Introduction to Finite Element Analysis and Design E-Book

Nam-Ho Kim

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 931

Veröffentlichungsjahr: 2018

Beschreibung

Introduces the basic concepts of FEM in an easy-to-use format so that students and professionals can use the method efficiently and interpret results properly

Finite element method (FEM) is a powerful tool for solving engineering problems both in solid structural mechanics and fluid mechanics. This book presents all of the theoretical aspects of FEM that students of engineering will need. It eliminates overlong math equations in favour of basic concepts, and reviews of the mathematics and mechanics of materials in order to illustrate the concepts of FEM. It introduces these concepts by including examples using six different commercial programs online.

The all-new, second edition of Introduction to Finite Element Analysis and Design provides many more exercise problems than the first edition. It includes a significant amount of material in modelling issues by using several practical examples from engineering applications. The book features new coverage of buckling of beams and frames and extends heat transfer analyses from 1D (in the previous edition) to 2D. It also covers 3D solid element and its application, as well as 2D. Additionally, readers will find an increase in coverage of finite element analysis of dynamic problems. There is also a companion website with examples that are concurrent with the most recent version of the commercial programs.

• Offers elaborate explanations of basic finite element procedures
• Delivers clear explanations of the capabilities and limitations of finite element analysis
• Includes application examples and tutorials for commercial finite element software, such as MATLAB, ANSYS, ABAQUS and NASTRAN
• Provides numerous examples and exercise problems
• Comes with a complete solution manual and results of several engineering design projects

Introduction to Finite Element Analysis and Design, 2^nd Edition is an excellent text for junior and senior level undergraduate students and beginning graduate students in mechanical, civil, aerospace, biomedical engineering, industrial engineering and engineering mechanics.

Leseprobe

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 THREE‐DIMENSIONAL 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 One‐Dimensional Problems

2.1 EXACT VS. APPROXIMATE SOLUTION

2.2 GALERKIN METHOD

2.3 HIGHER‐ORDER 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 RAYLEIGH‐RITZ 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 EQUATION

4.3 FINITE ELEMENT ANALYSIS – DIRECT METHOD

4.4 GALERKIN’S METHOD FOR HEAT CONDUCTION PROBLEMS

4.5 CONVECTION BOUNDARY CONDITIONS

4.6 TWO‐DIMENSIONAL HEAT TRANSFER

4.7 3‐NODE TRIANGULAR ELEMENTS FOR TWO‐DIMENSIONAL HEAT TRANSFER

4.8 FINITE ELEMENT MODELING PRACTICE FOR 2‐D 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 Two‐Dimensional Solid Mechanics

6.1 INTRODUCTION

6.2 TYPES OF TWO‐DIMENSIONAL 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 ONE‐DIMENSIONAL ISOPARAMETRIC ELEMENTS

7.3 TWO‐DIMENSIONAL ISOPARAMETRIC QUADRILATERAL ELEMENT

7.4 NUMERICAL INTEGRATION

7.5 HIGHER‐ORDER QUADRILATERAL ELEMENTS

7.6 ISOPARAMETRIC TRIANGULAR ELEMENTS

7.7 THREE‐DIMENSIONAL 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 VECTOR‐MATRIX CALCULUS

A.3 MATRIX EQUATIONS AND SOLUTION

A.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 structural 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 weights

Table 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 tuning 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

 = 1 GPa,

ν

 = 0.3)

Chapter 10

Table 10.1 Parametric study of a cantilevered beam

Table 10.2 Input data for ten‐bar 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 Free‐body diagram of node 3 in the example shown in figure 1.1. The external force,

F

3

,

and the forces,

, exerted by the springs attached to the node are shown. Note the forces

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 One‐dimensional structure with three uniaxial bar elements

Figure 1.9 Finite element model

Figure 1.10 Free‐body diagram of the structure

Figure 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 two‐bar truss

Figure 1.14 Definition of two‐dimensional truss element

Figure 1.15 Two‐bar 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 Three‐dimensional coordinates transformation

Figure 1.21 Three‐bar space truss structure

Figure 1.22 Effects of temperature change on the structure

Figure 1.23 A three‐element 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. This is called problem I. (c) The forces in problem I are reversed. No thermal stresses are involved in this problem. This is called problem II.

