Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics Volume 14, N2. 2, 1999 TORSION OF A HALLOW ELLIPTIC SHAFT Zirakashvili N. 1. Vekua Institute of Applied Mathematics In machine-building a.'! a consequence of depreciation during t.he work of circular cylindrical shafts often takes place there transformation into elliptic cylindrical ~dltllt.H, In connection with this the hollow shaft stress deformed condition is inversigated when on the outer surface of the cylindrical shaft the uniform distributed tangent st,t'C!!!! of P intencity is applied and normal stresses are zero. On the interior cylindrical surface of course the tangent stress is choosen such thaI. the main moment i~ ;l,PI'O and normal stresses are zero too. We consider menthioned problem, in orthogonal coordinate system 0,0, , (0 ::.; 0 < 00,0 ::; 0' < 211", -00 <(< 00) [I], for t.li<' bodies bounded by coordinate surfaces, while it is in plane deformable state [2J. 1f X,1/,Z are Cartesian coordinates, then x ::::c chii coso, y = c.,hO:,ina, z = (, while !to = ha = h == ~ Vch(20) - (:0..,(20), h, :::: 1, where c:::: consi is a scale constant [J J. Let us designate by u, 1), w components of displacement vector 0 along the tangent to the coordinate lines a, 0, (; No, No, N" SOo ;:: 5', 5'0(, 8(n components of sLrCMH tenzor. Because of plane deformable state the mentioned boundary value problem could he' found in the following form: tn::::: 0, 11.::: u(O, 0), 11== v(O, a). This conditions arc IIs('d in the domain bounded by elliptic coordinate surfaces 0, a: (0 ~ 0 < 00, 0~ 0' < 211"), The boundary value' problem owing to symmetry conditions could be set fOI' quarter of the elliptic domain. Let us find functions H; 13, u, 1) (11.(0, a), v(O, u) E (;3(fl) n = {()J < {} < (}'J' 0 l\' < -1})' which are satisfying to tile -vstcm of equilibrium equations 8K 011 ao - an = 0; iJJ< i)B 71('; + 00 -= O. fJfj _ oii _ ~I 'J8 BO 00. - It lo and following boundary condi tions an when 0' = 0: 11 = 0, No ::= 0 {::} 'It = 0, -;-- = 0; atl' ~ ~) when a - - . 1/ - 0 IV - 0 ¢} 1/, = O. _.- = (J. . - 2' -,' o- , do '