Harwood
Engineering Company, Inc.
Harwood
Engineering Company, Inc.
The method described below has been used by the author during a life-time experience of designing vessels and related equipment for high pressure. It is distinctly different from the usual design approach which is to make composite vessel cylinders carry more nearly equal shares of the stress burden, but that does lower the resistance of the outer cylinder to fatigue.
The subject of fatigue resistance in vessels with residual bore stresses developed by autofrettage and/or by shrinkage is too extensive to cover in detail in the allotted space for this paper.
The design fatigue data used in this method are presented here with scanty detail of their origin which will be given elsewhere in greater detail.
The fatigue data were gathered initially at Watertown Arsenal in World War II under the direction of the author1 and subsequently by a group of researchers at Watervliet Arsenal2,3 using apparatus designed by the author and colleagues at Harwood Engineering Company in Walpole, Massachusetts. Details of the equipment, tabulations of the data, physical properties of the specimens, etc. are to be found in the references cited here.
Briefly, in these studies, the mortal or low cycle fatigue end of the failure curves was explored. The specimens were monobloc cylinders of various wall ratios from 1.2 to 2.0. They were cyclically loaded by internal pressure until failure. In all, 264 tests were made, about equally divided between autofrettaged and nonautofrettaged specimens. They were tested at sufficiently high pressures so that failure occurred between 103 and 105 cycles. The tensile strength of materials tested varied from 110,000 to 180,000 psi. A few specimens were small in bore while others were sections of forged and centrifugal cast cylinders, some with smooth bore and others with ribbed or French rifling. The amount of autofrettage in the specimens varied.4 Some were tested with tap water plus rust inhibitor, others with synthetic oil for the pressure medium. There was a fair variation of surface finish.
In spite of the variations in geometry, metallurgy, surface finish and the small number of specimens (there are never enough), the data turned out to be surprisingly consistent. It was felt that the fatigue data could be used to develop design fatigue curves. These design curves (Figures 2 & 3) were obtained by introducing a margin of safety such that all but a few specimens (9 autofrettaged, 16 small bore nonautofrettaged) were on the safe side.
The resulting design curves appear to be acceptable from an engineering point of view, the way things are, while from an academic view, they undoubtedly leave much to be desired.
A reasonable estimate of the rupture pressure of the outer cylinder:
(2)
(3)The optimum wall ratio of the outer cylinder may be found as follows. The desirability of having Pro greater than Pb is obvious. If their difference is called Q then:
(4)
(5)
(6)Let us assume that a modified Goodman diagram5 (Figure 4) based on tangential stress at the bore of the outside cylinder can be used to determine the normalized tangential stress, at least in the case of a nonautofrettaged outer cylinder. Then cyclic stress amplitude plus the tangential stress resulting from shrink-fitting the outer and inner cylinders locates point (0') from which the diagram is drafted as shown with the stresses labeled in the usual way of modified Goodman diagrams.
The cyclically applied pressure at diameter (b) may be found from Equation (3). The cyclical stress at diameter (b) is:
(7)From similar triangles in Figure 4, the equivalent stress (ste), normalized, is shown to be:
(8)The pressure (Ps), resulting from an interface fit is a text book problem. If the interface (D) and (b) are diametric:
(9)
(10)
