Question 7.SP.5: Draw the shear and bending-moment diagrams for the beam and ......
Draw the shear and bending-moment diagrams for the beam and loading shown and determine the location and magnitude of the maximum bending moment.
STRATEGY: The load is a distributed load over part of the beam with no concentrated loads. You can use the equations in this section in two parts: for the load and no load regions. From the discussion in this section, you can expect the shear diagram will show an oblique line under the load, followed by a horizontal line. The bending-moment diagram should show a parabola under the load and an oblique line under the rest of the beam.
Final Answer
MODELING and ANALYSIS:
Free-Body, Entire Beam. Consider the entire beam as a free body (Fig. 1) to obtain the reactions
\mathbf{R}_A=80 ~\mathrm{kN} \uparrow \quad \mathbf{R}_C=40 ~\mathrm{kN} \uparrow
Shear Diagram. The shear just to the right of A is V_A=+80 kN. Because the change in shear between two points is equal to minus the area under the load curve between these points, you can obtain V_B by writing
V_B-V_A=-(20 ~\mathrm{kN} / \mathrm{m})(6 \mathrm{~m})=-120 ~\mathrm{kN}
V_B=-120+V_A=-120+80=-40 ~\mathrm{kN}
Since the slope dV/dx = -w is constant between A and B, the shear diagram between these two points is represented by a straight line. Between B and C, the area under the load curve is zero; therefore,
V_C-V_B=0 \quad V_C=V_B=-40 ~\mathrm{kN}
and the shear is constant between B and C (Fig. 1).
Bending-Moment Diagram. The bending moment at each end of the beam is zero. In order to determine the maximum bending moment, you need to locate the section D of the beam where V = 0. You have
V_D-V_A=-w x
0-80 ~\mathrm{kN}=-(20 ~\mathrm{kN} / \mathrm{m}) x
Solving for x: x = 4 m
The maximum bending moment occurs at point D, where we have dM/dx = V = 0. Calculate the areas of the various portions of the shear diagram and mark them (in parentheses) on the diagram (Fig. 1). Since the area of the shear diagram between two points is equal to the change in bending moment between those points, you can write
\begin{array}{ll}M_D-M_A=+160 ~\mathrm{kN} \cdot \mathrm{m} & M_D=+160 ~\mathrm{kN} \cdot \mathrm{m} \\M_B-M_D=-40 ~\mathrm{kN} \cdot \mathrm{m} & M_B=+120 ~\mathrm{kN} \cdot \mathrm{m} \\M_C-M_B=-120 ~\mathrm{kN} \cdot \mathrm{m} & M_C=0\end{array}
The bending-moment diagram consists of an arc of parabola followed by a segment of straight line; the slope of the parabola at A is equal to the value of V at that point.
The maximum bending moment is
M_{\max }=M_D=+160 ~\mathrm{kN} \cdot \mathrm{m}
REFLECT and THINK: The analysis conforms to our initial expectations. It is often useful to predict what the results of analysis will be as a way of checking against large-scale errors. However, final results can only depend on detailed modeling and analysis.
