
Problem 4.2
The simply
supported beam shown in figure 42(a) has overhanging portion on one
side. Find the reactions at the supports.
Figure 42(a)
Solution:
The given beam has a hinged support at
A and a roller support at B. The freebody diagram is given in
figure 42(b), which shows 2 reactions at A and one reaction at B. (The
xaxis and yaxis are as shown in the figure and the zaxis is
perpendicular to the xy plane.) There are three equations of
static equilibrium ΣF_{x} = 0, ΣF_{y} = 0, ΣM_{z}
= 0; available for this 2dimensional structure. The number of
unknown reaction components is equal to equations of static
equilibrium. Therefore this beam is statically determinate. 
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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 zaxis passing through A,
and taking moment of all the forces about zaxis (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 freebody diagram.
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