Civil Engineering Reference
In-Depth Information
P
iy
P
jy
∆
iy
M
jz
M
iz
EI
z
EI
z
Figure 4.31.
Example 4.18 Δ
iy
stiffness.
The conjugate beam can be constructed for the two basic assump-
tions. The shear in the conjugate is equal to the slope of the real beam, and
the moment of the conjugate is equal to the deflection of the real beam.
V
M
==
=
q
0
∆∆
iconj
ireal
=
iconj
ireal
iy
Since moment in the conjugate exists but shear is zero, the conjugate beam
is slotted in the
y
direction on the
i
i-end. This is a connection that is free to
move vertically, but restrained from rotation.
V
M
=
q
=
0
jconj
jreal
=
∆
=
0
jconj
jreal
Since both the shear and the moment in the conjugate do not exist, the con-
jugate beam is free on the
j
-end. The resulting conjugate beam is shown
in Figure 4.32.
The conjugate beam method can be applied to find the reactions at the
i
i-end of the conjugate beam, which are equal to deformations at the
i
-end
of the real beam. The load from Figure 4.31 is applied to the conjugate
beam in Figure 4.32.
2
P
L
EI
M
L
iy
iz
V
===−
q
0
+
iconj
ireal
EI
2
z
z
2
P
L
EI
M
L
L
2
3
L
iy
iz
M
=
∆∆
I
=
=−
+
iconj
ireal
iy
E
22
z
z
