Pressure Vessel Cylindrical Shells Under Internal Pressure
The most important case in vessel design is a thin shell surface of revolution subjected to internal pressure. The internal pressure can be a uniform gas pressure or a liquid pressure varying along the axis of rotation due to the liquid head. In the latter case usually two calculations are performed to determine the stresses due to the equivalent gas pressure plus the stress in the lowest part of the shell due to the liquid weight.
Stresses in a closed-end-cylindrical shell under internal pressure Pi, computed from the conditions of static equilibrium (Figure 100-2) are the longitudinal (meridional) stress:
As can be seen from these two equations, the hoop stress is always greater and determines the required thickness of the shell.
The equations above are accurate for thin wall cylinders (R/t > 10) under internal pressure. However, for thick wall cylinders (R/t < 10), the variation in stress from the inner to the outer surface becomes appreciable, and the above equations are not satisfactory.
The Lamé, or thick-cylinder equations, are used to calculate the stresses (radial and circumferential) at any radius, r, in a thick wall cylinder as shown in Figure 100-3.
where:
Pi = Internal pressure
a = Inside radius
b = Outside radius
These equations show that both stresses are maximum at the inner surface. The maximum tensile stress (Q) at the inner surface is
The radial stress (Q) is always a compressive stress and smaller than the maximum tensile stress (Qt max). The maximum tensile stress is always greater than the internal pressure, but approaches this value as the wall thickness increases. The difference between the minimum tensile stress at the outside surface and the maximum tensile stress at the inside surface is the magnitude of the internal pressure. Therefore, for very high internal pressures it is necessary to use comparably high-yield materials.
For thin walls, there is little difference between the maximum tensile stress given by the thick-cylinder equation and that given by the thin-cylinder or average-stress equation. For thick walls, however, the difference between the values of the two equations is significant. For example, at a wall thickness of 10% of the radius (R/t = 10), the maximum stress is only 5% higher than the average stress. However, at a ratio R/t = 6, the maximum stress is 37% higher than the average stress. For this reason the ASME-Code equations approximate the more accurate thick-wall equations, and are used for all thicknesses.
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