From eqn. (4.20) Differentiating,
ds Edr=r dou dar da i dr dr dr
dr do h
do i dr
Now, since El is constant, differentiating eqn. (4.18), dai dr
0 = (aH - ar)( 1 + v) + r( 1 - v2)—- - vr(l + v)—^
But the general equilibrium equation will be the same as that obtained in §4.2, eqn. (4.1), i.e. aH - ar
Therefore substituting for (an — ar), dar j 2
A 2 2 i dar daH dar 0 = pco r + r—--h r( 1 — v) —--vr-
dr dan dar dr dr dr dan dar
where 2A is a convenient constant of integration. This equation can now be compared with the equivalent equation of §4.2, when it is evident that similar results for an and ar can be obtained if (1 + v) is replaced by 1/(1 — v) or, alternatively, if v is replaced by v/(l — v), see §8.14.2. Thus hoop and radial stresses in rotating thick cylinders can be obtained from the equations for rotating discs provided that Poisson's ratio v is replaced by v/( 1 — v), e.g. the stress at the centre of a rotating solid shaft will be given by eqn. (4.11) for a solid disc modified as stated above, i.e.
4 J. Rotating disc of uniform strength
In applications such as turbine blades rotating at high speeds it is often desirable to design for constant stress conditions under the action of the high centrifugal forces to which they are subjected.
Consider, therefore, an element of a disc subjected to equal hoop and radial stresses, i.e.
The condition of equal stress can only be achieved, as in the case of uniform strength cantilevers, by varying the thickness. Let the thickness be t at radius r and (f + St) at radius
Then centrifugal force on the element
= mass x acceleration = (ptrSdSr)a)2r = pta?r2SdSr
The equilibrium equation is then pto)2r2mr + a(r + 8r)80(t + St) = lotSr sin \S6 + ortS6
i.e. in the limit atdr — pu>2r2tdr + atdr + ardt ordt = —pco2r2tdr dt pco2rt dr o
where logf A is a convenient constant.
where r = 0 t = A = to i.e. for uniform strength the thickness of the disc must vary according to the following equation, t = ioC(-^2)/<2-) (4.22)
4.6. Combined rotational and thermal stresses in uniform discs and thick cylinders
If the temperature of any component is raised uniformly then, provided that the material is free to expand, expansion takes place without the introduction of any so-called thermal or temperature stresses. In cases where components, e.g. discs, are subjected to thermal gradients, however, one part of the material attempts to expand at a faster rate than another owing to the difference in temperature experienced by each part, and as a result stresses are developed. These are analogous to the differential expansion stresses experienced in compound bars of different materials and treated in §2.3.+
Consider, therefore, a disc initially unstressed and subjected to a temperature rise T. Then, for a radial movements of any element, eqns. (4.2) and (4.3) may be modified to account for the strains due to temperature thus:
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