Preface
Glossary
Physical Description of Processes
Matrix– and Vector–Symbols at the Element Level
Matrix– and Vector–Symbols at the System Level
Denomination of Elements
Heat Conduction Elements
Plane Stress Elements
Kirchhoff Plate Elements
Reissner–Mindlin Plate Elements
Table of contents
FOUNDATIONS
1 Introduction
1.1 Governing Equations and Approximate Solution
1.2 General Aspects Concerning the Finite Element Method
1.3 A Comparison of Exact Solution and Approximate Solution
1.3.1 Analytically Exact Solutions
1.3.2 Approximate Solutions
1.3.3 The FE–Solution Regarding two Elements – Case 2
2 Discretization of the Work Equation
2.1 Principle of Virtual Displacements
2.1.1 Virtual Work
2.1.2 Fulfillment of Governing Equations and Boundary Conditions
2.1.3 Fulfillment of the Conditions at the Element Intersections
2.2 Principle of Virtual Forces
2.3 The General Procedure to set up the Element Matrices
2.3.1 Solution Procedure
2.3.2 Example
2.3.3 Matrix Notation
2.4 Shape Functions and Convergence Criteria
2.4.1 Shape Functions
2.4.2 Criteria of Convergence
2.4.3 Scaling Matrix
3 Structure and Solution of the System of Equations
3.1 Real and Virtual Nodal Displacements
3.2 Equations of Condition
3.3 Structure of the System of Equations
3.4 Assembly of the Stiffness Matrix of the Entire System
3.5 The Storage and Solution of the System of Equations
3.6 The Optimization of the Bandwidth
3.7 The Fulfillment of Dirichlet Boundary Conditions
4 Heat Conduction
4.1 Heat Conduction at One–Dimensional Description
4.1.1 Governing Equations for One Dimension
4.1.2 Boundary Conditions
4.1.3 Weak Form of the Energy Balance Equation
4.1.4 Example of use
4.2 Heat Conduction Regarding Two Spatial Dimensions
4.2.1 Governing Equations and the Weak Form
4.2.2 Matrix Representation of the Weak Form
4.2.3 Heat Conduction Matrix E and Operator Matrix D
4.2.4 Linear Shape Functions Regarding the Temperature
4.2.5 Differentiation of the Temperature Field
4.2.6 Element Matrix
4.2.7 Element Vector of Thermal Action
4.2.8 Subsequent Flux Analysis
4.3 Example of Use
5 Membrane Structures
5.1 Rectangular Elements Regarding Plane Stress Situation
5.1.1 Governing Equations and Work Equations
5.1.2 Matrix Notation of the Work Equations
5.1.3 Elasticity Matrix E and Operator Matrix D
5.1.4 Bi–linear Shape Functions Regarding Displacements
5.1.5 Differentiation of Displacement Fields
5.1.6 Element Stiffness Matrix
5.1.7 Element Load Vector
5.1.8 Subsequent Stress Analysis
5.1.9 Example of Use
5.2 Rectangular Element Comprising Modified Shear Strains
5.2.1 Selectively Reduced Integration
5.2.2 Example of Use
5.3 Plane Strain
6 Bending Structures
6.1 Element Matrices Regarding Euler–Bernoulli Beams
6.1.1 Governing Equations and Work Equations
6.1.2 Matrix Notation of the Work Equation
6.1.3 Elasticity Matrix E and Operator Matrix D
6.1.4 Shape Functions to Describe the Deflection
6.1.5 Differentiation of Shape Functions
6.1.6 Element Stiffness Matrix
6.1.7 Element Load Vector
6.1.8 Subsequent Stress Analysis
6.1.9 Example of Use
6.2 Kirchhoff Plate Element with 16 DOF
6.2.1 Governing Equations and Work Equations
6.2.2 Matrix Notation of the Work Equation
6.2.3 Elasticity Matrix E and Operator Matrix D
6.2.4 Shape Functions to Describe the Deflection
6.2.5 Differentiation of Shape Functions
6.2.6 Element Stiffness Matrix
6.2.7 Element Load Vector
6.2.8 Subsequent Stress Analysis
6.2.9 Example of Use
6.3 Kirchhoff Plate Element with 12 DOF
6.4 The 12 DOF Element Employing a Weak Conformity
6.4.1 Stiffness matrix
