Civil Engineering Reference
In-Depth Information
4
EI
z
L
=
k
(2)
24
K
46
=
96
EI
z
L
3
=
k
(2)
33
K
55
=
−
24
EI
z
L
2
=
k
(2)
34
K
56
=
8
EI
z
L
=
k
(2)
44
K
66
=
Using the general form
[
K
]
{
U
}={
F
}
we obtain the system equations as
96
24
L
−
96
24
L
0
0
v
1
1
v
2
2
v
3
3
F
1
M
1
F
2
M
2
F
3
M
3
8
L
2
−
24
L
4
L
2
24
L
0
0
EI
z
L
3
−
−
−
96
24
L
192
0
96
24
L
=
24
L
4
L
2
0
16
L
2
−
24
L
4
L
2
0
0
−
96
−
24
L
96
24
L
0
0
24
L
4
L
2
24
L
8
L
2
Invoking the boundary conditions
v
1
=
1
=
v
3
=
0
, the reduced equations become
=
192
0
24
L
v
2
2
3
−
P
0
0
EI
z
L
3
0
L
2
4
L
2
24
L
4
L
2
8
L
2
Yielding the nodal displacements as
−
7
PL
3
768
EI
z
PL
2
128
EI
z
−
PL
2
32
EI
z
v
2
=
2
=
3
=
The deformed beam shape is shown in superposition with a plot of the undeformed shape
with the displacements noted in Figure 4.7c. Substitution of the nodal displacement val-
ues into the constraint equations gives the reactions as
EI
z
L
3
11
P
16
F
1
=
(
−
96
v
2
+
24
L
2
)
=
EI
z
L
3
5
P
16
F
3
=
(
−
96
v
2
−
24
L
2
−
24
L
3
)
=
EI
z
L
3
3
PL
16
(
−
24
Lv
2
+
4
L
2
M
1
=
2
)
=
Checking the overall equilibrium conditions for the beam, we find
F
y
11
P
16
−
P
+
5
P
16
=
0
=
