Calculated the slope of the cantilever:

Calculated the slope of the cantilever given in Example 6 at its free end.

Solution

To find out rotation in a beam we must have a couple working at that point. Since at the free end, there is no couple, we introduce a fictitious couple M′ as illustrated in Figure

Figure

Now           M =- Px - M ′  &    ∂M / ∂M ′  = - 1

θA = PL²/ 2EI (Positive sign denoted rotation in the sense of M′, that means clockwise).

 Therefore, we see that the tip slope of a cantilever loaded along a concentrated vertical

force P at its free end is PL² /2EL