Civil Engineering Reference
In-Depth Information
bending moment diagram at a beam section is equal to minus the value of the shear
force at that section. For example, in Fig. 3.16(e) the bending moment in AB is a
mathematical maximum at the section where the shear force is zero.
Integrating Eq. (3.4) with respect to
x
we have
S
d
x
M
=−
+
C
2
(3.5)
in which
C
2
is a constant of integration. Substituting for
S
in Eq. (3.5) from Eq. (3.2)
gives
w
(
x
)d
x
C
1
d
x
M
=−
+
+
+
C
2
or
w
(
x
)d
x
M
=−
−
C
1
x
+
C
2
(3.6)
If
w
(
x
) is a uniformly distributed load of intensity
w
, Eq. (3.6) becomes
w
x
2
M
=−
2
−
C
1
x
+
C
2
which shows that the equation of the bending moment diagram on a length of beam
carrying a uniformly distributed load is parabolic.
In the case of a beam carrying concentrated loads only, then, between the loads,
w
(
x
)
=
0 and Eq. (3.6) reduces to
M
=−
C
1
x
+
C
2
which shows that the bending moment varies linearly between the loads and has a
gradient
−
C
1
.
The constants
C
1
and
C
2
in Eq. (3.6) may be found, for a given beam, from the loading
boundary conditions. Thus, for the cantilever beamof Fig. 3.12, we have already shown
that
C
1
=−
wx
2
/2
wL
so that
M
=−
+
wLx
+
C
2
. Also, when
x
=
L
,
M
=
0 which gives
wL
2
/2 and hence
M
wx
2
/2
wL
2
/2 as before.
C
2
=−
=−
+
wLx
−
Now integrating Eq. (3.4) over the length of beam between the sections X
1
and X
2
(Fig. 3.18(a))
x
2
x
2
d
M
d
x
d
x
=−
S
d
x
x
1
x
1
which gives
x
2
M
2
−
M
1
=−
S
d
x
(3.7)
x
1
where
M
1
and
M
2
are the bending moments at the sections X
1
and X
2
, respectively.
Equation (3.7) shows that the
change
in bending moment between two sections of a
