Question 4.3: Calculate the reactions at the built-in end of the cantileve...

In this example the load, W, produces reactions of vertical force, moment, and torque at the built-in end. The vertical force and moment are the same as in Example 4.1. To determine the torque reaction we impose a small, virtual displacement, \Delta_{V, C }, vertically downwards at C. This causes the beam AB to rotate as a rigid body through an angle, \theta_{v, AB }, which is given by

\theta_{v, AB }=\Delta_{v, C } / a  (i)

Alternatively, we could have imposed a small virtual rotation, \theta_{v, AB }, on the beam, which would have resulted in a virtual vertical displacement of C equal to a \theta_{v, AB }; clearly the two approaches produce identical results. The total virtual work done on the beam is then given by

W_{t}=W \Delta_{v, C }-T_{ A } \theta_{v, AB }=0  (ii)

since the beam is in equilibrium. Substituting for \theta_{v, AB }, in Eq. (ii) from Eq. (i), we have

which is the result which would have been obtained by considering the statical equilibrium of the beam.