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Aviation History
1932
1932 - 0504.PDF
36 SOTPLBMENT TO FLIGHT MAY 27, 1982 THE AIRCRAFT ENGINEER passing through the centre of area and normal to HK. Hence, *-«i (18) We have assumed in deriving the stress equation (2) or (14) that the section rotates through an angle 8<p. The deflection of the load is therefore b-8p or = A = b-^ where A is the deflection. Taking the value of 8^> given by (13) we get, E tan pni and substituting for pm from (14) we obtain Wfc» A = 2TCES y-Sa (19) If we adopt the approximate expression for 8^> given by (16) the corresponding deflection is which, using the corresponding equation (17) becomes W6»R " 27TEI V ' i F a • W/////, , :IG.7 4 C > r D2 , o, Di R I 1 „ '///////// >Y///?/M b * t ' t As an example of the foregoing let us calculate the maximum stress and deflection of a ring having a rect angular cross-section and dimensions as in Fig. 7. Since the section is symmetrical, OjO will pass through the centre of area hence, :*•- [ 3\8 8 K dr R + = — <* lOBo 12 Rn- also tan (im = 2R-6 inserting these expressions in equation (14) we get t W&&gt; 2R Pm = 27f-rtog, -I R- 6W6 ' TCP(2R - fc)log 3W(D2 - Dx) 2R + b 2R - b ^Dj log, D, (21) Using the above value of S -8r in equation (19) we get the deflection W62 R + 2TCE—rtege 12 B »-| 6W62 7iE«3 log,. D, (22) If we neglect the variation of r in the section the corresponding approximate expressions for maximum stress and deflection are easily obtained from equations (18) and (20) which in the special case under considera tion become 3W A =^r . . (23) A,. = 7TP 6W6R TtEJ3 (24) For the example of Fig. 7 it is possible to compare the results with the stress and deflection calculated by the usual flat plate formulae. Although this may be regarded as an extreme case the agreement is fairly good for a circular plate in which the outer diameter is twice the inner diameter, but the agreement improves as the ring narrows or the section thickens. The circumferential stress at radius r resulting from the flat plate theory is 3W p° = 2 %mF t (m+ l)logr — \(m — 1) + 1 f f calog c — a'log a (m — 1) a2 (m + 1) a*? log (a8 - c») e c_ 2(w +1) a (w+1) (25) where M is the reciprocal of Poisson's ratio and may be taken as «=, c and a are the inner and outer radii of the ring. The stress is a maximum when r — r, and for a ring in which C = 1, a = 2. pg = 1-48 W The corresponding expression for the deflection is given by 6W(m2-l)P( 3m + 1 a.2 , c r\ A = : log - + og T. -r1 TCE*3 m2 |_4| 2(m + l) a*-C« 8 a 8el m + 1 a?c2 a a m — 1 a? — <? c r , , «2c2 a (3m + 1) + 1 log - ^ ^tf-c? & c 8(m + 3*] The maximum deflection is at r = c, hence putting r = 1, a = 2 we get the deflection 2-68 W E<8 Expressions (21) and (22) give for the maximum stress and deflection of the same plate W P» = 1-375 W A = 2-75—- Er» 468rf
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