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Aviation History
1935
1935 - 0944.PDF
SUPPLEMENT TO FLIGHT 44 8/ APRIL 25, 1935 THE AIRCRAFT ENGINEER CONTINUOUS BEAMS The Graphical Solution of the Most General Problems By J. HANSON, B.Sc, D.I.C. Section 1.—Introduction SOLUTIONS of the problem of the laterally loaded beam continuous over more than two supports are generally dependent on the Theorem of Three Moments, which is associated with the names of Clapeyron, Bertot and Bresse. Solutions of the more complicated problem of the laterally loaded Continuous beam with end load were apparently first given by H. Booth and H. Bolas (1915) in England, and by H. Muller-Breslau (1915) in Germany. The work of Booth and Bolas was simplified and developed by Arthur Berry (1916), and in this form is at present the best-known procedure for the solution of continuous beam problems in aeronautics. The Berry Method is applicable only when in each of two adjacent sections of a beam, the lateral loading is uniformly distributed and the end load in the beam and its moment of inertia are constant. Thus, as well as points of support, where the end load usually changes in value, points at which isolated loads occur, points at which the uniformly distributed load changes in value and points at which the moment of inertia changes are all points of discontinuity, and the Theorem of Three Moments needs to be applied vto each pair of sections separated by such a point of dis continuity. In many practical examples the numerous points of discontinuity introduce an unwieldy multiplicity of simultaneous equations. Moreover, in many examples there is a continuously varying distributed loading and a continuously varying moment of inertia. In such examples, if the Berry procedure is to be adopted, the varying dis tributed loading and the varying moment of inertia have to be replaced, over short lengths of the beam, by constant values which are estimated to be equivalent. < X—• C Ik 1 i 1 » i i i F_KL± k 1 A more recent method of dealing with problems on continuous beams with end loads is the Polar Diagram Method due to H. B Howard (1928). This is a very elegant method which can be applied to obtain the essential Three Moments Equations, and from which, as the only real points of discontinuity are the points of support, the large number of simultaneous equations occurring in the Berry Method on account of changes in any of the variables are eliminated. As in the Berry procedure, however, continuous variations of distributed load and moment of inertia have to be represented by " steps " over which these parameters have constant values ; if the variations are great, then the number of " constant value steps " will have to be large to give a close approximation, and the diagrams will become complicated. Moreover, the method is not applicable at all when the end loads are tensile. My object in writing this note is to draw attention to a method in which the only points of discontinuity are the support points, in which continuous variations of lateral loading and moment of inertia are treated as such, and which is equally applicable to cases involving tensile and compressive end loads. This method, which, like previous methods, leads to Three Moment Equations, is based upon the successive * Mr. Hanson began his studirs under Dr. H. Roxbee Cox in the Department of Aecorrutics nt Imperial College. He is now engaged in the Ex.perimental Section of lb» Marine Aircraft Experimental Establishment at Felixstowe. approximation solution of differential equations due to E. Picard, and is put forward by H. Roxbee Cox in the form of Appendix III to R. and M. 1507 (1932). As R. and M. 1507 deals with the torsional distortion of aero plane wings, the enquirer into the theory and calculation of continuous beams is quite likely to miss the brief appended note on a method which is applicable to the most complicated form of continuous beam problem and which is very straightforward to apply. The remainder of this article is in three sections. § 2 gives the notation used, § 3 is devoted to the analysis by which the method of procedure is evolved, and § 4 describes the method of procedure in detail with reference to an example. There is no need to master the mathematics of § 3 before applying the method, so that the reader whose interests are mainly utilitarian may omit § 3. Section 2.—Notation Referring where necessary to Fig. i Let C, B, = Any two adjacent supports of a continuous beam, distant I apart ; # =?= distance of any point in CB from C ; y == deflection at x, positive upwards ; I = moment of inertia of beam section at x ; w = lateral loading per unit run at x, positive upwards ; P = constant end load in CB, positive when compressive ; M = bending moment at x, positive when the beam is concave upwards ; Y = shear at x, positive when M increases positively with x. Section 3.—Analytical Basis of Method The analysis here given is not in essentials different from that of Roxbee Cox in R. and M. 1507. The bending moment at any point of CB is M = - py + r \\>(dxy- + Y*-* J 0 J 0 M. .. (1) d"y Differentiating twice, and remembering that M = EI TJJ •• •• (2) .. (3) __==__M + . It is usual to write /J.2 El EI where |P| is the numerical value of the end load, so that when P is positive (i.e., compressive), d2M = — /^'M + to dx2 and when P is negative (i.e., tensile), = fi'M + CD . . dx2 We proceed to the solution of dm T)~ M -j- CU . . dx2 (4«) (5) where JJ2 can at will represent p2 or — ju.2, and is a function of x. By double integration of (5) M = ["J'tmi&l* + (" (^ dx)2 -f- Yr.x + V-4 (6> Jo.'i J 0 J 0 In the process of solving this equation by successive approximations, any value of M may be taken as a starting
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