518 Mechanics of Materials 2 §12.4 Also ds1 =r2 de2 and ds2 ri de o1tds1 .ds2+ot ds2 rI ds =p.dsds2 r2 and dividing through by ds ds2 t we have: 4+2= (12.18) r r2 t For a general shell of revolution,o and o2 will be unequal and a second equation is required for evaluation of the stresses set up.In the simplest application,i.e.that of the sphere,however,r=r2 =r and symmetry of the problem indicates that o1 =02=o. Equation (12.18)thus gives: a=pr 2t In some cases,e.g.concrete domes or dishes,the self-weight of the vessel can produce significant stresses which contribute to the overall failure consideration of the vessel and to the decision on the need for,and amount of,reinforcing required.In such cases it is necessary to consider the vertical equilibrium of an element of the dome in order to obtain the required second equation and,bearing in mind that self-weight does not act radially as does applied pressure,eqn.(12.18)has to be modified to take into account the vertical component of the forces due to self-weight. Thus for a dome of subtended arc 20 with a force per unit area g due to self-weight, egn.(12.18)becomes: +=±gcos9 (12.19) Combining this equation with one obtained from vertical equilibrium considerations yields the required values of o and o2. 12.4.Bending stresses at discontinuities in thin shells It is normally assumed that thin shells subjected to internal pressure show little resistance to bending so that only membrane (direct)stresses are set up.In cases where there are changes in geometry of the shell,however,such as at the intersection of cylindrical sections with hemispherical ends,the "incompatibility"of displacements caused by the membrane stresses in the two sections may give rise to significant local bending effects.At times these are so severe that it is necessary to introduce reinforcing at the junction locations. Consider,therefore,such a situation as shown in Fig.12.8 where both the cylindrical and hemispherical sections of the vessel are assumed to have uniform and equal thickness membrane stresses in the cylindrical portion are 1=0H=P" and o o whilst for the hemispherical ends pr σ1=02=0H=518 Mechanics of Materials 2 $12.4 Also dsl = r2d62 and ds2 = rI d81 .. 01 tdsl . and dividing through by dsl . ds2 . t we have: (12.18) For a general shell of revolution, al and a2 will be unequal and a second equation is required for evaluation of the stresses set up. In the simplest application, i.e. that of the sphere, however, rl = r2 = r and symmetry of the problem indicates that (TI = 02 = a. Equation (12.18) thus gives: P‘ 2t u=- In some cases, e.g. concrete domes or dishes, the self-weight of the vessel can produce significant stresses which contribute to the overall failure consideration of the vessel and to the decision on the need for, and amount of, reinforcing required. In such cases it is necessary to consider the vertical equilibrium of an element of the dome in order to obtain the required second equation and, bearing in mind that self-weight does not act radially as does applied pressure, eqn. (12.18) has to be modified to take into account the vertical component of the forces due to self-weight. Thus for a dome of subtended arc 28 with a force per unit area q due to self-weight, eqn . ( 12.18) becomes: (12.19) Combining this equation with one obtained from vertical equilibrium considerations yields the required values of a1 and 02. 12.4. Bending stresses at discontinuities in thin shells It is normally assumed that thin shells subjected to internal pressure show little resistance to bending so that only membrane (direct) stresses are set up. In cases where there are changes in geometry of the shell, however, such as at the intersection of cylindrical sections with hemispherical ends, the “incompatibility” of displacements caused by the membrane stresses in the two sections may give rise to significant local bending effects. At times these are so severe that it is necessary to introduce reinforcing at the junction locations. Consider, therefore, such a situation as shown in Fig. 12.8 where both the cylindrical and hemispherical sections of the vessel are assumed to have uniform and equal thickness membrane stresses in the cylindrical portion are whilst for the hemispherical ends Pr t 01 = 02 = OH = -