Problem

The solid disk of Example 11.7 is subjected to an angular velocity ω [rad/s] and is also e...

The solid disk of Example 11.7 is subjected to an angular velocity ω [rad/s] and is also exposed to a temperature change

where T0 is a positive constant.

a. Determine the additional stresses and resulting from T as functions of T0, b, and r.
b. Determine whether or not the stresses at r = 0 and r = b are increased because of T.

Example 11.7: Rotating Solid Disk

Consider a solid disk of radius b subjected to an angular velocity ω (Figure E11.7).

(a) Determine the polar coordinate stresses σrr and σθθ in the disk as functions of ρ, v, r, b and ω. Let T = 0.
(b) For temperature change T = 0, determine the maximum values of σrr and σθθ and their locations.

FIGURE E11.7

Solution:

(a) The boundary conditions for the disk are

By Eq. 11.47 the general solution .for the rotating disk is

By Eqs. 11.49 and 11.50, the stresses are

Hence, by the first of Eqs. (a) m1d Eq. (b),

Likewise, by Eq. (b) with C2 = 0, the first of Eqs. (c) and the second of Eqs. (a) yield

Consequently, by Eqs. (c)-(e), we obtain the stresses

(b)Since r ≤ b, then by Eqs. (f) σrr and σθθ are both positive and increase as r →0. Hence, at r = 0, the stresses approach their maximum values

Comparing Eq. (g) to Eq. 11.55, we see that the maximum stress in a solid disk, which occurs at its center, is one-half as large as the maximum stress resulting from the stress concentration at the edge of a small hole at the center of a disk. In other words, the stress concentration factor of the small hole is 2.0.

(11.47)