1st Edition

# The Finite Element Method for Mechanics of Solids with ANSYS Applications

**Also available as eBook on:**

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.

**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

1.7.2.1 General Beam Equations

1.7.3 Plates and Shells

1.7.4 Two- or Three-Dimensional Solids

1.8 Problems

References

Bibliography

**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

References

Bibliography

**3.1 Finite Element Approximation**

Chapter 3: Finite Element Method for Linear Elasticity

Chapter 3: Finite Element Method for Linear Elasticity

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

Bibliography

**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

**5.1 Four-Node Plane Rectangle**

Chapter 5: The Quadrilateral and the Hexahedron

Chapter 5: The Quadrilateral and the Hexahedron

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

**Term**

*λ*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

References

Bibliography

**6.1 General Remarks**

Chapter 6: Errors and Convergence of Finite Element Solution

Chapter 6: Errors and Convergence of Finite Element Solution

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

References

**7.1 Differential Equations and Virtual Work**

Chapter 7: Heat Conduction in Elastic Solids

Chapter 7: Heat Conduction in Elastic Solids

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

Bibliography

**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

Bibliography

**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

10.4.2.1 One-Element Model

10.4.2.2 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

10.7.3.1 Implicit Methods in General

10.7.3.2 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

Bibliography

**11.1 Fracture Criterion**

Chapter 11: Linear Elastic Fracture Mechanics

Chapter 11: Linear Elastic Fracture Mechanics

11.1.1 Analysis of Sheet

11.1.2 Fracture Modes

11.1.2.1 Mode I

11.1.2.2 Mode II

11.1.2.3 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

References

Bibliography

**12.1 Geometry of Deformation**

Chapter 12: Plates and Shells

Chapter 12: Plates and Shells

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

References

Bibliography

**13.1 Theory of Large Deformations**

Chapter 13: Large Deformations

Chapter 13: 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

13.1.10.1. Volumetric Data

13.1.10.2. Tensile Test

13.1.10.3. 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

References

Bibliography

**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

14.2.1.1 Load

*F*> 0 Is Applied with

**y***F*= 0

**x**14.2.1.2 Loads Are Ramped Up Together:

*F*= 27

**x***α*,

*F*= 12.8

**y***α*

14.2.2 Lagrange Multiplier, No Friction Force

14.2.2.1 Stick Condition

14.2.2.2 Slip Condition

14.2.3 Lagrange Multiplier, with Friction

14.2.3.1 Stick Condition

14.2.3.2 Slip Condition

14.2.4 Penalty Method

14.2.4.1 Stick Condition

14.2.4.2 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

References

Bibliography

**Chapter 15: ANSYS APDL Examples**

15.1 ANSYS Instructions

15.1.1 ANSYS File Names

15.1.2 Graphic Window Controls

15.1.2.1 Graphics Window Logo

15.1.2.2 Display of Model

15.1.2.3 Display of Deformed and Undeformed Shape White on White

15.1.2.4 Adjusting Graph Colors

15.1.2.5 Printing from Windows Version of ANSYS

15.1.2.6 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

Bibliography

**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

Bibliography

Index

Index

"… clearly written and addresses theory and solution of numerous example problems with the ANSYS software, including both ADPL and Workbench modules. … a useful reference for practicing engineers and scientists in industry and academia."

—John D. Clayton, Ph.D., A. James Clark School of Engineering, University of Maryland, College Park, USA