journal of engineering for industry tedmical briefs Deflections in Radially Loaded Thin Elastic Rings

This section consists of contributions of 1500 words or less (about 2 l /t double-spaced typewritten pages, including figures). Technical Briefs will be reviewed and approved by the specific division's reviewing committee prior to publication. After approval such contributions will be published as soon as possible, normally in the next issue of the Journal. Deflections in Radially Loade

Analysis of a Thin Elastic Ring Under Ar-bitrary Loading

] 3 are to be congratulated for their presentation and formulation of an excellent procedure for determining external displacements and internal loads in a thin elastic ring subjected to any self-equilibrating load system. The advantage of proceeding from displacement to load analysis (as was done in [1]), rather than vice versa, is certainly recognized. The authors of this discussion are particularly interested and gratified since the authors of [1] used one of the discussion authors' papers [2] among their references in extending the work to solve a class of external loadings not included in [2]. Furthermore, it is always welcome and fortifying to observe others who take another approach to selected problems and obtain the same correct results, as was demonstrated in [1] in applying the approach to some of the cases included in [2]. The mathematical equations in this paper are closed form expressions, a situation which could not be permitted in [2] since it was required in that procedure to assign absolute values to the angular arguments. The series expressions seem to eliminate some of the concern for sign and, possibly, magiiitude difficulties with regard to the internal load calculations. However, it is deemed necessary, even in this paper, to exercise a degree of caution particularly in applications to rings subjected to multiload sets and distributed loads acting over more than 90-deg arcs. It is noted that Liu and Chiu seemingly saw a need for this caution, as evidenced by their sign convention in Recently, it was brought to the attention of the discussion authors (by [3] 1 Liu, J. Y., and Chiu, Y. P., "Analysis of a Thin Elastic Ring Under Arbitrary Loading," presented at the Winter Annual Meeting, Detroit, Mich., Nov. 11-15,1973, ASME Paper No. 73-WA/DE-6. 2 Kurajian, G. M., and Na, T. Y., "Thin Elastic Rings Subjected to Ra-.dial Load Sets," JOURNAL OF ENGINEERING FOR INDUSTRY, TRANS. ASME, Series B, Vol. 94, No. 3, Aug. .1972, pp. 783-788. 3 Williams, H. E., Applied Mechanics Reviews, Vol. 26. No. 8, Aug. 1973, p. 947. 4 Wignot, J. E., Combs, H., and Ensrud, A. F., "Analysis of Circular Shell-Supported Frames," NACA TN-929,1944. 5 Prescott, J., Applied Elasticity, First Ed., Dover Publications, New York, 1961 6 Roark, R. J., Formulas for Stress and Strain, Fourth Ed., McGrawHill, New York, 1965. Authors' Closure The authors agree with the comments made by the discussers that in solving thin ring problems one must be very careful about' the sign convention used in the mathematical derivations in order to obtain correct solutions. This is especially so when the loading condition is complex. In the authors' paper the sign convention for the ring displacements as well as the applied loads is described in Though closed-form solutions have been presented for the several cases discussed by the authors, it is noted that when a distributed load function is complex, numerical integration by a digital computer may become necessary