Question 5.6: Draw the shear and bending moment diagrams and the qualitati...
Draw the shear and bending moment diagrams and the qualitative deflected shape for the beam shown in Fig. 5.10(a).
Reactions (See Fig. 5.10(b).)
+\longrightarrow \sum{F_x}=0 A_x=0
+\uparrow \sum{F_y}=0
A_y – 70 = 0
A_y = 70 kN A_y = 70 kN \uparrow
+\circlearrowleft \sum{M_A}=0
M_A – 70(6) – 200 = 0
M_A = 620 kN-m M_A = 620 kN-m \circlearrowleft
Shear Diagram
Point A S_{A,R} = 70 kN
Point B S_{B,L} = 70 + 0 = 70 kN
S_{B,R} = 70 – 70 = 0
Point C S_{C,L} = 0 + 0 = 0
S_{C,R} = 0 + 0 = 0 Checks
The numerical values of shear evaluated at points A, B, and C are used to construct the shear diagram as shown in Fig. 5.10(c). Because no load is applied to the beam between these points, the slope of the shear diagram is zero between these points. To facilitate the construction of the bending moment diagram, the area of the segment AB of the shear diagram has been computed and is shown in parentheses on the shear diagram (Fig. 5.10(c)).
Bending Moment Diagram
Point A Since a negative (counterclockwise) couple of 620 kN-m moment acts at point A, the bending moment diagram decreases abruptly from 0 to -620 kN-m at this point; that is,
M_{A,R} = -620 kN-mPoint B M_B = -620 + 420 = -200 kN-m
Point C M_{C,L} = -200 + 0 = -200 kN-m
M_{C,R} = -200 + 200 = 0 Checks
The bending moment diagram is shown in Fig. 5.10(d). The shape of this diagram between the ordinates just computed is based on the condition that the slope of the bending moment diagram at a point is equal to shear at that point. As the shear in the segments AB and BC is constant, the slope of the bending moment diagram must be constant in these segments. Therefore, the ordinates of the bending moment diagram are connected by straight lines. In segment AB, the shear is positive, and so is the slope of the bending moment diagram in this segment. In segment BC, the shear becomes zero; consequently, the slope of the bending moment diagram becomes zero, as shown in Fig. 5.10(d).
Qualitative Deflected Shape A qualitative deflected shape of the beam is shown in Fig. 5.10(e). As the bending moment is negative over its entire length, the beam bends concave downward, as shown.
