

1  (27) 


1  (1) 


2  (2) 

Introduction to Matrix Notation 


4  (2) 


6  (1) 

General Steps of the Finite Element Method 


7  (8) 

Applications of the Finite Element Method 


15  (4) 

Advantages of the Finite Element Method 


19  (4) 

Computer Programs for the Finite Element Method 


23  (5) 


24  (3) 


27  (1) 

Introduction to the Stiffness (Displacement) Method 


28  (37) 


28  (1) 

Definition of the Stiffness Matrix 


28  (1) 

Derivation of the Stiffness Matrix for a Spring Element 


29  (5) 

Example of a Spring Assemblage 


34  (3) 

Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) 


37  (2) 


39  (13) 

Potential Energy Approach to Derive Spring Element Equations 


52  (13) 


60  (1) 


61  (4) 

Development of Truss Equations 


65  (86) 


65  (1) 

Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates 


66  (6) 

Selecting Approximation Functions for Displacements 


72  (3) 

Transformation of Vectors in Two Dimensions 


75  (3) 


78  (4) 

Computation of Stress for a Bar in the xy Plane 


82  (2) 

Solution of a Plane Truss 


84  (8) 

Transformation, Matrix and Stiffness Matrix for a Bar in ThreeDimensional Space 


92  (8) 

Use of Symmetry in Structure 


100  (3) 

Inclined, or Skewed, Supports 


103  (6) 

Potential Energy Approach to Derive Bar Element Equations 


109  (11) 

Comparison of Finite Element Solution to Exact Solution for Bar 


120  (4) 

Galerkin's Residual Method and Its Use to Derive the OneDimensional Bar Element Equations 


124  (3) 

Other Residual Methods and Their Application to a OneDimensional Bar Problem 


127  (24) 


132  (1) 


132  (19) 

Development of Beam Equations 


151  (63) 


151  (1) 


152  (9) 

Example of Assemblage of Beam Stiffness Matrices 


161  (2) 

Examples of Beam Analysis Using the Direct Stiffness Method 


163  (12) 


175  (13) 

Comparison of the Finite Element Solution to the Exact Solution for a Beam 


188  (6) 

Beam Element with Nodal Hinge 


194  (5) 

Potential Energy Approach to Derive Beam Element Equations 


199  (2) 

Galerkin's Method for Deriving Beam Element Equations 


201  (13) 


203  (1) 


204  (10) 


214  (90) 


214  (1) 

TwoDimensional Arbitrarily Oriented Beam Element 


214  (4) 

Rigid Plane Frame Examples 


218  (19) 

Inclined or Skewed SupportsFrame Element 


237  (1) 


238  (17) 

Beam Element Arbitrarily Oriented in Space 


255  (14) 

Concept of Substructure Analysis 


269  (35) 


275  (1) 


275  (29) 

Development of the Plane Stress and Plane Strain Stiffness Equations 


304  (46) 


304  (1) 

Basic Concepts of Plane Stress and Plane Strain 


305  (5) 

Derivation of the ConstantStrain Triangular Element Stiffness Matrix and Equations 


310  (14) 

Treatment of Body and Surfaces Forces 


324  (5) 

Explicit Expression for the ConstantStrain Triangle Stiffness Matrix 


329  (2) 

Finite Element Solution of a Plane Stress Problem 


331  (19) 


342  (1) 


343  (7) 

Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis 


350  (48) 


350  (1) 


350  (13) 

Equilibrium and Compatibility of Finite Element Results 


363  (4) 


367  (1) 

Interpretation of Stresses 


368  (1) 


369  (5) 

Flowchart for the Solution of Plane Stress/Strain Problems 


374  (1) 

Computer Program Assisted StepbyStep Solution, Other Models, and Results for Plane Stress/Strain Problems 


374  (24) 


381  (1) 


382  (16) 

Development of the LinearStrain Triangle Equations 


398  (14) 


398  (1) 

Derivation of the LinearStrain Triangular Element Stiffness Matrix and Equations 


398  (5) 

Example LST Stiffness Determination 


403  (3) 


406  (6) 


409  (1) 


409  (3) 


412  (31) 


412  (1) 

Derivation of the Stiffness Matrix 


412  (10) 

Solution of an Axisymmetric Pressure Vessel 


422  (6) 

Applications of Axisymmetric Elements 


428  (15) 


433  (1) 


434  (9) 

