Downloaded from https://www.everyspec.com STRESS ANALYSIS ) When the attenuation length x., is small as Modulus of elasticity - compared to the undisturbed, uniform por- E = 29 x 106 psi tion of the length of the cylinder, the equa- Poisson's ratio tions given here are valid; if the attenuation = 0.3 length is larger the equations must be re- Internal pressure placed by more complex equations discussed p = 100 psi in the literature.
From Equations (3-1), (3-2) and Z3-5) one Experience has shown that satisfactory obtains accuracy is obtained for, engineering pur- Hoop stress in cylinder poses if the actual shell configurations are ,, = 20,000 psi I imagined to be replaced by equivalent circu- Axial stress in cylinder lar cylindrical shells in the calculation of the Ox 10,000 psi I disturbance stresses. In the case of the spher- Membrane stress in hemisphere ical shell, the equivalent cylindr:al shell has , = 10,000 psi the same radius as the spherical shell that it '.he changes in the length of the radius of replaces.
In the case of the arbitrary rota- cylinder and hemisphere are computed from tionally symmetric shell which joins the cyl- Equations (3-7) and (3-9) inder with a continuous tangent, the radius 1.466 X 10-1 in of the equivalent shell is R 0.603 X 10-2 in R, = R(r/R) 2 (3-17) The misfit between cylinder and hemisphere where R is the radius of the cylinder and r is therefore is the radius of the meridian of the rotation- AR,,, - AR,phe.. = 0.863 X 10-7 in ally symmetsc shell at the junction with the cylinder. Equations (3-10) to (3-16) retain This misfit must be eliminated with the aid their validity for the general rotationally of the bending stresses.
) symmetric shell joining the cylinder with a First, the hemisphere is replaced by the continuous tangent if R is replaced in them equivalent cylinder, and a coordinate x, is by R, and if x is measured along a meridian, introduced in it which is measured to the left This is true, however, only if the change in from the juncture between cylinder and the valves of the principal curvatures of the hemisphei-e. An edge moment M,. and an edge rotationally ,.ymmetric shell is insignificant shear force V., are applied to the cylinder as over the attenuation length x,t defined in shown in Figure 3-2 and equal and opposite Equation (3-16).
quantities are applied to the cylinder replac- V, hen the head of the pressure vessel has ing the hemisphere. The conditions must now a taagent which changes discontinuously be enforced that under the action of M,, and from the tangent of the cylinder, the equiva.. V. the ;relative displacement of the two shell lent cylinder concept is still valid, but the wails is 0.863 X 10- in., and that the slope of forces and displacements must be resolved in the gene.ators remains continuous. The l.t- the proper directions.
ter condi&;:ion is fulfilled if M,, = 0 because For illustrative purposes the numerical values the wall thicknesses are the same in cylinder characterizing the pressure vessel of Figure 3-1 and hemisphere. As the real and the equiva- may be taken as lent cylinder are eqaal in size, under the Radius of median surface of cylinder action of the same shear force V,, each one R,= 25 fn shows the same numerical value of change Radius of med.an 3urface of hemisphere in the length of the radius. Hence Equation R,,,h,.r = 2 in (3-10) becomes Length of cylinder - 0 EK h- L =100 in Wall thickness of cylinder and fiemisphere 3(1 - v:) R R h 0.125 in (J 18) 43 !I