Calculation of support reactions for 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 ΣF_x = 0,  ΣF_y = 0, ΣM_z = 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.

Figure 4-2(b) free body diagram

Applying equations of static equilibrium:

ΣF_x = 0;  A_x = 0;                                               (eq. 1) 

ΣF_y = 0; A_y + B_y – 25 – 5 × 4 = 0;     

             A_y + B_y  = 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);

ΣM_z = 0;  B_y × 6 – 25 × 10 – 5 × 4 × 4 = 0        (eq. 3)

 

Solving eq. 3 yields  B_y = 55 kN;

Substituting the value of B_y in eq. 2 gives A_y = –10 kN.

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

You can also use our overhanging beam calculator to determine the values of support reactions