Introduction to Finite Element Analysis and Design – 2nd Edition

Introduction to Finite Element Analysis and Design – 2nd Edition
اسم المؤلف
Nam-Ho Kim, Bhavani V. Sankar, Ashok V. Kumar
التاريخ
30 أغسطس 2023
المشاهدات
360
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Introduction to Finite Element Analysis and Design – 2nd Edition
Nam-Ho Kim, Bhavani V. Sankar, Ashok V. Kumar
Table of Contents
Cover
Preface
Chapter 1: Direct Method- Springs, Bars, and Truss Elements
1.1 ILLUSTRATION OF THE DIRECT METHOD
1.2 UNIAXIAL BAR ELEMENT
1.3 PLANE TRUSS ELEMENTS
1.4 THREEDIMENSIONAL TRUSS ELEMENTS (SPACE TRUSS)
1.5 THERMAL STRESSES
1.6 FINITE ELEMENT MODELING PRACTICE FOR TRUSS
1.7 PROJECTS
1.8 EXERCISES
Chapter 2: Weighted Residual Methods for OneDimensional Problems
2.1 EXACT VS. APPROXIMATE SOLUTION
2.2 GALERKIN METHOD
2.3 HIGHERORDER DIFFERENTIAL EQUATIONS
2.4 FINITE ELEMENT APPROXIMATION
2.5 ENERGY METHODS
2.6 EXERCISES
Chapter 3: Finite Element Analysis of Beams and Frames
3.1 REVIEW OF ELEMENTARY BEAM THEORY
3.2 RAYLEIGHRITZ METHOD
3.3 FINITE ELEMENT FORMULATION FOR BEAMS
3.4 PLANE FRAME ELEMENTS
3.5 BUCKLING OF BEAMS
3.6 BUCKLING OF FRAMES
3.7 FINITE ELEMENT MODELING PRACTICE FOR BEAMS
3.8 PROJECT
3.9 EXERCISES
Chapter 4: Finite Elements for Heat Transfer Problems
4.1 INTRODUCTION
4.2 FOURIER HEAT CONDUCTION EQUATION4.3 FINITE ELEMENT ANALYSIS- DIRECT METHOD
4.4 GAI.ERKIN’S METHOD FOR HEAT CONDUCTION PROBLEMS
4.5 CONVECTION BOUNDARY CONDITIONS
4.6 TWODIMENSIONAL HEAT TRANSFER
4.7 3NODE TRIANGULAR ELEMENTS FOR TWODIMENSIONAL HEAT
TRANSFER
4.8 FINITE ELEMENT MODELING PRACTICE FOR 2D HEAT TRANSFER
4.9 EXERCISES
Chapter 5: Review of Solid Mechanics
5.1 INTRODUCTION
5.2 STRESS
5.3 STRAIN
5.4 STRESS-STRAIN RELATIONSHIP
5.5 BOUNDARY VALUE PROBLEMS
5.6 PRINCIPLE OF MINIMUM POTENTIAL ENERGY FOR PLANE SOLIDS
5.7 FAILURE THEORIES
5.8 SAFETY FACTOR
5.9 EXERCISES
Chapter 6: Finite Elements for TwoDimensional Solid Mechanics
6.1 INTRODUCTION
6.2 TYPES OF TWODIMENSIONAL PROBLEMS
6.3 CONSTANT STRAIN TRIANGULAR (CST) ELEMENT
6.4 FOUR-NODE RECTANGULAR ELEMENT
6.5 AXISYMMETRIC ELEMENT
6.6 FINITE ELEMENT MODELING PRACTICE FOR SOLIDS
6.7 PROJECT
6.8 EXERCISES
Chapter 7: Isoparametric Finite Elements
7.1 INTRODUCTION
7.2 ONEDIMENSIONAL ISOPARAMETRIC ELEMENTS
7.3 TWODIMENSIONAL ISOPARAMETRIC QUADRILATERAL ELEMENT
7.4 NUMERICAL INTEGRATION
7.5 HIGHERORDER QUADRILATERAL ELEMENTS
7.6 ISOPARAMETRIC TRIANGULAR ELEMENTS7.7 THREEDIMENSIONAL ISOPARAMETRIC ELEMENTS
7.8 FINITE ELEMENT MODELING PRACTICE FOR ISOPARAMETRIC
ELEMENTS
7.9 PROJECTS
7.10 EXERCISES
Chapter 8: Finite Element Analysis for Dynamic Problems
8.1 INTRODUCTION
8.2 DYNAMIC EQUATION OF MOTION AND MASS MATRIX
8.3 NATURAL VIBRATION: NATURAL FREQUENCIES AND MODE SHAPES
8.4 FORCED VIBRATION: DIRECT INTEGRATION APPROACH
8.5 METHOD OF MODE SUPERPOSITION
8.6 DYNAMIC ANALYSIS WITH STRUCTURAL DAMPING
