Disk-Wheel Stress Determination

Abstract

Abstract This paper describes and applies a simplified method for determining the centrifugal stresses in a disk wheel of given irregular shapes of section. Stodola’s disk theory is assumed, together with his formula for disks of hyperbolic-section profile. The formulæ are then transformed so as to give the tangential stresses at the inner and outer radii in terms of the radial stresses, ratio of radii, and shape constant of disk section. For a given disk with a single hyperbolic outline the tangential stresses are expressed in terms of the radial stresses or loads; and if the latter are known, the stresses are determinate. In case the disk-section outline would require to be fitted by two hyperbolas meeting at the same thickness of disk, the wheel section can be divided at the meeting point of the curves into two imaginary rings. The rings at the meeting point, by the continuity of material and same thickness, are held together by a radial stress common to both and have the same elongation and tangential stress. One can then write the tangential stress of the one ring equal to the tangential stress of the other at their common radii, giving one equation with the one unknown quantity, the radial stress common to both, and all stresses are determinate. Thus any irregular-shaped disk can be fitted within hyperbolas, divided into n imaginary rings, and (n – 1) equations written for the common meeting points of the curves with (n – 1) unknown radial stresses, and the stress problem is solvable. The six coefficients whose values are given in Equation [4] as functions of the ratio of radii and hyperbolic outline of disk section, are laborious to calculate but should be used for accurate results. For commercial work approximate equations [6] are given which cover the practical disk proportions, and within the limits shown have a range of error of less than one per cent. As a further labor-saving device when a number of disks are to be estimated, the approximate equations can be placed in an alignment-chart form. An appendix gives the range and proportion of the five charts required which experience has shown to be the most useful. The number and size of these charts for accurate reading do not permit of reproduction. A practical example is given showing the actual application to the usual disk wheel.