The Finite Element Method for Mechanics of Solids with ANSYS Applications


The Finite Element Method for Mechanics of Solids with ANSYS Applications

While the finite element method (FEM) has become the standard technique used to solve static and dynamic problems associated with structures and machines, ANSYS software has developed into the engineer’s software of choice to model and numerically solve those problems.

An invaluable tool to help engineers master and optimize analysis, The Finite Element Method for Mechanics of Solids with ANSYS Applications explains the foundations of FEM in detail, enabling engineers to use it properly to analyze stress and interpret the output of a finite element computer program such as ANSYS.

Illustrating presented theory with a wealth of practical examples, this book covers topics including:

  • Essential background on solid mechanics (including small- and large-deformation elasticity, plasticity, and viscoelasticity) and mathematics
  • Advanced finite element theory and associated fundamentals, with examples
  • Use of ANSYS to derive solutions for problems that deal with vibration, wave propagation, fracture mechanics, plates and shells, and contact

Totally self-contained, this text presents step-by-step instructions on how to use ANSYS Parametric Design Language (APDL) and the ANSYS Workbench to solve problems involving static/dynamic structural analysis (both linear and non-linear) and heat transfer, among other areas. It will quickly become a welcome addition to any engineering library, equally useful to students and experienced engineers alike.


  • Provides strong coverage of FEM concepts, applications, and interpretation of output
  • Uses carefully explained ANSYS examples to show the use of a specific FEM program
  • Covers basic mechanics concepts as well as finite element methods
  • Includes an appendix providing ANSYS commands and procedures
  • Contains chapter problems
  • Demonstrates strong integration of mechanics concepts
  • Presents a chapter on FEM for fluid mechanics and heat transfer

Solutions manual available with qualifying course adoption

Table of Contents

Chapter 1: Finite Element Concepts
1.1 Introduction

1.2 Direct Stiffness Method
1.2.1 Merging the Element Stiffness Matrices
1.2.2 Augmenting the Element Stiffness Matrix
1.2.3 Stiffness Matrix Is Banded
1.3 The Energy Method
1.4 Truss Example
1.5 Axially Loaded Rod Example
1.5.1 Augmented Matrices for the Rod
1.5.2 Merge of Element Matrices for the Rod
1.6 Force Method
1.7 Other Structural Components
1.7.1 Space Truss
1.7.2 Beams and Frames General Beam Equations
1.7.3 Plates and Shells
1.7.4 Two- or Three-Dimensional Solids
1.8 Problems

Chapter 2: Linear Elasticity
2.1 Basic Equations

2.1.1 Geometry of Deformation
2.1.2 Balance of Momentum
2.1.3 Virtual Work
2.1.4 Constitutive Relations
2.1.5 Boundary Conditions and Initial Conditions
2.1.6 Incompressible Materials
2.1.7 Plane Strain
2.1.8 Plane Stress
2.1.9 Tensile Test
2.1.10 Pure Shear
2.1.11 Pure Bending
2.1.12 Bending and Shearing
2.1.13 Properties of Solutions
2.1.14 A Plane Stress Example with a Singularity in Stress
2.2 Potential Energy
2.2.1 Proof of Minimum Potential Energy
2.3 Matrix Notation
2.4 Axially Symmetric Deformations
2.4.1 Cylindrical Coordinates
2.4.2 Axial Symmetry
2.4.3 Plane Stress and Plane Strain
2.5 Problems

Chapter 3: Finite Element Method for Linear Elasticity
3.1 Finite Element Approximation

3.1.1 Potential Energy
3.1.2 Finite Element Equations
3.1.3 Basic Equations in Matrix Notation
3.1.4 Basic Equations Using Virtual Work
3.1.5 Underestimate of Displacements
3.1.6 Nondimensional Equations
3.1.7 Uniaxial Stress
3.2 General Equations for an Assembly of Elements
3.2.1 Generalized Variational Principle
3.2.2 Potential Energy
3.2.3 Hybrid Displacement Functional
3.2.4 Hybrid Stress and Complementary Energy
3.2.5 Mixed Methods of Analysis
3.3 Nearly Incompressible Materials
3.3.1 Nearly Incompressible Plane Strain

Chapter 4: The Triangle and the Tetrahedron
4.1 Linear Functions over a Triangular Region

4.2 Triangular Element for Plane Stress and Plane Strain
4.3 Plane Quadrilateral from Four Triangles
4.3.1 Square Element Formed from Four Triangles
4.4 Plane Stress Example: Short Beam
4.4.1 Extrapolation of the Solution
4.5 Linear Strain Triangles
4.6 Four-Node Tetrahedron
4.7 Ten-Node Tetrahedron
4.8 Problems

