Building Design and Construction Handbook - Merritt - Ventech!

5.32 SECTION FIVE
5.5.2 Reactions
As an example of the application of the equations of equilibrium (Art. 5.2.1) to the
determination of the reactions of a statically determinate beam, we shall compute
the reactions of the 60-ft-long beam
with overhangs in Fig. 5.17. This beam
carries a uniform load of 200 lb/lin ft
over its entire length and several concentrated
loads. The supports are 36 ft
apart.
To find reaction R1, we take moments
about R2 and equate the sum of the mo-
FIGURE 5.17 Beam with overhangs loaded
with both uniform and concentrated loads.
ments to zero (clockwise rotation is considered
positive, counterclockwise, negative):
�2000 � 48 � 36R1 � 4000 � 30 � 6000 � 18 � 3000 � 12
�200 � 60 � 18 � 0
R1 � 14,000 lb
In this calculation, the moment of the uniform load was found by taking the moment
of its resultant, which acts at the center of the beam.
To find R 2, we can either take moments about R 1 or use the equation �V � 0.
It is generally preferable to apply the moment equation and use the other equation
as a check.
3000 � 48 � 36R2 � 6000 � 18 � 4000 � 6 � 2000 � 12
� 200 � 60 � 18 � 0
R2 � 13,000 lb
As a check, we note that the sum of the reactions must equal the total applied
load:
14,000 � 13,000 � 2000 � 4000 � 6000 � 3000 � 12,000
27,000 � 27,000
5.5.3 Internal Forces
Since a beam is in equilibrium under the forces applied to it, it is evident that at
every section internal forces are acting to prevent motion. For example, suppose
we cut the beam in Fig. 5.17 vertically just to the right of its center. If we total
the external forces, including the reaction, to the left of this cut (see Fig. 5.18a),
we find there is an unbalanced downward load of 4000 lb. Evidently, at the cut
section, an upward-acting internal force of 4000 lb must be present to maintain
equilibrium. Again, if we take moments of the external forces about the section,
we find an unbalanced moment of 54,000 ft-lb. So there must be an internal moment
of 54,000 ft-lb acting to maintain equilibrium.
This internal, or resisting, moment is produced by a couple consisting of a force
C acting on the top part of the beam and an equal but opposite force T acting on