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Aviation History
1938
1938 - 0268.PDF
SUPPLEMENT TO FLIGHT JANUARY 27, 1938 THE AIRCRAFT ENGINEER RESULTANT STRESS TANGENTIAL FIG.I FIG.2 FIG.5 -G,^ = o .I- (5) Also, from the equilibrium conditions of the axial stresses at the elemental boundary prisms, the boundary condition (that the resultant cross-sectional stresses in the xy plane are tangential, (see Fig. 4) is = o (6) In order to arrive at equations similar to those of the _ torsion theory for isotropic bodies let (7)y/G which in effect means a change of co-ordinates such that the aeolotropic, now transformed, cross-sections behave like isotropic cross-sections. Substituting (7) in (5) and (6), equations (5a) and (6a) are obtained, exhibiting clearly the aimed at simplification. (5a) tr, = o . . (6a) The expression of the torsional moment, i.e., the integral of the moment of the cross-sectional stresses is M = — pvx Jdxdy, substituting (2) and (3) Substituting x^y-fa for xyG^G, from (7) „ f If liv . , J 3 — G\ \\ Viz xiz— \dx-,dVi — Gtol,, .. (8a) Thus we see that the " Torsion Formulae " for aeolo- tropic material (5a), (6a), (8a) show, by their similarity to those for isotropic material, that an aeolotropic bar of cross-sectional dimensions b, c, behaves exactly like an isotropic bar of dimensions fc, = b—zr-', c, = c- _ and modulus of rigidity G = i/Gx y/G „. This means that the aeolotropic bar has, for given torsional couple M, the same angle of twist wl and the axial displacement w. The stresses, however, are (like the co-ordinates) G= '• P -\ = G This applies to an ' shape of cross-section and, therefore, the available formula for isotropic materials can be used, after making the above transformations. An experimental determination of G* and G „ has been carried out on " Jablo " improved laminated wood. The specimens used in the experiment were all of length / = gin. and rectangular cross-section b = .8in., c = .27m. approxi- mately. The torsion-test apparatus was of ordinary type, giving sufficiently accurate readings of angle of twist 8 and torsional couple M. The tests were carried out several times and mean values taken. It is essential to have for each test two specimens, one with lamination parallel to b and one with the lamination perpendicular to b (see Fig. 5). For both, the couple M is plotted against angle of twist 0 and the straight line parts of the curves used in the calculation of Gj. and G „. The solution of equations (5a) and (6a) has been accom- plished very thoroughly by St. Venant with a complete tabulation of numerical results from which he has derived the following sufficiently exact approximative formulre for (- < < 1 V b .(1 -• / M"l 21 TV ~ c /G,\ •2IWG;) (9') \e"bcaj The solution of these equations is simplified by putting M7 e'bc3 ' 0"bc*
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