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MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
A - INTRODUCTION AND OVERVIEW
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 1
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Course Content:
A – INTRODUCTION AND OVERVIEW
Numerical method and Computer-Aided Engineering; Physical
problems; Mathematical models; Finite element method;.
B – REVIEW OF 1-D FORMULATIONS
Elements and nodes, natural coordinates, interpolation function, bar
elements, constitutive equations, stiffness matrix, boundary
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 2
elements, constitutive equations, stiffness matrix, boundary
conditions, applied loads, theory of minimum potential energy; Plane
truss elements; Examples.
C – PLANE ELASTICITY PROBLEM FORMULATIONS
Constant-strain triangular (CST) elements; Plane stress, plane
strain; Axisymmetric elements; Stress calculations; Programming
structure; Numerical examples.
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
COMPUTER-AIDED ENGINEERING (CAE)
The use of computers to analyze and simulate the
function (structural, motion, etc.) of mechanical,
electronic or electromechanical systems.
• Computer-Aided Design (CAD)
- Drafting
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 3
- Drafting
- Solid modeling
- Animation & visualization
- Dimensioning & tolerancing
• Engineering Analyses
- Analytical & numerical methods
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
The Process of FE analysis
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
Ref: K.J. Bathe, Finite Element Procedures, Prentice-Hall, 1996
4
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
02
2
=+ yEI
P
dx
yd
Buckling of Euler Column
PHYSICAL PROBLEM AND MATHEMATICAL MODEL
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 5
=
−
−+
2
1
2
12
2
1
22
221
0
0
x
x
m
m
x
x
kk
kkkω
Eigenvalue problem
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
EXAMPLE OF A MODEL
Steady-state heat conduction through a thick wall
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 6
)(:
0
xTSolution
Qdx
dTk
dx
d=+
+
=
+ ∞hTR
R
R
T
T
T
hkkk
kkk
kkk
LLLLLL
L
L
MM
L
M
L
L
2
1
2
1
21
22221
11211
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
REVIEW OF MATRIX ALGEBRA
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 7
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Matrix Algebra
In this course, we need to solve system of linear equations in the
form
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa
=+++
=+++
=+++
...
............................................
...
...
22222121
11212111
(2-1)
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
nnnnnn bxaxaxa =+++ ...2211
where x1, x2, …, xn are the unknowns.
Eqn. (2-1) can be written in a matrix form as
[ ]{ } { }bxA = (2-2)
where [A] is a (n x n) square matrix, {x} and {b} are (n x 1) vectors.
8
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
[ ] { } { }
=
=
= n
n
b
b
b
x
x
x
aaa
aaa
aaa
:b and ,
:x ,
...
::::
...
...
A2
1
2
1
22221
11211
The square matrix [A] and the {x} and {b} vectors are is given by,
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
… (2-3)
Note:
Element located at ith row and jth column of matrix [A] is denoted by aij. For
example, element at the 2nd row and 2nd column is a22.
nnnnnn bxaaa ...21
9
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Matrix Multiplication
The product of matrix [A] of size (m x n) and matrix [B] of size (n x
p) will results in matrix [C], with size (m x p).
Note: The (ij)th component of [C], i.e. cij, is obtained by taking the DOT
product,
(2-4)[ ] [ ] [ ] ) x ( ) x ( ) x (
pmpnnm
CBA =
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
product,
])[ ofcolumn th ( ])[ of rowth ( BjAicij ⋅= (2-5)
Example:
2) x (2 2) x (3 )3 x 2(
710-
157
30
25
41
120
312
=
−
−
10
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Matrix Transposition
If matrix [A] = [aij], then transpose of [A], denoted by [A]T, is given by
[A]T = [aji]. Thus, the rows of [A] becomes the columns of [A]T.
Example:
−
−
=32
60
51
][A
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
Note: In general, if [A] is of dimension (m x n), then [A]T has the dimension
of (n x m).
−
24
32
Then,
−
−=
2365
4201][ TA
11
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Transpose of a Product
The transpose of a product of matrices is given by the product of the
transposes of each matrices, in reverse order, i.e.
TTTT ABCCBA ][][][])][][([ = (2-6)
Determinant of a Matrix
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
Consider a 2 x 2 square matrix [x],
[ ]
=
2221
1211
xx
xxx
(2-7)
The determinant of this matrix is give by,
[ ] 12212211 xxxxxdet −=
12
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
EXAMPLE
Given that:
=
=
−=
−=
2
1
}{ 041
213
][
301
013][
41
01][
ED
CA
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
3302
Find the product for the following cases:
a) [A][C]
b) [D]{E}
c) [C]T[A]
13
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Solution of System of Linear Equations
System of linear algebraic equations can be solved for the unknown using
the following methods:
a) Cramer’s Rule
b) Inversion of Coefficient Matrix
c) Gaussian Elimination
d) Gauss-Seidel Iteration
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
Example: Solve the following SLEs using Gauss elimination method.
(iii) 6 122
(ii) 8324
(i) 11312
321
321
321
−=−+−
=+−
=−+
xxx
xxx
xxx
14
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Gauss Elimination Method
Reducing a set of n equations in n unknowns to an equivalent triangular
form (forward elimination). The solution is determined by back substitution
process.
Basic approach
-Any equation can be multiplied (or divided) by a nonzero scalar
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
-Any equation can be multiplied (or divided) by a nonzero scalar
-Any equation can be added to (or subtracted from) another equation
-The position of any two equations in the set can be interchanged
15
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Example: Solve the following SLEs using Gaussian elimination.
