Problem 42
1
AAE352Problem set #4 Solution to Problem 2
The configuration shown has four rod elements connected to three node points.
Node 1 is fixed
(u
1
=0).
Node moves only in the horizontal direction and has three of the four elements attached
to it.
All element
s have the same Young’s modulus.
Element 4 has one
half the crosssectional
area of the other elements.
Node 2 has only elements 2 and 4 attached to it.
a)
If a 1000 pound load is placed at node 2 as shown, find the system stiffness matrix for
the unrestrained system.
This will be in terms of a parameter EA/L.
b)
Find the system stiffness matrix for the restrained system.
c)
Find the deflections u
2
and u
3
and the reaction at node 1.
These will be in terms of a
parameter L/EA.
d)
Find the internal forces due to the 1000 pound load.
Both of the answers in (a) and (b)
will have the parameters E, A and L in the answers.
e)
Repeat parts (a), (b) and (c) when the temperature of element 1 is increased by an amount
T.
The coefficient of thermal expansion is
and is the same for all elements.
First we identify and number the three nodal displacements and then break the structure into four
component rods.
We then identify the internal rod forces at each end of each rod:
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Problem 42
2
The nodal forces are related to the element internal forces by writing force equilibrium equations
at each node point.
1
11
21
31
2
22
41
3
12
32
42
P
F
F
F
P
F
F
P
F
F
F
In matrix form this equation set reads as follows:
1
11
21
31
2
22
41
3
12
32
42
0
0
0
0
P
F
F
F
P
F
F
P
F
F
F
There are four 3x1 vectors on the right hand side of this equilibrium matrix because there are
four rod elements involved.
Each element has a stiffness matrix that relates the element forces to
displacements.
This is necessary in this problem because it is statically indeterminate.
Notice
also that we have not said anything yet about boundary conditions.
We have applied nodal
forces, the P’s, and have taken into account the possibility that any of the displacements, u, might
be an unknown.
The element stiffness matrices are:
1
1
1
11
1
3
3
3
12
1
1
1
1
1
1
1
2
2
1
1
1
1
1
1
2
2
2
u
u
u
F
EA
EA
EA
u
u
u
F
L
L
L
21
1
1
2
22
2
2
2
1
1
1
1
1
1
1
1
F
u
u
EA
EA
F
u
u
L
L
31
1
1
1
3
32
3
3
3
3
1
1
1
1
1
1
2
2
1
1
1
1
1
1
2
2
2
F
u
u
u
EA
EA
EA
F
u
u
u
L
L
L
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 Spring '08
 Chen
 Force, Chemical element, ea

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