220 Mechanics of Materials §10.4 Substituting the above conditions in eqn.(10.3), -P=A- B R 0=A B R PR i.e. A-7 R经-R) and B=1 PR2R3 R好-R) B radial stress a,=A- R-] PR2 []-P鬥 (10.5) where k is the diameter ratio D2/D=R2/R1 and bpms6“R[+] PR? R]-P[] (10.6) These equations yield the stress distributions indicated in Fig.10.2 with maximum values of both o,and oH at the inside radius. 10.4.Longitudinal stress Consider now the cross-section of a thick cylinder with closed ends subjected to an internal pressure P and an external pressure P2 (Fig.10.6). Closed ends Fig.10.6.Cylinder longitudinal section. For horizontal equilibrium: P,×πR子-P2×πR3=LXπ(R经-R)220 Mechanics of Materials 0 10.4 Substituting the above conditions in eqn. (10.3), i.e. B radial stress 6, = A - - r2 (10.5) where k is the diameter ratio D2 /Dl = R , f R , and hoop stress o,, = (10.6) PR: rZ+Ri wzm2 + 1 (R;-R:) [TI='[ k2-1 ] - These equations yield the stress distributions indicated in Fig. 10.2 with maximum values of both a, and aH at the inside radius. 10.4. Longitudinal stress Consider now the cross-section of a thick cylinder with closed ends subjected to an internal pressure PI and an external pressure P, (Fig. 10.6). UL - t t\ Closed ends Fig. 10.6. Cylinder longitudinal section. For horizontal equilibrium: P, x ITR: - P, x IT R$ = aL x n(R; - R:)