8.7 FINITE ELEMENT MODELING PRACTICE FOR DYNAMIC PROBLEMS
8.8 EXERCISES
Chapter 9: Finite Element Procedure and Modeling
9.6 INTRODUCTION
9.2 FINITE ELEMENT ANALYSIS PROCEDURES
9.3 FINITE ELEMENT MODELING ISSUES
9.4 ERROR ANALYSIS AND CONVERGENCE
9.5 PROJECT
9.6 EXERCISES
Chapter 10: Structural Design Using Finite Elements
10.8 INTRODUCTION
10.2 CONSERVATISM IN STRUCTURAL DESIGN
10.3 INTUITIVE DESIGN: FULLY STRESSED DESIGN
10.4 DESIGN PARAMETERIZATION
10.5 PARAMETRIC STUDY- SENSITIVITY ANALYSIS
10.6 STRUCTURAL OPTIMIZATION
10.7 PROJECTS
10.8 EXERCISES
Appendix Mathematical Preliminaries
A.1 VECTORS AND MATRICES
A.2 VECTORMATRIX CALCULUS
A.3 MATRIX EQUATIONS AND SOLUTIONA.4 EIGENVALUES AND EIGENVECTORS
A.5 QUADRATIC FORMS
A.6 MAXIMA AND MINIMA OF FUNCTIONS
A.7 EXERCISES
Index
End User License Agreement
List of Tables
Chapter 01
Table 1.1 Connectivity table for figure 1.1
Table 1.2 Connectivity table with element properties for example 1.5
Table 1.3 Nodal coordinates of space truss structure in example 1.6
Table 1.4 Element connectivity and direction cosines for truss structure in figure 1.21
Table 1.5 Element connectivity and direction cosines for truss structure in figure 1.23
Table 1.6 Solution of thermal stresses in a truss using the superposition method
Table 1.7 Nodal coordinates of space truss structure in example 1.10
Chapter 02
Table 2.1 Comparison of approximate and exact solutions
Table 2.2 Different types of finite elements
Chapter 04
Table 4.1 Analogy between strucUiral and heat conduction problems
Table 4.2 Connectivity table
Chapter 05
Table 5.1 Description of stress components
Table 5.2 Comparison of stress and strain
Table 5.3 Explanations of uniaxial tension test
Chapter 06
Table 6.1 Material property conversion between plane strain and plane stress problems
Chapter 07
Table 7.1 Element connectivity
Table 7.2 Gauss quadrature points and weightsTable 7.3 Gauss quadrature points and weights for triangles
Table 7.4 Results for the plate with holes
Chapter 08
Table 8.1 Newmark family of time integration algorithms
Table 8.2 First six natural frequencies of cantilever beam
Table 8.3 Modes shapes of vibration
Table 8.4 Natural frequencies (Hz) of the Pining fork
Table 8.5 Natural frequencies of the fully clamped beam
Chapter 09
Table 9.1 Different types of finite elements
Table 9.2 Patch tests for plane solids (E =1GPa, v = 0.3)
Chapter 10
Table 10.1 Parametric study of a cantilevered beam
Table 10.2 Input data for tenbar truss
Table 10.3 Lower and upper bounds of design parameters (unit mm)
List of Illustrations
Chapter 01
Figure 1.1 Rigid bodies connected by springs
Figure 1.2 Spring element (e) connected by node i and node j
Figure 1.3 Freebody diagram of node 3 in the example shown in figure 1.1. The
Ae)
external force, F3 and the forces, /3 , exerted by the springs attached to the node are
shown. Note the forces J3 act in the negative direction.
