Civil Engineering Reference
In-Depth Information
Since point
A
is pinned, we know that the deflection at point
A
, ∆
A
, is
equal to zero. We can superimpose Figures 4.10 and 4.11. Applying the sec-
ond area moment theorem, the following can be written. It should be noted
that the tangent to the deflected shape is at
B
and therefore the point is at
A
:
B
M
EI
xdx x
∫
∆
== =
0
t
=
Σ
A
AB
A
1
A
1
32
L
wL
EI
2
3
4
L
1
2
L
RL
2
3
L
3
wL
+
∴=
0
=−
A
R
A
EI
8
From statics on the original free-body diagram in Figure 4.9, the other
reactions at
B
can be found.
R R LR
wL
5
Σ
F
== +−∴=
0
y
A
B
B
8
2
2
R LM
wL
wL
Σ
M
==−−+∴=
0
M
B
A
B
B
2
8
4.5
cOnJugAtE bEAM MEtHOD
The conjugate beam method is based on the equation of beam bending
f = M/EI. The method was developed by Heinrich Müller-Breslau in 1865
(Müller-Breslau 1875). A
conjugate beam
can be summarized as an imag-
inary beam equal in length to the real beam. In the imaginary beam, the
shear at the conjugate support is equal to the slope of the real support.
Also, the moment at the conjugate support is equal to that of the deflection
at the real support. The conjugate beam is loaded with the
M
/
EI
diagram
from the real beam. A summary of the more common support conditions
is shown in Figure 4.12.
Take the pinned condition for example. In the real beam, the rotation is
unknown (exist) and deflection is equal to zero. Therefore, in the conjugate
beam, the shear is unknown (exist) and the moment is equal, thus creating
a pinned connection. This condition does not change from the real beam to
the conjugate beam. However, this is not true for all cases. For example,
if the real beam is fixed, the rotation is zero and the deflection is zero. The
result is zero moment and shear in the conjugate that yields a free end.
Example 4.6
Conjugate beam
Draw the conjugate beam for each of the real beams in Figure 4.13.