Figure 1.24 Force equilibrium at node 4

Figure 1.25 Three‐bar space truss structure

Figure 1.26 Statically indeterminate vertical bar

Figure 1.27 Thermally loaded three bars

Figure 1.28 Two‐bar 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 Ten‐bar 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 the Galerkin method for example 2.5

Figure 2.5 Comparison of

w

″ and

w

‴ for the beam problem in example 2.6

Figure 2.6 Boundary‐value problem in solid mechanics

Figure 2.7 Piecewise linear approximation of the solution for a one‐dimensional problem

Figure 2.8 Convergence of one‐dimensional finite element solution

Figure 2.9 Domain discretization of one‐dimensional problem

Figure 2.10 Interpolated solution and its gradient

Figure 2.11 Function

ϕ

i

(

x

) and its derivative

Figure 2.12 Trial function

ϕ

i

(

x

) for two equal‐length finite elements

Figure 2.13 Exact solution

u

(

x

) and finite element solution

ũ

(

x

)

Figure 2.14 Derivatives of the exact and finite element solutions

Figure 2.15 One‐dimensional 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 mass‐spring system

Figure 2.19 Uniaxial bar under body force

B

x

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 Euler‐Bernoulli 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 element

Figure 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 two‐member 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 Free‐body 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 A beam subjected to axial force and an end couple

Figure 3.29 Beam subjected to an axial tension and an end couple with a free‐body diagram to determine

M

(

x

)

Figure 3.30 End shortening of a cantilever beam under a compressive load

Figure 3.31 Non‐dimensional tip deflection as a function of non‐dimensional axial force

λL

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 element

Figure 3.34 Clamped‐hinged 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 Cross‐sectional dimensions for W 36 × 300 I‐beam section

Figure 3.43 Buckling of a bar with hinged ends

Figure 3.44 Bicycle frame structure

Chapter 04

Figure 4.1 Examples of one‐dimensional heat conduction problems; (a) heat conduction in a thin long rod; (b) a furnace wall with dimensions in the

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 One‐dimensional heat conduction of a long wire

Figure 4.4 Finite elements for one‐dimensional heat conduction problem

Figure 4.5 Balance in heat flow at node 2

Figure 4.6 Finite elements for one‐dimensional 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 the wall

Figure 4.10 Temperature distribution along the wall thickness

Figure 4.11 Heat transfer of a thermal protection system for a space vehicle

Figure 4.12 Finite element model of the thermal protection system

Figure 4.13 Temperature distribution in the thermal 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 wall

Figure 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 Two‐dimensional heat transfer analysis domain

Figure 4.21 Energy balance in an infinitesimal element

Figure 4.22 3‐node 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 Stress‐strain 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 solid

Figure 5.16 Boundary value problem

Figure 5.17 Cantilever beam bending problem

Figure 5.18 A plane solid under the distributed load {

T

x

,

T

y

} on the traction boundary

S

T

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 in‐plane applied forces

Figure 6.2 Dam structure with plane strain assumption

Figure 6.3 Constant strain triangular (CST) element

Figure 6.4 Inter‐element displacement compatibility of constant strain triangular element

Figure 6.5 Interpolation of displacements in triangular elements

Figure 6.6 Applied surface traction along edge 1‐2

Figure 6.7 Cantilevered plate

Figure 6.8 Four–node rectangular element

Figure 6.9 Four‐node rectangular element

Figure 6.10 Three‐dimensional surface plots of shape functions for a rectangular element; (a)

N

1

(

x

,

y

), (b)

N

2

(

x

,

y

)

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 Thick‐walled cylinder; (a) cross‐section, (b) plane strain model, (c) axisymmetric model

Figure 6.23 Comparison of results using plane strain and axisymmetric models for the thick‐walled 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 One‐dimensional 2‐node linear isoparametric element

Figure 7.2 3‐node 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 1‐D heat conduction model using 3‐node 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 two‐dimensional parent elements