Figure 1.4 Typical one dimensional bar problems
Figure 1.5 Uniaxial bar finite element
Figure 1.6 Force equilibrium at node i
Figure 1.7 Two clamped uniaxial bars
Figure 1.8 Onedimensional structure with three uniaxial bar elements
Figure 1.9 Finite element model
Figure 1.10 Freebody diagram of the structureFigure 1.11 A plane truss consisting of two members
Figure 1.12 Local and global coordinate systems
Figure 1.13 Local coordinate systems of the twobar truss
Figure 1.14 Definition of twodimensional truss element
Figure 1.15 Twobar truss structure
Figure 1.16 Local coordinates of element 1
Figure 1.17 Local coordinates of element 2
Figure 1.18 Element force for element 1 in local coordinates
Figure 1.19 Plane structure with three truss elements
Figure 1.20 Threedimensional coordinates transformation
Figure 1.21 Threebar space truss structure
Figure 1.22 Effects of temperature change on the structure
Figure 1.23 A threeelement truss: (a) The middle element is subjected to a
temperature rise. This is the given problem, (b) A pair of compressive forces is
applied to element 2 to prevent it from expanding. Titis is called problem I. (c) The
forces in problem I are reversed. No thermal stresses are involved in this problem.
Titis is called problem IL
Figure 1.24 Force equilibrium at node 4
Figure 1.25 Threebar space truss structure
Figure 1.26 Statically indeterminate vertical bar
Figure 1.27 Thermally loaded three bars
Figure 1.28 Twobar truss
Figure 1.29 25-member space truss
Figure 1.30 Plane truss and design domain for Project 1.2
Figure 1.31 Plane truss and design domain for Project 1.3
Figure 1.32 Tenbar truss structure for project 1.4
Chapter 02
Figure 2.1 Comparison of exact solution and approximate solutions for example 2.1
Figure 2.2 Weighted residual for differential equation in example 2.1
Figure 2.3 Comparison of exact solution and approximate solution and their derivatives
for example 2.4
Figure 2.4 Comparison of u(x) and its derivative obtained by die Galerkin method forexample 2.5
Figure 2.5 Comparison of w” and w”‘ for the beam problem in example 2.6
Figure 2.6 Boundaryvalue problem in solid mechanics
Figure 2.7 Piecewise linear approximation of the solution for a onedimensional
problem
Figure 2.8 Convergence of onedimensional finite element solution
Figure 2.9 Domain discretization of onedimensional problem
Figure 2.10 Interpolated solution and its gradient
Figure 2.11 Function (faM and its derivative
Figure 2.12 Trial function ^(x) for two equallength finite elements
Figure 2.13 Exact solution u(x) and finite element solution u(x)
Figure 2.14 Derivatives of the exact and finite element solutions
Figure 2.15 Onedimensional finite element with interpolation functions
Figure 2.16 Comparison of exact and approximate solution for example 2.8
Figure 2.17 A particle in equilibrium with four springs
Figure 2.18 Equilibrium of massspring system
Figure 2.19 Uniaxial bar under body force Bx and concentrated force F
Figure 2.20 Example of a discrete system with finite number of degrees of freedom
Figure 2.21 Uniaxial bar subject to distributed and concentrated forces
Chapter 03
Figure 3.1 Deflection of a plane EulerBernoulli beam
Figure 3.2 Positive directions for axial force, shear force, and bending moment of a
plane beam
Figure 3.3 Equilibrium of infinitesimal beam section under various loadings
Figure 3.4 Simply supported beam under uniformly distributed load
Figure 3.5 Comparison of finite element results with exact ones for a simply supported
beam; (a) deflection, (b) bending moment, and (c) shear force
Figure 3.6 Simply supported beam under a uniformly distributed load
Figure 3.7 Comparison of finite element results with exact ones for a cantilevered
beam; (a) deflection, (b) bending moment, and (c) shear force
Figure 3.8 Positive directions for forces and couples in a beam elementFigure 3.9 Nodal displacements and rotations for the beam element
Figure 3.10 Shape functions of the beam element
Figure 3.11 Cantilevered beam element with nodal displacements
