122 Mechanics of Materials 2 $4.3 At the outside of the disc,at r =R,the equations give =0 and on (1-v)20R2 4 (4.12) The complete distributions of radial and hoop stress across the radius of the disc are shown in Fig.4.3. Hoop stress o Rodial stress a Fig.4.3.Hoop and radial stress distributions in a rotating solid disc. 4.3.Rotating disc with a central hole (a)General equations The general equations for the stresses in a rotating hollow disc may be obtained in precisely the same way as those for the solid disc of the previous section, ie. ,=A-治-g+心以 8 m=A+g-1+3列心 8 The only difference to the previous treatment is the conditions which are required to evaluate the constants A and B since,in this case,B is not zero. The above equations are similar in form to the Lame equations for pressurised thick rings or cylinders with modifying terms added.Indeed,should the condition arise in service where a rotating ring or cylinder is also pressurised,then the pressure and rotation boundary conditions may be substituted simultaneously to determine appropriate values of the constants A and B. However,returning to the rotation only case,the required boundary conditions are zero radial stress at both the inside and outside radius, i.e.at r=R1, 0,=0 -3+)w2R B 0=A- 8 and at r=R2, 0,=0 B 0=A- -3+)w2 8
122 Mechanics of Materials 2 $4.3 At the outside of the disc, at r = R, the equations give po2R2 a, = 0 and UH = (1 - u)- 4 (4.12) The complete distributions of radial and hoop stress across the radius of the disc are shown in Fig. 4.3. kootal streis u, Fig. 4.3. Hoop and radial stress distributions in a rotating solid disc 43. Rotating disc with a central hole (a) General equations The general equations for the stresses in a rotating hollow disc may be obtained in precisely the same way as those for the solid disc of the previous section, i.e. B pW2r2 r2 8 (T, = A - - - (3 + u)- The only difference to the previous treatment is the conditions which are required to evaluate the constants A and B since, in this case, B is not zero. The above equations are similar in form to the Lam6 equations for pressurised thick rings or cylinders with modifying terms added. Indeed, should the condition arise in service where a rotating ring or cylinder is also pressurised, then the pressure and rotation boundary conditions may be substituted simultaneously to determine appropriate values of the constants A and B. However, returning to the rotation only case, the required boundary conditions are zero radial stress at both the inside and outside radius, i.e. at r = RI, a, = 0 and at r = R2, a, = 0 .. B po2 R: R: 8 0 = A - - - (3 + u)-
§4.3 Rings,Discs and Cylinders Subjected to Rotation and Thermal Gradients 123 Subtracting and simplifying, B=(3+)Aw2R程 8 and A=(3+)w2R+R3) 8 Substituting in eqns.(4.7)and(4.8)yields the final equation for the stresses =(3+)8 R+殴- r2 -2 (4.13) g=+(+对+)-a+3r (4.14) (b)Maximum stresses The maximum hoop stress occurs at the inside radius where r =R1, ie. CHmat= 0w2 =8[【3+R++R)-(1+3DR 4[3+R经+1-R别 (4.15) As the value of the inside radius approaches zero the maximum hoop stress value approaches D 4(3+3 This is twice the value obtained at the centre of a solid disc rotating at the same speed.Thus the drilling of even a very small hole at the centre of a solid disc will double the maximum hoop stress set up owing to rotation. At the outside of the disc when r =R2 4[3+R+(1-vR】 0 Hmit- The maximum radial stress is found by consideration of the equation =a+g+-- (4.13)(bis) dor This will be a maximum when =0, dr i.e. when 0 0=R号-2r 4=RR7 r=V(RiR2) (4.16)
94.3 Rings, Discs and Cylinders Subjected to Rotation and Thermal Gradients 123 Subtracting and simplifying, pW2 R: R; 8 B = (3 + U) po2(R: + R;) 8 and A = (3 + U) Substituting in eqns. (4.7) and (4.8) yields the final equation for the stresses (b) Maximum stresses The maximum hoop stress occurs at the inside radius where r = R1, OH,,, = - [(3 + u)(R: + R: + RZ) - (1 + 3u)R:] PW2 i.e. 8 (4.13) (4.14) (4.15) As the value of the inside radius approaches zero the maximum hoop stress value approaches PO2 -(3 + v)RZ This is twice the value obtained at the centre of a solid disc rotating at the same speed. Thus the drilling of even a very small hole at the centre of a solid disc will double the maximum hoop stress set up owing to rotation. 4 At the outside of the disc when r = R2 The maximum radial stress is found by consideration of the equation dOr This will be a maximum when - = 0, dr i.e. dr (4.13)(bis) 0 = R,R27 2 2L - 2r (4.16)
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 dr
124 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