- •Copyright Notice
- •Table of Contents
- •Chapter 1. Introduction
- •I. Basic Concepts
- •Examples:
- •Why Finite Element Method?
- •Applications of FEM in Engineering
- •Examples:
- •A Brief History of the FEM
- •FEM in Structural Analysis (The Procedure)
- •Example:
- •Available Commercial FEM Software Packages
- •Objectives of This FEM Course
- •II. Review of Matrix Algebra
- •Linear System of Algebraic Equations
- •Matrix Addition and Subtraction
- •Scalar Multiplication
- •Matrix Multiplication
- •Transpose of a Matrix
- •Symmetric Matrix
- •Unit (Identity) Matrix
- •Determinant of a Matrix
- •Singular Matrix
- •Matrix Inversion
- •Examples:
- •Solution Techniques for Linear Systems of Equations
- •Positive Definite Matrix
- •Differentiation and Integration of a Matrix
- •Types of Finite Elements
- •III. Spring Element
- •One Spring Element
- •Spring System
- •Checking the Results
- •Notes About the Spring Elements
- •Example 1.1
- •Chapter 2. Bar and Beam Elements
- •I. Linear Static Analysis
- •II. Bar Element
- •Stiffness Matrix --- Direct Method
- •Stiffness Matrix --- A Formal Approach
- •Example 2.1
- •Example 2.2
- •Distributed Load
- •Bar Elements in 2-D and 3-D Space
- •2-D Case
- •Transformation
- •Stiffness Matrix in the 2-D Space
- •Element Stress
- •Example 2.3
- •Example 2.4 (Multipoint Constraint)
- •3-D Case
- •III. Beam Element
- •Simple Plane Beam Element
- •Direct Method
- •Formal Approach
- •3-D Beam Element
- •Example 2.5
- •Equivalent Nodal Loads of Distributed Transverse Load
- •Example 2.6
- •Example 2.7
- •FE Analysis of Frame Structures
- •Example 2.8
- •Chapter 3. Two-Dimensional Problems
- •I. Review of the Basic Theory
- •Plane (2-D) Problems
- •Stress-Strain-Temperature (Constitutive) Relations
- •Strain and Displacement Relations
- •Equilibrium Equations
- •Exact Elasticity Solution
- •Example 3.1
- •II. Finite Elements for 2-D Problems
- •A General Formula for the Stiffness Matrix
- •Constant Strain Triangle (CST or T3)
- •Linear Strain Triangle (LST or T6)
- •Linear Quadrilateral Element (Q4)
- •Quadratic Quadrilateral Element (Q8)
- •Example 3.2
- •Transformation of Loads
- •Stress Calculation
- •I. Symmetry
- •Types of Symmetry:
- •Examples:
- •Applications of the symmetry properties:
- •Examples:
- •Cautions:
- •II. Substructures (Superelements)
- •Physical Meaning:
- •Mathematical Meaning:
- •Advantages of Using Substructures/Superelements:
- •Disadvantages:
- •III. Equation Solving
- •Direct Methods (Gauss Elimination):
- •Iterative Methods:
- •Gauss Elimination - Example:
- •Iterative Method - Example:
- •IV. Nature of Finite Element Solutions
- •Stiffening Effect:
- •V. Numerical Error
- •VI. Convergence of FE Solutions
- •Type of Refinements:
- •Examples:
- •VII. Adaptivity (h-, p-, and hp-Methods)
- •Error Indicators:
- •Examples:
- •Chapter 5. Plate and Shell Elements
- •Applications:
- •Forces and Moments Acting on the Plate:
- •Stresses:
- •Relations Between Forces and Stresses
- •Thin Plate Theory ( Kirchhoff Plate Theory)
- •Examples:
- •Under uniform load q
- •Thick Plate Theory (Mindlin Plate Theory)
- •II. Plate Elements
- •Kirchhoff Plate Elements:
- •Mindlin Plate Elements:
- •Discrete Kirchhoff Element:
- •Test Problem:
- •Mesh
- •III. Shells and Shell Elements
- •Example: A Cylindrical Container.
