Essentials of the Finite Element Method,

Edition 2

Editors: By Dimitrios G. Pavlou

Publication Date: 04 Sep 2026

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Essentials of the Finite Element, Second Edition aims to simplify search by offering a comprehensive yet concise resource suitable for newcomers to FEM or those in need of a refresher. This edition begins by explaining the basics of FEM and gradually introduces advanced topics, while also illustrating the practical applications of the theory in engineering. The book covers various specific subjects, including linear spring elements, bar elements, trusses, beams and frames, plates, heat transfer, structural dynamics, and buckling. Throughout the text, readers are provided with step-by-step detailed analyses for the development of finite element equations.

Moreover, the book demonstrates the programming aspect of FEM, offering examples in MATLAB, EXCEL, CALFEM, and ANSYS. This allows readers to gain insights into developing their own computer code. Designed for a wide range of readers, from first-time BSc/MSc students to experienced researchers and practicing mechanical/structural engineers, the book serves as a comprehensive reference text suitable for modern engineers.

Key Features

Includes step-by-step instructions for developing finite element equations with detailed analysis procedures
Presents Excel exercises with dynamic inputs and static tests, allowing for innumerable exercise possibilities
Contains comprehensive coverage of Higher-order beam models in Finite Element Analysis that are often absent in classical textbooks
Covers stiffness matrices that are provided for commonly used engineering elements in practice
Provides theoretical resources for conducting FE analysis on isotropic and orthotropic materials

About the author

By Dimitrios G. Pavlou, Professor, University of Stavanger, Stavanger, Norway

1: AN OVERVIEW OF THE FINITE ELEMENT METHOD
1.1 What Are Finite Elements?
1.2 Why Finite Element Method Is Popular
1.3 Main Advantages of Finite Element Method
1.4 Main Disadvantages of Finite Element Method
1.5 What Is Structural Matrix? o Stiffness Matrix o Transfer Matrix
1.6 What Are the Steps to be Followed for Finite Element Method Analysis of Structure?
Step 1. Discretize or Model the Structure
Step 2. Define the Element Properties
Step 3. Assemble the Element Structural Matrices
Step 4. Apply the Loads
Step 5. Define Boundary Conditions
Step 6. Solve the System of Linear Algebraic Equations
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
1.11 Introduction to Excel
1.12 Merging dynamic equations and static text: a powerful learning tool.
1.13 Basic knowledge required to write a personal calculation code.
1.14 Introductory examples
1.15 Introduction to CALFEM
•Spring elements
•Bar Elements for Two-Dimensional Analysis
•Bar Elements for Three-Dimensional Analysis
•Beam Elements for Two-Dimensional Analysis
•Beam Elements for Three-Dimensional Analysis
•System Functions
•Statement Functions
•Graphic Functions
•Working Environment in ANSYS References

2: MATHEMATICAL BACKGROUND
2.1 Vectors
•Definition of Vector
•Scalar Product
•Vector Product
•Rotation of Coordinate System
•The Vector Differential Operator (Gradient)
•Green’s Theorem
2.2 Coordinate Systems
•Rectangular (or Cartesian) Coordinate System
•Cylindrical Coordinate System
•Spherical Coordinate System
•Component Transformation
•The Vector Differential Operator (Gradient) in Cylindrical
•Spherical Coordinates
2.3 Elements of Matrix Algebra
•Basic Definitions
•Basic Operations
2.4 Variational Formulation of Elasticity Problems
•Definition of the Variation of a Function
•Properties of Variations
•Derivation of the Functional from the Boundary Value Problem
2.5 Excel Exercise References

3: BAR, SPRING, HYDRAULIC ELEMENTS, AND CORRESPONDING NETWORKS
3.1 Displacement Interpolation Functions
3.1.1 Functional Form of Displacement Distribution
3.1.2 Derivation of the Element Equation
3.2 Alternative Procedure Based on the Principle of Direct Equilibrium.
•The Mechanical Behavior of the Material
•The Principle of Direct Equilibrium
3.3 Finite Element Method Modeling of a System of Bars
3.3.1 Derivation of Element Matrices
3.3.2 Expansion of Element Equations to the Degrees of Freedom of the Structure
3.3.3 Assembly of Element Equations
3.3.4 Derivation of the Field Values
3.3.5 Linear Spring Elements
3.4 Finite Elements Method Modeling of a Piping Network
3.5 The Element Equation for Plane Truss Members
3.6 The Element Equation for 3D Trusses
3.7 Excel Examples References

4: EULER-BERNOULLI, EHRENFEST-TIMOSHENKO AND REDDY BEAM MODELS
Introduction
4.1 Bernoulli beam model
•Element Equation of a Two-Dimensional Beam Subjected to Nodal Forces
•The Displacement Function
•The Element Stiffness Matrix
•The Role of Shape Functions: Approximate vs Exact Solutions
•Engineering Applications of the Element Equation of the Beam on Elastic Foundation
4.2 Timoshenko beam model
•Element Equation of a Two-Dimensional Beam Subjected to Nodal Forces
•The Displacement Function
•The Element Stiffness Matrix
•The Role of Shape Functions: Approximate vs Exact Solutions
4.3 Reddy beam model
•Element Equation of a Two-Dimensional Beam Subjected to Nodal Forces
•The Displacement Function
•The Element Stiffness Matrix
•The Role of Shape Functions: Approximate vs Exact Solutions
4.4 Element Equation for a Beam Subjected to Torsion
•The Mechanical Behavior of the Material
•The Principle of Direct Equilibrium
•Two-Dimensional Element Equation for a Beam Subjected To Nodal Axial Forces, Shear Forces, Bending Moments, and Torsional Moments
4.5 Three-Dimensional Element Equation for a Beam Subjected to Nodal Axial Forces, Shear Forces, Bending Moments, and Torsional Moments
4.6 Functionally Graded Beams
4.7 Excel Examples References

