Membrane stresses in shells

ADVANCED STRENGTH OF MATERIALS, J.P. Den Hartog. Superbly written advanced text covers torsion, rotating disks, membrane stresses in shells, much more. Many problems and answers. 388pp. 54 x 84. 65407-9 Pa. 8.95... 

Calculate the maximum membrane stress in the wall of shells having the shapes listed below. The vessel walls are 2 mm thick and subject to an internal pressure of 5 bar. 

This is of course based on the assumption that the longitudinal stress is a form of membrane stress in that there is no variation across the thickness of the shell. Thus we have... 

The force due to internal pressure is resisted by the membrane stress in the shell (see Figure 6.2). Because of the geometrical symmetry, the membrane stresses in the circumferential and the meridional directions are the same, and are denoted by S. We have... 

Computed stresses are based on test thickness at test temperature. Since water pressure is a short-term condition, the allowable stresses for structural parts such as supports are frequently increased by a factor of 1.2. The upper limit of stress in the vessel shell during a hydrotest of pressure parts is not specified by the Code. However, it is a good engineering practice to limit the maximum membrane stress in any part of the vessel during a hydrotest to 80 percent of the yield strength. Thermal expansion stresses and local mechanical stresses will be absent and need not be considered. 

Internal-pressure design rules and formulas are given for cylindrical and spherical shells and for ellipsoidal, torispherical (often called ASME heads), hemispherical, and conical heads. The formulas given assume membrane-stress failure, although the rules for heads include consideration for buckling failure in the transition area from cylinder to head (knuckle area). 

The analysis of the membrane stresses induced in shells of revolution by internal pressure gives a basis for determining the minimum wall thickness required for vessel shells. The actual thickness required will also depend on the stresses arising from the other loads to which the vessel is subjected. 

In conclusion, membrane stress analysis is not completely accurate but allows certain simplifying assumptions to be made while maintaining a fair degree of accuracy. The main simplifying assumptions are that tire stress is biaxial and that the stresses are uniform across the shell wall. For thin-walled vessels these assumptions have proven themselves to be reliable. No vessel meets the criteria of being a true membrane, but we can use this tool with a reasonable degree of accuracy. 

This issue was addressed in 1979 by McBride and Jacobs. Jacobs was from Fluor in Houston. The principle was to calculate stresses in two distinct areas, membrane and bending. Membrane stresses are based on pressure area times metal urea. Bending is based on AISC beam formulas. The neck-and-shell section (and sometimes the flange as well) is assumed as bent on the hard axis. This is not a beam-on-elastic-foundation calculation. It is more of a brute-force approach. 

This procedure determines the bending stress in the stiffener only. The stresses in the vessel shell should be checked by an appropriate local load procedure. These local stresses are secondary bending stresses and should be combined with primary membrane and bending stresses. 

The stresses found from these charts will be reduced by the effect of internal pressure, but this reduction is small and can usually be neglected in practice. Bijlaard found that for a spherical shell with R y T = 100, and internal pressure causing membrane stress of 13,000 psi, the maximum deflection was decreased by only 4%-5% and bending moment by 2%. In a cylinder with the same Rm/T ratio, these reductions were about 10 times greater. This small reduction for spherical shells is caused by the smaller and more localized curvatures caused by local loading of spherical shells. 

The thickness of the RPV shell is chosen to minimize the stress in the wall of the RPV. The major concern in RPV design is the pressure stress that the RPV must withstand during operation. The design pressure of the RPV is chosen to be higher than the operating pressure, and is used to establish the main safety valve set points limiting maximum achievable pressure in the RPV. The pressure stress, sometimes called the hoop or membrane stress, is simply the maximum stress produced by the design pressure as determined by (Tm = P rit, where is the membrane (pressure) stress, P is the maximum pressure, r is the RPV mean radius and t is the RPV wall thickness. 

In addition to the simple membrane stress of the cyhnder, the shell is subjected to a radial stress due to the direct apphcation of the pressure against the wall. This is a compressive stress and is insignificant for thin walled pressure vessels when compared to the other principal stresses. But the radial stress becomes more significant as the pressure and thus the thickness is increased. 

If one judges on the basis of membrane theory, the hemispherical head has twice the st rength of a cylindrical shell of the same diameter, as indicated by Eqs. 3.13, 3.14 (note that the hoop stress in a hemisphere is the same as the axial stress in a cylinder). This Ls. shown in Fig. 7,19 where the curve I,un is equal to 1.0 when Ih/ts = 0.5. The maximuin where E — joint eflitaency... 

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