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Aviation History
1934
1934 - 1433.PDF
45 JUNE 21, 1934 THE AIRCRAFT ENGINEER SUPPLEMENT TO FLIGHT need for a more practical account of the method. This paper is intended to be such an outline. Although the chief applications of the method are to be found in the design of Aircraft Structures, a sufficient number of cases arise in general engineering to make it worthy of study by others who are not aeronautical men. Tke great advantage of the polar diagram is that it replaces complex and laborious mathematical analysis by simple graphical constructions. During research on beams under combined lateral and compressive end load, it was recognised that if the bending moment diagrams previously plotted on a straight base were plotted as polar diagrams about a pole, great simplifications in the nature of the curves involved would result. The complicated functions of sines and cosines represented graphically on a straight base by sinuous curves could be represented in a polar diagram by arcs of circles amenable to elementary geometrical methods. In the first place, we must show how to define a point on a beam by an angle. This is done quite simply as follows :— Take some arbitrary point on the beam, the midpoint for example, and assume some constant fi. To start with, /J. will be assumed to be the same for every point on the beam. Then, if x is the distance from the assumed point to any other point which has to be specified, fix is the angle to this point, and if 2a is the length of the beam the angle 2 \M represents the length 2a. Fig. 1 explains this. A FIG.I • < Y j i X III — \ A 11/ >\ B 7A / MOMENTS POSITIVE IN THIS DIRECTION Sign Convention. Take any line OX with 0 as pole. This represents the arbitrary origin e.g., mid-point of beam). To each side of OX draw the angles AOX and BOX. OA and OB represent the ends A and B of the beam, and any other radius vector OY at an angle fix with OX, represents the point Y distant x from X on the beam. Distances and angles to the right of OX are considered positive ; those to the left negative. The following notation will be used :— 2a = length of beam. I = moment of inertia of beam. P = compressive end load. w = distributed load per unit run. W = concentrated load. E = Young's Modulus. / a = ix a (called angle of beam). MA, MB = end moments. In the case of single bays where I is constant throughout the bay a must always be less than (or 90°). We can proceed now to the various cases. CASE I. Beam with end moments MA, MB and end load P ; no lateral load. First calculate p and a. Set out the angle AOB equal to 2a. (See Fig. 2a). Along OA and OB measure off to some scale MA and MB respectively. The sign convention for moments is indicated in Fig. 1. In Fig. 2a, both moments are assumed positive. Through A and B draw perpendiculars to OA and OB respec- tively to meet in Y. Join OY. On OY as diameter draw the circle OAYB. Polar Diagram for Beam subjected to End Load and End Moments only. The point Y at the end of the diameter OY will be referred to as the vertex of the circle. The bisector OX of the angle AOB may be inserted to represent the mid-point of the beam. From this line, angles can be measured to any other point on the spar. The radius vector at any given angle to OX drawn from O to the circle OAYB equals, to the scale of the diagram, the bending moment at the corresponding point of the beam. Fig. 2b shows the form of the diagram when MA is positive and MB negative. CASE II. Beam, with or without end moments MA and MB, with end load P, and uniform distributed load w. This case must be introduced by a more complete sign convention than was required for Case I. Sign Convention.—Distance x along beam positive to the right. Deflection y of any point on beam positive upwards. Slope i of beam positive if upwards to the right. POSITIVE EXTERNM ^ MOMENT „ MB POSITIVE EXTERNAL MOMENT FIG.3 tt t tmtt11 tiT ' UT/unil- run Sign Convention.
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