Civil Engineering Reference
In-Depth Information
2. Given that the “first” shape function of a 4-node plane stress rectangular element of
width
a
and height
b
is:
1
1
x
a
y
b
N
1
=
−
−
use analytical integration to show that the element stiffness and mass matrices include
the following terms:
b
3
a
+
E
1
−
ν
2
a
3
b
k
11
=
−
ν
2
1
ρab
9
m
11
=
3. For the problem shown in Figure 5.38, estimate the force necessary to displace the
loaded node horizontally by 0.015 units.
Ans: 0.8
P
4-node element
plane strain
E= 100
u
= 0.25
1
1
Figure 5.38
4. Derive the vertical nodal forces that are equivalent to the triangular stress distribution
acting on the 4-node element shown in Figure 5.39. Given that the stiffness matrix
of this element (assuming local freedom numbering in the order
u
1
v
1
u
2
v
2
u
3
v
3
u
4
v
4
)is:
57
.
69 24
.
04
9
.
62
−
4
.
81
−
28
.
85
−
24
.
04
−
38
.
46
4
.
81
57
.
69
4
.
81
−
38
.
46
−
24
.
04
−
28
.
85
−
4
.
81
9
.
62
57
.
69
−
24
.
04
−
38
.
46
−
4
.
81
−
28
.
85
24
.
04
57
.
69
4
.
81
9
.
62
24
.
04
−
28
.
85
57
.
69
24
.
04
9
.
62
−
4
.
81
57
.
69
4
.
81
−
38
.
46
57
.
69
−
24
.
04
57
.
69
compute the vertical displacement of the top two nodes.
Ans: 0.55, 0.19
