

1  (25) 


1  (1) 


2  (1) 

Introduction to Matrix Notation 


3  (3) 


6  (1) 

General Steps of the Finite Element Method 


6  (7) 

Applications of the Finite Element Method 


13  (5) 

Advantages of the Finite Element Method 


18  (1) 

Computer Programs for the Finite Element Method 


19  (7) 


22  (3) 


25  (1) 

Introduction to the Stiffness (Displacement) Method 


26  (37) 


26  (1) 

Definition of the Stiffness Matrix 


26  (1) 

Derivation of the Stiffness Matrix for a Spring Element 


27  (5) 

Example of a Spring Assemblage 


32  (3) 

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


35  (2) 


37  (13) 

Potential Energy Approach to Derive Spring Element Equations 


50  (13) 


58  (1) 


59  (4) 

Development of Truss Equations 


63  (74) 


63  (1) 

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


63  (6) 

Selecting Approximation Functions for Displacements 


69  (2) 

Transformation of Vectors in Two Dimensions 


71  (3) 


74  (4) 

Computation of Stress for a Bar in the xy Plane 


78  (2) 

Solution of a Plane Truss 


80  (7) 

Transformation Matrix and Stiffness Matrix for a Bar in ThreeDimensional Space 


87  (5) 

Use of Symmetry in Structure 


92  (3) 

Inclined, or Skewed, Supports 


95  (6) 

Potential Energy Approach to Derive Bar Element Equations 


101  (11) 

Comparison of Finite Element Solution to Exact Solution for Bar 


112  (4) 

Galerkin's Residual Method and Its Application to a OneDimensional Bar 


116  (21) 


119  (1) 


120  (17) 

Development of Beam Equations 


137  (51) 


137  (1) 


138  (5) 

Example of Assemblage of Beam Stiffness Matrices 


143  (2) 

Examples of Beam Analysis Using the Direct Stiffness Method 


145  (9) 


154  (11) 

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


165  (6) 

Beam Element with Nodal Hinge 


171  (5) 

Potential Energy Approach to Derive Beam Element Equations 


176  (3) 

Galerkin's Method for Deriving Beam Element Equations 


179  (9) 


181  (1) 


181  (7) 


188  (76) 


188  (1) 

TwoDimensional Arbitrarily Oriented Beam Element 


188  (4) 

Rigid Plane Frame Examples 


192  (19) 

Inclined or Skewed SupportsFrame Element 


211  (1) 


212  (17) 

Beam Element Arbitrarily Oriented in Space 


229  (5) 

Concept of Substructure Analysis 


234  (30) 


240  (1) 


240  (24) 

Development of the Plane Stress and Plane Strain Stiffness Equations 


264  (43) 


264  (1) 

Basic Concepts of Plane Stress and Plane Strain 


265  (5) 

Derivation of the ConstantStrain Triangular Element Stiffness Matrix and Equations 


270  (14) 

Treatment of Body and Surface Forces 


284  (5) 

Explicit Expression for the ConstantStrain Triangle Stiffness Matrix 


289  (2) 

Finite Element Solution of a Plane Stress Problem 


291  (16) 


301  (1) 


301  (6) 

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


307  (37) 


307  (1) 


308  (10) 

Equilibrium and Compatibility of Finite Element Results 


318  (2) 


320  (1) 

Interpretation of Stresses 


321  (2) 


323  (4) 

Flowchart for the Solution of Plane Stress/Strain Problems 


327  (1) 

Computer Program Results for Some Plane Stress/Strain Problems 


328  (16) 


331  (1) 


332  (12) 

Development of the LinearStrain Triangle Equations 


344  (14) 


344  (1) 

Derivation of the LinearStrain Triangular Element Stiffness Matrix and Equations 


344  (5) 

Example LST Stiffness Determination 


349  (3) 


352  (6) 


354  (1) 


355  (3) 


358  (28) 


358  (1) 

Derivation of the Stiffness Matrix 


358  (10) 

Solution of an Axisymmetric Pressure Vessel 


368  (8) 

Applications of Axisymmetric Elements 


376  (10) 


380  (1) 


