Cylindrical Coordinates

20 So far all equations have been written in either vector, indicial, or engineering notation. The last two were so far restricted to an othonormal cartesian coordinate system.

21 We now rewrite some of the fundamental relations in cylindrical coordinate system, Fig. 9.5, as this would enable us to analytically solve some simple problems of great practical usefulness (torsion, pressurized cylinders, ...). This is most often achieved by reducing the dimensionality of the problem from 3 to 2 or even to 1.

1 This theorem is attributed to Kirchoff (1858).

Figure 9.5: Cylindrical Coordinates

Figure 9.5: Cylindrical Coordinates

9.8.1 Strains

22 With reference to Fig. 9.6, we consider the displacement of point P to P*. the displacements can be y r.

Figure 9.6: Polar Strains expressed in cartesian coordinates as ux,uy, or in polar coordinates as Uy, uq. Hence, ux = ur cos 9 — Uq sin 9 Uy = ur sin 9 + uq cos 9

substituting into the strain definition for exx (for small displacements) we obtain dux dx exx dux d9 dux dr dux

dux dr d9 dx dr dx d9 dx dr dx dur a ■ a duQ ■ a a

d9 d9

dr dr sin 9

r cos 9

dur duQ

sin 9

Noting that as 0 ^ 0, £xx ^ £rr, sin. 0 ^ 0, and cos 0 ^ 1, we obtain exx I

duT dr

23 Similarly, if 0 ^ n/2, exx ^ eee, sin 0 ^ 1, and cos 0 ^ 0. Hence, eee = exxIe^n/2 = _ dugdO +—T

finally, we may express exy as a function of ur, ug and 0 and noting that £xy ^ erg as 0 ^ 0, we obtain ere =

due ue 1 duT

24 In summary, and with the addition of the z components (not explicitely derived), we obtain

0 0

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