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Aviation History
1934
1934 - 1443.PDF
JULY 26, 1934 65 THE AIRCRAFT ENGINEER SUPPLEMENT TO FLIGHT 766^ This does not mean, however, that a change of end load in the bay can be dealt with graphically by the polar diagram. Such a change cannot be allowed for by pure graphics ; if it is necessary to take account of it recourse must be had to a combination of analytical and graphical methods outside the scope of this article. When the end loads in the various parts of the beam do not differ greatly it is usually sufficient to use the mean end load throughout the bay. It is, of course, evident that in no case is the method applicable to a tensile end load. The angles a would, for negative values of P become imaginary, and therefore could not be drawn. A sudden change in material along the span can be dealt with quite easily by exactly the same construction as for a change in moment of inertia. In fact all the observations which have been made, or will be made about the latter, apply equally to a change in the modulus of elasticity. With the preceding proviso we proceed to an example. ' Example (5) A beam is subjected to the system of loading shown in Fig. 13. It will be noticed that there is a change in the '. moment of inertia 40 in. from the end A, and 25 in. from the end B. In addition there is a change in the distributed loading and a concentrated load. It will be found convenient in this and all complicated cases to tabulate the necessary quantities as follows :— A X Y Z W B 1 in.4 .. Plb. .. fi . . . . W lb./in. w a •. W 1.2 9,5°° .000264 .01625 13 49,250 37.25 deg. • 9 9,500 .000352 .01875 13 36.95O 43 deg. •9 9,500 .000352 .01875 10.5 29,850 215 deg. 1 <• • 9 9.5°° .000352 .01875 10.5 29,850 26.85 deg. 60 1.2 9,500 .000264 .01625 10.5 39,800 23.3 deg. The various vertices for the parts AX, XY, YZ, ZW and WB will be denoted by Xt, X2, X3, X4 and X5. Having drawn all the radial dividing and bounding lines and arcs of radius —j the end moments may be set orf. This gives the points A! and Br On the normal to OA at Aj any assumed vertex X, for AX taken, from which by the construction for change in moment of inertia, we arrive at the vertex X8' for XY, X3[ is displaced from X2> parallel to OY by an amount —£¥- ~ in the positive direction. . r* While X4l is displaced from X.,1 normal to OZ in the positive direction. Having arrived at X4J the construction for moment of inertia is again performed, and, by drawing a line through X6] parallel to the final adjusted locus EF meeting the normal to OB at B1 in X6; we obtain the correct vertex X5 for WB. Starting now from OB with the correct X5 the con- struction is repeated in the reverse direction, and the remaining correct vertices found. A very interesting special case arises if a change of shear, due to a concentrated lateral load, occurs at the same place as the change in moment of inertia, or more generally the change in the quantity ft. It becomes, in this cxse impossible to say from the general construction which is the correct value of /x to take at the discontinuity. An obvious way of overcoming the difficulty, is to separate the two changes by a small distance along the spar. If this is done, there will be no appreciable change in the diagram, and the ordinary methods previously described can be used. It is much better, however, not to introduce more different points of change than are necessary : a large number of changes calls for much more careful drawing and sometimes repeat diagrams to improve the accuracy of the first attempt. Consider in particular the case in which a concentrated lateral load W occurs at X where the moment of inertia of the beam changes from I, to I2 and, consequently, ix changes from /xt to it2. It is easily seen that a change in the magnitude of the distributed loading has no influence on the solution to the problem. The difficulty is this : are we to use fj.l or fi,2 in calculating w — for use in the construction at the point X ? /" Since the solution in the general case is unique, it is evident that there can be no ambiguity in the solution for this special case, because it is simply the limiting case arrived at by reducing the distance, originally finite, between W and the change in I until it is zero. Assuming that I always changes at X there are two ways of looking at this apparent anomaly. The case may lie assumed to be either the result of moving a concentrated load W initially a small distance to the left of X up to Xr or, the result of moving a concentrated load W initially a small distance to the right of X back to X,. Looking at the problem from this point of view, we infer that it is the sequence of the construction for the load \V and the change in moment of inertia which decides the value of p to be taken. If in our construction we are crossing a dividing line from a part A where n = 11, to a part H where n = ii2 we should expect the following rule to hold. If the construction for W is performed first, the value of (i for the part A should be used (viz., p,) to calculate thew value of — : if the construction for W is performed after the construction for the change in moment of inertia, the value of y. for the part B should be used (viz., ii2) to w.calculate the value of — These rules may be proved if the reader so desires by considering all the sequences which can be obtained by combining the different values of /x with the alternative sequences oi the constructions. ' It is then easily proved that the rule gives the only sequences compatible with a unique and consistent solution when the constructions have been performed from both ends of the beam as described above. It appears unnecessary to include a proof in the text of this article The descriptions will enable anyone to use the method for any single bay beam without an understanding of the principles underlying it. Once the reader is conversant with the method he will gradually appreciate its mechanism and will realise that every construction has for its basis a known change in shear at each discontinuity in the bending moment diagram. In the case of a concentrated load W there is a change W in the shear : in all other cases there is no change in shear By stating whichever condition is applicable in any case the reader should have no difficult ' in proving by simple geometry the construction given. THE INSTITUTE OF METALS ; This year's Annual Autumn Meeting ot the Institute oi Metals will be held in Manchester from September 3 to Septem- ber 6. The thirteenth Autumn Leclure will be delivered by Dr. J. L. Haughton on September 3, the subject being "The Work of Walter Kosenbain." Of particular interest to air- craft engineers should be the paper on " The Influence ol Pickling on the Fatigue Strength of Duralumin," by H. Suttos: and W. J. Taylor. Those wishing to take part in the proceed- ings should communicate with the Secretary of the Institute of Metals, 36, Victoria Street, London, S.W.I.
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