Chapter 5: The Quadrilateral and the Hexahedron
5.1 Four-Node Plane Rectangle

5.1.1 Stress Calculations
5.1.2 Plane Stress Example: Pure Bending
5.1.3 Plane Strain Example: Bending with Shear
5.1.4 Plane Stress Example: Short Beam
5.2 Improvements to Four-Node Quadrilateral
5.2.1 Wilson–Taylor Quadrilateral
5.2.2 Enhanced Strain Formulation
5.2.3 Approximate Volumetric Strains
5.2.4 Reduced Integration on the κ Term
5.2.5 Reduced Integration on the λ Term
5.2.6 Uniform Reduced Integration
5.2.7 Example Using Improved Elements
5.3 Numerical Integration
5.4 Coordinate Transformations
5.5 Isoparametric Quadrilateral
5.5.1 Wilson–Taylor Element
5.5.2 Three-Node Triangle as a Special Case of Rectangle
5.6 Eight-Node Quadrilateral
5.6.1 Nodal Loads
5.6.2 Plane Stress Example: Pure Bending
5.6.3 Plane Stress Example: Bending with Shear
5.6.4 Plane Stress Example: Short Beam
5.6.5 General Quadrilateral Element
5.7 Eight-Node Block
5.8 Twenty-Node Solid
5.9 Singularity Element
5.10 Mixed U–P Elements
5.10.1 Plane Strain
5.10.2 Alternative Formulation for Plane Strain
5.10.3 3D Elements
5.11 Problems

Chapter 6: Errors and Convergence of Finite Element Solution
6.1 General Remarks

6.2 Element Shape Limits
6.2.1 Aspect Ratio
6.2.2 Parallel Deviation for a Quadrilateral
6.2.3 Large Corner Angle
6.2.4 Jacobian Ratio
6.3 Patch Test
6.3.1 Wilson–Taylor Quadrilateral

Chapter 7: Heat Conduction in Elastic Solids
7.1 Differential Equations and Virtual Work

7.2 Example Problem: One-Dimensional Transient Heat Flux
7.3 Example: Hollow Cylinder
7.4 Problems

Chapter 8: Finite Element Method for Plasticity
8.1 Theory of Plasticity

8.1.1 Tensile Test
8.1.2 Plane Stress
8.1.3 Summary of Plasticity
8.2 Finite Element Formulation for Plasticity
8.2.1 Fundamental Solution
8.2.2 Iteration to Improve the Solution
8.3 Example: Short Beam
8.4 Problems

Chapter 9: Viscoelasticity
9.1 Theory of Linear Viscoelasticity

9.1.1 Recurrence Formula for History
9.1.2 Viscoelastic Example
9.2 Finite Element Formulation for Viscoelasticity
9.2.1 Basic Step-by-Step Solution Method
9.2.2 Step-by-Step Calculation with Load Correction
9.2.3 Plane Strain Example
9.3 Problems

Chapter 10: Dynamic Analyses
10.1 Dynamical Equations

10.1.1 Lumped Mass
10.1.2 Consistent Mass
10.2 Natural Frequencies
10.2.1 Lumped Mass
10.2.2 Consistent Mass
10.3 Mode Superposition Solution
10.4 Example: Axially Loaded Rod
10.4.1 Exact Solution for Axially Loaded Rod
10.4.2 Finite Element Model One-Element Model Two-Element Model
10.4.3 Mode Superposition for Continuum Model of the Rod
10.5 Example: Short Beam
10.6 Dynamic Analysis with Damping
10.6.1 Viscoelastic Damping
10.6.2 Viscous Body Force
10.6.3 Analysis of Damped Motion by Mode Superposition
10.7 Numerical Solution of Differential Equations
10.7.1 Constant Average Acceleration
10.7.2 General Newmark Method
10.7.3 General Methods Implicit Methods in General Explicit Methods in General
10.7.4 Stability Analysis of Newmark’s Method
10.7.5 Convergence, Stability, and Error
10.7.6 Example: Numerical Integration for Axially Loaded Rod
10.8 Example: Analysis of Short Beam
10.9 Problems

Chapter 11: Linear Elastic Fracture Mechanics
11.1 Fracture Criterion

11.1.1 Analysis of Sheet
11.1.2 Fracture Modes Mode I Mode II Mode III
11.2 Determination of K by Finite Element Analysis
11.2.1 Crack Opening Displacement Method
11.3 J-Integral for Plane Regions
11.4 Problems

