beam on Elastic Foundation
Subjected to Uniform Loading . 176
6.5 Engineering Applications of the Element Equation of the Beam on
Elastic Foundation 181
6.5.1 Beam Supported on Equispaced Elastic Springs . 181
6.5.2 Cylindrical Shells Under Axisymmetric Loading 181
6.6 Element Equation for a Beam Subjected to Torsion . 192
6.6.1 The Mechanical Behavior of the Material . 192
6.6.2 The Principle of Direct Equilibrium 193
6.7 Two-Dimensional Element Equation For a Beam Subjected To Nodal
Axial Forces, Shear Forces, Bending Moments, and Torsional
Moments . 194
Contents ix6.8 Three-Dimensional Element Equation for a Beam Subjected to Nodal Axial
Forces, Shear Forces, Bending Moments, and Torsional Moments . 196
References 212
CHAPTER 7 Frames . 213
7.1 Framed Structures . 213
7.2 Two-Dimensional Frame Element Equation Subjected to Nodal Forces . 213
7.3 Two-Dimensional Frame Element Equation Subjected to Arbitrary
Varying Loading . 217
7.4 Three-Dimensional Beam Element Equation Subjected to Nodal Forces 230
7.5 Distribution of Bending Moments, Shear Forces, Axial Forces, and Torsional
Moments of Each Element . 234
References 278
CHAPTER 8 The Principle of Minimum Potential Energy for
One-Dimensional Elements . 279
8.1 The Basic Concept . 279
8.2 Application of the MPE Principle on Systems of Spring Elements . 280
8.3 Application of the MPE Principle on Systems of Bar Elements 281
8.4 Application of the MPE Principle on Trusses . 284
8.5 Application of the MPE Principle on Beams 284
References 288
CHAPTER 9 From “Isotropic” to “Orthotropic” Plane Elements: Elasticity
Equations for Two-Dimensional Solids . 289
9.1 The Generalized Hooke’s Law 289
9.1.1 Effects of Free Thermal Strains . 292
9.1.2 Effects of Free Moisture Strains 293
9.1.3 Plane Stress Constitutive Relations 295
9.2 From “Isotropic” to “Orthotropic” Plane Elements . 296
9.2.1 Coordinate Transformation of Stress and Strain Components
for Orthotropic Two-Dimensional Elements . 298
9.3 Hooke’s Law of an Orthotropic Two-Dimensional Element, with
Respect to the Global Coordinate System . 299
9.4 Transformation of Engineering Properties . 300
9.4.1 Elastic Properties of an Orthotropic Two-Dimensional Element
in the Global Coordinate System . 300
9.4.2 Free Thermal and Free Moisture Strains in Global Coordinate System . 303
9.5 Elasticity Equations for Isotropic Solids . 305
x Contents9.5.1 Generalized Hooke’s Law for Isotropic Solids 305
9.5.2 Correlation of Strains with Displacements 307
9.5.3 Correlation of Stresses with Displacements 307
9.5.4 Differential Equations of Equilibrium . 308
9.5.5 Differential Equations in Terms of Displacements . 308
9.5.6 The Total Potential Energy 308
References 309
CHAPTER 10 The Principle of Minimum Potential Energy for Two-Dimensional
and Three-Dimensional Elements 311
10.1 Interpolation and Shape Functions . 311
10.1.1 Linear Triangular Elements (or CST Elements) 316
10.1.2 Quadratic Triangular Elements (or LST Elements) . 318
10.1.3 Bilinear Rectangular Elements (or Q4 Elements) . 321
10.1.4 Tetrahedral Solid Elements 322
10.1.5 Eight-Node Rectangular Solid Elements . 326
10.1.6 Plate Bending Elements 328
10.2 Isoparametric Elements 332
10.2.1 Definition of Isoparametric Elements 332
10.2.2 Lagrange Polynomials 332
10.2.3 The Bilinear Quadrilateral Element . 333
10.3 Derivation of Stiffness Matrices 337
10.3.1 The Linear Triangular Element (or CST Element) . 337
10.3.2 The Quadratic Triangular Element (or LST Element) 339
10.3.3 The Bilinear Rectangular Element (or Q4 Element) . 339
10.3.4 The Tetrahedral Solid Element 339
10.3.5 Eight-Node Rectangular Solid Element . 339
