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(oH - o>)( 1 + v) + r but, from the equilibrium eqn. (4.1), don dor dT

Therefore substituting for (oH — or) in eqn. (4.25),

/•I , \( 22, d°r\ , doH (1 + v) Ipr o) + r— J + r-j- -

dor dT

dr dr

dor don dT

dr dr dr dT

Integrating, do h dor j

dr dr dr

where, again, 2A is a convenient constant. Subtracting eqn. (4.1), dor pr2u>2 2or + r—~ = -Î-—— (3 + v) - EaT + 2A dr 2

7 pr4co f 2Ar2

where, as in eqn. (4.7), — B is a second convenient constant of integration.

B pr2co2 „ Ea f or=A-~- ^¿—(3 + v) - — J Trdr (4.27)

B pr2a>2 ^ „ Ea a„ = A + — - (1 + 3») --EaT + ( Trdr (4.28)

i.e. the expressions obtained for the hoop and radial stresses are those of the standard Lamé equations for simple pressurisation with (a) modifying terms for rotational effects as obtained in previous sections of this chapter, and (b) modifying terms for thermal effects.

A solution to eqns. (4.27) and (4.28) for discs may thus be obtained provided that the way in which T varies with r is known. Because of the form of the equations it is clear that, if required, pressure, rotational and thermal effects can be considered simultaneously and the appropriate values of A and B determined.

For thick cylinders with an axial length several times the outside diameter the above plane stress equations may be modified to the equivalent plane strain equations (see §8.14.2) by replacing v by v/(l — v), E by E/( 1 — v2) and a by (1 + v)a.

In the absence of rotation the equations simplify to

With a linear variation of temperature from T = 0 at r = 0, i.e. with T = Kr

B EaKr

rL 3

r2 3

With a steady heat flow, for example, in the case of thick cylinders when Ea becomes Ea/( 1 - v)-see p. 125.

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