- •Shell Theory:
- •Shell Elements:
- •Curved shell elements:
- •Test Cases:
- •Chapter 6. Solid Elements for 3-D Problems
- •I. 3-D Elasticity Theory
- •Stress State:
- •Strains:
- •Stress-strain relation:
- •Displacement:
- •Strain-Displacement Relation:
- •Equilibrium Equations:
- •Stress Analysis:
- •II. Finite Element Formulation
- •Displacement Field:
- •Stiffness Matrix:
- •III. Typical 3-D Solid Elements
- •Tetrahedron:
- •Hexahedron (brick):
- •Penta:
- •Element Formulation:
- •Solids of Revolution (Axisymmetric Solids)
- •Axisymmetric Elements
- •Applications
- •Chapter 7. Structural Vibration and Dynamics
- •I. Basic Equations
- •A. Single DOF System
- •B. Multiple DOF System
- •Example
- •II. Free Vibration
- •III. Damping
- •IV. Modal Equations
- •V. Frequency Response Analysis
- •VI. Transient Response Analysis
- •B. Modal Method
- •Cautions in Dynamic Analysis
- •Examples
- •Chapter 8. Thermal Analysis
- •Further Reading
Lecture Notes: Introduction to Finite Element Method |
Chapter 7. Structural Vibration and Dynamics |
III. Damping
Two commonly used models for viscous damping.
A. Proportional Damping (Rayleigh Damping)
C =αM + βK |
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where the constants α & β are found from |
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ξ2 |
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β |
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with ω1, ω2 ,ξ1 & ξ2 (damping ratio) being selected.
Damping ratio
B. Modal Damping
Incorporate the viscous damping in modal equations.
© 1997-2003 Yijun Liu, University of Cincinnati |
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Lecture Notes: Introduction to Finite Element Method |
Chapter 7. Structural Vibration and Dynamics |
IV. Modal Equations
Use the normal modes (modal matrix) to transform the coupled system of dynamic equations to uncoupled system of equations.
We have
[K − ω i |
2 M ]ui |
= 0 , |
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i = 1,2,..., n |
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where the normal mode |
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satisfies: |
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u |
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u iT |
K u j = |
0 , |
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for i ≠ j, |
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M u j |
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u iT |
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and |
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u iT M u i |
= 1, |
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for i = 1, 2, …, n. |
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ω i2 |
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u iT K u i = |
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Form the modal matrix: |
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Φ ( n × n ) |
= [u 1 u 2 L u n ] |
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We can verify that
ω12
ΦT KΦ= Ω= 0M0
ΦT MΦ= I.
0 |
L 0 |
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ω2 |
M |
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2 |
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(Spectral matrix), |
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L 0 ω2 |
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n |
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Transformation for the displacement vector,
u = z1 u1 + z 2 u 2 + L + z n u n = Φ z ,
(18)
(19)
(20)
(21)
© 1997-2003 Yijun Liu, University of Cincinnati |
168 |
Lecture Notes: Introduction to Finite Element Method Chapter 7. Structural Vibration and Dynamics
where
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z1 |
(t ) |
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z = |
z2 |
(t ) |
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M |
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zn |
(t ) |
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are called principal coordinates. Substitute (21) into the dynamic equation:
M Φ &z& + C Φ z& + K Φ z = f ( t ).
Pre-multiply by ΦT, and apply (20):
z + C φ |
z + Ω z = p ( t ), |
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(22) |
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&& |
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& |
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where C φ |
= α I + β Ω |
(for proportional damping), |
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p |
= Φ T f ( t ) . |
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Introduce modal damping: |
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2 ξ 1ω 1 |
0 |
L |
0 |
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C φ |
= |
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2 ξ 2 ω 2 |
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M |
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M |
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. (23) |
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© 1997-2003 Yijun Liu, University of Cincinnati |
169 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 7. Structural Vibration and Dynamics |
Equation (22) becomes,
&& |
+ |
& |
2 |
z i |
= p i ( t ), i = 1,2,…,n. (24) |
z i |
2 ξ iω i z i |
+ ω i |
Equations in (22) or (24) are called modal equations. These equations are uncoupled, second-order differential equations, which are much easier to solve than the original dynamic equation (a coupled system).
To recover u from z, apply transformation (21) again, once z is obtained from (24).
Notes:
•Only the first few modes may be needed in constructing the modal matrix Φ (i.e., Φ could be an n×m rectangular matrix with m<n). Thus, significant reduction in the size of the system can be achieved.
•Modal equations are best suited for problems in which higher modes are not important (i.e., structural vibrations, but not for structures under a shock load).
© 1997-2003 Yijun Liu, University of Cincinnati |
170 |