10.3.6 Plate Bending Element . 339
10.3.7 Isoparametric Formulation . 340
References 371
CHAPTER 11 Structural Dynamics 373
11.1 The Dynamic Equation 373
11.2 Mass Matrix 374
11.2.1 Bar Element 374
11.2.2 Two-Dimensional Truss Element . 376
11.2.3 Three-Dimensional Truss Element . 379
11.2.4 Two-Dimensional Beam Element 382
11.2.5 Three-Dimensional Beam Element 383
Contents xi11.2.6 Inclined Two-Dimensional Beam Element (Two-Dimensional
Frame Element) 385
11.2.7 Linear Triangular Element (CST Element) . 387
11.3 Solution Methodology for the Dynamic Equation 388
11.3.1 Central Difference Method . 388
11.3.2 Newmark-Beta Method 389
11.4 Free Vibration—Natural Frequencies 390
References 412
CHAPTER 12 Heat Transfer 413
12.1 Conduction Heat Transfer 413
2D Steady-State Heat Conduction Equation in Cartesian Coordinates 415
3D Steady-State Heat Conduction Equation in Cartesian Coordinates 415
3D Steady-State Heat Conduction Equation in Cylindrical Coordinates 416
3D Steady-State Heat Conduction Equation in Spherical Coordinates 417
Heat conduction of orthotropic materials 417
12.2 Convection Heat Transfer 420
12.3 Finite Element Formulation . 420
12.3.1 One-Dimensional Heat Transfer Modeling Using a Variational
Method 420
12.3.2 Two-Dimensional and Three-Dimensional Heat Transfer Modeling
Using a Variational Method . 435
References 477
Index 479
xii ContentsPref
Index
Note: Page numbers followed by b indicate boxes, f indicate figures and t indicate tables.
A
ANSYS, 16–17
axial force calculation, bars, 122b
beams, 165b
chip-cooling problem, 460b
plane frame analysis, 257b
plane stress problem using, 358b
Axial force and torsional moment distributions, 234–277
B
Bars, 11–12, 374–376
axial forces calculation
ANSYS implementation, 122b
boundary conditions, 92–94, 102–105, 116–117
CALFEM/MATLAB computer code, 121–122
degrees of freedom, 89–91, 100–101, 115–116
direction cosines calculation, 97–99
expanded stiffness matrices, 105b
global stiffness matrix, 117–119
internal forces, 94–96
load vector, 116
local stiffness matrices, 86–89, 99–100
nodal displacements, 85–133
structure equation, 91–92, 101–102
displacement interpolation functions
derivation of element equation, 59–60
functional form, 57–58
mechanical behavior of material, 60
principle of direct equilibrium, 60–61
dynamic loading, 394b
finite element method modeling
piping network, 67–79
system of bars, 61–67
free axial vibration, 391, 391f
free longitudinal vibration, 393, 393f
MPE principle on, 281–283
Beams, 13–15
arbitrary varying loading
algebraic operations, 156
ANSYS, 165b
boundary conditions, 154
derivation, 153
load matrix, 156–176, 157t
MATLAB/CALFEM, 157b
parameters, 156–176, 157t
physical quantities, 154–155
shear force, 154
bending moments, 194–211
cylindrical shells under axisymmetric loading
axisymmetric loads, 181, 182f
equilibrium, 181–183, 182f
FE analysis, 184b
flexural rigidity, 183
foundation modulus, 183
Hooke’s law, 183
equispaced elastic springs, 181
MPE principle, 284–287
nodal axial forces, 194–211
nodal forces
displacement function, 135–137
element stiffness matrix, 137–149
shear forces, 194–211
three-dimensional, 383–385
torsion
mechanical behavior, 192–193
principle of direct equilibrium, 193
torsional moments, 194–211
two-dimensional, 382–383
uniform loading
elastic foundation, 176–181
physical quantities, 151
shear forces, 150
Bending moment and shear force distributions, 234
Bilinear quadrilateral heat transfer isoparametric
element, 441
Bilinear rectangular elements
shape functions, 321f
stiffness matrices, 339
Boundary value problem, 35–39
C
