The elasticity and resistance of the materials of engineering . s ofArt. 7 which are given in terms of semi-polar co-ordinates. Let Fig. I represent a cylindrical piece of material, withany cross section, fixed in the plane ZY^ and let the origin ofco-ordinates be taken at O. Letit be twisted, also, by a couple = PI, the plane of which is parallel toZY. The material will thus besubjected to no bending, but topure torsion. The axis of the piece is sup-posed to be parallel to the axis ofA^as well as the axis of the sections of the piece, orig-inally parallel to ZO Y, will not
The elasticity and resistance of the materials of engineering . s ofArt. 7 which are given in terms of semi-polar co-ordinates. Let Fig. I represent a cylindrical piece of material, withany cross section, fixed in the plane ZY^ and let the origin ofco-ordinates be taken at O. Letit be twisted, also, by a couple = PI, the plane of which is parallel toZY. The material will thus besubjected to no bending, but topure torsion. The axis of the piece is sup-posed to be parallel to the axis ofA^as well as the axis of the sections of the piece, orig-inally parallel to ZO Y, will not re- / PS*main plane after torsion takes place. But the tendency totwist any elementary portion of the piece about an axis pass-ing through its centre and parallel to the axis of Jf will be verysmall compared with the tendency to twist it about either theaxis of r or ^ ; consequently the first will be neglected. Inthe notation of Art. /, this condition is equivalent to makingTr^ = o. As the piece is acted upon by a couple only, all normalstresses will be 44 TORSION IN EQUILIBRIUM. [Art. 10. Eqs. (7), (8), (9) and (i i), of Art. 7, then become : iV, = d -Y2G~- =0 (i) \ — 2X dx R =^^^+2C^ = o . (2) \ — 2X dr ^ ^ ^^ 2GV n , r-f dw , P\ r . f dp . dw . w\ . , After introducing the values of T^^ and Zif,^, from Eqs. (10)and (12) of Art. 7, in Eqs. (2), (3) and (4) of the same Article,at the same time making the external forces and second mem-bers of those equations equal to zero, and bearing in mind theconditions given above, there will result: dT^x dT^x T^x _ ^f d^tc d^p d^w d^ti dr r dcp r \ dr^ dr dx r dqj dx rdqy^ . du . dp \ , . dT^x ^[ d^ii . d^p\ ,^. dx ~ W ^ r dcp dx) - o . . I7j Also by Eq. (6) of Art. 7: Art. 10.] TORSION IN EQUILIBRIUM, 45 e — -^i Jr-^ Jr —— 4- - fs^i dx dr r dcp r ^ ^ The cylindrical piece of material is supposed to be of suchlength, that the portion to which these equations apply is notaffected by the manner of application of the couple.
Size: 1408px × 1774px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., book, bookcentury1800, booksubjectbuildingmaterials, bookyear1883
