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Aviation History
1931
1931 - 1038.PDF
SUPPLEMENT TOFLIGHT SEPTEMBER 25, 1931 THE AIRCRAFT ENGINEER increased, and modifications would therefore have to be introduced. The graph will be found in " Aeronautics in Theory and Experiment," by W. L. Cowley, A.R.C.Sc, D.I.C., and H. Levy, M.A., B.Sc, F.R.S.E. It must be remembered that the mathematical theory of fluid motion does not take into consideration mole- cular velocities; hence the phenomena occurring when the velocity of sound is reached cannot be explained by it. N.B.—From the formula R = 0.0032 S V2, where R is the resistance in lb. wt.; S the area in sq. ft., and V the speed in m.p.h., the resistance on a sq. ft. should be = 0.0032 x (1,135)2 = 4,130 lb. at speed of mole- cules. Now from the curve connecting V/v and the resistance coefficient, we see that the upper limit of the resistance coefficient is approximately 2.25 times the lower limit. But 9,540 : 4,130 = 2.3 approx., in good agreement with the previous value 2.25. Therefore the kinetic theory of gases explains the increase in resist- ance coefficient. TORSION IN THIN CYLINDERS. By E. H. ATKXN, B.SC. (LOND).* A considerable volume of work has been done in the investi- gation of the stresses and strains in clastic cylindrical prisms under pure torsion. The theoretical work of Saint-Venant1 and others has been supplemented experimentally by the work of Prandtl, whose membrane analogy was developed by Griffith & Taylor1. Hence it is now possible, theoretically or experimentally, to determine the stresses in any torsionally loaded prism of any section whatever. For most engineering purposes, however, the experimental FIG. B STREAMLINE TUBE SQUARE. TUBE c SPAR SECTION (DOUBLE WEB) SPAR SECTION (SINGLE WEB)STK • Mr. Atkin is on the Technical Staff of A. V. Eoe & Co. Ltd., at Manchester. method is out of the question as the technique is too involved and the apparatus too expensive. Unless, therefore, the cross- section is one of the simple geometrical forms which can be dealt with by a mathematical formula, or one which can be dealt with by the semi-empirical rules due to Griffith and Taylor3, the designer is compelled to rely on his experience. The only other method available, a step by step method of integration due to Thom*, is too difficult to be of general use. Among the sections which cannot be dealt with by either of the two practicable methods must be included a series of sections of a type common to modern all-metal aircraft construction. This is the drawn strip steel closed section. Hence some simple method of determining the shear stresses in a thin-walled hollow cylinder under torsion has become very- desirable. But, while the solution of this problem in its simplest form is fairly well known, the general solution does not appear to have been put down in a form suitable for routine calculation. Indeed, as far as the writer is aware, the problem has not been worked out in detail at all. It is, therefore, proposed to give, in this article a method of determining the shear stresses in any thin walled section under pure torsion. Four such sections are shown in Fig. 1. The method of calculation for the stream line tube A, and the square tube B is, as already stated, well known. Let fs be the mean shear stress at any point round the section and t the thickness at that point, Am the area en- closed by the mean contour (see Fig. 2) of the section, and^I1 the applied torque. It can be shown6 that This formula holds round any thin section such as A or B in Fig. 1, even though the thickness varies round the cross- section, provided that the curvature of the cross-section is / / MEAN CONTOUR // FIG.2. LINES OF CONSTANT^ small at the point considered. But, at a point of large cur- vature, the maximum shear stress on the concave side is much higher than the mean. Points such as x in section A, and y in section B therefore, require special consideration. The calculation of the stress at these points will be discussed later. Formula (1) can be obtained in a simple manner by an inde- pendent investigation ; but as such a proof can be found elsewhere, we shall confine ourselves to deducing it as a par- ticular case of a more general theory. It will perhaps be as well, however, to give the compamon formula for the. angle of twist 6 of a length I of the section. The work done on the section = J T8 . And the strain energy of the section is given by the integra taken right round the mean contour of the section. G is the modulus of rigidity, and ds an element of the m contour (see Fig. 2). 970/
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