§10.3 Thick Cylinders 219 Substituting in (10.1)for oH, dor 2A-0,-0,= 'dr Multiplying through by r and rearranging, 2a,7+r2dg-24=0 dr o,2-Ar内=0 d i.e. Therefore,integrating, a,r2-Ar2 constant =-B (say) 6,=A- B (10.3) and from eqn.(10.2) B GH=A+ (10.4) The above equations yield the radial and hoop stresses at any radius r in terms of constants A and B.For any pressure condition there will always be two known conditions of stress (usually radial stress)which enable the constants to be determined and the required stresses evaluated. 10.3.Thick cylinder-internal pressure only Consider now the thick cylinder shown in Fig.10.5 subjected to an internal pressure P,the external pressure being zero. Fig.10.5.Cylinder cross-section. The two known conditions of stress which enable the Lame constants A and B to be determined are: Atr=R1 ,=-P and at r=R2 G,=0 N.B.-The internal pressure is considered as a negative radial stress since it will produce a radial compression(i.e.thinning)of the cylinder walls and the normal stress convention takes compression as negative.$10.3 Thick Cylinders 219 Substituting in (10.1) for o~, dor 2A-ar-ar = r- dr Multiplying through by r and rearranging, dor 2orr + r2 - - 2Ar = 0 dr i.e. Therefore, integrating, .. d -(or? -A?) = 0 dr orrZ - Ar2 = constant = - B (say) and from eqn. (10.2) (10.4) B UH=A+- rz The above equations yield the radial and hoop stresses at any radius r in terms of constants A and B. For any pressure condition there will always be two known conditions of stress (usually radial stress) which enable the constants to be determined and the required stresses evaluated. 10.3. Thick cylinder - internal pressure only Consider now the thick cylinder shown in Fig. 10.5 subjected to an internal pressure P, the external pressure being zero. Fig. 10.5. Cylinder cross-section. The two known conditions of stress which enable the Lame constants A and B to be determined are: At r = R, or= -P and at r = R, or =O N.B. -The internal pressure is considered as a negative radial stress since it will produce a radial compression (i.e. thinning) of the cylinder walls and the normal stress convention takes compression as negative