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Aviation History
1931
1931 - 1160.PDF
SUPPLEMENT TO FLIGHT 76 THE AIRCRAFT ENGINEER OCTOBER 30, 1931 TORSION IN THIN CYLINDERS. By E. H. ATKIS, B.Sc. (LOND).* (Concluded from p. 72) (iii) The Stresses in multiply-connected Cress Sections. The following theory, as will be seen, can be extended to sections of any degree of connectedness, but to avoid too " highbrow " a notation, the quadruply connected section of Fig. 5 will be considered. The significance of the symbols used is indicated on the diagram. To calculate the torsional resistance of the section, take moments of the resultant shear tractions all round the section about some point 0. This part of the integral is therefore, where AB X c is the area of the section B X C of the complete section. Forming all such terms for the sections, and adding them together, we have If _j_ Vp Uy" l If Vir I XL" -~H ' AB X° ^ —4 " AA<: + ' 2 AAB XL? i vrp XL" • VLl XL" L_ XL" 1 2 I * a . I 4 "^ * 9 . . T3~TId. Areas contained by $•,,<&.$3 &y4 denoted by A, A2 A3 & A+ respectively. CONNECTED SECTION FIG.5. There is a certain degree of indefiniteness at points such as A, B, D, C, where the walls meet, but as they are assumed thin, no great error can arise. We may write therefore —- (Area circuit AB x C) -\ -^ (Area circuit B x CD) •j (Area circuit ABD) H (Area circuit ADC). Writing AY, for compactness, in place of " Area circuit AB x C," and so on, the complete expression for the torque becomes, T = G6[2Y4Ad 4 2Y3A3 4 2 Y2A2 - 2 YXAX 4 Y,AY2 4 Y2AY2 4 Y3AY3 4 YAYJ (A) It is quite easy to see that equations (1) to (3) are to be derived as simple cases of (A). In the general case, however, equation (A), containing as it does four out of the five quanti- ties 6, Yj, Y2, Y3, Y4, at present undetermined, is not sufficient to solve our problem. Hence three more relation- ships between these quantities must be found. (Any one of the four quantities Y,, Y2, Y3, Y4 may be assigned the value 0, and can therefore be considered as known.) It is obvious that the integral -~ ds taken right round any boundary must be zero, because the relative displacement of any point in relation to itself is of necessity zero. This integral may also be written andTo do this, we remember the expressions given above for and, remembering the relationships between <^, the components of shear parallel to the x and y axes, viz.:— we mf)y also write dV.dx I^—, — G0 -IT- . Hence the torque T is at once given by y + -v~ §y ) ds 3Y DY\x —- + y —- dxdy. dx ciy I T= —' By Green's Theorem, we may write or ] dn dx ds 4 pds \: = -G6 f Y dr>_' ds 4 2 dxdy. The double integral is to be taken all over the section, and the single integral all round the boundaries of the section. 8w is measured positively outwards. Evaluate the single integral first: for the external boundary Ti, the corresponding part of J 0™ where p is the perpendicular from the origin to a tangent to the boundary. The first integral is easily expressed in terms of the boundary \ir \T/> values of Y by writing — 2, etc., as the approximate 1 <3VF i.value of —, hence similar terms. f d (T ) f drY • ds becomes V, \ r dx = - 2 T^A, ; J dn J dr> (see Fig. 5 for notation) for the internal boundary Y2 is added the term 2^^s, hence the single integral may be written 2 G8 (YtA4 + T3A3 4 Y.A, - Y^J. Now to evaluate the double integral: the contribution of the part, say, B x C, may be written down if it is assumed that the mean value of Y for this part may be taken. The second integral is evidently twice the area enclosed by the contour, therefore the complete expression may be equated to zero thus :— 1- 2A» = 0 (B> • Mr. Atkin is on the Technical Staff of A. V. Roe & Co. Ltd., at Manchester. Each boundary gives us one equation of this form. It would appear at first sight that Y1( Y8, Y3, Y4 can be determined from four equations such as (B). This is, of course, true, but, owing to the approximate nature ol the solution to the problem, it is sufficient to put 11 tiie constant of the external boundary, equal to 0, and evaluate the other three constants from the equations corresponding to the internal boundaries. 1086 <*
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