Though the viewport penetrations in the cylinder wall are, strictly speaking,
fully three dimensional, nonsymmetric problems, and should be treated
accordingly, an approximate method can be employed that renders conservative
results. This method is possible because of the small ratio of the
penetration diameter, d, to the mean diameter of the cylinder D, in this case
1:12.3. This method involves the use of an axisymmetric shell analysis which
includes a Fourier series loading capability. Specifically, a flat circular
annular plate of thickness equal to the cylinder wall thickness, with the
inner radius equal to the penetration radius and reinforced as actually
reinforced, with an outer radius of at least 10 times the inner radius, is
subjected to an in-plane edge load, P([theta]) of
P([theta]) = - (ST) - - (ST) cos 2 [theta]
S = the hoop stress found in an unpenetrated cylinder, away from
any stress riser, psi
T = thickness of shell, away from reinforcement, inches
[theta] = angle from longitundinal plane.
To include the effect of the resultant pressure from the viewport itself, the
penetration reinforcement as actually developed is modeled into the center of
a hemispherical shell with twice the inner radius of the cylinder. The
resultant pressure of the viewport is then applied as a pressure band over
the actual area of contact. This three component model is shown in Figure
2-32. Fourier loading was used to analyze this approximate model. The hoop
stresses in the vicinity of the penetration, for both the circumferential and
longitudinal sections, [theta] = 90 and [theta] = 0 degrees are shown plotted
in Figure 2-33. As can be seen, the stresses developed around the
penetration decay quite rapidly along the shell.
The axial stress distribution was also developed but not plotted.
maximum stress condition is shown in this figure.
(5) Categorization of Stress. At this point the stresses should
be broken into the stress categories, as fully defined in the previous
(6) Stress Intensities. As in the previous example, the stresses
are now converted into stress intensities. We will consider only one point
in the vessel (all other stress intensities are satisfactory). Examining
Figures 2-29 and 2-30 we note that the maximum stress intensity in the
tori-spheroidal head, on the inner surface is
Smax = 34,800 - (-9,800) = 44,600 psi.
Now in the knuckle (torus) of a tori-spheroidal head the stress must be
categorized as a primary local membrane plus a secondary bending stress.
Thus the stress intensity limit for this configuration is (see Figure 2-7)
PL + Pb + Q
/ = 3 Sm = 69,600 psi