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2.3.2 Concentrated Force at Intermediate
Point January 4, 2005 16:18 STRENGTH OF MACHINES
The cantilevered beam shown in Fig. 2.90 has a concentrated force (F) acting vertically
downward at an intermediate point, a distance (a) from the left end of the beam. The
cantilever reaction is on the right end of the beam, at point B. The length of the beam is
labeled (L).
F
a b
A B
L
FIGURE 2.90 Concentrated force at intermediate point.

Reactions. The reactions at the support are shown in Fig. 2.91—the balanced free-body-
diagram. Notice that the vertical reaction (B y ) is equal to the force (F), and because the
force (F) is acting straight down, the horizontal reaction (B x ) is zero. If the force (F) had a
horizontal component, either left or right, then the horizontal reaction (B x ) would be equal,
but opposite in direction to this horizontal component. The couple reaction (C B ) is in a
negative direction, meaning clockwise (cw), and equal to a negative of the force (F) times
the length (b) that is the distance from the force to the wall at B.

F
B = 0
x
C = –Fb
B
B y = F
FIGURE 2.91 Free-body-diagram.

U.S. Customary SI/Metric
Example 1. Determine the reactions for a can- Example 1. Determine the reactions for a can-
tilevered beam of length (L) with a concentrated tilevered beam of length (L) with a concentrated
force (F) acting at an intermediate point, where force (F) acting at an intermediate point, where
F = 150 lb F = 700 N
L = 8ft L = 2.5 m
a = 3 ft, b = 5ft a = 1m, b = 1.5 m
solution solution
From Fig. 2.91 calculate the reactions (B x , B y , From Fig. 2.91 calculate the reactions (B x , B y ,
and C B ) at the right end of the beam. and C B ) at the right end of the beam.
Step 1. As the force (F) is vertical, Step 1. As the force (F) is vertical,
B x = 0 B x = 0
and and
B y = F = 150 lb B y = F = 700 N