381  (5) 

Isoparametric Formulation 


386  (35) 


386  (1) 

Isoparametric Formulation of the Bar Element Stiffness Matrix 


386  (6) 

Rectangular Plane Stress Element 


392  (3) 

Isoparametric Formulation of the Plane Element Stiffness Matrix 


395  (9) 

Gaussian Quadrature (Numerical Integration) 


404  (3) 

Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature 


407  (6) 

HigherOrder Shape Functions 


413  (8) 


417  (1) 


417  (4) 

ThreeDimensional Stress Analysis 


421  (20) 


421  (1) 

ThreeDimensional Stress and Strain 


421  (2) 


423  (7) 

Isoparametric Formulation 


430  (11) 


436  (1) 


436  (5) 


441  (17) 


441  (1) 

Basic Concepts of Plate Bending 


441  (4) 

Derivation of a Plate Bending Element Stiffness Matrix and Equations 


445  (5) 

Some Plate Element Numerical Comparisions 


450  (2) 

Computer Solution for a Plate Bending Problem 


452  (6) 


454  (1) 


455  (3) 

Heat Transfer and Mass Transport 


458  (50) 


458  (1) 

Derivation of the Basic Differential Equation 


459  (3) 

Heat Transfer with Convection 


462  (1) 

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


463  (1) 

OneDimensional Finite Element Formulation Using a Variational Method 


464  (14) 

TwoDimensional Finite Element Formulation 


478  (9) 


487  (3) 

OneDimensional Heat Transfer with Mass Transport 


490  (1) 

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


491  (4) 

Flowchart and Examples of a HeatTransfer Program 


495  (13) 


499  (1) 


499  (9) 


508  (24) 


508  (1) 

Derivation of the Basic Differential Equations 


508  (5) 

OneDimensional Finite Element Formulation 


513  (8) 

TwoDimensional Finite Element Formulation 


521  (5) 

Flowchart and Example of a FluidFlow Program 


526  (6) 


527  (1) 


528  (4) 


532  (27) 


532  (1) 

Formulation of the Thermal Stress Problem and Examples 


532  (27) 


553  (1) 


554  (5) 

Structural Dynamics and TimeDependent Heat Transfer 


559  (57) 


559  (1) 

Dynamics of a SpringMass System 


559  (2) 

Direct Derivation of the Bar Element Equations 


561  (4) 

Numerical Integration in Time 


565  (12) 

Natural Frequencies of a OneDimensional Bar 


577  (4) 

TimeDependent OneDimensional Bar Analysis 


581  (5) 

Beam Element Mass Matrices and Natural Frequencies 


586  (5) 

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


591  (4) 

TimeDependent Heat Transfer 


595  (7) 

Computer Program Example Solutions for Structural Dynamics 


602  (14) 


609  (1) 


610  (6) 
Appendix A Matrix Algebra 

616  (14) 


616  (12) 

A.1 Definition of a Matrix 


616  (1) 


617  (7) 

A.3 Cofactor or Adjoint Method to Determine the Inverse of a Matrix 


624  (2) 

A.4 Inverse of a Matrix by Row Reduction 


626  (2) 


628  (1) 


628  (2) 
Appendix B Methods for Solution of Simultaneous Linear Equations 

630  (22) 


630  (19) 

B.1 General Form of the Equations 


630  (1) 

B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solution 


631  (1) 

B.3 Methods for Solving Linear Algebraic Equations 


632  (11) 

B.4 BandedSymmetric Matrices, Bandwidth, Skyline, and Wavefront Methods 


643  (6) 


649  (1) 


650  (2) 
Appendix C Equations from Elasticity Theory 

652  (8) 


652  (7) 

C.1 Differential Equations of Equilibrium 


652  (2) 

C.2 Strain/Displacement and Compatibility Equations 


654  (2) 

C.3 Stress/Strain Relationships 


656  (3) 


659  (1) 
Appendix D Equivalent Nodal Forces 

660  (3) 


660  (3) 
Appendix E Principle of Virtual Work 

663  (4) 


666  (1) 
Answers to Selected Problems 

667  (22) 
Index 

689  