Civil Engineering Reference
In-Depth Information
From statics on the real beam considering
q
jy
:
MM
EI
L
y
=−
=
q
iy
jy
jy
k
EI
L
PP
k
y
=
q
11 11
,
jy
==
==
0
iz
jz
k
0
311
,
9 11
,
The resulting stiffness matrix is shown in Equation 5.2 with only the
affected terms replaced with the new values.
k
000 00 00000
0 0 00 0 0 00
0 000 00000 000
000
k
11
,
17
,
∆
∆
∆
P
P
P
M
M
M
P
P
P
M
M
M
k
k
k
k
ix
ix
22
,
26
,
28
,
212
,
iy
iy
k
00000
k
0
0
iz
iz
44
,
410
,
q
q
q
EI
L
EI
L
ix
ix
y
y
0 000 00000
−
0
iy
iy
0
k
0 00 0
k
k
0 00
k
62
,
66
,
68
,
6
12
iz
iz
=
k
000 00 00000
0 0 00 0 0 00
0 000 000000 0
k
∆
∆
∆
71
,
77
,
jx
jx
k
k
k
k
82
,
86
,
88
,
812
,
jy
jy
0
jz
jz
000
k
0 0000 00
k
q
q
q
10 4
,
1010
,
jx
jx
EI
L
EI
L
jy
y
y
jy
0 000 00000
−
0
jz
jz
0
k
0 00 0
k
k
00 0
k
12 2
,
126
,
12 8
,
1212
,
(5.2)
Example 5.2
q
iy
end release
Derive the local member stiffness for a
q
iy
member end release using the
conjugate beam method.
A free-body diagram of the released beam is shown in Figure 5.2.
Since the beam is allowed to rotate at the
i
i-end in the
y
direction, the
reaction
M
iy
is equal to zero. The loaded conjugate beam is also shown in
Figure 5.2. Note that the shear in the conjugate beam is equal to the rota-
tion in the real beam and the moment in the conjugate beam is equal to the
deflection in the real beam.
If a motion ∆
iz
is imposed, there is resistance and therefore forces. The
resulting forces are derived using conjugate beam.
