Civil Engineering Reference
In-Depth Information
P
jz
P
iz
M
jy
M
iy
θ
iy
EI
y
EI
y
Figure 4.27.
Example 4.15
q
iy
stiffness.
positive
y
direction using the right-hand rule and the Cartesian right-hand
coordinate system. The forces are shown consistent with the deformation.
The moment diagram divided by
EI
is shown for the reaction forces on the
i
i-end of the member.
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
q
iconj
ireal
iy
∆
=
0
iconj
ireal
Since moment in the conjugate does not exist but shear does, the conjugate
beam is pinned on the
i
-end.
V
M
=
q
=
0
jconj
jreal
=
∆
=
0
jconj
jreal
Since both the shear and moment in the conjugate do not exist, the conju-
gate beam is free on the
j
-end. The resulting conjugate beam is shown in
Figure 4.28.
Figure 4.28.
Example 4.15
q
iy
stiffness.
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 4.27 is applied to the conjugate
beam in Figure 4.28.