5: FRAMES
5.1 Framed Structures
5.2 Two-Dimensional Frame Element Equation Subjected to Nodal Forces
5.3 Two-Dimensional Frame Element Equation Subjected to Arbitrary Varying Loading
5.4 Three-Dimensional Beam Element Equation Subjected to Nodal Forces
5.5 Distribution of Bending Moments, Shear Forces, Axial Forces, and Torsional Moments of Each Element
5.6 Excel Examples References

6: KIRCHHOFF, MINDLIN AND REDDY PLATE MODELS

Introduction
6.1 Kirchhoff plate model
•Element Equation of a Two-Dimensional Beam Subjected to Nodal Forces
•The Displacement Function
•The Element Stiffness Matrix
6.1 Mindlin plate model
•Element Equation of a Two-Dimensional Beam Subjected to Nodal Forces
•The Displacement Function
•The Element Stiffness Matrix
6.1 Reddy plate model
•Element Equation of a Two-Dimensional Beam Subjected to Nodal Forces
•The Displacement Function
•The Element Stiffness Matrix

7: THE PRINCIPLE OF MINIMUM POTENTIAL ENERGY
7.1 The Basic Concept
7.2 Application of the MPE Principle on Systems of Spring Elements
7.3 Application of the MPE Principle on Systems of Bar Elements
7.4 Application of the MPE Principle on Trusses
7.5 Application of the MPE Principle on Euler-Bernoulli Beams
7.6 Application of the MPE Principle on Ehrenfest-Timoshenko Beams
7.7 Application of the MPE Principle on Reddy Beams
7.8 Application of the MPE Principle on Kirchhoff Plates
7.9 Application of the MPE Principle on Mindlin Plates
7.10 Application of the MPE Principle on Reddy Plates
7.11 Excel Examples References

8: FROM “ISOTROPIC” TO “ORTHOTROPIC” PLANE ELEMENTS: ELASTICITY EQUATIONS FOR TWO-DIMENSIONAL SOLIDS
8.1 The Generalized Hooke’s Law o Effects of Free Thermal Strains
•Effects of Free Moisture Strains
•Plane Stress Constitutive Relations
8.2 From “Isotropic” to “Orthotropic” Plane Elements
•Coordinate Transformation of Stress and Strain Components for Orthotropic Two-Dimensional Elements
8.3 Hooke’s Law of an Orthotropic Two-Dimensional Element, with Respect to the Global Coordinate System
8.4 Transformation of Engineering Properties
•Elastic Properties of an Orthotropic Two-Dimensional Element in the Global Coordinate System
•Free Thermal and Free Moisture Strains in Global Coordinate System
8.5 Elasticity Equations for Isotropic Solids
•Generalized Hooke’s Law for Isotropic Solids
•Correlation of Strains with Displacements
•Correlation of Stresses with Displacements
•Differential Equations of Equilibrium
•Differential Equations in Terms of Displacements
•The Total Potential Energy
8.6 Excel Examples References

9: THE PRINCIPLE OF MINIMUM POTENTIAL ENERGY FOR TWO-DIMENSIONAL AND THREE-DIMENSIONAL ELEMENTS
9.1 Interpolation and Shape Functions
•Linear Triangular Elements (or CST Elements)
•Quadratic Triangular Elements (or LST Elements
•Bilinear Rectangular Elements (or Q4 Elements)
•Tetrahedral Solid Elements
•Eight-Node Rectangular Solid Elements
•Plate Bending Elements
9.2 Isoparametric Elements
9.3 Derivation of Stiffness Matrices
•The Linear Triangular Element (or CST Element)
•The Quadratic Triangular Element (or LST Element)
•The Bilinear Rectangular Element (or Q4 Element)
•The Tetrahedral Solid Element
•Eight-Node Rectangular Solid Element
•Plate Bending Element
•Isoparametric Formulation
9.4 Excel Examples References

10: STRUCTURAL DYNAMICS AND ELASTIC STABILITY
10.1 The Dynamic Equation
10.2 Mass Matrix
10.2.1 Definition of Isoparametric Elements
10.2.2 Lagrange Polynomials
10.2.3 The Bilinear Quadrilateral Element
•Bar Element o Two-Dimensional Truss Element
•Three-Dimensional Truss Element
•Two-Dimensional Beam Element
•Three-Dimensional Beam Element
•Inclined Two-Dimensional Beam Element (Two-Dimensional Frame Element)
•Linear Triangular Element (CST Element)
10.3 Solution Methodology for the Dynamic Equation
10.4 Free Vibration—Natural Frequencies
10.5 Elastic Buckling of Euler-Bernoulli beams
10.6 Excel Examples References

11: HEAT TRANSFER
11.1 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
11.2 Convection Heat Transfer
11.3 Finite Element Formulation
11.3.1 Central Difference Method
11.3.2 Newmark-Beta Method
•One-Dimensional Heat Transfer Modeling Using a Variational Method
•Two-Dimensional and Three-Dimensional Heat Transfer Modeling Using a Variational Method
References
Index

Essentials of the Finite Element Method,

Editors: By Dimitrios G. Pavlou

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