Concept explainers
Calculate the support reactions for the given structures.

Answer to Problem 29P
The horizontal reaction at A is Ax=0_.
The vertical reaction at A is Ay=7.5 k↑_.
The vertical reaction at B is By=90 k↑_.
The vertical reaction at C is Cy=90 k↑_.
The vertical reaction at D is Dy=7.5 k↑_.
Explanation of Solution
Given information:
The structure is given in the Figure.
Apply the sign conventions for calculating reaction forces and moments using the three equations of equilibrium as shown below.
- For summation of forces along x-direction is equal to zero (∑Fx=0), consider the forces acting towards right side as positive (→+) and the forces acting towards left side as negative (←−).
- For summation of forces along y-direction is equal to zero (∑Fy=0), consider the upward force as positive (↑+) and the downward force as negative (↓−).
- For summation of moment about a point is equal to zero (∑Mat a point=0), consider the clockwise moment as negative and the counter clockwise moment as positive.
Calculation:
Let Ax and Ay be the horizontal and vertical reactions at the hinged support A.
Let By, Cy, and Dy be the vertical reaction at the roller support B, C, and D.
Sketch the free body diagram of the given structure as shown in Figure 1.
Use equilibrium equations:
Summation of forces along x-direction is equal to 0.
+→∑Fx=0Ax=0
Therefore, the horizontal reaction at A is Ax=0_.
For the member AE, the summation of moments about E is equal to 0.
∑MAEE=0−Ay(15)+12(3)(15)(13×15)=0−Ay(15)+112.5=0Ay=7.5 k↑
Therefore, the vertical reaction at A is Ay=7.5 k↑_.
For the member DF, the summation of moments about F is equal to 0.
∑MDFF=0Dy(15)−12(3)(15)(13×15)=0Dy(15)−112.5=0Dy=7.5 k↑
Therefore, the vertical reaction at D is Dy=7.5 k↑_.
Summation of moments about C is equal to 0.
∑MC=0−Ay(50)−By(20)+12(3)(15)(153+35)+3(50)(10)−12(3)(15)(20)+Dy(30)=0
Substitute 7.5 k for Ay and 7.5 k for Dy.
−7.5(50)−By(20)+12(3)(15)(153+35)+3(50)(10)−12(3)(15)(153+15)+7.5(30)=0−20By+1,800=0By=90 k↑
Therefore, the vertical reaction at B is By=90 k↑_.
Summation of forces along y-direction is equal to 0.
+↑∑Fy=0Ay+By+Cy+Dy−12(3)(15)−3(50)−12(3)(15)=0
Substitute 7.5 k for Ay, 90 k for By, and 7.5 k for Dy.
7.5+90+Cy+7.5−12(3)(15)−3(50)−12(3)(15)=0Cy−90=0Cy=90 k↑
Therefore, the vertical reaction at C is Cy=90 k↑_.
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Chapter 3 Solutions
Structural Analysis, 5th Edition
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