1st Edition

The Finite Element Method for Mechanics of Solids with ANSYS Applications

By Ellis H. Dill Copyright 2011
    508 Pages 213 B/W Illustrations
    by CRC Press

    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 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



    Ellis H. Dill

    "… 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