Stress Analysis of Nondevelopable Semimonocoque Structures Considered as Beams of Variable Section

Abstract

T E PURPOSE of this paper is to present a rational, although approximate, method of analysis of structures consti tuted by a shell of elongated shape, reinforced by frames. Conventionally, beam formulas are applied to such shells and define axial stresses. Shearing stresses are computed, either applying shear distribution formulas for beams—in which case, the forces and torques are corrected by the so-called shearing components of the axial forces—or, more accurately, considering forces as equal to the variations of axial loads between consecutive sections, summed along the circumference of the shell. Shear centers are determined in same manner as in cylinders. Stress distributions obtained in t ha t manner do not fulfill the conditions of equilibrium and the conditions of compatibility of deformation. In the case of elongated closed shells with a small taper, the errors introduced by these approximate methods can be neglected for practical purposes, although the equilibrium of frames, resulting from the shear distribution, may prove unsatisfactory. When the taper of a shell is large, this taper being roughly defined as the difference of average cross dimensions at two sections, divided by their distance, and especially when open shells are considered, the error may be as high as 100 per cent. Such large errors generally result from incorrect distributions in torsion bending which are not in equilibrium with the applied loads. In view of the inaccurate results obtained by conventional methods, these methods must be revised. In the following, only conditions of equilibrium will be satisfied, and the s tudy of the conditions of compatibility will be disregarded. More precisely, conventional beam distribution for normal stresses will be used to determine stresses in the shell along the frames. This means t ha t the influence of shear, of the local distr ibution of loads, and of the flexibility of the frames are disregarded. For t ha t reason the method is considered as approximate. In many cases, however, the results are not less accurate than for cylindric shells, for which the abovementioned effects exist, with the exception of the influence of shear, which does not affect normal stresses at great distance of points of load application. When the differences between a cylindric shell and a noncylindric elongated shell are examined, they can be reduced to th ree 'main factors: the taper, defined as indicated previously; the twist, i.e., the effect of the nonsimilar shape of consecutive cross sections; and the curvature of the shell in the direction of its great dimension. The effect of the taper has been investigated, in relationship with the equilibrium of developable surfaces, in a previous paper by the author, and its physical significance is not examined in this paper. The effect of the curvature is disregarded, because it is assumed to be small in the types of shells under consideration. Thus, the main object of this paper is the s tudy of the effect of the twist and of the combined effects of twist and taper.