Figure 3.12 Finite element models using four beam elements
Figure 3.13 Finite element models of stepped cantilevered beam
Figure 3.14 Work equivalent nodal forces for the distributed load
Figure 3.15 Finite element models of stepped cantilevered beam
Figure 3.16 Cantilevered beam under uniformly distributed load and couple
Figure 3.17 Comparison of beam deflection and rotation with exact solutions; (a)
deflection, (b) slope
Figure 3.18 Comparison of bending moment and shear force with exact solutions; (a)
bending moment, (b) shear force
Figure 3.19 One element model with distributed force p
Figure 3.20 Transverse displacement of the beam element
Figure 3.21 Comparison of FE and analytical solutions for the beam shown in figure
3.19; (a) bending moment, (b) shear force
Figure 3.22 Frame structure and finite elements
Figure 3.23 Local degrees of freedom of plane frame element
Figure 3.24 A twomember plane frame
Figure 3.25 Deformed shape of the frame in figure 3.24. The displacements are
magnified by a factor of 200
Figure 3.26 Freebody diagrams of elements 1 and 2 of the frame in example 3.10
Figure 3.27 Support reactions for the frame in example 3.10
Figure 3.28 Abeam subjected to axial force and an end couple
Figure 3.29 Beam subjected to an axial tension and an end couple with a freebody
diagram to determine M(x)
Figure 3.30 End shortening of a cantilever beam under a compressive load
Figure 3.31 Nondimensional tip deflection as a function of nondimensional axial
force AL for a given end couple in a cantilever beam
Figure 3.32 Deflection curve of a cantilever beam subjected to an end couple and
different values of the axial force P
Figure 3.33 Buckling mode shapes of a cantilever beam obtained using one beam finite
elementFigure 3.34 Clampedhinged beam subjected to an axial force
Figure 3.35 Buckling mode shapes for the beam in example 3.13 with two elements
Figure 3.36 Degrees of freedom of plane portal frame
Figure 3.37 A portal frame subjected to two axial forces
Figure 3.38 First mode (assymteric or swaying mode) and second mode (symmteric
mode) buckling of the portal frame in example 3.14
Figure 3.39 Beam bending with distributed loads
Figure 3.40 Deflection curve of the beam
Figure 3.41 Portal frame under symmetric loading
Figure 3.42 Crosssectional dimensions for W 36 x 300 Ibeam section
Figure 3.43 Buckling of a bar with hinged ends
Figure 3.44 Bicycle frame structure
Chapter 04
Figure 4.1 Examples of onedimensional heat conduction problems: (a) heat
conduction in a thin long rod; (b) a furnace wall with dimensions in die y and z
directions much greater than the thickness in the x direction
Figure 4.2 Energy balance in an infinitesimal volume
Figure 4.3 Onedimensional heat conduction of a long wire
Figure 4.4 Finite elements for onedimensional heat conduction problem
Figure 4.5 Balance in heat flow at node 2
Figure 4.6 Finite elements for onedimensional heat conduction problem
Figure 4.7 Network of heat conduction elements
Figure 4.8 Heat transfer problem for insulated wall
Figure 4.9 Finite element approximation of die wall
Figure 4.10 Temperature distribution along the wall thickness
Figure 4.11 Heat transfer of a diermal protection system for a space vehicle
Figure 4.12 Finite element model of the thermal protection system
Figure 4.13 Temperature distribution in die diermal protection system
Figure 4.14 Finite element approximation of the furnace wall
Figure 4.15 Heat transfer problem of an insulated wall
Figure 4.16 Finite element approximation of the furnace wallFigure 4.17 Heat conduction and convection in a long rod
Figure 4.18 Heat flow through a thin fin and finite element model
Figure 4.19 Temperature distribution in a thin fin
Figure 4.20 Twodimensional heat transfer analysis domain
Figure 4.21 Energy balance in an infinitesimal element
Figure 4.22 3node triangular element
Figure 4.23 Plot of linear shape function for triangular element
Figure 4.24 Linear shape functions for interpolation along edge
Figure 4.25 Heat conduction example
Figure 4.26 Finite element mesh for heat conduction analysis
