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

Question

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 [latex]l[/latex] = 10 in, and load location [latex]a[/latex] = 4 in. The magnitude of the applied force is [latex]F[/latex] = 40 lb.

Assumptions: The weight of the beam is negligible compared to the applied load and so can be ignored.

Accepted Answer

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 [latex]y[/latex] direction gives

[latex] \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} [/latex] (3.3a)

[latex] \sum F_x=0 \quad \sum F_y=0 \quad \sum M_z=0 [/latex] (3.3b)

[latex] \begin{aligned} \sum M_z &=0=F a-M_1 \ M_1 &=F a=40(4)=160 \end{aligned} [/latex] (a)

[latex] \begin{aligned} \sum F_y &=0=R_1-F \ R_1 &=F=40 \end{aligned} [/latex] (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 [latex]R_{1}[/latex] acting upward. This shear force remains constant until the downward force [latex]F[/latex] at [latex]x = a[/latex] 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 ([latex]a[/latex]) and ([latex]b[/latex]) above.

Question 3.3A: Shear and Moment Diagrams of a Cantilever Beam Using a Graph...

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