Question 3.3A: Shear and Moment Diagrams of a Cantilever Beam Using a Graph...
Shear and Moment Diagrams of a Cantilever Beam Using a Graphical Method
Problem: Determine and plot the shear and moment functions for the cantilever beam with a concentrated load as shown in Figure 3-22b.
Given: Beam length l = 10 in, and load location a = 4 in. The magnitude of the applied force is F = 40 lb.
Assumptions: The weight of the beam is negligible compared to the applied load and so can be ignored.
See Figures 3-22b and 3-25.
1 Solve for the reaction forces using equations 3.3 (p. 78). Summing moments about the left end and summing forces in the y direction gives
\begin{array}{lll} \sum F_x=0 & \sum F_y=0 & \sum F_z=0 \\ \sum M_x=0 & \sum M_y=0 & \sum M_z=0 \end{array} (3.3a)
\sum F_x=0 \quad \sum F_y=0 \quad \sum M_z=0 (3.3b)
\begin{aligned} \sum M_z &=0=F a-M_1 \\ M_1 &=F a=40(4)=160 \end{aligned} (a)
\begin{aligned} \sum F_y &=0=R_1-F \\ R_1 &=F=40 \end{aligned} (b)
2 By the sign convention, the shear is positive and the moment is negative in this example. To graphically construct the shear and moment diagrams for a cantilever beam take an imaginary “backward walk” starting at the fixed end of the beam and moving toward the free end (from left to right in Figure 3-24).
In this example, that results in the first observed force being the reaction force R_{1} acting upward. This shear force remains constant until the downward force F at x = a is reached, which closes the shear diagram to zero.
3 The moment diagram is the integral of the shear diagram, which in this case is a straight line of slope = 40.
4 Both the shear and moment are maximum at the wall in a cantilever beam. Their maximum magnitudes are as shown in equations (a) and (b) above.
