- •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 |
VI. Transient Response Analysis
(Dynamic Response/Time-History Analysis)
• Structure response to arbitrary, time-dependent loading.
f(t) |
t |
u(t) |
t |
Compute responses by integrating through time:
u 1
u n u n+1
u 2
t 0 t 1 t 2 |
t n t n+1 |
t |
© 1997-2003 Yijun Liu, University of Cincinnati |
172 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 7. Structural Vibration and Dynamics |
Equation of motion at instance tn , n = 0, 1, 2, 3, :
Mu&&n +Cu&n +Kun = fn.
Time increment: ∆t=tn+1-tn, n=0, 1, 2, 3, .
There are two categories of methods for transient analysis.
A.Direct Methods (Direct Integration Methods)
•Central Difference Method
Approximate using finite difference:
& |
= |
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+ 1 |
− |
u n − 1 ), |
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u n |
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∆ |
t |
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&& |
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( u |
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− 2 u n + u n − 1 ) |
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u n |
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∆ |
t ) 2 |
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Dynamic equation becomes,
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(un+1 |
− 2un + un−1 ) |
+ C |
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− un−1 ) |
+ Kun = fn , |
(∆t) |
2 |
2∆t |
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which yields,
Aun+1 = F(t) where
F(t )
= fn − K
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A = |
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M + |
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C, |
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2∆t |
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u n − |
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u n −1. |
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2 ∆t |
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un+1 is calculated from un & un-1, and solution is marching from t0 ,t1, L tn ,tn +1, L , until convergent.
© 1997-2003 Yijun Liu, University of Cincinnati |
173 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 7. Structural Vibration and Dynamics |
This method is unstable if ∆t is too large.
•Newmark Method: Use approximations:
un+1 |
& |
+ |
(∆t)2 |
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&& |
&& |
&& |
= L) |
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≈ un + ∆tun |
2 |
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[(1 − 2β )un + 2 |
βun+1 |
],→ (un+1 |
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un+1 |
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+γun+1 ], |
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≈ un + ∆t[(1 −γ )un |
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& |
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&& |
&& |
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where β & γ are chosen constants. These lead to
Au n +1 = F ( t ) where
A = K + |
γ |
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C + |
1 |
M , |
β ∆ t |
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β ( ∆ t ) 2 |
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F ( t ) = f ( f n + 1 |
, γ , β , ∆ t , C , M , u n , u n , u n ). |
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& && |
This method is unconditionally stable if
2 β ≥ |
γ |
≥ |
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e . g ., |
γ |
= |
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which gives the constant average acceleration method.
Direct methods can be expensive! (the need to compute A-1, often repeatedly for each time step).
© 1997-2003 Yijun Liu, University of Cincinnati |
174 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 7. Structural Vibration and Dynamics |
B. Modal Method
First, do the transformation of the dynamic equations using the modal matrix before the time marching:
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m |
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u = ∑ u i z i ( t ) =Φ z , |
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i =1 |
i = 1,2, , m. |
z i + 2ξ iω i z i + ω i z i |
= p i ( t ), |
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&& |
& |
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Then, solve the uncoupled equations using an integration method. Can use, e.g., 10%, of the total modes (m= n/10).
•Uncoupled system,
•Fewer equations,
•No inverse of matrices,
•More efficient for large problems.
Comparisons of the Methods
Direct Methods |
Modal Method |
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Small model |
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Large model |
• More accurate (with small ∆t) |
• Higher modes ignored |
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Single loading |
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Multiple loading |
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Shock loading |
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Periodic loading |
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… |
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… |
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© 1997-2003 Yijun Liu, University of Cincinnati |
175 |