124 Mechanics of Materials 2 $4.4 Substituting for r in eqn.(4.13). 0r=(3+y 8 Ri+-R1R:-RRa] =3+ 8 (R2-R12 (4.17) The complete radial and hoop stress distributions are indicated in Fig.4.4. Hoop折tsW OH Rodial stress Fig.4.4.Hoop and radial stress distribution in a rotating hollow disc. 4.4.Rotating thick cylinders or solid shafts In the case of rotating thick cylinders the longitudinal stress oL must be taken into account and the longitudinal strain is assumed to be constant.Thus,writing the equations for the strain in three mutually perpendicular directions(see $4.2), 1 eL=EoL-oH-o） (4.18) Er= E(,-WOH -voL)= 1 ds (4.19) dr 1 EH =E(OH-vO,-VOL)= (4.20) From eqn.(4.20) Es r[oH-v(a,+L)] Differentiating, +1 [OH-vo,-vOL] Substituting for E(ds/dr)in eqn.(4.19), [doH dor doL Or-VOH VOL r dr dr dr +OH VO,-VOL doL 0=(oH-o)1+)+r- dr dr dr124 Mechanics of Materials 2 94.4 Substituting for r in eqn. (4.13). PO2 = (3 + u)- [R2 - R112 8 (4.17) The complete radial and hoop stress distributions are indicated in Fig. 4.4. Fig. 4.4. Hoop and radial stress distribution in a rotating hollow disc. 4.4. Rotating thick cylinders or solid shafts In the case of rotating thick cylinders the longitudinal stress OL must be taken into account and the longitudinal strain is assumed to be constant. Thus, writing the equations for the strain in three mutually perpendicular directions (see $4.2), (4.19) S (4.20) 1 E r &H = -(cH - war - u0L) = - From eqn. (4.20) Differentiating, ES = r[O,y - ~(0,. + OL)] + 1 [OH - ua, - UOL] Substituting for E(ds/dr) in eqn. (4.19), 0,. - vaH - v0L = r [% - + OH - u(T, - u0L d0H dor dot 0 = (ojl - 0,.)(1 + w) + r- - wr- - ur- dr dr dr