812.4 Miscellaneous Topics 519 M Frresrsrsssergn Unit width 15anaam M d8 （0} (b) Fig.12.8.Loading conditions at discontinuities in thin shells. The radial displacements set up by these stress systems are;for the cylinder: 8=E(OH-WL) pr2 2rE2-) and for the hemispherical ends: 8=E(n-WoH) 2E1-以. There will thus be a difference in deformation radially of: 6-8= Dr2 2E2-)-21-叨 upr2 2tE which can only be reacted by the introduction of shear forces and moments as shown in Fig.12.8(a)where o=shear force and M moment,both per unit length. Because of the total symmetry of the cylinder about its axis we may now consider bending of a small element of the cylinder of unit width as shown in Fig.12.8(b). The shear stress O produces inward bending of the elemental strip through a radial displacement 8,and a compressive hoop or circumferential strain given by: EH =$12.4 Miscellaneous Topics 519 +!LL ?- 0, (a) Fig. 12.8. Loading conditions at discontinuities in thin shells. The radial displacements set up by these stress systems are; for the cylinder: and for the hemispherical ends: There will thus be a difference in deformation radially of Pr2 6 - 6’ = --[(2 - u) - 2(1 - v)] 2tE - vpr2 2tE - which can only be reacted by the introduction of shear forces and moments as shown in Fig. 12.8(a) where Q = shear force and M = moment, both per unit length. Because of the total symmetry of the cylinder about its axis we may now consider bending of a small element of the cylinder of unit width as shown in Fig. 12.8 (b). The shear stress Q produces inward bending of the elemental strip through a radial displacement 15, and a compressive hoop or circumferential strain given by: