Civil Engineering Reference
In-Depth Information
X
1
X
2
w
(
x
)
w
(
x
)
S
S
A
B
M
M
M
x
1
S
x
F
IGURE
3.18
Load, shear force
and bending
moment
relationships
x
A
B
x
2
x
(a)
(b)
small the distributed load may be regarded as constant over the length
δ
x
. For vertical
equilibrium of the element
S
+
w
(
x
)
δ
x
−
(
S
+ δ
S
)
=
0
so that
+
w
(
x
)
δ
x
− δ
S
=
0
Thus, in the limit as
δ
x
→
0
d
S
d
x
=+
w
(
x
)
(3.1)
From Eq. (3.1) we see that the rate of change of shear force at a section of a beam,
in other words the gradient of the shear force diagram, is equal to the value of the
load intensity at that section. In Fig. 3.12(c), for example, the shear force changes
linearly from
wL
at A to zero at B so that the gradient of the shear force diagram at
any section of the beam is
−
w
where
w
is the load intensity. Equation (3.1)
also applies at beam sections subjected to concentrated loads. In Fig. 3.13(a) the load
intensity at B, theoretically, is infinite, as is the gradient of the shear force diagram at
B (Fig. 3.13(d)). In practice the shear force diagram would have a finite gradient at
this section as illustrated in Fig. 3.14.
+
wL
/
L
=+
Now integrating Eq. (3.1) with respect to
x
we obtain
w
(
x
)d
x
S
=+
+
C
1
(3.2)
in which
C
1
is a constant of integration which may be determined in a particular case
from the loading boundary conditions.
If, for example,
w
(
x
) is a uniformly distributed load of intensity
w
, i.e., it is not a
function of
x
, Eq. (3.2) becomes
S
=+
wx
+
C
1
