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).

simply supported beam

Figure 4-1(a)


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.)

free-body diagram

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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