Essentials of the finite element method : for mechanical and structural engineers /
Dimitrios G. Pavlou.
- xv, 484 pages : illustrations (chiefly color) ; 23 cm.
Includes bibliographical references and index.
Machine-generated contents note: ch. 1 An Overview of the Finite Element Method -- 1.1. What Are Finite Elements? -- 1.2. Why Finite Element Method Is Very Popular? -- 1.3. Main Advantages of Finite Element Method -- 1.4. Main Disadvantages of Finite Element Method -- 1.5. What Is Structural Matrix? -- 1.5.1. Stiffness Matrix -- 1.5.2. Transfer Matrix -- 1.6. What Are the Steps to be Followed for Finite Element Method Analysis of Structure? -- 1.6.1. Step 1. Discretize or Model the Structure -- 1.6.2. Step 2. Define the Element Properties -- 1.6.3. Step 3. Assemble the Element Structural Matrices -- 1.6.4. Step 4. Apply the Loads -- 1.6.5. Step 5. Define Boundary Conditions -- 1.6.6. Step 6. Solve the System of Linear Algebraic Equations -- 1.6.7. Step 7. Calculate Stresses -- 1.7. What About the Available Software Packages? -- 1.8. Physical Principles in the Finite Element Method -- 1.9. From the Element Equation to the Structure Equation -- 1.10. Computer-Aided Learning of the Finite Element Method -- 1.10.1. Introduction to CALFEM -- 1.10.2. Spring elements -- 1.10.3. Bar Elements for Two-Dimensional Analysis -- 1.10.4. Bar Elements for Three-Dimensional Analysis -- 1.10.5. Beam Elements for Two-Dimensional Analysis -- 1.10.6. Beam Elements for Three-Dimensional Analysis -- 1.10.7. System Functions -- 1.10.8. Statement Functions -- 1.10.9. Graphic Functions -- 1.10.10. Working Environment in ANSYS -- References -- ch. 2 Mathematical Background -- 2.1. Vectors -- 2.1.1. Definition of Vector -- 2.1.2. Scalar Product -- 2.1.3. Vector Product -- 2.1.4. Rotation of Coordinate System -- 2.1.5. The Vector Differential Operator (Gradient) -- 2.1.6. Green's Theorem -- 2.2. Coordinate Systems -- 2.2.1. Rectangular (or Cartesian) Coordinate System -- 2.2.2. Cylindrical Coordinate System -- 2.2.3. Spherical Coordinate System -- 2.2.4. Component Transformation -- 2.2.5. The Vector Differential Operator (Gradient) in Cylindrical and Spherical Coordinates -- 2.3. Elements of Matrix Algebra -- 2.3.1. Basic Definitions -- 2.3.2. Basic Operations -- 2.4. Variational Formulation of Elasticity Problems -- 2.4.1. Definition of the Variation of a Function -- 2.4.2. Properties of Variations -- 2.4.3. Derivation of the Functional from the Boundary Value Problem -- References -- ch. 3 Linear Spring Elements -- 3.1. The Element Equation -- 3.1.1. The Mechanical Behaviour of the Material -- 3.1.2. The Principle of Direct Equilibrium -- 3.2. The Stiffness Matrix of a System of Springs -- 3.2.1. Derivation of Element Matrices -- 3.2.2. Expansion of Element Equations to the Degrees of Freedom of the Structure -- 3.2.3. Assembly of Element Equations -- 3.2.4. Derivation of the Field Values -- References -- ch. 4 Bar Elements and Hydraulic Networks -- 4.1. Displacement Interpolation Functions -- 4.1.1. Functional Form of Displacement Distribution -- 4.1.2. Derivation of the Element Equation -- 4.2. Alternative Procedure Based on the Principle of Direct Equilibrium -- 4.2.1. The Mechanical Behaviour of the Material -- 4.2.2. The Principle of Direct Equilibrium -- 4.3. Finite Element Method Modelling of a System of Bars -- 4.3.1. Derivation of Element Matrices -- 4.3.2. Expansion of Element Equations to the Degrees of Freedom of the Structure -- 4.3.3. Assembly of Element Equations -- 4.3.4. Derivation of the Field Values -- 4.4. Finite Elements Method Modelling of a Piping Network -- References -- ch. 5 Trusses -- 5.1. The Element Equation for Plane Truss Members -- 5.2. The Element Equation for 3D Trusses -- 5.3. Calculation of the Bar's Axial Forces (Internal Forces) -- References -- ch. 6 Beams -- 6.1. Element Equation of a Two-Dimensional Beam Subjected to Nodal Forces -- 6.1.1. The Displacement Function -- 6.1.2. The Element Stiffness Matrix -- 6.2. Two-Dimensional Element Equation of a Beam Subjected to a Uniform Loading -- 6.3. Two-Dimensional Element Equation of a Beam Subjected to an Arbitrary Varying Loading -- 6.4. Two-Dimensional Element Equation of a Beam on Elastic Foundation Subjected to Uniform Loading -- 6.5. Engineering Applications of the Element Equation of the Beam on Elastic Foundation -- 6.5.1. Beam Supported on Equi-spaced Elastic Springs -- 6.5.2. Cylindrical Shells Under Axisymmetric Loading -- 6.6. Element Equation for a Beam Subjected to Torsion -- 6.6.1. The Mechanical Behaviour of the Material -- 6.6.2. The Principle of Direct Equilibrium -- 6.7. Two-Dimensional Element Equation for a Beam Subjected To Nodal Axial Forces, Shear Forces, Bending Moments, and Torsional Moments -- 6.8. Three-Dimensional Element Equation for a Beam