Cable bridge analysis, MATLAB/CALFEM
axial forces, 272–273
bar element matrices, 270–271
beam element matrices, 270
bending moments, 273
boundary conditions, 269
computer code, 275–277
deformed elements, 272
displacement field and support reactions, 271
479Cable bridge analysis, MATLAB/CALFEM (Continued)
geometric data and elastic properties, 270
material and cross-section data, 269
shear forces, 273
stiffness matrix and load vector, 269
topology matrix, 269–271
undeformed elements, 271–272
CALFEM. See Computer aided learning of the finite element
method (CALFEM)
CALFEM/MATLAB
bar’s axial forces calculation, 121–122
cable bridge analysis (see Cable bridge analysis,
MATLAB/CALFEM)
hydraulic network analysis, 51b
plane frame analysis
axial forces, 250–254
bending moments, 250–254
boundary conditions, 249
computer code, 256–257
data, 247–248
degrees of freedom, 248
displacement field, 249–250
global stiffness matrix, 249
load vector, 248–249
shear forces, 250–254
C0 continuous function, 314, 315f
C1 continuous function, 314, 315f
Central difference method, 388–389
Chip-cooling problem, ANSYS implementation, 460b
Classical Kirchhoff thin plate theory, 328
Compliance matrix, 292
Component transformation, 26–28
Computer aided learning of the finite element method
(CALFEM)
advantage, 7–8
ANSYS, 16–17
bar elements, 11–12
beam elements, 13–15
element functions, 10
general purpose commands, 8
graphic functions, 16
material functions, 9
matrix functions, 8–9
spring elements, 10–11
statement functions, 15
system functions, 15
Conduction heat transfer
2D steady-state heat conduction equation, 415
3D steady-state heat conduction equation
cartesian coordinates, 415, 416f
cylindrical coordinates, 416, 416f
spherical coordinates, 416f, 417–419
Fourier’s law for heat conduction, 312
Consistent-mass matrix, 376
beam element, 383
2D inclined beam element, 385–386
linear triangular element, 387
modified, 378
Convection heat transfer, 420, 420f
Coordinate systems
component transformation, 26–28
cylindrical, 25
rectangular, 24
spherical, 25
vector differential operator, 28
Cylindrical coordinate system, 25
Cylindrical shells under axisymmetric loading
axisymmetric loads, 181, 182f
equilibrium, 181–183, 182f
FE analysis, 184b
flexural rigidity, 183
foundation modulus, 183
Hooke’s law, 183
D
D’Alambert’s principle, 375
Diagonal matrix, 29
Displacement function, 135–137
Displacement interpolation functions
derivation of element equation, 59–60
functional form, 57–58
mechanical behavior of material, 60
principle of direct equilibrium, 60–61
E
Eight-node isoparametric heat transfer 2D element, 442
Eight-node isoparametric heat transfer 3D element, 444
Eight-node rectangular solid elements
shape functions, 326f
stiffness matrices, 339
Eight-nodes brick element, 326, 326f
Elasticity equations, for isotropic solids
differential equations of equilibrium, 308
displacements
differential equations, 308
strain correlation with, 307
stress correlation with, 307
generalized Hooke’s law, 305–307
total potential energy, 308–309
Element equation
3D trusses, 83–85
mechanical behavior, 41
plane truss members, 81–83
principle of direct equilibrium, 42–43
Element matrices, 43
480 IndexElement stiffness matrix, 137–149
Equispaced elastic springs, 181
Expanded stiffness matrix, 223–226
F
Finite element method (FEM)
advantages, 1
analysis of structure
assembling, 4
boundary conditions, 4
discretization, 4
element properties, 4
linear algebraic equations, 4
loads, 4
stress calculation, 4
CALFEM
advantage, 7–8
ANSYS, 16–17
bar elements, 11–12
beam elements, 13–15
element functions, 10