Isoparametric Formulation 


443  (47) 


443  (1) 

Isoparametric Formulation of the Bar Element Stiffness Matrix 


444  (5) 

Rectangular Plane Stress Element 


449  (3) 

Isoparametric Formulation of the Plane Element Stiffness Matrix 


452  (11) 

Gaussian and NewtonCotes Quadrature (Numerical Integration) 


463  (6) 

Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature 


469  (6) 

HigherOrder Shape Functions 


475  (15) 


484  (1) 


484  (6) 

ThreeDimensional Stress Analysis 


490  (24) 


490  (1) 

ThreeDimensional Stress and Strain 


490  (3) 


493  (8) 

Isoparametric Formulation 


501  (13) 


508  (1) 


509  (5) 


514  (20) 


514  (1) 

Basic Concepts of Plate Bending 


514  (5) 

Derivation of a Plate Bending Element Stiffness Matrix and Equations 


519  (4) 

Some Plate Element Numerical Comparisons 


523  (1) 

Computer Solution for a Plate Bending Problem 


524  (10) 


528  (1) 


529  (5) 

Heat Transfer and Mass Transport 


534  (59) 


534  (1) 

Derivation of the Basic Differential Equation 


535  (3) 

Heat Transfer with Convection 


538  (1) 

Typical Units; Thermal Conductivities, K; and HeatTransfer Coefficients, h 


539  (1) 

OneDimensional Finite Element Formulation Using a Variational Method 


540  (15) 

TwoDimensional Finite Element Formulation 


555  (9) 


564  (2) 

ThreeDimensional Heat Transfer Finite Element Formulation 


566  (3) 

OneDimensional Heat Transfer with Mass Transport 


569  (1) 

Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method 


569  (5) 

Flowchart and Examples of a HeatTransfer Program 


574  (19) 


577  (1) 


577  (16) 


593  (24) 


593  (1) 

Derivation of the Basic Differential Equations 


594  (4) 

OneDimensional Finite Element Formulation 


598  (8) 

TwoDimensional Finite Element Formulation 


606  (5) 

Flowchart and Example of a FluidFlow Program 


611  (6) 


612  (1) 


613  (4) 


617  (30) 


617  (1) 

Formulation of the Thermal Stress Problem and Examples 


617  (30) 


640  (1) 


641  (6) 

Structural Dynamics and TimeDependent Heat Transfer 


647  (61) 


647  (1) 

Dynamics of a SpringMass System 


647  (2) 

Direct Derivation of the Bar Element Equations 


649  (4) 

Numerical Integration in Time 


653  (12) 

Natural Frequencies of a OneDimensional Bar 


665  (4) 

TimeDependent OneDimensional Bar Analysis 


669  (5) 

Beam Element Mass Matrices and Natural Frequencies 


674  (7) 

Trusts, Plane Frame, Plane Stress/Strain Axisymmetric, and Solid Element Mass Matrices 


681  (5) 

TimeDependent Heat Transfer 


686  (7) 

Computer Program Example Solutions for Structural Dynamics 


693  (15) 


702  (1) 


702  (6) 

Appendix A Matrix Algebra 


708  (14) 


708  (1) 


708  (1) 


709  (7) 

Cofactor or Adjoint Method to Determine the Inverse of a Matrix 


716  (2) 

Inverse of a Matrix by Row Reduction 


718  (4) 


720  (1) 


720  (2) 

Appendix B Methods for Solution of Simultaneous Linear Equations 


722  (22) 


722  (1) 

General Form of the Equations 


722  (1) 

Uniqueness, Nonuniqueness, and Nonexistence of Solution 


723  (1) 

Methods for Solving Linear Algebraic Equations 


724  (11) 

BandedSymmetric Matrices, Bandwidth, Skyline, and Wavefront Methods 


735  (9) 


741  (1) 


742  (2) 

Appendix C Equations from Elasticity Theory 


744  (8) 


744  (1) 

Differential Equations of Equilibrium 


744  (2) 

Strain/Displacement and Compatibility Equations 


746  (2) 

Stress/Strain Relationships 


748  (4) 


751  (1) 

Appendix D Equivalent Nodal Forces 


752  (3) 


752  (3) 

Appendix E Principle of Virtual Work 


755  (4) 


758  (1) 

Appendix F Properties of Structural Steel and Aluminum Shapes 


759  (14) 
Answers to Selected Problems 

773  (26) 
Index 

799  