Calculation of Reactions in Statically Determinate Systems

For statically determinate systems, reactions can be determined from equilibrium equations
[Eq. (3.11) or (3.12)]. For example, in the planar system shown in Fig. 3.66, reactions R1, H1, and R2 can be calculated from the three equilibrium equations. The beam with overhang carries a uniform load of 3 kips / ft over its 40-ft horizontal length, a vertical 60-kip concentrated load at C, and a horizontal 10-kip concentrated load at D. Support A is hinged; it can resist vertical and horizontal forces. Support B, 30 ft away, is on rollers; it can resist only vertical force. Dimensions of the member cross sections are assumed to be small relative to the spans.

Only support A can resist horizontal loads. Since the sum of the horizontal forces must
equal zero and there is a 10-kip horizontal load at D, the horizontal component of the reaction at A is H1 = 10 kips.

The vertical reaction at A can be computed by setting the sum of the moments of all forces about B equal to zero:

3 x 40 x 10 + 60 x 15 - 10 x 6 - 30R1 = 0
from which R1  68 kips. Similarly, the reaction at B can be found by setting the sum of
the moments about A of all forces equal to zero:
3 x 40 x 20 + 60 x 15 - 10 x 6 - 30R1 = 0
from which R2 = 112 kips. Alternatively, equilibrium of vertical forces can be used to obtain R2, given R1  68:
R + R - 3 x 40 - 60 = 0
Solution of this equation also yields R2 = 112 kips.

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