Wymiarowanie Prętów o Krzywiźnie Pierwotnej na Gruncie Teorii Sprężystości - Du-Ching, Wang; Życzkowski, M. (Engineering Transactions, 1965)
This paper is devoted to the problem of direct evaluation of the dimensions of a circularly curved bar loaded with a concentrated moment and a concentrated force of any direction, Fig. 1. By contrast with Ref. [4] accurate equations of the theory of elasticity are used instead of the elementary formulae of strength of materials. This necessitates the considerations to be confined to the case of a rectangular cross-section. The maximum effective stress at the contour may occur at the points A1, A2, B1, B2, Ci and C2, Fig. 1. The safety conditions are established for all of these points and solved by the method of power series for the unknown height h of the profile. The highest of them determines the absolutely necessary value of h, because, as shown later (in some particular cases, however), the upper bound of the effective stress cannot take place inside the region. In order to establish the rules for selecting the approximate formula, their results are compared directly. They depend on the direction of the concentrated force (angle q) and the force-to-moment ratio described by the parameter ψ, (3.2). The validity regions of the equations are determined in the ψ - ψ plane (Fig. 5), thus showing that the point C2 is never dangerous. The case in which the force is normal to the free section, with simultaneous action of the moment, and the case of a concentrated force acting alone in an arbitrary direction are discussed in detail. The accuracy of the equations is illustrated by a numerical example.