Civil Engineering Reference
In-Depth Information
q
(
x
)
y
d
M
d
x
M
M
d
x
x
V
d
x
d
V
d
x
V
d
x
Figure 5.7
Differential section of a loaded beam.
longitudinal axis of a loaded beam as depicted in Figure 5.7 where
q
(
x
) repre-
sents a distributed load expressed as force per unit length. Whereas
q
may vary
arbitrarily, it is assumed to be constant over a differential length
d
x
. The condi-
tion of force equilibrium in the
y
direction is
V
d
x
d
V
d
x
−
V
+
+
+
q
(
x
)d
x
=
0
(5.38)
from which
d
V
d
x
=−
q
(
x
)
(5.39)
Moment equilibrium about a point on the left face is expressed as
V
d
x
d
x
d
M
d
x
d
V
d
x
d
x
2
=
M
+
d
x
−
M
+
+
+
[
q
(
x
)d
x
]
0
(5.40)
which (neglecting second-order differentials) gives
d
M
d
x
=−
V
(5.41)
Combining Equations 5.39 and 5.41, we obtain
d
2
M
d
x
2
=
q
(
x
)
(5.42)
Recalling, from the elementary strength of materials theory, the
flexure formula
corresponding to the sign conventions of Figure 5.7 is
d
2
v
d
x
2
M
=
EI
z
(5.43)
(where in keeping with the notation of Chapter 4,
v
represents displacement in
the
y
direction), which in combination with Equation 5.42 provides the govern-
ing equation for beam flexure as
EI
z
d
2
v
d
x
2
d
2
d
x
2
=
q
(
x
)
(5.44)