Figure 7.15 Numerical integration of a square element

Figure 7.16 Three rigid‐body modes of plane solids

Figure 7.17 Two extra zero‐energy modes of plane solids

Figure 7.18 Polynomial triangle

Figure 7.19 9‐node Lagrange element in parametric space

Figure 7.20 8‐node serendipity element in parametric space

Figure 7.21 Shape function for node 1 of a 4‐node element

Figure 7.22 5‐node transition element

Figure 7.23 Triangular element by collapsing a 4‐node quadrilateral

Figure 7.24 Triangular element in physical space

Figure 7.25 3‐node isoparametric triangular element

Figure 7.26 Six‐node isoparametric triangular element

Figure 7.27 Four‐node isoparametric tetrahedral element

Figure 7.28 Ten‐node isoparametric tetrahedral element

Figure 7.29 Eight‐node isoparametric hexahedral element

Figure 7.30 U‐shaped beam

Figure 7.31 Plane stress model of U‐shaped beam

Figure 7.32 Deflection and stress distribution in U‐shaped beam

Figure 7.33 Plate with holes

Figure 7.34 Pressure distribution

p

(

ϕ

) 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

2

) 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 h‐adaptive 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 1D spring‐mass system

Figure 8.4 Free vibration of 1D mass‐spring system

Figure 8.5 Vibration of a clamped‐free bar modeled using two elements

Figure 8.6 Free vibration of a clamped‐clamped beam using two beam elements

Figure 8.7 Mode shapes of a clamped‐clamped beam of length 2 m

Figure 8.8 Rigid bodies connected by springs

Figure 8.9 A clamped‐free uniaxial bar subjected to a tip force

F

(

t

)

Figure 8.10 Tip‐displacement of a uniaxial bar in figure 8.9 using the central difference method. The quasi‐static 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) ∆t = 42 μsec, and (b) ∆t = 75 μsec

Figure 8.14 (Top) Impact of a mass on a simply supported beam; (bottom) one‐element FE model of one half of the beam.

Figure 8.15 Impact response of a simply supported beam subjected to central impact. The figure shows the impact 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 Impact response of a beam in figure 8.15 obtained using the mode superposition method

Figure 8.18 One‐dimensional spring‐mass‐dashpot 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 the 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 clamped 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 impact problem

Figure 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 impact problem with different time‐step sizes

Figure 8.32 Stress history of elastic rod impact 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

σ

xx

of the plate model (element size = 0.2 in)

Figure 9.23 Contour plot of

σ

xx

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

σ

xx

Figure 9.40 Patch test for bar elements

Figure 9.41 Stresses at integration points versus node‐averages stresses

Figure 9.42 Converging to the exact solution with mesh refinement

Figure 9.43 Design domain and boundary and loading condition for the bracket

Chapter 10

Figure 10.1 Histogram of failure strengths and allowable strengths

Figure 10.2 Knockdown factor for the B‐basis allowable strength

Figure 10.3 Cantilevered beam design

Figure 10.4 Three‐bar 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 with a hole

Figure 10.7 Design perturbation using isoparametric mapping method

Figure 10.8 Parametric study plot for the cantilevered beam

Figure 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 four‐bar truss

Figure 10.16 List of the submenu in Tools menu (Solver appears in Tools menu)

Figure 10.17 Add‐in dialog box with installed Solver add‐in

Figure 10.18 Excel worksheet for minimum weight design of the four‐bar 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 Ten‐bar truss

Figure 10.25 Geometry of a bracket (unit mm)

Appendix

Figure A.1 Three‐dimensional geometric vector

Figure A.2 Illustration of vector product

Guide

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Introduction to Finite Element Analysis and Design

Second Edition

Nam H. Kim, Bhavani V. Sankar, and Ashok V. Kumar

University of Florida

This edition first published 2018© 2018 John Wiley & Sons Ltd

Edition HistoryJohn Wiley & Sons Ltd (1e, 2008)

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

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Library of Congress Cataloging‐in‐Publication Data

Names: Kim, Nam H., author. | Sankar, Bhavani V., author. | Kumar, Ashok V., author.Title: Introduction to finite element analysis and design / by Nam H. Kim, Bhavani V. Sankar, Ashok V. Kumar.Description: Second edition. | Hoboken, NJ : John Wiley & Sons, 2018. | Includes bibliographical references and index. |Identifiers: LCCN 2018002473 (print) | LCCN 2018007147 (ebook) | ISBN 9781119078746 (pdf) | ISBN 9781119078739 (epub) | ISBN 9781119078722 (cloth)Subjects: LCSH: Finite element method. | Engineering mathematics.Classification: LCC TA347.F5 (ebook) | LCC TA347.F5 K56 2018 (print) | DDC 620.001/51825–dc23LC record available at https://lccn.loc.gov/2018002473