Subjected to Nodal Axial Forces, Shear Forces, Bending Moments, and Torsional Moments -- References -- ch. 7 Frames -- 7.1. Framed Structures -- 7.2. Two-Dimensional Frame Element Equation Subjected to Nodal Forces -- 7.3. Two-Dimensional Frame Element Equation Subjected to Arbitrary Varying Loading -- 7.4. Three-Dimensional Beam Element Equation Subjected to Nodal Forces -- 7.5. Distribution of Bending Moments, Shear Forces, Axial Forces, and Torsional Moments of Each Element -- References -- ch. 8 The Principle of Minimum Potential Energy for One-Dimensional Elements -- 8.1. The Basic Concept -- 8.2. Application of the MPE Principle on Systems of Spring Elements -- 8.3. Application of the MPE Principle on Systems of Bar Elements -- 8.4. Application of the MPE Principle on Trusses -- 8.5. Application of the MPE Principle on Beams -- References -- ch. 9 From "Isotropic" to "Orthotropic" Plane Elements: Elasticity Equations for Two-Dimensional Solids -- 9.1. The Generalized Hooke's Law -- 9.1.1. Effects of Free Thermal Strains -- 9.1.2. Effects of Free Moisture Strains -- 9.1.3. Plane Stress Constitutive Relations -- 9.2. From "Isotropic" to "Orthotropic" Plane Elements -- 9.2.1. Coordinate Transformation of Stress and Strain Components for Orthotropic Two-Dimensional Elements -- 9.3. Hooke's Law of an Orthotropic Two-Dimensional Element, with Respect to the Global Coordinate System -- 9.4. Transformation of Engineering Properties -- 9.4.1. Elastic Properties of an Orthotropic Two-Dimensional Element in the Global Coordinate System -- 9.4.2. Free Thermal and Free Moisture Strains in Global Coordinate System -- 9.5. Elasticity Equations for Isotropic Solids -- 9.5.1. Generalized Hooke's Law for Isotropic Solids -- 9.5.2. Correlation of Strains with Displacements -- 9.5.3. Correlation of Stresses with Displacements -- 9.5.4. Differential Equations of Equilibrium -- 9.5.5. Differential Equations in Terms of Displacements -- 9.5.6. The Total Potential Energy -- References -- ch. 10 The Principle of Minimum Potential Energy for Two-Dimensional and Three-Dimensional Elements -- 10.1. Interpolation and Shape Functions -- 10.1.1. Linear Triangular Elements (or CST Elements) -- 10.1.2. Quadratic Triangular Elements (or 1st Elements) -- 10.1.3. Bilinear Rectangular Elements (or Q4 Elements) -- 10.1.4. Tetrahedral Solid Elements -- 10.1.5. Eight-Node Rectangular Solid Elements -- 10.1.6. Plate-Bending Elements -- 10.2. Isoparametric Elements -- 10.2.1. Definition of Isoparametric Elements -- 10.2.2. Lagrange Polynomials -- 10.2.3. The Bilinear Quadrilateral Element -- 10.3. Derivation of Stiffness Matrices -- 10.3.1. The Linear Triangular Element (or CST Element) -- 10.3.2. The Quadratic Triangular Element (or 1st Element) -- 10.3.3. The Bilinear Rectangular Element (or Q4 Element) -- 10.3.4. The Tetrahedral Solid Element -- 10.3.5. Eight-Node Rectangular Solid Element -- 10.3.6. Plate-Bending Element -- 10.3.7. Isoparametric Formulation -- References -- ch. 11 Structural Dynamics -- 11.1. The Dynamic Equation -- 11.2. Mass Matrix -- 11.2.1. Bar Element -- 11.2.2. Two-Dimensional Truss Element -- 11.2.3. Three-Dimensional Truss Element -- 11.2.4. Two-Dimensional Beam Element -- 11.2.5. Three-Dimensional Beam Element -- 11.2.6. Inclined Two-Dimensional Beam Element (Two-Dimensional Frame Element) -- 11.2.7. Linear Triangular Element (CST Element) -- 11.3. Solution Methodology for the Dynamic Equation -- 11.3.1. Central Difference Method -- 11.3.2. Newmark-Beta Method -- 11.4. Free Vibration -- Natural Frequencies -- References -- ch. 12 Heat Transfer -- 12.1. Conduction Heat Transfer -- 2D Steady-State Heat Conduction Equation in Cartesian Coordinates -- 3D Steady-State Heat Conduction Equation in Cartesian Coordinates -- 3D Steady-State Heat Conduction Equation in Cylindrical Coordinates -- 3D Steady-State Heat Conduction Equation in Spherical Coordinates -- Heat conduction of orthotropic materials -- 12.2. Convection Heat Transfer -- 12.3. Finite Element Formulation -- 12.3.1. One-Dimensional Heat Transfer Modelling Using a Variational Method -- 12.3.2. Two-Dimensional and Three-Dimensional Heat Transfer Modelling Using a Variational Method -- References.
Fundamental coverage, analytic mathematics, and up-to-date software applications are hard to find in a single text on the finite element method (FEM). Dimitrios Pavlou's Essentials of the Finite Element Method: For Structural and Mechanical Engineers makes the search easier by providing a comprehensive but concise text for those new to FEM, or just in need of a refresher on the essentials. Essentials of the Finite Element Method explains the basics of FEM, then relates these basics to a number of practical engineering applications. Specific topics covered include linear spring elements, bar.