Chapter 12: Plates and Shells
12.1 Geometry of Deformation

12.2 Equations of Equilibrium
12.3 Constitutive Relations for an Elastic Material
12.4 Virtual Work
12.5 Finite Element Relations for Bending
12.6 Classical Plate Theory
12.7 Plate Bending Example
12.8 Problems

Chapter 13: Large Deformations
13.1 Theory of Large Deformations

13.1.1 Virtual Work
13.1.2 Elastic Materials
13.1.3 Mooney–Rivlin Model of an Incompressible Material
13.1.4 Generalized Mooney–Rivlin Model
13.1.5 Polynomial Formula
13.1.6 Ogden’s Function
13.1.7 Blatz–Ko Model
13.1.8 Logarithmic Strain Measure
13.1.9 Yeoh Model
13.1.10 Fitting Constitutive Relations to Experimental Data Volumetric Data Tensile Test Biaxial Test
13.2 Finite Elements for Large Displacements
13.2.1 Lagrangian Formulation
13.2.2 Basic Step-by-Step Analysis
13.2.3 Iteration Procedure
13.2.4 Updated Reference Configuration
13.2.5 Example I
13.2.6 Example II
13.3 Structure of Tangent Modulus
13.4 Stability and Buckling
13.4.1 Beam–Column
13.5 Snap-Through Buckling
13.5.1 Shallow Arch
13.6 Problems

Chapter 14: Constraints and Contact
14.1 Application of Constraints

14.1.1 Lagrange Multipliers
14.1.2 Perturbed Lagrangian Method
14.1.3 Penalty Functions
14.1.4 Augmented Lagrangian Method
14.2 Contact Problems
14.2.1 Example: A Truss Contacts a Rigid Foundation Load Fy > 0 Is Applied with Fx = 0 Loads Are Ramped Up Together: Fx = 27α, Fy = 12.8α
14.2.2 Lagrange Multiplier, No Friction Force Stick Condition Slip Condition
14.2.3 Lagrange Multiplier, with Friction Stick Condition Slip Condition
14.2.4 Penalty Method Stick Condition Slip Condition
14.3 Finite Element Analysis
14.3.1 Example: Contact of a Cylinder with a Rigid Plane
14.3.2 Hertz Contact Problem
14.4 Dynamic Impact
14.5 Problems

Chapter 15: ANSYS APDL Examples
15.1 ANSYS Instructions

15.1.1 ANSYS File Names
15.1.2 Graphic Window Controls Graphics Window Logo Display of Model Display of Deformed and Undeformed Shape White on White Adjusting Graph Colors Printing from Windows Version of ANSYS Some Useful Notes
15.2 ANSYS Elements SURF153, SURF154
15.3 Truss Example
15.4 Beam Bending
15.5 Beam with a Distributed Load
15.6 One Triangle
15.7 Plane Stress Example Using Triangles
15.8 Cantilever Beam Modeled as Plane Stress
15.9 Plane Stress: Pure Bending
15.10 Plane Strain Bending Example
15.11 Plane Stress Example: Short Beam
15.12 Sheet with a Hole
15.12.1 Solution Procedure
15.13 Plasticity Example
15.14 Viscoelasticity Creep Test
15.15 Viscoelasticity Example
15.16 Mode Shapes and Frequencies of a Rod
15.17 Mode Shapes and Frequencies of a Short Beam
15.18 Transient Analysis of Short Beam
15.19 Stress Intensity Factor by Crack Opening Displacement
15.20 Stress Intensity Factor by J-Integral
15.21 Stretching of a Nonlinear Elastic Sheet
15.22 Nonlinear Elasticity: Tensile Test
15.23 Column Buckling
15.24 Column Post-Buckling
15.25 Snap-Through
15.26 Plate Bending Example
15.27 Clamped Plate
15.28 Gravity Load on a Cylindrical Shell
15.29 Plate Buckling
15.30 Heated Rectangular Rod
15.31 Heated Cylindrical Rod
15.32 Heated Disk
15.33 Truss Contacting a Rigid Foundation
15.34 Compression of a Rubber Cylinder between Rigid Plates
15.35 Hertz Contact Problem
15.36 Elastic Rod Impacting a Rigid Wall
15.37 Curve Fit for Nonlinear Elasticity Using Blatz–Ko Model
15.38 Curve Fit for Nonlinear Elasticity Using Polynomial Model

Chapter 16: ANSYS Workbench
16.1 Two- and Three-Dimensional Geometry

16.2 Stress Analysis
16.3 Short Beam Example
16.3.1 Short Beam Geometry
16.3.2 Short Beam, Static Loading
16.3.3 Short Beam, Transient Analysis
16.4 Filleted Bar Example
16.5 Sheet with a Hole



The Finite Element Method for Mechanics of Solids with ANSYS Applications
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