Finite Element Analysis in Structural Mechanics
Lecture 1: Introduction
Department of Civil Engineering Module code: CIVE4061
Contents
Introduction (Analytical methods Vs Numerical methods)
Differential equations in Civil Engineering
Finite Element Analysis
- Definition
- History
- Procedure
- Characteristics
Finite Elements applications
Course Contents
Introduction to MATLAB
Finite Element Packages
Assessment of course
References
Summary
1
Analytical methods VS Numerical methods
dy
Example: For y ( x) = x 2 , calculate at x = 2.
dx
Analytical method Numerical method
dy dy ∆y y ( x + ∆x) − y ( x)
= 2x ≈ =
dx dx ∆x ∆x
dy Take ∆x = 0.01
= 2*2 = 4
dx x=2
∆y y (2.01) − y (2)
=
∆x x=2 ∆x
∆y 4.0401 − 4.0000
= = 4.01
∆x x=2 0.0100
2
Analytical methods VS Numerical methods
- What if function is complicated?
cosh( x) cos( x) − sinh( x) sin( x)
y=
x
- What if we have to integrate rather than differentiate?
∞
eiξx
y= ∫ c ξ 4 + c2 dξ , where c1 and c2 are constants.
−∞ 1
- What if we are trying to find roots of a function?
tan x
y=
x
Analytical solutions are our first option but they are not
always possible or they may be too complicated or require
a significant knowledge
Numerical methods can provide a quick and easy solution
for Engineering problems
Most of engineering applications are described by
differential equations in closed domain with boundary
conditions
3
Classical Plate Theory
∂4w ∂4w ∂4w 12(1 −ν 2 )
+ 2 + = p z
∂x 4 ∂x 2 ∂y 2 ∂y 4 Et
3
w( x, y )
E ,ν
y
x L y
h
L
x
Boundary Conditions: clamped plate
∂w
w= = 0 @ x = 0, L x
∂x
∂w
w= =0 @ y = 0, L y
∂y
Seepage Flow
∂ ∂φ ∂ ∂φ ∂φ
[k ] + [k ]+Q = n
∂x ∂x ∂y ∂y ∂t
x y
φ =fluid head, k , k =coefficients of permeability in the x and y directions
x y
Q =specific fluid flux and n =effective porosity
φ=z
d1 y
z
x
d2
dφ
φ=d 1 =0 φ=d 2
dz
4
Heat conduction
∂U ∂U 2
=k 1-D
∂t ∂x 2
where k is constant depending on the material
U (x,t < 0) = U 1
L U ( x = L , t >= 0 ) = U
U (x = 0, t >= 0) = U 2
3
Remember that it is almost always the case that problems are described
by differential equations in closed domain with boundary conditions
The Finite Element (FE) method
Definition:
It is a numerical method that solves the governing differential
equations of a system through a discretization process
The method has wide range of applicability such as for linear
and non-linear structural analysis, dynamics of complex
structures, soil mechanics, fluid mechanics, thermodynamics,
electromagnatisim, etc…
5
History of Finite Element
1941 Hrennikoff and Courant, use of mesh discretization of a
continuous domain into a set of discrete sub-domains solution of
second order elliptic partial differential equations
1956 Turner, Clough, Martin, and Topp established a broader
definition of numerical analysis in a paper on the "stiffness and
deflection of complex structures".
By the early 70's, Finite Element Analysis was limited to expensive
mainframe computers generally owned by the aeronautics,
automotive, defence, and nuclear industries.
Since the rapid decline in the cost of computers and the phenomenal
increase in computing power, the Finite Element method has been
widely used in Engineering applications
More details: Logan, A First Course in the Finite Element Method,
Sec. 1.1
General Steps of the Finite Element Method
1- Discretize and select the element types
2- Select a displacement function
3- Define the Strain/Displacement relationships
4-Derive the element stiffness matrix and equations
direct equilibrium method, work or energy methods, methods of
weighted residuals
5- Assemble the element equations to obtain the global or total equations and
introduce boundary conditions
6- Solve for the unknown degrees of freedom (or generalized displacements)
7- Solve for the element stresses and strains
8- Interpret the results
6
7
Finite Element Characteristics
The FE method has the capability to account for:
- Change in properties within the domain
- Irregular shaped boundaries
- Different types, shapes, and sizes of elements can be used within
the same region
General rule: the number of elements should be large enough to
give useful results and small enough to reduce computations
Advantages of the Finite Element Method
Logan, A First Course in the Finite Element Method, Sec. 1.6
Applications of the Finite Element method
The FEM can be used to analyze both structural and nonstructural
problems.
