/ 4 -DE{ Stress Concentrations at a. Cut-out. R. & M. No (14,~9) (A.R.C. Technical Repot0 MINISTRY OF SUPPLY AERONAUTICAL RESEARCH COUNCIL

Transcription

1 ...?& /"",,, a ' V'-. '.' &,q.w ~,7~,~a: u ;,~ R. & M. No (14,~9) (A.R.C. Technical Repot {, /_ e> a.<7'-:;.-~: al.%~v$t-2 MINISTRY OF SUPPLY AERONAUTICAL RESEARCH COUNCIL / 4 -DE{ REPORTS AND MEMORANDA Stress Concentrations at a in a Swept Wing Cut-out E. H. MANSFIELD, M.A. Crown CopyrigAt Reserved LONDON: HER MAJESTY'S STATIONERY OFFICE 1954 SIX SHILLINGS NET

2 Stress Concentrations at a Cut-out in a Swept Wing E. H. MaNSFIB~D, M.A. COMMUNICATED BY THE PRINCIPAL DIRECTOR OF SCIENTIFIC RESEARCH (AIR), MINISTRY OF SUPPLY _Reports and _Memoranda NO. 2823* ]g]y, I95I i I ~ ~ ~ "-DEC 198! Summary.~The stress concentrations are determined for a panel, bounded by main load-carrying members and an oblique edge, such as might occur at a cut-out in a swept wing. The solutions given are exact and cover the effects of a member along the oblique edge and of closely-spaced stringers attached to the panel. 1. Introductibn.--The determination of the stress distribution in a panel bounded by two main members and an oblique edge between the members is complicated by difficulties in satisfying the boundary conditions along the oblique edge (R. & M ). The use of oblique co-ordinates (R. & M ) does not help since these still give rise to stress functions which are not orthogonal. It can be shown, however, that the stress distribution in the immediate vicinity of either main member and the oblique edge is independent of the stress distribution elsewhere. This means fhat such a localised stress distribution may be determined by ignoring the other main member and regarding the structure as an infinite wedge, bounded on the one side by one main member and on the other by the oblique edge. Furthermore, since the peak values of the shear stress in the actual panel occur where a main member and the oblique edge meet, the peak values of the shear stress can be determined exactly, and the distribution of stress near the peak values can be determined approximately*, if the structure is treated as an infinite wedge. OBLIQUE ~, FIG. 1. Panel at cut-out in swept wing. * R.A.E. Report Structures 114, received 1st October, (61so) A

3 2. Assum])tions.--In determining the stress distribution in the vicinity of the apex of the infinite wedge the following assumptions are made. (a) stress-strain relations are linear (b) buckling does not take place " (c) rivet flexibility is negligible (d) the flexural rigidity of the main member and the oblique edge member, if any, is negligible (e) if stringers (parallel to the main member) are present their stiffening effect may be adequately represented by assuming them to be spread out into an elastic sheet with equivalent directional properties. Of these assumptions (d) is the most open to objection. 3. Plain sheet.--the analysis for the case when the sheet is not reinforced by stringers is simple and will be considered in detail. It is shown in Appendix I that the stress distribution in the immediate vicinry of the apex of the wedge is independent of the boundary conditions away from the apex; these boundary conditions may therefore be chosen to have the most convenient values to suit the analysis. They are chosen so that the stresses along the edges of the wedge are constant and equal to the values at the apex. This implies that the stress distribution in the wedge has a pattern which depends only on. See Fig. 2. \ FIG. 2. The infinite wedge. MAIN MEMBER. The most general form for the stress-function ~ which gives rise to such a stress pattern is 1 2 = ~r (al -- a2 sin 2 -- a3 cos 2 + a~o) (1) where the a's are at present arbitrary. This function determines the following set of stresses, ar = al + assin 2 -b a~ cos 2 + a~o z = al -- a~ sin 2 -- aa cos 2 + a~o (2) ~r = as cos 2 -- a~ sin 2 -- ½a4 * The values of 8a/Or and 8~/Or at a corner in the actual panel are not determined by this analysis, but if themain members are tapered so as to have constant stress characteristics it can be shown that these derivatives are zero, 2