Figure 4.27 Finite element model for slab with pipes: (a) periodicity and symmetry, (b)
model
Figure 4.28 Temperature distribution in the slab
Figure 4.29 Heat flux components in the slab
Chapter 05
Figure 5.1 Surface traction acting on a plane at a point
Figure 5.2 Equilibrium of a uniaxial bar under axial force
Figure 5.3 Normal and shear stresses at a point P
Figure 5.4 Stress components in a Cartesian coordinate system
Figure 5.5 Surface traction and stress components acting on faces of an infinitesimal
tetrahedron, at a given point P
Figure 5.6 Equilibrium of a square element subjected to shear stresses
Figure 5.7 Coordinate transformation of stress
Figure 5.8 Coordinate transformation of example 5.6
Figure 5.9 Maximum shear stress
Figure 5.10 Deformation of line segments
Figure 5.11 Deformation in the principal directions
Figure 5.12 Uniaxial tension test
Figure 5.13 Stressstrain diagram for a typical ductile material in tension
Figure 5.14 Stress variations in infinitesimal components
Figure 5.15 Traction boundary condition of a plane solidFigure 5.16 Boundary value problem
Figure 5.17 Cantilever beam bending problem
Figure 5.18 A plane solid under the distributed load {Tv Ty} on the traction boundary
Sj
Figure 5.19 Material failure due to relative sliding of atomic planes
Figure 5.20 Stress-strain curve and the strain energy
Figure 5.21 Failure envelope of the distortion energy theory
Figure 5.22 Failure envelope of the maximum shear stress theory
Figure 5.23 Failure envelope of the maximum principal stress theory
Figure 5.24 Bracket structure
Chapter 06
Figure 6.1 Thin plate with inplane applied forces
Figure 6.2 Dam structure with plane strain assumption
Figure 6.3 Constant strain triangular (CST) element
Figure 6.4 Interelement displacement compatibility of constant strain triangular
element
Figure 6.5 Interpolation of displacements in triangular elements
Figure 6.6 Applied surface traction along edge 12
Figure 6.7 Cantilevered plate
Figure 6.8 Four-node rectangular element
Figure 6.9 Fournode rectangular element
Figure 6.10 Threedimensional surface plots of shape functions for a rectangular
element; (a)JV^y), (b) N^x^
Figure 6.11 A square element under a simple shear condition
Figure 6.12 Simple shear deformation of a square element
Figure 6.13 A square element under pure bending condition
Figure 6.14 Pure bending deformation of a square element
Figure 6.15 Axisymmetric geometry; (a) revolved geometry, (b) section- plane of
deformation
Figure 6.16 Circumferential strain due to radial displacement
Figure 6.17 Triangular axisymmetric element; (a) axisymmetric model of ring, (b)element e
Figure 6.18 Beam model using plane stress CST elements
Figure 6.19 Beam deflection computed using CST elements
Figure 6.20 Computed normal strain component without smoothing
Figure 6.21 Normal stress component after smoothing
Figure 6.22 Thickwalled cylinder; (a) crosssection, (b) plane strain model, (c)
axisymmetric model
Figure 6.23 Comparison of results using plane strain and axisymmetric models for the
thickwalled cylinder; (a) displacement magnitude with plane strain model, (b)
displacement magnitude with axisymmetric model, (c) von Mises stress with plane
strain model, (d) von Mises stress with axisymmetric model
Figure 6.24 Cantilever beam model
Chapter 07
Figure 7.1 Onedimensional 2node linear isoparametric element
Figure 7.2 3node quadratic isoparametric element
Figure 7.3 Regular versus irregular quadratic element
Figure 7.4 Mapping and interpolation for the regular element
Figure 7.5 Irregular versus irregular quadratic element
Figure 7.6 ID heat conduction model using 3node elements
Figure 7.7 Four-node quadrilateral element for plane solids
Figure 7.8 Mapping of a quadrilateral element
Figure 7.9 Four-node quadrilateral element
Figure 7.10 Isoparametric lines of a quadrilateral element
Figure 7.11 An example of invalid mapping
Figure 7.12 Recommended ranges of internal angles in a quadrilateral element
Figure 7.13 Mapping of a rectangular element
Figure 7.14 Gauss integration points in twodimensional parent elements
