Question 12.3: Draw the shear and bending-moment diagrams for the beam and ...
Draw the shear and bending-moment diagrams for the beam and loading shown.
STRATEGY: The beam supports two concentrated loads and one distributed load. You can use the equations in this section between these loads and under the distributed load, but you should expect changes in the diagrams at the concentrated load points.
MODELING and ANALYSIS:
Reactions. Consider the entire beam as a free body as shown in Fig. 1.
Note that at both A and E the bending moment is zero. Thus, two points (indicated by dots) are obtained on the bending-moment diagram.
Shear Diagram. Since dV/dx = –w, between concentrated loads and reactions the slope of the shear diagram is zero (i.e., the shear is constant). The shear at any point is determined by dividing the beam into two parts and considering either part to be a free body. For example, using the portion of beam to the left of section 1, the shear between B and C is
Also, the shear is +12 kips just to the right of D and zero at end E. Since the slope dV/dx = –2w is constant between D and E, the shear diagram between these two points is a straight line.
Bending-Moment Diagram. Recall that the area under the shear curve between two points is equal to the change in bending moment between the same two points. For convenience, the area of each portion of the shear diagram is computed and indicated in parentheses on the diagram in Fig. 1.
Since the bending moment M_A at the left end is known to be zero,
M_B-M_A=+108 \quad \quad \quad M_B=+108 \text{ kip.ft} \\ M_C-M_B=-16 \quad \quad \quad M_C=+92 \text{ kip.ft} \\ M_D-M_C=-140 \quad \quad M_D=-48 \text{ kip.ft} \\ M_E-M_D=+48 \quad \quad \quad M_E=0
Since M_E is known to be zero, a check of the computations is obtained.
Between the concentrated loads and reactions, the shear is constant.
Thus, the slope dM/dx is constant, and the bending-moment diagram is drawn by connecting the known points with straight lines. Between D and E where the shear diagram is an oblique straight line, the bending-moment diagram is a parabola.
From the V and M diagrams, note that V_{max} = 18 \text{ kips and } M_{max} = 108 \text{ kip . ft}.
REFLECT and THINK: As expected, the shear and bending-moment diagrams show abrupt changes at the points where the concentrated loads act.
