Question 12.6: The simple beam AC in Fig. 1 is loaded by a couple of moment...
The simple beam AC in Fig. 1 is loaded by a couple of moment T applied at point B. Draw the shear and bending-moment diagrams of the beam.
STRATEGY: The load supported by the beam is a concentrated couple.
Since the only vertical forces are those associated with the support reactions, you should expect the shear diagram to be of constant value. How–ever, the bending-moment diagram will have a discontinuity at B due to the couple.
MODELING and ANALYSIS:
The entire beam is taken as a free body.
\pmb{\text{R}}_A=\frac{T}{L}\uparrow \quad \quad \pmb{\text{R}} _C=\frac{T}{L} \downarrow
The shear at any section is constant and equal to T/L. Since a couple is applied at B, the bending-moment diagram is discontinuous at B. It is represented by two oblique straight lines and decreases suddenly at B by an amount equal to T. This discontinuity can be verified by equilibrium analysis. For example, considering the free body of the portion of the beam from A to just beyond the right of B as shown in Fig. 1, M is
+\circlearrowleft \sum{M_B}=0: \quad \quad -\frac{T}{L}a+T+M=0 \quad \quad M=-T\left(1-\frac{a}{L} \right)
REFLECT and THINK: Notice that the applied couple results in a sudden change to the moment diagram at the point of application in the same way that a concentrated force results in a sudden change to the shear diagram.