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Aviation History
1939
1939 - 1739.PDF
SUPPLEMENT TO FLIGHT 576d 20 THE AIRCRAFT ENGINEER JUNE I, 1939 . . calculated stiffness giving a percentage ratio ; of 07 per cent. s & v 6 measured stiffness ' r Torsional Stiffness (Covered Framework) : When torque is applied to the framework the ply sheets develop diagonal tension fields as indicated in Fig. 4. The strain energy of one side beam and one sheet will be considered. The side beam bends as a cantilever under load F and constraining moment Q supplied by the skin covering. This constraint prevents the twisting of the end rib. The cantilever deflects, but the direction of the plane of the end rib does not change. The bending moment on the beam is represented by the equation rf2y EI-> = F(*-,)- G in which the end conditions of the beam are o and x — I. (6) —• = o when x ax On integrating (6) and inserting the end conditions it ¥1 is found that Q = — 2 Strain Energy of Side Beam. U,= 2EI F 2/3 substituting Q = 3 Fl + QH - FQ/2 3 F 2/ 5GA (2a) Ux F2/3 24EI Strain Energy of Sheet. The 3 F2/ 5GA deflection of where a diagonal , p; tension field sheet with rigid boundary girders is — Ex dt is the shear load, is the length of the sheet, is the width of the sheet, is the thickness of the sheet, and E, is the elastic constant of the field tension diagonals. along the Strain Energy of Sheet 4 PI AE^ 'It 2 2P2/ AE^ •where A is a constant representing the effect of non-uni formity of the tension fields due to the non-rigidity of the boundary girders. It is unfortunate that the introduction of this constant A is necessary, since it will be observed that a standard framework of non-rigid boundary girders will deflect different amounts under the influence of tension diagonal fields of different E s. By the above approximate analysis a different A is required for each tension field. The problem is stated completely by means of differential equations in the paper on "Plane Panel Frames of Very Thin Sheet" by H. Wagner. These equations have been solved by approximate methods. Wagner's analysis shows that with non-rigid boundary girders only part of the panel is stressed and acts as diagonal ties. The use of AEj is an attempt to represent this. .•. Total Strain Energy of one side beam and one sheet is U^U. + TJ, FH3 3 F2/ 2P2/ U = + 4 24EI 5 GA AFLydt The externa] torque acting on the box wing test specimen is T = \\d f Ph where W is the load on the side beam and P the load on the panels T — W'd that is P = so that h — d (7) (8) dP 7w The tension field exerts end loads on the boundary girders so that at distance x from the stiff rib, the end load" Px on one boundary girder is —- as shown in Fig. 4. The d Px boundary girder adjacent also has an end load — in the d opposite distance direction, from the other sheet. Therefore at x, the side beam comprising these adjacent boundary girders has a moment on it of Ph d That is, the at its end of side beam may be supposed to have a load Ph opposite in direction to W. This is represented by d so that and W Ph\ d ) F dF -(,w-I) (9) (10) rfW 'It is to be noted that the side beams have additional end loads downwards of P, since they act as supports to the cross beam which is loaded with components from the tension field. The strain energy in the side beams due to these loads P will be neglected in this discussion. The analysis of the Strain Energy of a Cantilever with load F and end constraining moment Q has been performed above. (See equation (2a).) By minimum strain energy j&U _ MJj dF_ ill, 3\V- 0F "dW+~W . F/3 **?-. 6Fl_ dF_ dW + s GA ' dW &>>(*&>]- ' 12EI substituting (7), (8), (9) W and T is found, viz. \5 GA J T HE***.. The deflection of the side beam is given by t>U - / I3 6 I \ 8 = — = F ( A ) iF \12EI 5GA/ (11) and (10) in (11) a relation between 4/ I3 2 6dEl + d .. (12) where F == I 2W -1; d/ In the case of tension fields at 450 to the grain, that is in specimens 2 and 3, the value of the symbols in expression (12) are :— E GjB = A I A = 1.75 X io4 lb./in.* = .50 x 10* lb./in.* 135 X io6 lb./in.* = .325 ins.2 = 5.8 ins.4 — -753 f°r this particular frame only. W 8500 , 305 0 T" [265-5 IO« + I9Q-5" AE,_ 104 ' AE» from which W = 93 lb. and F = 61 lb. .". 8 = F(2i25) = .13 ins. calculated deflection 8 = io* = .128—.132 measured deflection. This exact agreement between calculated deflection and measured deflection is factitious in the foregoing analysis. But a check may be performed by using Batho's method for thin walled sections in conjunction with bending theory. That is, the box wing test specimen is regarded as resisting torsion partly by twisting as a tube and partly by opposite bending of the side beams. The shear constant AE„ G used in Batho's formula is replaced by where AE, is the elastic constant of the field along the tension diagonals. A correction to take into account the composite material must be made in the matter of the ply webs, however. Here T = Q + W.d (13) where Q is the torque resisted by the tube and W is the load on the side beams. Q = T - Wd so that ° d\X Strain Energy of Tube. where K =
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