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Example 4.4

(a) Derive expressions for the hoop and radial stresses developed in a solid disc of radius R when subjected to a thermal gradient of the form T = Kr. Hence determine the position and magnitude of the maximum stresses set up in a steel disc of 150 mm diameter when the temperature rise is 150°C. For steel, a = 12 x 10"6 per °C and E = 206.8 GN/m2.

(b) How would the values be changed if the temperature at the centre of the disc was increased to 30°C, the temperature rise across the disc maintained at 150°C and the thermal gradient now taking the form T = a + brl

Solution

(a) The hoop and radial stresses are given by eqns. (4.29) and (4.30) as follows:

r2 r2

the constant of integration being incorporated into the general constant A.

rl 3

rL 3

Now in order that the stresses at the centre of the disc, where r = 0, shall not be infinite, B must be zero and hence B/r2 is zero. Also o> = 0 at r — R. Therefore substituting in (3), aEKR aEKR

Substituting in (3) and (4) and rearranging, aEK ,n or = ~(R - r)

The variation of both stresses with radius is linear and they will both have maximum values at the centre where r — 0.

aEKR

0 0

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