Cover Design: WileyCover Image: Courtesy of Nam H. Kim

To our wives, Jeehyun, Mira, and Gouri

Preface

Finite Element Method (FEM) is a numerical method for solving differential equations that describe many engineering problems. One of the reasons for FEM's popularity is that the method results in computer programs versatile in nature that can solve many practical problems with a small amount of training. Obviously, there is a danger in using computer programs without proper understanding of the theory behind them, and that is one of the reasons to have a thorough understanding of the theory behind FEM.

Many universities teach FEM to students at the junior/senior level. One of the biggest challenges to the instructor is finding a textbook appropriate to the level of students. In the past, FEM was taught only to graduate students who would carry out research in that field. Accordingly, many textbooks focus on theoretical development and numerical implementation of the method. However, the goal of an undergraduate FEM course is to introduce the basic concepts so that the students can use the method efficiently and interpret the results properly. Furthermore, the theoretical aspects of FEM must be presented without too much mathematical niceties. Practical applications through several design projects can help students to understand the method clearly.

This book is suitable for junior/senior level undergraduate students and beginning graduate students in engineering mechanics, mechanical, civil, aerospace, biomedical and industrial engineering as well as researchers and design engineers in the above fields.

The textbook is organized into ten chapters. The Appendix at the end summarizes most mathematical preliminaries that are repeatedly used in the text. The Appendix is by no means a comprehensive mathematical treatment of the subject. Rather, it provides a common notation and the minimum amount of mathematical knowledge that will be required in using the book effectively. This includes basics of matrix algebra, minimization of quadratic functions, and techniques for solving linear equations that are commonly used in commercial finite element programs.

The book begins with the introduction of finite element concepts via the direct stiffness method using spring elements. The concepts of nodes, elements, internal forces, equilibrium, assembly, and applying boundary conditions are presented in detail. The spring element is then extended to the uniaxial bar element without introducing interpolation. The concept of local (elemental) and global coordinates and their transformations and element connectivity tables are introduced via two– and three–dimensional truss elements. Four design projects are provided at the end of the chapter, so that students can apply the method to real life problems. The direct method in Chapter 1 provides a clear physical insight into FEM and is preferred in the beginning stages of learning the principles. However, it is limited in its application in that it can be used to solve one–dimensional problems only.

The direct stiffness method becomes impractical for more realistic problems especially multi‐dimensional problems. In Chapter 2, we introduce more general approaches, such as, the Weighted Residual Methods and, in particular, the Galerkin Method. Similarity to energy methods in solid and structural mechanics problems is discussed. We include a simple 1–D variational formulation in Chapter 2 using boundary value problems. The concept of polynomial approximation and domain discretization is introduced. The formal procedure of finite element analysis is also presented in this chapter. Chapter 2 is written in such way that it can be left out in elementary level courses.

The 1–D formulation is further extended to beams and plane frames in Chapter 3. At this point, the direct method is not useful because the stiffness matrix generated from the direct method cannot provide a clear physical interpretation. Accordingly, we use the principle of minimum potential energy to derive the matrix equation at the element level. The 1–D beam element is extended to 2–D frame element by using coordinate transformation. A 2–D bicycle frame design project is included at the end of this chapter. Buckling of beams and plane frames is included in the revised second edition. First, the concepts of linear buckling of beam is introduced using the Rayleigh-Ritz method. Then the corresponding energy terms are derived in the finite element context.

The finite element formulation is extended to the steady–state heat transfer problem in Chapter 4. Both direct and Galerkin's methods along with convective boundary conditions are included. Two-dimensional heat transfer problems are discussed in the second edition. Practical issues in modeling 2D heat transfer problems are also discussed.

