370 SHELLS 个五 Figure 8.8:The loads and the membrane forces on a cylinder. about the x-axis yield (Fig.8.8) N2nR=N 2Ny P:2R (8.10) (Nzy R)2nR=T. where R is the radius of the wall's reference surface.From these equations,the membrane forces are N Nx=2R Ny P:R Nsy =2R (8.11) The strains corresponding to these membrane forces are calculated by Eq.(8.4). The axial u°，radial w°，and circumferential v°displacements are °=ed=xe+ugw°=Regv°= yo,dx xy+vo. (8.12) where ue and ve represent rigid-body motion. 8.2.2 Built-In Ends As we noted previously,near boundary supports the membrane theory is inaccu- rate,and the forces,moments,and displacements of the shell must be calculated by other means.In the following,we consider thin-walled circular cylinders built-in at each end.The cylinder is subjected to pressure pa,axial load N.and torque Figure 8.9:Cylinder built-in at both ends subjected to pressure p:,axial load N,and torque T. k 2R370 SHELLS Nxy Nx N pz Ny x x T Figure 8.8: The loads and the membrane forces on a cylinder. about the x-axis yield (Fig. 8.8) Nx2π R = N
2Ny = pz2R (8.10) (NxyR) 2π R = T
, where R is the radius of the wall’s reference surface. From these equations, the membrane forces are Nx = N
2π R Ny = pzR Nxy = T
2π R2 . (8.11) The strains corresponding to these membrane forces are calculated by Eq. (8.4). The axial uo, radial wo, and circumferential vo displacements are uo = ) o xdx = xo x + uo o wo = Ro y vo = ) γ o xydx = xγ o xy + vo o, (8.12) where uo o and vo o represent rigid-body motion. 8.2.2 Built-In Ends As we noted previously, near boundary supports the membrane theory is inaccu￾rate, and the forces, moments, and displacements of the shell must be calculated by other means. In the following, we consider thin-walled circular cylinders built-in at each end. The cylinder is subjected to pressure pz, axial load N
, and torque T
pz L N N T T 2R x Figure 8.9: Cylinder built-in at both ends subjected to pressure pz, axial load N
, and torque T