SIGMA PHI STRESSES
SIGMA THETA STRESSES
Membrane
Inner
Outer
Membrane
Inner
Outer
9,905
-1,948
21,757
29,371
25,815
32,926
For this point the stresses can be categorized as follows:
Pm = 9.905 psi in [PHI] direction
PL = 29,371 psi in [theta] direction*
Pb = 0
Q
=
11,852 psi on the inner surface in the [PHI] direction
=
-3,555 psi on the inner surface in the [theta] direction
=
+11,852 psi on the inner surface in the [PHI] direction
=
+3,555 psi on the inner surface in the [theta] direction
We must also consider that there is a radial stress, [sigma]r, equal to the
pressure and considered as a compressive stress acting on the inner surface
and perpendicular to it. This is a surface stress and some thought must be
given as to where it is placed. For instance, at any point being considered
where there exists only a membrane stress, Pm, the [sigma] should be put
into the Q category. The same reasoning applies to where there are only Pm
and Pb stress components, as in the case of a flat head or some knuckle
positions. In such a case, it would be best to consider [sigma]r or a Pb
stress. For the example being discussed, we have additional Q stress
Q = 1000 psi on the inner surface in the r direction
Finally, the peak stress components must be considered. This is usually done
by denoting these as stress concentration factors, KPHI], and K[theta]B,
acting in the [PHI] direction and the [theta] direction, respectively. F
components are developed by multiplying the sum of the principal stresses in
that location in each direction by the factor (K[PHI] - 1.0) or (K[theta]
- 1.0), respectively. For instance, assume that the stresses as shown in the
computer printout line above, were calculated by hand or by some cruder
computer program incapable of accurately modeling the fillet radius. The
effect of the reentrant corner on the outside surface would have to be
determined in some manner. Assume that this geometry was estimated to induce
a K[PHI] of 1.6 and K[theta] of 0.0. Then
F = (K[PHI] - 1) (total principal stress in [PHI] direction
F = (1.6 - 1) (21,757) = 13,054 in the [PHI] direction on the
outside surface.
* This figure is a local membrane stress because the computer printout
shows that this stress decays very rapidly as distance is measured from
the geometric discontinuity.
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