Chapter 3, Problem 90P

The figure shows a simple model of the loading of a square thread of a power screw transmitting an axial load F with an application of torque T. The torque is balanced by the frictional force F_f acting along the top surface of the thread. The forces on the thread are considered to be distributed along the circumference of the mean diameter d_m over the number of engaged threads, n_t. From the figure, d_m = d_r + p/2, where d_r is the root diameter of the thread and p is the pitch of the thread.

(a)    Considering the thread to be a cantilever beam as shown in the cutaway view, show that the nominal bending stress at the root of the thread can be approximated by

${\sigma }_b=\pm \frac{6F}{\pi d_r{n}_{t}p}$

(b)    Show that the axial and maximum torsional shear stresses in the body of the shaft can be approximated by

${\sigma }_a=-\frac{4F}{\pi d_r^2}$ and ${\tau }_t=-\frac{16T}{\pi d_r^3}$

(c)    For the stresses of parts (a) and (b) show a three-dimensional representation of the state of stress on an element located at the intersection of the lower thread root base and the thread body. Using the given coordinate system label the stresses using the notation given in Fig. 3–8a.

Problem 3–90

(d)    A square-thread power screw has an outside diameter d = 1.5 in, pitch p = 0.25 in, and transmits a load F = 1 500 lbf through the application of a torque T = 235 lbf · in. If n_t = 2, determine the key stresses and the corresponding principal stresses (normal and shear).