4 3.1. Oblique Edge Free.---The conditions.along the free edge, = c, are O" = Tr z and if ~... is the direct stress in the main member the conditions at = are (~r --?'(TO ~--~ ~r,~ and by virtue of assumption (d) " ~ O. These four conditions are sufficient to determine the four constants of equation (1) and thence the complete stress distribution, which has been plotted in Figs. 6 to 1 for values of ~ the wedge angle equal to 45 deg, 6 deg, 9 deg. 12 deg and 135 deg. The peak value of the shear stress is given by z,,... = sin 2c~ -- 2~ cos 2~ ~,, (3) and the direct stress along the free edge is [ sin2c~--2~ ]^ a~,~ = sin 2~ Z 2~ cos 2~ a~,'~" (4) The peak value of the shear stress has been plotted against c~ in Fig. 11. The factors in the brackets in expressions (3) and (4) above become infinite when ~ = 129 deg and change their sense when ~ > 129 deg. The reason for this becomes clearer if the problem is considered from the more direct approach o5 applying a known shear stress ~o,,,~ to the wedge along --- and then determining ~... (= a...). The ratio 8,,... : T,o,,,~ has been plotted in Fig. 12. With the shear stress acting in the sense shown in Fig. 12 ~... is a tensile stress from deg < < 129 deg and a compressive stress for > 129 deg, which might be expected as that part of the wedge for which > 9 deg tends to act in the nature of a buffer to the applied load. In an actual construction the load is applied through a boom and equations (3) and (4) are no longer valid for ~ > 129 deg since they would then necessitate negative boom areas. In practice it cart be concluded that very high shear stresses will be developed when the wedge angle exceeds about 12 deg Oblique Edge S~pported.--When there is a member along the oblique edge and the load is applied along the line of the main member, as in Fig. 3, there will be no load in the oblique edge member at the apex and the conditions along that edge in the, simplified wedge structure will be G~---, b,,,~ I FIG. 3. Load applied along line of main member. 3 (618) 'A*

5 The four constants of equation (1) are now determined and thence the complete stress distribution, which has been plotted in Figs. 13 to 17 for values of c~ the wedge angle equal to 45 deg, 6 deg, 9 deg, 12 deg and 135 deg. The peak shear stresses occur at each edge of the wedge and are given by and... = (5) T ro, c =-- [' 4~ 2 sin '12c~ &... " (6) The factors in the brackets of equations (5) and (6) become infinite when c~ = 9 deg and change their sense when ~ > 9 deg. In an actual construction the load is applied through a boom and equations (5) and (6) are no longer valid for a > 9 deg since they would then necessitate negative boom areas. In practice flexural rigidity of the edge members will prevent very high shear stresses developing, but it can be concluded that high shear stresses are likely when the wedge angle exceeds about 8 deg Oblique edge SUlSported loads in both edge members.--if the applied load is at an angle/~ to the direction of the main member as in Fig. 4 below the stresses in the edge members are given by P sin (~ -- /3) b,,... = A,,, sin c~ P sin /~ &,,,e -- A, sin (7) FIG. 4. Load applied at angle to main member. The stress distribution is now obtained from a combination of those considered in section 3.2. In particular, ~... = 4c~ -- ~ sin 2c~ 4-c~ &r,~ (8) ~y,e = 4-c~ 2 sin 2~ b,, c~ &~," " (9) 4. Stringer-reinforced Sheet.--The stress function corresponding to equation (1) differs only in that the last term a~o becomes a~ho(o), where Ho(O) has been derived in Appendix II. This stress function determined a set of stress resultants* (~, ao,?~o) identical with that of equation (2) * Stress resultants are here defined as (the resultant force in the stiffened sheet per unit length) -- t. They therefore have the dimensions of a stress, and when there is no reinforcement the stress resultants are the actual stresses in the sheet. 4

6 except that the functions appropriate to a~ become H~(O), Hs(O), H3(O). These functions, which have been derived in Appendix II, have been tabulated (Tables 1 to 5) for different values of the stringer reinforcement parameter X Oblique Edge Free.--The peak value of the shear stress, adjacent to the main member, is given by (1 + x) sin cos + sin... (1) r,,o,,, = H,(~) cos 2c< q- H3(~) sin 2~ ~'"~ " " These values have been plotted against c~ in Fig Oblique Edge Suflported.--When there is a member along the oblique edge and the load is applied along the line of the main member, as in Fig. 3, the shear stress adjacent to the main m.2mber is given by HI(~) sin" ~ {1 + X sin" ~ (1 + ~,)(1-4- cos s ~ -- v s ins cz)}- -- H,(~) cos s c,. (1 + X sin * c~ (1-4- v)s} TrO,m q- Hd~ ) sin s cz sin 2~ X(1 -ff ~,)(cos s c~ -- v sin s ~) HI(~){1 + X sin" ~ (1 + v)(1 + cos s ~ -- v sin" ~)} (1 + x)e... sin 2cz (11) + Hdc~){1 -~ X(1 + v)[1 -- sin" c~ cos s ~ (1 + ~,)]} + H~(~) sin 2c~ X(1 + v)(cos s ~ -- ~, sin" ~) 5. Range of Validity.--The present analysis gives only the peak values of the stresses with no suggestion as to the rate at which these die away. Some indication of this rate may, however, be obtained from a consideration of Fig. 18. Fig. 18 shows contours of constant shear stress T~y in a rectangular panel with the booms tapered so as to be uniformly stressed. It will be seen that the ' -distribution ' considered in this report has an approximate range of validity extending over regions within about ½-panel width from each corner. If the booms are untapered the range of validity will be somewhat sma]ler. Further, the integral of the shear stress along each edge must equal the total load transferred to the sheet, so that in a swept panel bounded by untapered booms the greater shear stress will die away at a greater rate than the smaller shear stress. Thus we expect the range of validity of the -distributions to be increased at an acute angle and decreased at an obtuse angle. 6. Comlusions.--Exact solutions have been obtained for the stress concentrations which occur at a cut-out in a swept wing. The solutions include the effects of a member along the oblique edge and of closely spaced stringers attached to the panel. The analysis has been simplified by using the fact that the stress distribution in the immediate vicinity of either main member and the oblique edge is independent of the stress distribution elsewhere. (618) A* 2