general purpose commands, 8
graphic functions, 16
material functions, 9
matrix functions, 8–9
spring elements, 10–11
statement functions, 15
system functions, 15
cylindrical shells under axisymmetric loading, 184b
3D beam problem
bending moment distributions, 207–209
boundary conditions, 204–206
expanded local element equations, 202–203
global stiffness matrix, 203–204
local element equations, 200–202
shear force distributions, 209–211
system of equations, 206–207
torsional moment distributions, 211
description, 1
disadvantages, 1–2
element equation, 7
physical principles, 5–7
piping network
boundary conditions, 71
CALFEM/MATLAB, 72b
degrees of freedom, 69
fluid viscosity, 67–68
local element equations, 68
nodal variables, 67
submatrix, 70–71
software packages, 4–5
structural matrix
stiffness matrix, 2
transfer matrix, 3
system of bars
assembly of element equations, 63
derivation of element matrices, 62
derivation of field values, 63–67
expansion of element equations, 62–63
Fourier’s law for heat conduction, 312
Four-node thin plate element, 328, 328f
Frames. See also Plane frame analysis
beams, 213, 214f
MATLAB/CALFEM, 406b
two-dimensional element equation
arbitrary varying loading, 217–229
nodal forces, 213–217
Free axial vibration, bar, 391, 391f
Free longitudinal vibration, bar, 393, 393f
Free moisture strains
generalized Hooke’s law, 293–294
transformation, 304–305
Free thermal strains
generalized Hooke’s law, 292–293
transformation, 304–305
Free vibration, 390–412
natural circular frequency, 373
G
Gauss Quadrature method, 340–371
Generalized Hooke’s law, 289–296
effects of
free moisture strains, 293–294
free thermal strains, 292–293
isotropic solids, 305–307
plane stress constitutive relations, 295–296
Global stiffness matrix, 226
Green’s theorem, 23–24
H
Heat exchanger, 429b
Heat transfer
conduction (see Conduction heat transfer)
convection heat transfer, 420, 420f
2D and 3D modeling, 435–436
bilinear quadrilateral heat transfer isoparametric element,
441
eight-node isoparametric heat transfer 2D
element, 442
eight-node isoparametric heat transfer 3D
element, 444
linear triangular heat transfer element, 436
1D finite element modeling, 420–435
in heat exchanger, 429b
Index 481Heat transfer (Continued)
modes of, 413, 414f
temperature distribution, in 2D problem, 446b
through multilayered wall, 424b
I
Inclined bar element, 385–387
Isoparametric elements
bilinear quadrilateral element, 333–336
definition of, 332
Lagrange polynomials, 332–333
stiffness matrices, 340–371
Isotropic solids, elasticity equations
differential equations of equilibrium, 308
displacements
differential equations, 308
strain correlation with, 307
stress correlation with, 307
generalized Hooke’s law, 305–307
total potential energy, 308–309
Isotropic-to-orthotropic plane elements, 296–298
L
Linear interpolation, shape functions, 313, 313f
Linear spring elements
element equation
mechanical behavior, 41
principle of direct equilibrium, 42–43
stiffness matrix
assembly of element equations, 44
derivation, 43
derivation of field values, 44–55
expansion of element equations, 44
Linear triangular elements, 341, 387
algebraic equations and results, 347–348
boundary conditions, 345–347
global matrix, 345
heat transfer element
applied heat flux, 440
convection, 439
coordinates vector, 436
coordinate system, 436, 437f
elements area, 437
heat convection losses, 440
load vectors, 439–440
matrix [B] derivation, 437
stiffness matrix, 437–438
thermal conductivity matrix, 436–437
vector of nodal temperatures, 437
shape functions, 316–318, 316f
stiffness matrix, 337–338, 342–345
stresses, 348
