Question 4.SP.2: Three loads are applied to a beam as shown. The beam is supp......
Three loads are applied to a beam as shown. The beam is supported by a roller at A and by a pin at B. Neglecting the weight of the beam, determine the reactions at A and B when P = 15 kips.
STRATEGY: Draw a free-body diagram of the beam, then write the equilibrium equations, first summing forces in the x direction and then summing moments at A and at B
MODELING:
Free-Body Diagram. The reaction at A is vertical and is denoted by A (Fig. 1). Represent the reaction at B by components B_{x}~and~B_{y}. Assume that each component acts in the direction shown.
ANALYSIS:
Equilibrium Equations. Write the three equilibrium equations and solve for the reactions indicated:
\underrightarrow{+}\Sigma F_{x}=0\colon\qquad \qquad \qquad B_{x} = 0\qquad \qquad \qquad B_{x} = 0◂
+↺ΣM_{A} = 0\colon
−(15 kips)(3 ft) + B_{y}(9 ft) − (6 kips)(11 ft) − (6 kips)(13 ft) = 0
B_{y} = +21.0 kips B_{y} = 21.0 kips ↑ ◂
+↺ΣM_{B} = 0\colon
−A(9 ft) + (15 kips)(6 ft) − (6 kips)(2 ft) − (6 kips)(4 ft) = 0
A = +6.00 kips A = 6.00 kips ↑ ◂
REFLECT and THINK: Check the results by adding the vertical components of all of the external forces:
+ ↑ ΣF_{y} = +6.00 kips − 15 kips + 21.0 kips − 6 kips − 6 kips = 0
REMARK. In this problem, the reactions at both A and B are vertical; however, these reactions are vertical for different reasons. At A, the beam is supported by a roller; hence, the reaction cannot have any horizontal component. At B, the horizontal component of the reaction is zero because it must satisfy the equilibrium equation ΣF_{x} = 0 and none of the other forces acting on the beam has a horizontal component.
You might have noticed at first glance that the reaction at B was vertical and dispensed with the horizontal component B_{x}. This, however, is bad practice. In following it, you run the risk of forgetting the component B_{x} when the loading conditions require such a component (i.e., when a horizontal load is included). Also, you found the component B_{x} to be zero by using and solving an equilibrium equation, ΣF_{x} = 0. By setting B_{x} equal to zero immediately, you might not realize that you actually made use of this equation. Thus, you might lose track of the number of equations available for solving the problem.
