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Aviation History
1938
1938 - 0270.PDF
SUPPLEMENT TOFLIGHT 96/ JANUARY 27, 1938 THE AIRCRAFT ENGINEER AERODYNAMIC SHEAR FORCE and BENDING MOMENT in WINGS Simple Formulce Which Eliminate the Necessity for Graphical :s. i....-.-: Integration of Span-loading Curves By C. F. TOMS* Introduction •-•••I N order to stress an aeroplane wing, two of the quantitiesrequired to be known are the shear force and bending moment at any point, due to the aerodynamic loadson the wing. The current method of estimating these quantities isto integrate, graphically, the span-loading curve (a plot of /CLC\(~r—) aSamst yls> where CL and C are the local lift co- efficient and chord, y and s are defined hereu rider) from the tip of the wing, into the point in question, multiplying the result by appropriate factors to allow for scale, speed, etc., and then to repeat the process, with the curve ob- tained by the above operation. The first operation gives the shear force diagram, the second the bending moment diagram. This process is rather tedious, however, and liable to cumulative error. Therefore, an analytical method has been sought. FIG.I • PAIRED CURVE USED IN PRACTICE •CURVE REPRESENTED BY EON:(I.) [lorz method) •7 -8 1-0 This paper develops two simple groups of coefficients, called respectively, Shear force coefficient (S.F )c, and Bending moment coefficients (B.M.)c. The span-loading curve ior an aerofoil is given by theequation :— *"* i - CLC ——= A, sin HA3 sin 56 + A, sin 58 + A, sin 76 (1)where :— C = chord at point considered Co = chord at axis of symmetry of aerofoil. The substitution made is :— y = — s COS 8 where :— y — dist. of point considered from axis of symmetry s = semi-span of aerofoil, so that 8 varies from JT/2 to w, across the starboard half of the aerofoil. •Mr. Toms is on the Technical Stafl ol the Bristol Aeroplane Co., Ltd. It must be noted here that it may be unsatisfactory to use these formula; with the coefficients A,, etc., in (I) (more especially in the case of twisted, e.g. flapped wings), if the said coefficients have been calculated to conform with the characteristics of the wing at a few points only. This is because the curve obtained by smoothly joining the values given by (I) at the particular points at which the coefficients A1; etc., apply, is not identical with the locus of all points along the semi-span represented by (I). This is graphically explained in Fig. 1. '2 •1 \ / \ / \ \ N N FIG.2. \ > \ •2 -3 •6 7 8 -9 1-0 As the number of points along the semi-span with which the coefficients A,, etc., are made to conform increases, the magnitude of this variation decreases. In the course of a calculation by the Lotz method, ten such points are satisfied, and the variation is quite small. The points are :— y/s 0° 0 90 .156 99 •3°9 108 •454 117 .588 126 .707 135 .809 144 .891 153 .951 162 .988 171 and these have been retained as the points at which to make the arithmetical substitutions. (a) Aerodynamic Shear Force The aerodynamic shear force at a point distant as from the axis of symmetry is given by :— S.F. = s — 2 (i) may be written 2' -.sin Substituting this in (2) we get, putting 8 at yjs = o, equal to 0O:— S.F.. s.C0.-".V>J « ft- ij S _, sin (2»- '16. —(4)
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