Before proceeding to solid elements in Chapter 6, a review of solid mechanics is provided in Chapter 5. The concepts of stress and strain are presented followed by constitutive relations and equilibrium equations. We limit our interest to linear, isotropic materials in order to make the concepts simple and clear. However, advanced concepts such as transformation of stress and strain, and the eigen value problem for calculating the principal values, are also included. Since, in practice, FEM is used mostly for designing a structure or a mechanical system, failure/yield criteria are also introduced in this chapter.

In Chapter 6, we introduce 2–D solid elements. The governing variational equation is developed using the principle of minimum potential energy. The finite element concepts are explained in detail using only triangular and rectangular elements. Numerical performance of each element is discussed through examples. A new addition to the second edition is the axisymmetric element as it is essentially a plane problem.

The concept of isoparametric mapping is introduce in a separate chapter (Chapter 7) as most practical problems require irregular elements such as linear or higher order quadrilateral elements. Three-dimensional solid elements are introduced in this chapter. Numerical integration and FE modeling practices for isoparametric elements are also included.

Dynamic problems is another addition to the second edition. The concept of free vibration, calculation of natural frequencies and mode shapes, various time integration methods and mode superposition method, are all explained using 1-D structural elements such as uniaxial bars and beams.

In Chapter 9, we discuss traditional finite element analysis procedures, including preliminary analysis, pre-processing, solving matrix equations, and post-processing. Emphasis is on selection of element types, approximating the part geometry, different types of meshing, convergence, and taking advantage of symmetry. A design project involving 2–D analysis is provided at the end of the chapter.

As one of the important goals of FEM is to use the tool for engineering design, the last chapter (Chapter 10) is dedicated to the topic of structural design using FEM. The basic concept of design parameterization and the standard design problem formulation are presented. This chapter is self contained and can be skipped depending on the schedule and content of the course.

Each chapter contains a comprehensive set of homework problems, some of which require commercial FEA programs. A total of nine design projects are provided in the book.

We are thankful to several instructors across the country who used the first edition and provided feedback. We are grateful for their valuable suggestions especially regarding example and exercise problems.

September 2017Nam H. Kim, Bhavani V. Sankar and Ashok V. Kumar

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/kim/finite_element_analysis_design

The website includes:

Programs

Exercise problems

Chapter 1Direct Method – Springs, Bars, and Truss Elements

An ability to predict the behavior of machines and engineering systems in general is of great importance at every stage of engineering processes, including design, manufacture, and operation. Such predictive methodologies are possible because engineers and scientists have made tremendous progress in understanding the physical behavior of materials and structures and have developed mathematical models, albeit approximate, in order to describe their physical behavior. Most often the mathematical models result in algebraic, differential, or integral equations or combinations thereof. Seldom can these equations be solved in closed form, and hence numerical methods are used to obtain solutions. The finite difference method is a classical method that provides approximate solutions to differential equations with reasonable accuracy. There are other methods of solving mathematical equations that are covered in traditional numerical methods courses1.

The finite element method (FEM) is one of the numerical methods for solving differential equations. The FEM, originated in the area of structural mechanics, has been extended to other areas of solid mechanics and later to other fields such as heat transfer, fluid dynamics, and electromagnetism. In fact, FEM has been recognized as a powerful tool for solving partial differential equations and integro‐differential equations, and it has become the numerical method of choice in many engineering and applied science areas. One of the reasons for FEM’s popularity is that the method results in computer programs versatile in nature that can solve many practical problems with the least amount of training. Obviously, there is a danger in using computer programs without proper understanding of the theory behind them, and that is one of the reasons to have a thorough understanding of the theory behind the FEM.

The basic principle of FEM is to divide or discretize the system into a number of smaller elements called finite elements (FEs), to identify the degrees of freedom (DOFs) that describe its behavior, and then to write down the equations that describe the behavior of each element and its interaction with neighboring elements. The element‐level equations are assembled to obtain global equations, often a linear system of equations, which are solved for the unknown DOFs. The phrase finite element refers to the fact that the elements are of a finite size as opposed to the infinitesimal or differential element considered in deriving the governing equations of the system. Another interpretation is that the FE equations deal with a finite number of DOFs as opposed to the infinite number of DOFs of a continuous system.

