120 Mechanics of Materials 2 §4.2 Now for equilibrium of the element radially 66 2awor sin+r(+8)(r+5r6-pr280 8r If 80 is small, 6868 sin radian Therefore in the limit,as r-0(and therefore 8o,-0)the above equation reduces to OH -O-r- rdor =pr2o (4.1) dr If there is a radial movement or"shift"of the element by an amount s as the disc rotates, the radial strain is given by ds I Er= -=(O,-voH) (4.2) dr E Now it has been shown in $9.1.3(a)that the diametral strain is equal to the circumferential strain. 5 EoH-or） (4.3) 5= E(OH-vo,) ds I Differentiating, dr=E(on -va,)+ (4.4) dr dr Equating eqns.(4.2)and(4.4)and simplifying, (oH-o,)1+v)+r -ur do.=0 (4.5) dr dr Substituting for (oH-o,)from eqn.(4.1), (0+n）1++r dr dr don dor =-pro?(1+v) dr dr Integrating, pr2w2 OH十O,三- 2(1+)+24 (4.6) where 24 is a convenient constant of integration. Subtracting egn.(4.1), 2o,+r dor pr202 dr -2(3+)+2A But 2+20=品× EJ.Hearn,Mechanics of Materials 1.Butterworth-Heinemann,1997.120 Mechanics of Materials 2 $4.2 Now for equilibrium of the element radially If SO is small, 68 68 22 sin - = - radian Therefore in the limit, as Sr + 0 (and therefore Sa, + 0) the above equation reduces to d ur 22 UH -ar - r- = pr o dr (4.1 ) If there is a radial movement or “shift” of the element by an amount s as the disc rotates, (4.2) Now it has been shown in $9.1.3(a)’ that the diametral strain is equal to the circumferential strain. 1 E Differentiating, dr E E [dr-dr] Equating eqns. (4.2) and (4.4) and simplifying, s = --(OH - war) ds 1 r dDH Vdo, - = -(OH - va,.) + - dOH do,. (CH - Gr)(l + V) + r- - vr- = 0 dr dr Substituting for (OH - a,.) from eqn. (4.1), dcTH do,. (I + v) + r- - vr- = o dr dr dCJH do,. 2 .. - + - = -prw (1 + v) dr dr Integrating, OH +ar = -- pr2w2 (1 + ”) + 2A 2 where 2A is a convenient constant of integration. Subtracting eqn. (4.1), But pr2w2 (3 + v) + 2A do, 2ar + r- = -- dr 2 dar d 2 I 20, +r- = - [(r a,)] x - dr dr r (4.3) (4.4) (4.5) (4.6) E.J. Hearn, Mechanics ofMatericrls 1. Butterworth-Heinemann, 1997