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Finite Element Method –An Introduction (One Dimensional Problems)
by
Tarun Kant
tkant@civil.iitb.ac.in
www.civil.iitb.ac.in/~tkant
Department of Civil Engineering Indian Institute of Technology Bombay
Powai, Mumbai – 400076
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Elastic Spring
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Elastic Spring
One Spring: 2 degrees of freedom (dofs)
We wish to establish a relationship between nodal forces and nodal displacements as:
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Elastic Spring
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Elastic Spring
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Linear response – superpose two independent solutions
In matrix form,
Nodal force vector;
Nodal displacement vector
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Two Springs
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Two Springs
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Two Springs
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Two Springs
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Two Springs
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Some Properties of K
1. Sum of elements in any column is 0 equilibrium
2. K is symmetric
3. singular : no BC’s
4. All terms on main diagonal positive.
If this were not so, a +ive nodal force Pi could produce a corresponding –ive ui.
5. If proper node numbering is done, K is banded.
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An Alternative Procedure
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where,
1, 2, 3 :Global and Local node numbers
, :Element numbers
1 2
Our aim is to compute a 3x3 K matrix of the two spring assemblage
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Continuity (Compatibility) conditions
Equilibrium of nodal forces
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Direct Stiffness Method -Assembly
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Axial Rod
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Axial Rod
Where k = stiffness of spring
Direct Method – Simple ‘discrete’ elements
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Variational Method -Energy Method
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Strains
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Constitutive Relation
Minimum Potential Energy The Total Potential Energy pi is given by:
where, U is strain energy stored in the body during deformation and
W is the work done by the external loads
We have established In the above, we have discretized displacement and strain expressions
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Variational Statement
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Uniform Load
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Axial Rod To determine Governing Equation,
1. Equilibrium equation:
2. Strain-displacement relation:
3. Constitutive relation:
Using the above relations, we have
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Weighted Residual Method
We need to minimize the weighted residue in order to develop our appropriate method
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Thus, we obtain the discrete governing equation.
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Transformation
• Local Coordinate
p1
u1
p2
u2
x
1 1
2 2
AE AE
p uL L
p uAE AE
L L
A, E, L
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In many instances it is convenient to introduce both local and global coordinates.
The local coordinates are always chosen to represent an individual element. Global coordinates on other hand, are chosen for the entire system.
Usually, applied loads, boundary conditions, etc. are described in global coordinates
Transformation (contd.)
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Stiffness Equation in Local Coordinates in Expanded Form
y
Y
q2 , v2
Q2 , V2
p2 u2
x
P2 , U2 2
1
Q1 , V1
P1 , U1
q1 , v1
We have introduced q1 and q2 and v1 and v2
q1 and q2 do not exist since a truss element can not withstand a force normal to its axis.
p1 u1
θ x
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Stiffness Equation in Local Coordinates in Expanded Form (Contd.)
1 1
1 1
2 2
2 2
1 0 1 0
0 0 0 0
1 0 1 0
0 0 0 0
p u
q vAE
p uL
q v
y
Y
q2 , v2
Q2 , V2
p2 u2
x
P2 , U2 2
1
Q1 , V1
P1 , U1
q1 , v1
p1 u1
θ x
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In a compact form the matrix equation in a local coordinates can be expressed as
1 1 2 2
1 1 2 2
,
t
t
in which
p q p q
u v u v
p ku
p
u
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Stiffness Equation in Local Coordinates in Expanded Form (Contd.)
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Transformation of Coordinates
Transformation is required when local coordinates for description of elements change from element to element, e.g.,
1 6 11
5
2
10
7
3 4
8 9
1
2
4
3
5
6
6 – Nodes 11 – Elements
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1 1 1
1 1 1
-1
cos sin
- sin cos
At Node
p P Q
q P Q
2 2 2
2 2 2
- 2
cos sin
- sin cos
At Node
p P Q
q P Q
y
Y
q2 , v2
Q2 , V2
p2 u2
x
P2 , U2 2
1
Q1 , V1
P1 , U1
q1 , v1
p1 u1
θ x
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Transformation of Coordinates (Contd.)
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1 1
1 1
2 2
2 2
sin cos ,
, ,
0 0
0 0
0 0
0 0
:
.
Let s and c
then combining the above equations we can write
p Pc s
q Qs c
p Pc s
q Qs c
in compact form
p T P
Transformation of Coordinates (Contd.)
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1 1 2 2
1 1 2 2
,
,
, , .
,
t
t
in which
p q p q
P Q P Q
and T is called which transforms
the global nodal forces into local nodal forces
Since displacements are also ve
p
P
transformation matrix
P p
1 1 2 2
1 1 2 2
,
, . .,
=
,
t
t
ctors like forces
a similiar transformation rule exists for them too i e
in which
u v u v
U V U V
u TU
u
U
Transformation of Coordinates (Contd.)
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,
We can also write
Since is an orthogonal matrix
-1
-1 t
P = T p
T
T = T
Transformation of Coordinates (Contd.)
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Stiffness Equation in Global Coordinates
2 2
2 2
2 2
2 2
,
in which
c cs c cs
cs s cs sAE
L c cs c cs
cs s cs s
t
t
t
t
T
T k u
T k T U
K U
K T kT
P = p
=
=
=
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2 2
2 2
2 2
2 2
, .
c cs c cs
cs s cs sAE
L c cs c cs
cs s cs s
This is stiffness matrix of the truss element as shown in the
figure with reference to the global X Y coordinates
The above formulation for fi
K
K :rst appeared in
Stiffness Equation in Global Coordinates (Contd.)
Turner MJ, Clough RW, Martin HC and Topp LJ (1956), Stiffness and deflection analysis of complex structures,
J. Aeronautical Sciences, (9), 805 - 824.23
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,
,
A derivation of without involving transformation matrix
is given in
K T
Martin HC (1958), Truss analysis by stiffness consideration,
Stiffness Equation in Global Coordinates (Contd.)
, 1182 - 1194.Trans. ASCE,123
End
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