In general, solutions of practical engineering problems are quite complex, and they cannot be represented using simple mathematical expressions. An important concept of the FEM is that the solution is approximated using simple polynomials, often linear or quadratic, within each element. Since elements are connected throughout the system, the solution of the system is approximated using piecewise polynomials. Such approximation may contain errors when the size of an element is large. As the size of element reduces, however, the approximated solution will converge to the exact solution.

There are three methods that can be used to derive the FE equations of a problem: (a) direct method, (b) variational method, and (c) weighted residual method. The direct method provides a clear physical insight into the FEM and is preferred in the beginning stages of learning the principles. However, it is limited in its application in that it can be used to solve one‐dimensional problems only. The variational method is akin to the methods of calculus of variations and is a powerful tool for deriving the FE equations. However, it requires the existence of a functional, whose minimization results in the solution of the differential equations. The Galerkin method is one of the popular weighted residual methods and is applicable to most problems. If a variational function exists for the problem, then the variational and Galerkin methods yield identical solutions.

In this chapter, we will illustrate the direct method of FE analysis using one‐dimensional elements such as linear spring, uniaxial bar, and truss elements. The emphasis is on construction and solution of the finite element equations and interpretation of the results, rather than the rigorous development of the general principles of the FEM.

1.1 ILLUSTRATION OF THE DIRECT METHOD

Consider a system of rigid bodies connected by springs as shown in figure 1.1. The bodies move only in the horizontal direction. Furthermore, we consider only the static problem and hence the mass effects (inertia) will be ignored. External forces, F2, F3, and F4, are applied on the rigid bodies as shown. The objectives are to determine the displacement of each body, forces in the springs, and support reactions.

Figure 1.1 Rigid bodies connected by springs

We will introduce the principles involved in the FEM through this example. Notice that there is no need to discretize the system as it already consists of discrete elements, namely, the springs. The elements are connected at the nodes. In this case, the rigid bodies are the nodes. Of course, the two walls are also the nodes as they connect to the elements. Numbers inside the little circles mark the nodes. The system of connected elements is called the mesh and is best described using a connectivity table that defines which nodes an element is connected to as shown in table 1.1. Such a connectivity table is included in input files for finite element analysis software to describe the mesh.

Table 1.1Connectivity table for figure 1.1

Element

LN1 (

i

)

LN2 (

j

)

1

1

2

2

2

4

3

2

3

4

1

3

5

3

4

6

4

5

Consider the free‐body diagram of a typical element (e) as shown in figure 1.2. It has two nodes, nodes i and j. They will also be referred to as the first and second node or local node 1 (LN1) and local node 2 (LN2), respectively, as shown in the connectivity table. Assume a coordinate system going from left to right. The convention for first and second nodes is that . The forces acting at the nodes are denoted by and . In this notation, the subscripts denote the node numbers and the superscript the element number. This notation is adopted because multiple elements can be connected at a node, and each element may have different forces at the node. We will refer to them as internal forces. In figure 1.2, the forces are shown in the positive direction. The unknown displacements of nodes i and j are ui and uj, respectively. Note that there is no superscript for u, as the displacement is unique to the node denoted by the subscript. We would like to develop a relationship between the nodal displacements ui and uj and the internal forces and .

Figure 1.2 Spring element (e) connected by node i and node j

The elongation of the spring is denoted by . Then the force of the spring is given by

(1.1)

where k(e) is the spring rate or stiffness of element (e). In this text, the force in the spring, P(e), is referred to as element force. If , then the spring is elongated, and the force in the spring is positive (tension). Otherwise, the spring is in compression. The spring element force is related to the internal force by

(1.2)

Note that the sign of and is determined based on the direction that the force is applied, while the sign of P(e) is determined based on whether the element is in tension or compression. For equilibrium, the sum of the forces acting on element (e) must be equal to zero, i.e.,

(1.3)

Therefore, the two forces are equal, and they are applied in opposite directions. When is positive, the element is in tension, and thus, P(e) is positive.

From eqs. (1.1)–(1.3), we can obtain a relation between the internal forces and the displacements as

(1.4)

Equation (1.4) can be written in matrix forms as:

(1.5)

We also write eq. (1.5) in a shorthand notation as:

or,

(1.6)

where [k(e)] is the element stiffness matrix, {q(e)} is the vector of DOFs associated with element (e), and {f(e)} is the vector of internal forces. Sometimes we will omit the superscript (e) with the understanding that we are dealing with a generic element. Equation (1.6) is called the element equilibrium equation.

