
Concept explainers
Calculate the support reactions for the given structure.

Answer to Problem 42P
The horizontal reaction at A is Ax=309.25 kN←_.
The vertical reaction at A is Ay=170.25 kN↓_.
The moment at A is MA=1,138.5 kN⋅m_ acting in the counterclockwise direction.
The horizontal reaction at B is Bx=50.75 kN←_.
The vertical reaction at B is By=129.75 kN↓_.
The moment at B is MB=76.5 kN⋅m_ acting in the counterclockwise direction.
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, Ay, and MA be the horizontal reaction, vertical reaction, and moment at the fixed support A.
Let Bx, By, and MB be the horizontal reaction, vertical reaction, and moment at the fixed support B.
Let Cx and Cy be the horizontal and vertical reaction at the internal hinge C.
Let Dx and Dy be the horizontal and vertical reaction at the internal hinge D.
Sketch the free body diagram of the structure as shown in Figure 1.
Find the angle of the inclined member with respect to horizontal axis:
Refer Figure 1.
tanθ=310θ=tan−1(310)=16.7°
Convert the
Use Figure 1 to find the length of the inclined member:
Use equilibrium equations:
Find the forces at the internal hinges C and D:
Consider the free body diagram of the portion CED.
Summation of moments about D is equal to 0.
For the member CE, the summation of moments about E is equal to 0.
Substitute
Summation of forces along x-direction is equal to 0.
Substitute
Summation of forces along y-direction is equal to 0.
Substitute
Find the reactions at the support A:
Consider the equilibrium of the portion AC.
Summation of forces along x-direction is equal to 0.
Substitute
Therefore, the horizontal reaction at A is
Summation of forces along y-direction is equal to 0.
Substitute
Therefore, the vertical reaction at A is
Summation of moments about A is equal to 0.
Substitute
Therefore, the moment at A is
Find the reactions at the support B:
Consider the equilibrium of the portion BD.
Summation of forces along x-direction is equal to 0.
Substitute
Therefore, the horizontal reaction at B is
Summation of forces along y-direction is equal to 0.
Substitute
Therefore, the vertical reaction at B is
Summation of moments about B is equal to 0.
Substitute
Therefore, the moment at B is
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