(iii) 6 122
(ii) 8324
(i) 11312
321
321
321
−=−+−
=+−
=−+
xxx
xxx
xxx
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
Eliminate x1 from eq.(ii) and eq.(iii). Multiply eq.(ii) by 0.5 we get,
(iii) 6 122
*(ii) 45.112
(i) 11312
321
321
321
−=−+−
=+−
=−+
xxx
xxx
xxx
16
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Subtract eq.(ii)* from eq.(i), we obtain
Add eq.(iii) with eq.(i), yields
(iii) 6 122
**(ii) 75.420
(i) 11312
321
321
321
−=−+−
=−+
=−+
xxx
xxx
xxx
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
Add eq.(iii) with eq.(i), yields
*(iii) 5 430
**(ii) 75.420
(i) 11312
321
321
321
=−+
=−+
=−+
xxx
xxx
xxx
17
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Eliminate x2 from eq.(iii)*. Multiply eq.(ii)** by 3 and eq.(iii)* by 2
we get
Subtract eq.(iii)** from eq.(ii)***, we obtain
(i) 11312 =−+ xxx
**(iii) 10 860
***(ii) 215.1360
(i) 11312
321
321
321
=−+
=−+
=−+
xxx
xxx
xxx
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
***(iii) 11 5.500
**(ii) 75.420
(i) 11312
321
321
321
=−+
=−+
=−+
xxx
xxx
xxx
From eq.(iii)*** we determine the value of x3, i.e.
25.5
113 −=
−=x
18
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Back substitute value of x3 into eq.(ii)** and solve for x2, we get
12
)2(5.472 −=
−+=x
Back substitute value of x2 and x3 into eq.(i) and solve for x1, we
get
6=x
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
61 =x
19
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Example
Solve the following systems of linear equations by using the
Gaussian elimination method.
340
1242
223
321
321
321
=++
=+−
=−+−
xxx
xxx
xxxa)
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM
6222
8324
11312
340
321
321
321
321
−=−+−
=+−
=−+
=++
xxx
xxx
xxx
xxx
b)
20
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Steps in solving a continuum problem by FEM
� Identify and understand the problem(This essential step is not FEM)
� Select the solution domainSelect the solution region for analysis.
� Discretize the continuumDivide the solution region into finite number of elements,
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 21
Divide the solution region into finite number of elements,
connected to each other at specified points / nodes.
� Select interpolation functionsChoose the type of interpolation function to represent the
variation of the field variables over the element.
� Derive element characteristic matrices and vectorsEmploy direct, variational, weighted residual or energy
balance approach.
[k](e) {φ}(e) = {f}(e)
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
Steps > (Continued)
� Assemble the element characteristic matrices and vectors
� Combine the element matrix equations and form the matrix
equations expressing the behavior of the entire solution region /
system.
� Modify the system equations to account for the boundary conditions
of the problem.
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 22
of the problem.
� Solve the system equations
Solve the set of simultaneous equations to obtain the unknown
nodal values of the field variables.
� Make additional computations, if desired
Use the resulting nodal values to calculate other important
parameters.
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
What is the problem?
Solution region of
interest
P
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 23
Task:
To simulate stress and strain fields in the
vicinity of a sharp notch under tensile load
Other examples:
-Scratches on a tensile surface
-Oil groove on a shaft
-Threaded connections
-Rivet holes under tension
Stress concentration along a shaft
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
� Select the solution domain
� Discretize the continuum
� Choose inerpolation functions
� Derive element characteristic
matrices and vectors
� Draw the model geometry
� Mesh the model geometry
� Select element type
� (The FEA software was written to do this)
Input material properties
FE procedures: FE software user steps:
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 24
matrices and vectors
� Assemble element
characteristic matrices and
vectors
� Solve the system equations
� Make additional
computations, if desired
Input material properties
� (The computer will assemble it)
Input specified load and boundary conditions
� (The compiler will solve it)
Request for output
� Post-process the result file
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
� Draw the model geometry
� Mesh the model geometry
� Select element type
[20-node Hexahedral elements]
� Input material properties
EXAMPLE: Stresses in a C(T) specimen
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 25
� Input material properties
[Elastic modulus, Poisson’s ratio]
� Input specified load and BC
[Pin loading, displacement rate,
zero displacement at lower pin]
� Request for output
[Displacement, strain and stress
components]
� Post-process the result file
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
EXAMPLE: Stresses in a C(T) specimen
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 26
� Check for obvious mistakes/errors
� Extract useful information
� Explain the physics/ mechanics
� Validate the results
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
MODELING CAPABILITIES – Microelectronic device reliability
- Prediction of spatial distribution of physical parameters
Silicon Die
Heat sink
Solder
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 27
MT1
TC1
TD1
0.00
0.02
0.04
0.06
0.08
0.10
0 1 2 3 4 5
N
εin
-40
125
183
25
Re-flow
T
Time
- Prediction of damage evolution characteristics
Substrate
Motherboard PCB
Solder joints
Solder
mask
MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS
w
MODELING CAPABILITIES – Deflection of composite laminates beam
INTRODUCTION AND OVERVIEW M.N. Tamin, CSMLab, UTM 28
• 14 layers [0/45/90/-45/45/-45/02]s
• Layer thickness = 0.3571 mm
Al Comp-0º Comp-90º
E E11 E22
70 GPa 44.74 GPa 12.46 GPa
)L(6
EI3
−= xwx
y