EML 5526 Finite Element Analysis and Design Examination #2

Last Name:_Anirudha _____
First Name: _Mandal______

EML 5526 Finite Element Analysis and Design
Examination #2

Problem 1: 1
For the truss-like structure shown in figure, assume that all
elements have the same length L and the same properties: Young’s 1
modulus = E and area of cross section = A. Py
45o
(a) Write the connectivity table for this model assuming that node 2 Px
is the first node for each element (10 pts) 2 2
Ans ; Connectivity table of the given truss is shown as: 3
Element Local node 1 Local node 2 θ l=cosθ m=sinθ 3
1 2 1 135 -0.707 0.707
2 2 3 180 -1 0 4
3 2 4 270 0 -1

(b) Write stiffness matrix of each element. (20 pts)
we know that,
l2 lm −l 2 −lm
K e=
EA (e) lm
( )
L 2
−l −lm
−lm −m
[
m2 −lm −m2

2
l
lm
Stiffness matrix for Element 1
2
lm
m
2 ]
0.4998 −0.4998 −0.4998 0.4998
K 1=
EA
( )
L
[
(1)
−0.4998 0.4998
−0.4998 0.4998
0.4998 −0.4998
0.4998 −0.4998
0.4998 −0.4998 −0.4998 0.4998
Stiffness matrix for Element 2
]
1 0 −1 0

)[ ]
(2)
EA 0 0 0 0
K 2= (
L −1 0 1 0
0 0 0 0
Stiffness matrix for Element 3

0 0 0 0

)[ ]
(3 )
EA 0 1 0 −1
K 3= (
L 0 0 0 0
0 −1 0 1




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, Last Name:_Anirudha _____
First Name: _Mandal______
(c) After applying the boundary conditions, the stiffness matrix reduces to a 2x2 matrix.
Directly assemble the 2x2 stiffness matrix to write the global equation for this truss
structure. (10 pts)
Ans ;
Since E, A and L are same for all the elements

We get Assembly matrix as

Here the applied boundary conditions are as follows ;
Since the nodes are
u1=v 1=u3=v 3=u 4=v 4=0 ( ¿ )

EA 0.4998+1+ 0 −0.4998+0+ 0 u 2 P
[
L −0.4998+ 0+0 0.4998+0+1 v 2
= x
Py ]{ } { }
EA 1.4998 −0.4998 u2 P
[
L −0.4998 1 .4998 v 2
= x
Py ]{ } { }
Problem 2:
A beam of length L is subjected to a uniformly distributed load p and a concentrated
bending moment C at node 2 as shown in figure. Solve using just the one beam element
shown in the figure. Write your answers in terms of Young’s modulus E, Moment of inertia
I, C , p and the length of the beam L.

y
C
x
1 2



A. Model the structure using one beam element. Write the system of equations [K]{X}
= {F} for the structure and then apply boundary conditions. (15 pts)
Ans ;
Here E= youngs modulus, I = moment of inertia , l=¿ Length of the beam
We get the stiffness matrix as
12 6 l −12 6 l
K= 3
EI 6 l
[ 4 l 2 −6 l 2l 2
l −12 −6 l 12 −6 l
6l 2 l 2 −6 l 4 l 2
The concentrated load is given as
]
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