The element stiffness matrix [k(e)] has the following properties:

It is square as it relates to the same number of forces as the displacements;

It is symmetric (a consequence of the Betti–Rayleigh Reciprocal theorem in solid and structural mechanics

2

);

It is singular,

i.e

., its determinant is equal to zero, so it cannot be inverted; and

It is positive semidefinite.

Properties 3 and 4 are related to each other, and they have physical significance. Consider eq. (1.6). If the nodal displacements ui and uj of a spring element in a system are given, then it should be possible to predict the force P(e) in the spring from its change in length , and hence the forces {f(e)} acting at its nodes can be predicted. In fact, the internal forces can be computed by performing the matrix multiplication [k(e)]{q(e)}. On the other hand, if the two spring forces are given (they must have equal magnitudes but opposite directions), the nodal displacements cannot be determined uniquely, as a rigid body displacement (equal ui and uj) can be added without affecting the spring force. If [k(e)] were to have an inverse, then it would have been possible to solve for uniquely in violation of the physics. Property 4 has also a physical interpretation, which will be discussed in conjunction with energy methods.

In the next step, we develop a relationship between the internal forces and the known external forces Fi. For example, consider the free‐body diagram of node 3 (or the rigid body in this case) in figure 1.1. The forces acting on the node are the external force F3 and the internal forces from the springs connected to node 3 as shown in figure 1.3.

Figure 1.3 Free‐body diagram of node 3 in the example shown in figure 1.1. The external force, F3, and the forces, , exerted by the springs attached to the node are shown. Note the forces act in the negative direction.

For equilibrium of the node, the sum of the forces acting on the node should be equal to zero:

or,

(1.7)

where ie is the number of elements connected to node i, and ND is the total number of nodes in the model. Equation (1.7) is the equilibrium between externally applied forces at a node and internal forces from connected elements. If there is no externally applied force at a node, then the sum of internal forces at the node must be zero. Such equations can be written for each node including the boundary nodes, such as nodes 1 and 5 in figure 1.1. The internal forces in eq. (1.7) can be replaced by the unknown DOFs {q} by using eq. (1.6). For example, the force equilibrium for the springs in figure 1.1 can be written as

(1.8)

This will result in ND number of linear equations for the ND number of DOFs:

(1.9)

Or, in shorthand notation where [Ks] is the structural stiffness matrix, {Qs} is the vector of displacements of all nodes in the model, and {Fs} is the vector of external forces, including the unknown reactions. The expanded form of eq. (1.9) is given in eq. (1.10) below:

or,

(1.10)

The properties of the structural stiffness matrix [Ks] are similar to that of the element stiffness matrix: square, symmetric, singular, and positive semi‐definite. In addition, when nodes are numbered properly, [Ks] will be a banded matrix. It should be noted that when the boundary displacements in {Qs} are known (usually equal to zero3), the corresponding forces in {Fs} are unknown reactions. In the present illustration, , and corresponding forces (reactions) F1 and F5 are unknown. It should also be noted that when displacements in {Qs} are unknown, the corresponding forces in {Fs} should be known (either a given value or zero when no force is applied).

We will impose the boundary conditions as follows. First, we ignore the equations for which the RHS forces are unknown and strike out the corresponding rows in [Ks]. This is called striking the rows. Then we eliminate the columns in [Ks] that are multiplied by the zero values of displacements of the boundary nodes. This is called striking the columns. It may be noted that if the nth row is eliminated (struck), then the nth column will also be eliminated (struck). This process results in a system of equations given by , where [K] is the global stiffness matrix, {Q} is the vector of unknown DOFs, and {F} is the vector of known forces. The global stiffness matrix will be square, symmetric, and positive definite and hence nonsingular. Usually [K] will also be banded. In large systems, that is, in models with large numbers of DOFs, [K] will be a sparse matrix with a small proportion of nonzero numbers in a diagonal band.

After striking the rows and columns corresponding to zero DOFs (u1 and u5) in eq. (1.10), we obtain the global equations as follows:

or,

(1.11)