We wish to reformulate the torsion problem using cylindrical coordinates. First show that the general form of the displacements can be expressed as u_r = 0, u_θ = αrz, u_z = u_z(r,θ). Next show that this leads to the following strain and stress fields:

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er = e =e = ero = 0, erz 1 uz 2 dr' duz or=000 = Tro= 0, Trz= . r egz Toz = 1/2 (ar + a1 ap 182 - + + ar rar 00 1 trz_dtez tez, 2 r 30 r ar + 1 duz Verify that with no body forces the equilibrium equations reduce to otrz /r + (1/r)atoz /00 + Trz/r= 0. Following the same scheme as used in Section 9.3.1, develop the compatibility relation -2 r 30 1 uz r 30 Finally, choosing a stress-stress function relation of the form Trz= (1/r)0p/30, tez = -d/ar, show that the equilibrium is identically satisfied and the compatibility relation gives the expected result