Lumped-mass matrix, 375
M
Mapped plane element, 333f, 334
Mass matrix, 373
bar element, 374–376
inclined bar element, 385–387
linear triangular element, 387
three-dimensional beam element, 383–385
three-dimensional truss element, 379–382
two-dimensional beam element, 382–383
two-dimensional truss element, 376–379
Mass-spring system, dynamic response, 373, 374f
MATLAB/CALFEM. See also CALFEM/MATLAB
beams
boundary conditions, 158–159
computation of displacements, 161
computer code, 164
degrees of freedom, 158
displacement field, 160–161
global stiffness matrix, 159–160
graphical representation of displacements, 161–163
load vector, 158
frames dynamic response, 406b
Matrix algebra
definitions, 28–29
operations, 29–34
Maxwell-Betti Reciprocal Theorem, 291
Mindlin bending theories, 328
Minimum potential energy (MPE) principle, 34
on beams, 284–287
mathematical expression of, 279
on systems of bar elements, 281–283
on systems of spring elements, 280–281
on trusses, 284
Modified consistent-mass matrix, 378
Moisture expansion coefficients transformation, in global
coordinate system, 303–304
MPE principle. See Minimum potential energy (MPE)
principle
N
Newmark-Beta method, 389–390
O
1D heat transfer finite element modeling, 420–435
Orthotropic elasticity, plane structural elements, 296–297, 297f
Orthotropic thermal conductivity coefficients, 419–420, 419f
Orthotropic two-dimensional elements
elastic properties of, 300–302
Hooke’s law, 299
stress and strain components, coordinate transformation of,
298, 298f
482 IndexP
Physical plane element, 333, 333f
Plane frame analysis
axial forces, 250–254
bending moments, 250–254
boundary conditions, 249
computer code, 256–257
data, 247–248
degrees of freedom, 248
displacement field, 249–250
global stiffness matrix, 249
load vector, 248–249
shear forces, 250–254
Plane stress problem, using ANSYS, 358b
Plane truss members, 81–83
Plate bending elements
shape functions, 328–331
stiffness matrices, 339
Poisson’s ratio, orthotropic 2D elements, 300
Principle of direct equilibrium, 42–43, 60–61, 193
Q
Quadratic interpolation, shape functions, 314, 314f
Quadratic triangular elements
shape functions, 318–320, 319f
stiffness matrices, 339
R
Rectangular/cartesian coordinate system, 24
S
Sationary potential energy principle, 279
Shape functions, 311–331
bilinear rectangular elements, 321–322, 321f
eight-node rectangular solid elements, 326–328, 326f
for linear interpolation, 313, 313f
linear triangular elements, 316–318, 316f
plate bending elements, 328–331
for quadratic interpolation, 314, 314f
quadratic triangular elements, 318–320, 319f
tetrahedral solid elements, 322–326, 323f
Spherical coordinate system, 25
Spring elements, 10–11
MPE principle on, 280–281
Stiffness matrix, 2, 292, 348b, 354b
assembly of element equations, 44
bilinear rectangular element, 339
derivation, 43
derivation of field values
boundary conditions, 45
final solution, 45–55
simple structure composed of springs, 51b
structural system, 46b
eight-node rectangular solid element, 339
expansion of element equations, 44
isoparametric formulation, 340–371
linear triangular element, 337–338
plate bending element, 339
quadratic triangular element, 339
tetrahedral solid element, 339
Structural dynamics
dynamic equation, 373
central difference method, 388–389
Newmark-Beta method, 389–390
Structural matrix
stiffness matrix, 2
transfer matrix, 3
Symmetric matrix, 29
T
Temperature distribution, in 2D heat transfer problem, 446b
Tetrahedral solid elements
shape functions, 322–326, 323f
stiffness matrices, 339
Thermal equilibrium, in one-dimensional element, 413, 414f
