Civil Engineering Reference
In-Depth Information
observing that the shear in the conjugate beam equals the rotations in the
real beam and the moment in the conjugate equals the deflection in the
real beam. Therefore, the fixed-end of the real beam becomes free in the
conjugate beam and the free end in the real beam becomes fixed in the
conjugate beam. The conjugate beam is loaded with the
M
/
EI
diagram of
the real beam. This is a triangular moment diagram due to the applied load
P divided by EI. The last beam in Figure 4.14 is a free-body diagram of the
conjugate beam with the area of the
M
/
EI
diagram equated to a point load
located at the centroid of the diagram. Static equilibrium can then be per-
formed on the conjugate beam to find shear and moment at point B, which
represents rotation and deflection in the real beam shown in Figure 4.15.
P
B
A
L/2
L/2
P
P
M=PL/2
L/2
L/2
L/2
L/2
A′
B′
PL
2EI
M
B′
L/6
5L/6
V
B′
PL
2
8EI
Figure 4.14.
Example 4.6 Conjugate beam.
2
2
PL
EI
VV
PL
Σ
F
==−−∴=−
0
y
B
B
8
8
EI
2
PL
EI
q
=−
B
8
2
3
PL
EI
5
6
L
MM
5
48
PL
EI
Σ
M
== +∴
0
=−
B
B
B
8
3
5
48
PL
EI
∆
=−
B
