Downloaded from https://www.everyspec.com STRUCTURES 3-3. DISCONTINUITY STRESSES IN Under the action of the edge moment and THIN SHELLS the edge shear force the region of the shell close to the edge deforms. The deformations vie with twoylind erical pressure vessel p- of the edges of cylinder and hemisphere are vided with two hem'spherical piressure ve.el of such magnitude that the diicontinuity in heads, and the vesel is filled with a gas at the deformations as caused by the membrane high pressure, the change in the length of the stresses alone is eliminated and cylinder and radius of the cylinder is more than twice the hemisphere fit together.
The bending stresses change in the length of the radius of the ,, . caused by M. and V.. are known as dЈsconti- sphere in accordance with Equations (3-7) and (3-9) if the wall thicknesses are equal. When M. and V° are applied to the edge of Such deformations are obviou.ly impossibhle the cylindrical shell shown in Figure 3-2, the when cylinder and hemisphere are welded th cyidicalhiell csw in Fighe 3-. h together (see Figure 3-1). For this reason bending stresses develop in the walls of the by the following equation vessel in addition to the membrane stresses 2V3(1- (R VR) discussed in Section 3-2.
When the material = Eh + is perfectly elastic, these bending stresses The positive directions of M, V and w are as vary linearly across the wall and attain their shown in the figure. The slope of the de- maximum values on thp 'juter and inner sur- forwed shape of the shell is faces of the wall. Th 'bending stresses add up to a bending moment of M., per unit length (dw/dx). = 3(1 - (2MoK + VoR) of the circumference at the edge of the cylin- (3-1) der of Figure 3-2 and the shearing stresses (3-11) yields a negative slope when accrmpanying them to a shear force V, per M,, and V., are positive; correspondingly the unit length of the circumference.
radial dispiacement w decreases with in- creasing x in Figure 3-2, that is the slope X--oo-X shown in the figure is negative. The value of the constant K in Equations (3-10) and (3-11) is K = '/3(1 - 9) V(R/hj (3-1? The bending moment M,, is not necessarily the largest bending moment in the wall of the shell; variation in the value of the bend- ing moment is given by the equation Figure 3-1. Theoretical Deformations if Pressure M = e-Kx / [mcos -R + sin Kx Vessel under Membrane Strestset Alone +i V,.R .K x + sirlK1c (313) V.
t Mo h The location of the maximum bending mo- ment can be computed from the formula xR Kx 2MK Kcot 1 2+ (3-14) oX _ o R -- VtR The connection between the maximum bend- ing stress and the bending moment is M o = 6M/h2 (3-15) The'attenuafion of the bending stresses is characterized hy the factor e-Kx"/ ; this is Figure 3-2. Doformations of Thin Circular unity at x - 0 and e-- I 0.368 when Cylindrical Shell under Bending S$resses i,, = R/K (3-16) ( 42