Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.35
The bending moment under the 6000-lb load in Fig. 5.20a considering only the
force to the left is 7000 � 10, or 70,000 ft-lb. The bending-moment diagram, then,
between the left support and the first concentrated load is a straight line rising from
zero at the left end of the beam to 70,000 ft-lb, plotted to a convenient scale, under
the 6000-lb load.
The bending moment under the 9000-lb load, considering the forces on the left
of it, is 7000 � 20 � 6000 � 10, or 80,000 ft-lb. (It could have been more easily
obtained by considering only the force on the right, reversing the sign convention:
8000 � 10 � 80,000 ft-lb.) Since there are no loads between the two concentrated
loads, the bending-moment diagram between the two sections is a sloping straight
line.
If the bending moment and shear are known at any section of a beam, the
bending moment at any other section may be computed, providing there are no
unknown forces between the two sections. The rule is:
The bending moment at any section of a beam is equal to the bending
moment at any section to the left, plus the shear at that section times the
distance between sections, minus the moments of intervening loads. If the section
with known moment and share is on the right, the sign convention must
be reversed.
For example, the bending moment under the 9000-lb load in Fig. 5.20a could
also have been obtained from the moment under the 6000-lb load and the shear to
the right of the 6000-lb load given in the shear diagram (Fig. 5.20b). Thus,
80,000 � 70,000 � 1000 � 10. If there had been any other loads between the two
concentrated loads, the moment of these loads about the section under the 9000-lb
load would have been subtracted.
Bending-moment diagrams for commonly encountered loading conditions are
given in Figs. 5.30 to 5.41. These may be combined to obtain bending moments
for other loads.
5.5.6 Moments in Uniformly Loaded Beams
When a bean carries a uniform load, the bending-moment diagram does not consist
of straight lines. Consider, for example, the beam in Fig. 5.21a, which carries a
uniform load over its entire length. As shown in Fig. 5.21c, the bending-moment
diagram for this beam is a parabola.
The reactions at both ends of a simply supported, uniformly loaded beam are
both equal to wL/2 � W/2, where w is the uniform load in pounds per linear foot,
W � wL is the total load on the beam, and L is the span.
The shear at any distance x from the left support is R 1 wx � wL/2 � wx (see
Fig. 5.21b). Equating this expression to zero, we find that there is no shear at the
center of the beam.
The bending moment at any distance x from the left support is
��
2
x wLx wx w
M � Rx� 1 wx � � � x(L � x) (5.51)
2 2 2 2
Hence:
The bending moment at any section of a simply supported, uniformly loaded
beam is equal to one-half the product of the load per linear foot and the
distances to the section from both supports.
The maximum value of the bending moment occurs at the center of the beam.
It is equal to wL 2 /8 � WL/8.