MESF593 Finite Element Methods

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MESF593 Finite Element Methods
HW 2 Solutions
2
Prob. 1 (25)
q(x)
Q1 , v1
Q3 , v3
Q2 , v2
Q4 , v4
x L
x 0
Beam Width b, Beam Thickness a(c-a)x/L
The element equations of a general tapered beam
with a rectangular cross-section are given above.
Derive the explicit expression of each term of
Kij in terms of the dimensions (a, b, c, L) and
Youngs modulus (E) of the beam.
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Prob. 1 Solution
q(x)
Q1 , v1
Q3 , v3
Q2 , v2
Q4 , v4
x L
x 0
Beam Width b, Beam Thickness a(c-a)x/L
4
Prob. 1 Solution
5
Prob. 2 (15)
For the truss structure shown above at the left
hand side, is it possible to use the finite
elements shown at the right hand side for
modeling? Why?
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Prob. 2 Solution
Connection to the mid node of a higher order bar
element is not allowed. This is because the
degree of freedom (i.e., displacement) of the mid
node must be along the axial direction of the bar
element as shown in Fig. 1. However, if the mid
node is tied to the end node of another linear
element, that means this common node can move in
any direction. As a result, the higher order bar
element may not be a straight element.
Fig. 1
Fig. 2
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Prob. 3 (30)
Y2 , v2
X2 , u2
E, A, I, L
M2 , q2
Y1 , v1
-
yx
X1 , u1
M1 , q1
Note The plane coordinate transformation matrix
for a rotation about the z-axis is
Derive the explicit form of the stiffness matrix
Kij of a combined bar-beam element arbitrarily
oriented on a 2-D plane.
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Prob. 3 Solution
From the lecture notes
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Prob. 3 Solution
10
Prob. 3 Solution
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Prob. 4 (30)
(All three segments have the same E, A, I, L)
Use the finite element method to solve for all
reaction forces and moments at the boundaries of
the anti-symmetric Z-frame structure given above.
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Prob. 4 Solution
For Element I
(from the lecture notes)
13
Prob. 4 Solution
For Element II
(use the result of Prob. 4, let ? -p/2)
For Element III
(from the lecture notes)
14
Prob. 4 Solution
15
Prob. 4 Solution
(due to anti-symmetry, u2 -u3, v2 v3, q2
-q3)
16
Prob. 4 Solution