Gere & Goodno — Torsion: Problem 3.9-8

3.9-8 A statically indeterminate stepped shaft \(ACB\) is fixed at ends \(A\) and \(B\) and loaded by a torque \(T_0\) at point \(C\) (see figure). The two segments of the bar are made of the same material, have lengths \(L_A\) and \(L_B\), and have polar moments of inertia \(I_{pA}\) and \(I_{pB}\). Determine the angle of rotation \(\phi\) of the cross section at \(C\) by using strain energy. Hint: Use Eq. (3-55b) to determine the strain energy \(U\) in terms of the angle \(\phi\). Then equate the strain energy to the work done by the torque \(T_0\). Compare your result with Eq. (3-52) of Example 3-9.