Question 3.11: A prismatic bar AB, fixed at one end and free at the other, ...
A prismatic bar AB, fixed at one end and free at the other, is loaded by a distributed torque of constant intensity t per unit distance along the axis of the bar (Fig. 3-42).
(a) Derive a formula for the strain energy of the bar.
(b) Evaluate the strain energy of a hollow shaft used for drilling into the earth if the data are as follows:
(a) Strain energy of the bar. The first step in the solution is to determine the internal torque T(x) acting at distance x from the free end of the bar (Fig. 3-42). This internal torque is equal to the total torque acting on the part of the bar between x = 0 and x = x. This latter torque is equal to the intensity t of torque times the distance x over which it acts:
T(x) = tx (a)
Substituting into Eq. (3-58), we obtain
U=\int_{0}^{L} \frac{[T(x)]^{2} d x}{2 G I_{P}}=\frac{1}{2 G I_{P}} \int_{0}^{L}(t x)^{2} d x=\frac{t^{2} L^{3}}{6 G I_{P}} (3-64)
This expression gives the total strain energy stored in the bar.
(b) Numerical results. To evaluate the strain energy of the hollow shaft, we substitute the given data into Eq. (3-64):
This example illustrates the use of integration to evaluate the strain energy of a bar subjected to a distributed torque.