Typical structural areas include, e.g., Stress analysis, including
truss and frame analysis, and stress concentration problems
(associated with holes, fillets, changes in geometry), vibration
analysis
Nonstructural problems include: Heat transfer, Fluid flow
(including seepage through porous media), Distribution of electric
or magnetic potential
Logan, A First Course in the Finite Element Method, Sec. 1.5
8
Examples of FE applications
Studying the state of noise and vibration in a car
http://www.msc.commas.uni-stuttgart.de/lectures.html
Examples of FE applications
Studying the human head impact tolerance
From Krabbel and Muller, DEVELOPMENT OF A FINITE ELEMENT MODEL OF THE HEAD USING THE VISIBLE HUMAN DATA
9
Examples of FE applications
Studying the Dynamics of a High-Rise Building
Courtesy of Weng Cheong Yip and John Owen, Department of Civil Engineering, University of Nottingham
Examples of FE applications
10
Examples of FE applications
Summary of Contents
Introduction to Matrices
Introduction to CIVE4061
and FE Formulations
FE Formulation of Bar
FE Package (ANSYS)
and Beam Elements
Programming to Solving Isoparametric FE
Discrete & 2D Formulation of 2D
Continuum Problems Elements
CW1 10 %
CW2 20 %
Final Exam 70 %
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Course Contents
- Introduction to matrices and FE formulations
-The Stiffness method
- Isoparametric Finite Element method
- Applications of Finite Element method in Structural
Mechanics
MATLAB
MATLAB is a high-level language and
interactive environment that enables
performing computationally intensive
tasks.
It is a high-performance language
that integrates computation,
visualization, and programming in an
easy-to-use environment.
See getting-started Demo at:
www.mathworks.com/products/demos/shipping/matlab/GettingStarte
dwithMATLAB.html
12
MATLAB
See getting-started Demo at:
https://www.mathworks.com/videos/getting-started-with-matlab-
68985.html
https://www.mathworks.com/help/matlab/getting-started-with-
matlab.html
https://www.mathworks.com/products/matlab/videos.html?s_iid=ovp
_vids_4221048365001-101161_rr
Finite Element Packages
Numerous Finite Element computer programs are
commercially available for specific or general applications
Examples
Computer Programs for the Finite Element Method
General and special purpose programs
Logan, A First Course in the Finite Element Method, Sec. 1.7
13
ANSYS
ANSYS is a software package which is used in finite element
analysis. Its field of use is large including structural work,
electromagnetics, fluid dynamics, thermal analysis, etc.
Assessment of course
Coursework (30%)
- Coursework 1: Displacement based finite element method (10%)
- Coursework 2: The Isoparametric finite element method (20%)
Exam (70%)
2 hrs Exam at the end of semester
14
References
1. Bathe, Finite Element Procedures, Prentice Hall
2. Zienkiewicz, Taylor and Zhu, The Finite Element method, its
basis and fundamentals, Elsevier
3. Becker, An Introductory Guide to Finite Element Analysis,
Professional Engineering Publications
4. Cheung, Lo and Leung, Finite Element Implementation,
Blackwell Science Ltd
5. Rao, The Finite Element method in Engineering, Elsevier
6. Asghar Bhatti, Advanced topics in Finite Element Analysis of
Structures, John Wiley & Sons, inc.
7. Daryl Logan , A First Course in the Finite Element Method,
Thompson
Summary
The Finite Element method is used to solve differential
equations in closed domain
The method will be used in this course for applications in
Structural Mechanics
MATLAB, ANSYS as essential packages for understanding the
course contents
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