6.4.2 Example of Use
TRIANGULAR ELEMENTS
7 Triangular Elements – Description of Geometry
7.1 Local ξ–η–Coordinate System
7.1.1 Transformation of Coordinates
7.1.2 Transformation of Partial Derivatives
7.1.3 The Integration of the Element Area
7.1.4 Linear Shape Functions and Nodal Degrees of Freedom
7.2 Description Employing Area Coordinates
7.2.1 Transformation between Cartesian and Area Coordinates
7.2.2 Transformation of Partial Derivatives
7.2.3 Integration with Respect to Area Coordinates
7.2.4 Shape Functions Employing Area Coordinates
8 Triangular Elements to Describe Heat Conduction
8.1 A Linear Approach Related to the ξ–η–Coordinate System
8.1.1 Linear Shape Functions and Nodal Unknowns
8.1.2 Differentiation of the Temperature Field
8.1.3 Element Matrix
8.1.4 Vector of Thermal Action
8.1.5 Subsequent Flux Analysis
8.2 Example of Use
9 Triangular Elements for Membrane Structures
9.1 Linear Shape Functions with Respect to ξ–η–Coordinates
9.1.1 Shape Functions and Nodal Unknowns
9.1.2 Differentiation of Displacement Fields
9.1.3 Element Stiffness Matrix
9.1.4 Load Vector
9.1.5 Subsequent Stress Analysis
9.2 Description Employing Area Coordinates
9.2.1 Linear Shape Functions Employing Area Coordinates
9.3 Quadratic Approach Employing ξ–η–Coordinates
9.3.1 Quadratic Shape Functions
9.3.2 Element Stiffness Matrix
9.4 Quadratic Shape Functions Using Area Coordinates
9.4.1 Stiffness Matrix
9.4.2 Load Vector
9.4.3 Subsequent Stress Analysis
9.5 A Comparison of Standard Elements
10 Triangular Elements for Kirchhoff Plates
10.1 Choice of Shape Functions and Nodal Unknowns
10.2 Complete Approach of the 5th Order
10.2.1 The Process to Assemble the Element Stiffness Matrix
10.2.2 General Polynomial Employing Area Coordinates
10.2.3 Derivatives with Respect to x–y–Coordinates
10.2.4 Scaling Matrix
10.2.5 Element Stiffness Matrix
10.2.6 Element Load Vector
10.2.7 Subsequent Analysis for Stress Resultants
10.3 A Plate Element with 18 DOF
10.4 A Comparison of Standard Plate Elements
ISOPARAMETRIC ELEMENTS
11 Numerical Integration
11.1 Numerical Integration Using Gauss–Legendre Quadrature
11.2 Numerical Integration Applied to Membrane Elements
11.2.1 Shape Functions Employing Local Coordinates
11.2.2 Derivatives with respect to local coordinates
11.2.3 Element Stiffness Matrix and Load Vector
11.2.4 Element Load Vector
11.2.5 The Subsequent Stress Analysis
11.3 Triangular Elements
11.3.1 Triangular Elements for Membranes
11.3.2 Triangular Elements for Plates
12 Isoparametric Elements
12.1 Description of the Element Geometry
12.2 Isoparametric Elements Regarding Membranes
12.2.1 The Stiffness Matrix and the Stress Matrix
12.2.2 The Selectively Reduced Integration
12.2.3 Comparison of Standard Membrane Elements
12.3 Quadrilateral Plate Elements
12.3.1 The 16 DOF Plate Element
12.3.2 The 12 DOF Plate Element
HYBRID QUADRILATERAL ELEMENTS
13 Hybrid Finite Elements
13.1 Mixed Formulation of Governing Equations
13.2 Mixed Formulation of Work Equations
13.2.1 Shape Functions, Element Matrix, Load Vector
13.2.2 Matrix Notation of the Work Equations
13.2.3 Test
13.3 Hybrid Discretization of Work Equations
13.3.1 Work Equation at Element Level
13.3.2 Stiffness Matrix and Load Vector of the Hybrid Element
14 Hybrid–Mixed Plane Stress Elements
14.1 Mixed Principle of Work for Plane Stress Structures
14.1.1 Governing Equations
14.1.2 Work Equations in Mixed Formulation
14.2 Work Equations of a Hybrid Plane Stress Element
14.2.1 Work Equations at Element Level
14.2.2 Conditions of Deformation at the Element Interface