Figure 7.15 Numerical integration of a square element
Figure 7.16 Three rigidbody modes of plane solids
Figure 7.17 Two extra zeroenergy modes of plane solids
Figure 7.18 Polynomial triangleFigure 7.19 9node Lagrange element in parametric space
Figure 7.20 8node serendipity element in parametric space
Figure 7.21 Shape function for node 1 of a 4node element
Figure 7.22 5node transition element
Figure 7.23 Triangular element by collapsing a 4node quadrilateral
Figure 7.24 Triangular element in physical space
Figure 7.25 3node isoparametric triangular element
Figure 7.26 Sixnode isoparametric triangular element
Figure 7.27 Fournode isoparametric tetrahedral element
Figure 7.28 Tennode isoparametric tetrahedral element
Figure 7.29 Eightnode isoparametric hexahedral element
Figure 7.30 Ushaped beam
Figure 7.31 Plane stress model of Ushaped beam
Figure 7.32 Deflection and stress distribution in Ushaped beam
Figure 7.33 Plate with holes
Figure 7.34 Pressure distribution p(0) for bearing load
Figure 7.35 Finite element model of plate with holes
Figure 7.36 Deflection (mm) due to uniform load versus bearing load
Figure 7.37 von Mises stress (N/m^) due to uniform load versus bearing load
Figure 7.38 Plate with normal forces
Figure 7.39 A 3D bracket drawing
Figure 7.40 A 3D bracket loads and boundary conditions
Figure 7.41 hadaptive mesh refinement for 3D bracket
Figure 7.42 Cantilever beam model
Figure 7.43 Dimensions of torque arm model
Chapter 08
Figure 8.1 Uniaxial bar element in dynamic analysis
Figure 8.2 Lumped mass idealization of a uniaxial bar element
Figure 8.3 Free vibration of ID springmass system
Figure 8.4 Free vibration of ID massspring systemFigure 8.5 Vibration of a clampedfree bar modeled using two elements
Figure 8.6 Free vibration of a dampedclamped beam using two beam elements
Figure 8.7 Mode shapes of a clampedclamped beam of length 2 m
Figure 8.8 Rigid bodies connected by springs
Figure 8.9 A clampedfree uniaxial bar subjected to a tip force F(t)
Figure 8.10 Tipdisplacement of a uniaxial bar in figure 8.9 using the central
difference method. The quasistatic response is shown in a dashed line.
Figure 8.11 Instability of the central finite difference method due to a large time step
Figure 8.12 Equivalence of dynamic and static solution under slowly applied load
Figure 8.13 Tip displacement of the uniaxial bar in figure 8.9 using the Newmark
method: (a) At = 42 psec, and (b) At = 75 psec
Figure 8.14 (Top) Inpact of a mass on a simply supported beam; (bottom) one
element FE model of one half of the beam.
Figure 8.15 Inpact response of a simply supported beam subjected to central inpact.
The figure shows the inpact force history. The dotted line represents the approximate
single DOF solution.
Figure 8.16 Tip displacement of the uniaxial bar in figure 8.9 using the modal
superposition method
Figure 8.17 Inpact response of a beam in figure 8.15 obtained using the mode
superposition method
Figure 8.18 Onedimensional springmassdashpot element
Figure 8.19 Tip displacement of a uniaxial bar in figure 8.9 using the central difference
method when structural damping is included
Figure 8.20 Tip displacement of a uniaxial bar in figure 8.9 using die Newmark method
when structural damping is included
Figure 8.21 Finite element models for beam modal analysis
Figure 8.22 Tuning fork
Figure 8.23 Finite element mesh for the tuning fork
Figure 8.24 Mode shapes of the tuning fork with no boundary conditions
Figure 8.25 Beam with a harmonic distributed load and clanped at both ends
Figure 8.26 Beam with deflection at A and B due to load (a)
Figure 8.27 Beam with deflection at A and B due to load (b)
Figure 8.28 Elastic rod inpact problemFigure 8.29 Analytical solutions of elastic rod impact problem; (a) displacements and
(b) stresses
Figure 8.30 Stress history of elastic rod impact problem with explicit time integration
(superconvergent solution)
Figure 8.31 Stress history of element 10 of elastic rod inpact problem with different
timestep sizes