Thermal expansion coefficients transformation, in global
coordinate system, 303–304, 304f
3D beam problem, FE analysis
bending moment distributions, 207–209
boundary conditions, 204–206
expanded local element equations, 202–203
global stiffness matrix, 203–204
local element equations, 200–202
shear force distributions, 209–211
system of equations, 206–207
torsional moment distributions, 211
Three-dimensional beam element, 383–385
Three-dimensional beam element equation, 230–233
Three-dimensional truss element, 379–382
3D steady-state heat conduction equation
cartesian coordinates, 415, 416f
cylindrical coordinates, 416, 416f
spherical coordinates, 416f, 417–419
3D trusses, 83–85
Transfer matrix, 3
Translational nodal motions, beam element, 383–384, 383f
Trusses
bar’s axial forces
ANSYS implementation, 122b
boundary conditions, 92–94, 102–105, 116–117
CALFEM/MATLAB computer code, 121–122
degrees of freedom, 89–91, 100–101, 115–116
direction cosines calculation, 97–99
expanded stiffness matrices, 105b
global stiffness matrix, 117–119
Index 483Trusses (Continued)
internal forces, 94–96
load vector, 116
local stiffness matrices, 86–89, 99–100
nodal displacements, 85–133
structure equation, 91–92, 101–102
element equation
3D system, 83–85
plane members, 81–83
MPE principle on, 284
three-dimensional, 83–85, 376–382
two-dimensional, 376–379
2D and 3D heat transfer modeling, 435–436
bilinear quadrilateral heat transfer isoparametric
element, 441
eight-node isoparametric heat transfer 2D element, 442
eight-node isoparametric heat transfer 3D element, 444
linear triangular heat transfer element, 436
Two-dimensional beam element, 382–383
Two-dimensional frame element. See Inclined two-dimensional
beam element
Two-dimensional frame element equation
arbitrary varying loading
algebraic system, 228–229
ANSYS, 257b
axial force and torsional moment distributions, 234–277
bending moment and shear force distributions, 234
boundary conditions, 226–228
cable bridge analysis, 270
CALFEM/MATLAB, 247b
equivalent nodal forces, 219, 219f
expanded stiffness matrix, 223–226
force matrix, 217, 218t
geometric parameters, 235
global stiffness matrix, 226
inclined element, 217, 217f
local load vectors, 222–223
local stiffness matrices, 221–222
slopes and lengths calculation, 220–221
three-dimensional beam element equation, 230–233
nodal forces
axial deflections, 215–216
global coordinate system, 213
moments, 215
nodal parameters, 217
transformation of displacements, 213, 214f
2D orthotropic material, principal coordinate system for, 417, 417f
2D steady-state heat conduction equation, 415
U
Unit matrix, 29
V
Variational method
2D and 3D heat transfer modeling, 435–436
bilinear quadrilateral heat transfer isoparametric element,
441
eight-node isoparametric heat transfer 2D element, 442
eight-node isoparametric heat transfer 3D element, 444
linear triangular heat transfer element, 436
formulation of elasticity problems
boundary value problem, 35–39
definition, 34
properties, 35
variational principle, 279 (see also Minimum potential energy
(MPE) principle)
Vectors
definition, 19
Green’s theorem, 23–24
rotation of coordinate system, 22–23
scalar product, 19–20
vector differential operator, 23, 28
vector product, 21
W
Winkler foundation, 176
Z
Zero matrix, 29
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رابط من موقع عالم الكتب لتنزيل كتاب Essentials of the Finite Element Method - For Mechanical and Structural Engineers
رابط مباشر لتنزيل كتاب Essentials of the Finite Element Method - For Mechanical and Structural Engineers 
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