Equilibrium

25 Whereas the equilibrium equation as given In Eq. 6.24 was obtained from the linear momentum principle (without any reference to the notion of equilibrium of forces), its derivation (as mentioned) could have been obtained by equilibrium of forces considerations. This is the approach which we will follow for the polar coordinate system with respect to Fig. 9.7.

26 Summation of forces parallel to the radial direction through the center of the element with unit thickness in the z direction yields:

dTr,

we approximate sin(d0/2) by d0/2 and cos(d0/2) by unity, divide through by rdrd0,

r dr

27 Similarly we can take the summation of forces in the 0 direction. In both cases if we were to drop the dr/r and d0/r in the limit, we obtain

Figure 9.7: Stresses in Polar Coordinates

dr

, 1 dTer + r dO

H—(Trr — Tee) r

H fr

= o (9.33)

dTre dr

, 1 dTee + r dO

H— (Tre — Ter ) r

H fe

= 0 (9.34)

28 It is often necessary to express cartesian stresses in terms of polar stresses and vice versa. This can be done through the following relationships yielding

T Tx

xx

Txy

Txy

Tyy

xx

=

Tyy

= Trr cos2 O H Tee sin2 O — Tre sin 2 O Trr sin2 O H Tee cos2 O H Tre sin 2 O (Trr — Tee) sin O cos O H Tre(cos2 O — sin2

(recalling that sin2 9 = 1/2 sin 2 9, and cos2 9 = 1/2(1 +cos2 9)).

0 0

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