Figure 8.32 Stress history of elastic rod inpact problem with implicit time integration
Chapter 09
Figure 9.1 Finite element analysis procedures
Figure 9.2 Frame structure under a uniformly distributed load
Figure 9.3 Plate with a hole under tension
Figure 9.4 Stress concentration factor of plate with a hole
Figure 9.5 Singularity in finite element model
Figure 9.6 Solid model of plate with a hole
Figure 9.7 Automatically generated elements in a plate with hole
Figure 9.8 Bad quality elements
Figure 9.9 Quick transition of element size
Figure 9.10 Shrink plot of elements to find missing elements
Figure 9.11 Error in element connection
Figure 9.12 Finite element modeling using different element types
Figure 9.13 Convergence of finite element analysis results
Figure 9.14 Applying displacement boundary conditions at a hole in a plate
Figure 9.15 Applying displacement boundary conditions on truss
Figure 9.16 Concentrated and distributed forces in a finite element model
Figure 9.17 Stress distribution due to concentrated force
Figure 9.18 Applying a couple to different element types
Figure 9.19 Modeling a shaft force using assumed pressure and bar elements
Figure 9.20 Displacement and forces of the plate model
Figure 9.21 Deformed shape of the plate model
Figure 9.22 Contour plot of of the plate model (element size = 0.2 in)
Figure 9.23 Contour plot of 0^ in the refined model (element size = 0.1 in)Figure 9.24 Averaging stresses at nodes
Figure 9.25 Detail model of a wheel cover
Figure 9.26 Mesh generation using mapping
Figure 9.27 Mapped and free meshes
Figure 9.28 Full-sized model of a plate with a hole
Figure 9.29 An example of a symmetric model with a symmetric load
Figure 9.30 Symmetric models of a plate with a hole
Figure 9.31 Singularity in connecting a plane solid with a frame
Figure 9.32 Connecting a plane solid with frame
Figure 9.33 Modeling bolted joints
Figure 9.34 Finite element models of stepped cantilevered beam
Figure 9.35 Illustration of linear systems
Figure 9.36 Structural linear systems
Figure 9.37 Fatigue analysis of airplane wing structure
Figure 9.38 A patch of quadrilateral elements
Figure 9.39 Generalized patch test for constant (Jxx
Figure 9.40 Patch test for bar elements
Figure 9.41 Stresses at integration points versus nodeaverages stresses
Figure 9.42 Converging to the exact solution with mesh refinement
Figure 9.43 Design domain and boundary and loading condition for die bracket
Chapter 10
Figure 10.1 Histogram of failure strengths and allowable strengdis
Figure 10.2 Knockdown factor for the Bbasis allowable strength
Figure 10.3 Cantilevered beam design
Figure 10.4 Threebar truss for fully stressed design
Figure 10.5 Sizing design variables for cross sections of bars and beams; (a) Solid
circular cross section; (b) Rectangular cross section; (c) Circular tube; (d) Rectangular
tube; (e) I-section
Figure 10.6 Shape design variables in a plate widi a hole
Figure 10.7 Design perturbation using isoparametric mapping method
Figure 10.8 Parametric study plot for the cantilevered beamFigure 10.9 Influence of step size in the forward finite difference method
Figure 10.10 Structural design optimization procedure
Figure 10.11 Design parameters for beam cross section
Figure 10.12 Design of a beer can
Figure 10.13 Local and global minima of a function
Figure 10.14 Graphical optimization of the beer can problem
Figure 10.15 Minimum weight design of fourbar truss
Figure 10.16 List of the submenu in Tools menu (Solver appears in Tools menu)
Figure 10.17 Addin dialog box with installed Solver addin
Figure 10.18 Excel worksheet for minimum weight design of the fourbar truss and
Solver Parameters dialog box
Figure 10.19 Add Constraint dialog box
Figure 10.20 Solver Options dialog box
Figure 10.21 Show Trial Solution dialog box
Figure 10.22 Solver Results dialog box
Figure 10.23 Answer Report worksheet
Figure 10.24 Tenbar truss
Figure 10.25 Geometry of a bracket (unit mm)
Appendix
Figure A.1 Threedimensional geometric vector
Figure A.2 Illustration of vector product

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