Civil Engineering Reference
In-Depth Information
relationships for stiffness methods of analysis. The
area moment method
of analysis is based on the basic equation of beam bending,
f
=
M
/
EI
,
and consists of two theorems. The theorems were developed by Christian
Otto Mohr in 1874 (Timoshenko 1953). The first theorem of area moment
states the change in slope between two points on the elastic curve, 1 and 2,
is equal to the area of the
M
/
EI
diagram between the points 1 and 2.
2
=
∫
M
∆
q
12
EI
dx
1
The second theorem of area moment states that the tangential deviation
of a point, 1, from a tangent to the elastic curve at point 2 (the tangent)
is equal to the moment of the area of the
M
/
EI
diagram between points
1 and 2, taken about point 1 (the point). Note that the moment of the area
is the area times a distance,
x
1
, from point 1.
2
M
EI
xdx
=
∫
t
12
1
1
Example 4.5
Area moment
Find the reactions on the following propped end, uniformly loaded beam
shown in Figure 4.8 using the area moment method.
Z
W
X
A
B
L
Figure 4.8.
Example 4.5 Area moment.
A free-body diagram of the beam is drawn in Figure 4.9.
W
M
B
R
A
R
B
Figure 4.9.
Example 4.5 Area moment.
