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Calculation of Support Reactions of overhanging beam

Problem 4.2

The simply supported beam shown in figure 4-2(a) has overhanging portion on one side. Find the reactions at the supports.

Figure 4-2(a)

Solution:

The given beam has a hinged support at A and a roller support at B. The free-body diagram is given in figure 4-2(b), which shows 2 reactions at A and one reaction at B. (The x-axis and y-axis are as shown in the figure and the z-axis is perpendicular to the x-y plane.) There are three equations of static equilibrium ΣFx = 0,  ΣFy = 0, ΣMz = 0; available for this 2-dimensional structure. The number of unknown reaction components is equal to equations of static equilibrium. Therefore this beam is statically determinate.

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Figure 4-2(b) free body diagram

Applying equations of static equilibrium:

ΣFx = 0;  Ax = 0;                                               (eq. 1)

ΣFy = 0; Ay + By – 25 – 5 × 4 = 0;

Ay + By  = 45 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 × 6 – 25 × 10 – 5 × 4 × 4 = 0        (eq. 3)

Solving eq. 3 yields  By = 55 kN;

Substituting the value of By in eq. 2 gives Ay = –10 kN.

The –ve sign of reaction indicates that Ay will be in the downward direction instead of upward as shown in the free-body diagram.

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