14.2.3 Stress Conditions at the Element Interface
14.3 Element Stiffness Matrix
14.4 Subsequent Stress Analysis
14.5 Linear Shape Functions
14.6 Quadratic Shape Functions
14.7 Linear Approaches with Coupling of Degrees of Freedom
14.7.1 Approach Following Pian and Sumihara
14.7.2 Approach with Transformation of Coordinates
14.8 Convergence Behavior of the Elements
15 Hybrid–Mixed Euler–Bernoulli Beam Elements
15.1 Mixed Formulation Employing Forces and Displacements
15.1.1 Governing Equations
15.1.2 Work Equations
15.1.3 Boundary and Element Interface Conditions
15.2 Work Equation in Hybrid Formulation
15.3 Element Stiffness Matrix
16 Hybrid–Mixed Kirchhoff Plate Elements
16.1 Mixed Principles of Work Concerning Kirchhoff Plates
16.1.1 Governing Equations
16.1.2 Work Equations
16.1.3 Element Matrix and Load Vector
16.1.4 Convergence Behavior Concerning the Mixed Element
16.2 Hybrid–Mixed Rectangular Plate Element
16.2.1 Work Equations
16.2.2 Element Stiffness Matrix
16.2.3 Convergence Behavior of the Hybrid–Mixed Element
HYBRID TRIANGULAR PLATE ELEMENTS
17 Hybrid Triangular Plate Elements
17.1 Cubic Approach for Triangular Plate Elements
17.1.1 Elimination of wd at Element Level
17.1.2 Subsequent Analysis Concerning the Stress Resultants
17.1.3 Hybrid Elements to Ensure C1 - Conformity
17.2 Hybrid–Displacement Elements Employing 10+3 DOF
17.2.1 Version A Employing Lagrange Multipliers
17.2.2 Version B Employing Rotational Degrees of Freedom
17.3 Displacement–based Element with Weak Conformity
18 Hybrid–Mixed Triangular Plate Elements
18.1 Work Equations
18.2 Approaches Related to the Deflection and the Stresses
18.3 Element Stiffness Matrix and Load Vector
18.4 Fulfillment of the Continuity Conditions at the Interface
18.5 Subsequent Stress Analysis
18.6 Test
19 Discrete Kirchhoff –Theory Element
19.1 The Discrete Kirchhoff Triangular Element
19.2 Stiffness Matrix, Load Vector and Stress Analysis
19.3 Test
20 Benchmark Concerning Triangular Plate Elements
20.1 Square Plate Subjected to Distributed Loading
20.2 Trapezoidal Plate Subjected to Distributed Loading
SHEAR–DEFORMATION BEAM AND PLATE ELEMENTS
21 Timoshenko Beam Elements
21.1 Elements Employing Displacements as Primary Variables
21.2 Mixed Elements
21.3 Hybrid–Mixed Elements
21.4 Convergence Behavior Concerning the Beam Elements
22 Plate Elements Including Shear Deformations
22.1 Kirchhoff and Reissner–Mindlin Theories by Comparison
22.2 Governing Equations of the Reissner–Mindlin Theory
22.3 Weak Formulation of the Governing Equations
22.4 Quadrilateral Element Employing a Bi–linear Approach
22.4.1 Approaches to Describe the Deflection and Rotations
22.4.2 Element Stiffness Matrix and Load Vector
22.5 Hybrid–Displacement Quadrilateral Element
22.5.1 Element Stiffness Matrix
22.5.2 Subsequent Stress Analysis
22.6 Comparison of the Elements
22.6.1 Test 1
22.6.2 Test 2
22.6.3 Remarks
22.6.4 Benchmark
22.7 Displacement–Based Triangular Element
22.7.1 Approaches to Describe the Displacement Variables
22.7.2 Element Stiffness Matrix and Load Vector
22.7.3 Subsequent Stress Analysis
22.7.4 Behabior of Convergence Concerning the Triangular Element
22.8 Mixed Quadrilateral Element
22.9 Hybrid–Mixed Quadrilateral Element
EVALUATION OF RESULTS
23 Error Estimation
23.1 The Least Squares Method
23.2 Error Estimation by Applying the Principle of VirtualWork
23.3 Mesh–Adaptation
24 Quality of Elements
24.1 The Eigenvalue Analysis
24.2 The Locking Phenomena
24.3 Improvement of Elements Suffering from Locking
24.4 The Patch–Test
References
Index