$4.2 Rings,Discs and Cylinders Subjected to Rotation and Thermal Gradients 121 品=++网 r2a=- pr4 2Ar2 83+)+ 2 B where -B is a second convenient constant of integration, ,=A-克-B+A B 8 (4.7) and from eqn.(4.5), %=A+,月-(1+3a2r2 、B 8 (4.8) For a solid disc the stress at the centre is given when r=0.With r equal to zero the above equations will yield infinite stresses whatever the speed of rotation unless B is also zero, i.e.B=0 and hence B/r2=0 gives the only finite solution. Now at the outside radius R the radial stress must be zero since there are no external forces to provide the necessary balance of equilibrium if o,were not zero. Therefore from egn.(4.7), 0r=0=A-(3+) Dw2R2 8 A=(3+)Pw2R2 8 Substituting in eqns.(4.7)and (4.8)the hoop and radial stresses at any radius r in a solid disc are given by o4=(3+)w 8-1+3w)0w2r2 8 83+R2-(1+3r2] (4.9) 0=(3+yw2 8-(3+v)eo 8 8+ 8R2-内 (4.10) (b)Maximum stresses At the centre of the disc,where r =0,the above equations yield equal values of hoop and radial stress which may also be seen to be the maximum stresses in the disc,i.e.maximum hoop and radial stress (at the centre) =3+吵w2R2 8 (4.11)$4.2 Rings, Discs and Cylinders Subjected to Rotation and Thermal Gradients 2 pr4w2 2Ar2 r a, = -- (~+v)+--B 8 2 where -B is a second convenient constant of integration, B po2r2 r2 8 ~r = A - - -. (3 + u)- and from eqn. (4.3, 121 (4.7) (4.8) For a solid disc the stress at the centre is given when r = 0. With r equal to zero the above equations will yield infinite stresses whatever the speed of rotation unless B is also zero, i.e. B = 0 and hence B/r2 = 0 gives the only finite solution. forces to provide the necessary balance of equilibrium if a,. were not zero. Now at the outside radius R the radial stress must be zero since there are no external Therefore from eqn. (4.7), Substituting in eqns. (4.7) and (4.8) the hoop and radial stresses at any radius r in a solid disc are given by = .o‘ [(3 + v)R2 - (1 + 3u)r2] 8 pw2 R2 pw2r2 Or = (3 + u)- 8 - (3 + u)---- 8 (4.9) (4.10) (b) Maximum stresses At the centre of the disc, where r = 0, the above equations yield equal values of hoop and radial stress which may also be seen to be the maximum stresses in the disc, i.e. maximum hoop and radial stress (at the centre) (4.1 I)