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Aviation History
1936
1936 - 0192.PDF
SUPPLEMENT TO FLIGHT Hf JANUARY 23, T936 THE AIRCRAFT ENGINEER If 1* ft'j is the ordinate representing unbalanced bending moment at I', make p </T equal to a\ b'x on another diagram E. I, . IR and draw^> h' at right angles to it to represent — —-——-=— / lr -p J-B at 1' to the scale chosen. It is naturally convenient for projection to draw p qx horizontally in line with OB, as shown in Fig. 6. Also, by projection or otherwise mark off r\ r\, etc., such that p . r's represents E . I, I, / I, I, at 2', p r'3 denotes it at 3', etc.; on the opposite side of p 9, mark vlf v.,. v3, etc., such that p vx represents to the chosen scale the value of /K — at the point 1, while p v2, p v9, etc., represent it for points 2, 3, etc., respectively. Join h\ to <7J and produce this line to meet at j\ the line through vt parallel to p qv Let the projection of qx jx on to p qt produced be qx ev as shown. This represents to the same scale as />, y, a bending moment difference ; transferring this to the other diagram, with due regard to scales, mark «', on 2'b'\ such that i'a\ is less than i'a'3 by the moment in question. This may easily be done by making a\ f, in a't 1' represent the moment, and drawing f\ a'2 parallel to OB to meet 2' b\ in a'2. Let the moment thus transferred be Cj units. Make qx q„ equal to a's b\ by producing p c/, to q2- Let the parallel to p qx through r.2 meet //', qt in //'.,. Join h'0_q„ and produce it to meet in j2 the parallel to p q2 through. i„. Let q2 e2 be the projection of q2 /., on p2 q2. This represents a moment which = C2, say. Transfer this moment as before, making the difference between the moments shown by za\ and ^a'3 equal to C2. Make qz q3 in p q2 produced represent the moment shown by fl'3 b\ and proceed as before, joining qs to /;'.. the intersection of h\ q and the parallel to p q?> through r'3. Continue a similar construction progressivelv to obtain points a\, etc., till a point (in this case a&) is obtained in the vertical for the centre line of the outermost element. If this point is such that the smooth curve through a'x, a\, etc., passes through B, when extended approxi mately in a straight line, this curve then is to a very good degree of approximation the curve of transferred bending moment, from which the actual bending moment curves may be obtained as explained in the section on " principles involved." Trial and Error If this condition is not satisfied, other series of points, such as au, a\, a'2, etc., must be found, starting with different positions of au, until one of them gives the required result. A curve of error may be plotted after two nearly correct series of point have been found. If further accuracy is needed the construction may be continued to give the last line of the series a0 a\, axa',, etc. (i.e. a line which in the case shown is to the right of a'-). IK This may be done bv using the value of — at the tip (represented by B) to give a final projection on p-qtqz . . . Let the value of the couple represented by these be C5. Then if the ordinate 5a'., denotes a bending moment greater than half C-, the lure to be drawn by the con struction would obviously pass above B, while if it is less it will pass below. If the quantities are equal the line will pass through B ; and trial lines a0 a'x . . . a's should be drawn until the condition is satisfied. When a sufficiently accurate line has been drawn for «0 a\ • • • B, the relationships explained in the section on " principles involved " may be used to give the actual bending moments, and hence the stresses in the spars. It will be seen, by reference to Fig. 4, that in the most usual case (which this figure represents) of the unbalanced ending moment curve having the same sign throughout the span, the addition or subtraction of the transferred bending moment to or from this to give the actual may be memorised by the simple rule that where the independent bending moment is greater than the balanced the trans ferred bending moment is subtracted from it to give the actual and vice versa. When there is reversal of sign of the unbalanced bending moment curve the relationship is eas.ily seen by drawing out a set of curves such as shown in Fig. 4. The stresses in the skin are given from the torque taken by the skin. This at any point is equal to the slope of the transferred bending moment curves, expressed in units of force (e.g. lb.-ins. per lb., i.e. lbs.), multiplied by the distance between the spars. If this torque is T units the shear stress in the skin at any point is equal to —7-., where A is the mean area of the and t is its thickness at the 2 At tube formed by the skin point considered. Proof of Qraphical Method • Assume that in Fig. 6 the points a„ a'x a\ ... . B are the final series satisfying the conditions of construction. Let Fig. 7 represent the right-hand construction of Fig. 6. Drop a perpendicular from h2 to pqx to meet it at g. E I„ I„ Then since p h x represents to some scale — • — for the element Oi (assuming the average value to be suffi ciently accurately shown by the value at the centre of the element) and p qx' represents to another scale the average differential bending moment in the element (being equal or proportional to «/ bx') the tangent of the angle qxhxp represents to a derivative scale the increase in the relative slope of the spars from the root to the point represented by 1. This follows from the relationship \ expressed by equation (6). ; Since the relative slope at the root, must be zero, it follows that this tangent, or the slope of h1'q1 relative to the vertical, represents to this scale the relative slope at I, assuming the transferred bending moment to be correctly represented by Oan. If we now assume that i'a\ correctly represents the transferred bending moment at 2, so that a'2 b'2 and qx q2 also represent the differential bending moment at 2 (by construction) we have :— Tangent of angle q2h.,'g = gq.,igh2 = giilsK + gigz/gV = pqiiPK' + ?rf*/£V = constant for scale x M(il H i.^2 • -IR2 where Md denotes differential bending moment and the suffixes 1 and 2 after I denote its average value in the element 0-1 and 1-2 respectively. This follows from the fact that gh2 was made equal to 1 L + L- E' I, x 1, for the point 2' (which is within sufficient accuracy equal to the average value for the element 1-2) and from the relationship of equation (6). Thus the slope of h2'q2 relative to the vertical (or tangent of the angle q2h./ q) represents to the same scale as for the slope in the first element the sum of the changes in relative slope of the spars in the elements 0-1 and'1—2 ; that is, it represents the relative slope of the spars at the point on them represented by " 2 " on the diagram. Further, it can be shown in similar fashion that, assuming the bending moments represented fl/fr/, a./b2, a3ba', etc., correctly to show the differential bending moments, the slope relative to the vertical of an - line in the series I 'E' El + la El • IR1
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