Calculation of support reactions of a beam
Problem 4-1
Find the reactions at the support of the following beam shown in figure 4-1(a).
Figure 4-1(a)
Solution:
The given beam is considered as 2-dimensional structure. The free-body diagram of the beam is given in figure 4-1 (b) which shows all the reaction components (Ax, Ay, By) and the applied loads. This beam is statically determinate as it can have only 3 reaction components; 2 at the hinged support at A and 1 at the roller support at B, and there are 3 possible equations of static equilibrium ΣFx = 0, ΣFy = 0, ΣMz = 0; (The x-axis and y-axis are as shown in the figure and the z-axis is perpendicular to the x-y plane.)
Figure 4-1(b) free-body diagram
Applying equations of static equilibrium:
ΣFx = 0; Ax = 0; (eq. 1)
ΣFy = 0; Ay + By – 10 – 20 × 4 = 0;
Ay + By = 90 kN; (eq. 2)
Considering z-axis passing through A, and taking moment of all the forces about z-axis (taking clockwise –ve and anticlockwise +ve);
ΣMz = 0; By × 10 -10 × 8 – 20 × 4 × 2 = 0 (eq. 3)
Solving eq. 3 yields By = 24 kN;
Substituting the value of By in eq. 4-2 gives Ay = 66 kN.
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