7 LIST OF SYMBOLS ~ o- r FIG. 5. Direction of positive stresses. 4o), r,o al, a2~ 6~a~ a ~Yr (7 TrO A CZ t X ~r~ ~~ ~ro Ho(O) H (o) Suffices,,, and ~ refer to the TrO,m A~ Polar co-ordinates Stress function Arbitrary constants Direct radial stress Direct tangential stress Shear stress Direct radial stress in tension member attached to sheet Angle of wedge Offset angle of applied load Section area of member Sheet thickness Relative section area of stringer reinforcement Stringer area + (t stringer pitch) Stress resultants for reinforced sheet Stress function for reinforced sheet Stress resultants appropriate to Ho(O) main member and ed.ge member respectively ; e.g., Direct. stress in main member Shear stress in sheet adjacent to main member Section area of edge member No. Author 1 E.H. Mansfield... 2 W.S. Hemp... 3 S. Timoshenko... REFERENCES Title, etc... Elasticity of a sheet reinforced by stringers and skew ribs, with applications to swept wings. R. & M December, On the application of oblique co-ordinates to problems of plane elasticity and swept-back wings. R. & M January, Theory of Elasticity. p McGraw-Hill Book Company. 6

8 APPENDIX I To show that the Stress Distribution in the Immediate Vicinity of the Apex of a Wedge is Independent of the Boundary Conditions away from the Apex If we preclude the possibility of singularities at the apex the most general form for the stresses in the wedge (in plain sheet) can be expressed as ~ co ~=--2DoO + X r'~{(n+2)a, cos(n+2) +(n--2)b. cosn ~b= + (n +2)C. sin (n + 2) + (n -- 2)D,, sin no} co ~o = -- 2DoO -- E r"(n + 2)(A. cos (n + 2) -+- B,~ cos no + C. sin (n + 2) + D,~ sin no} oo z~ -- + Do -- E r'~((n + 2)A,, sin (n + 2) + rib,, sin no -- (n + 2)C. cos (n + 2) -- nd. cos n} The most general form for the boundary conditions expressed in terms of the stresses is (12) oo = EKv" co = E L,,r" oo Y~ M,y' (13) where the 2's are constants. co E N,,r" ~1,= Equating coefficients of powers of r in equations (12) and (13) gives sets of simultaneous equations from which A,. B,,, C,. D,, may be determined. In particular taking n = shows that Ao, Bo, Co, Do are.functions of Ko, Lo, Mo, No, and are independent of the other K, L, M, N's. Now at the apex r is zero so that Ko, L, Mo, No are the boundary values at the apex. It follows from equation (12) that the stress distribution in the immediate vicinity of the apex of a wedge is independent of the boundary conditions away from the apex. 7

9 APPENDIX II Stress F,mctions for Stringer-reinforced Sheet The sheet is reinforced by stringers of relative section area X in the Ox direction,.i.e., parallel to =. In Cartesian co-ordinates the stress-function (R. & M ) equation is where ~+ ~y~/\,~ ~+~) k,z= 1 -~ (1 -~- ~))EX-~- ~/(X -{- X2)~ l k:= 1 + (1 +,)EX- ~/(X + X~)~ / and ~ is Poisson's ratio. =o (14).. (15) If we search for a solution Of equation (14) in the form = x~f(y/.) --x~f(~), say the equation for F reduces to * (16) so that and d~f -~z a constant dz (k?z ~ + 1)(k~z ~ + 1) fxf, f~ (~,~z + 1)(k# ~ + 1)-d Fl~z j = (17) a constant F= o o (da)a" (k,2z ~-~- 1)(k~=Z 2-t- 1) -}-a+b2+c2 = (18) the three constants of integration corresponding to the three simple solutions of equation (14), namely -- ax ~ + bxy + cyl The integral of equation (18) may be integrated to give (k:z- 1 (~:~- 1 ~ (k#~+ 1 F \ ~ )tan-lh,z \ 2k~ ) tan--* k# -- ~log 1) -- \ k~z ~ + " (19) The appropriate stress function in polar co-ordinates is therefore r~i-2 ( k~ sin~kl-- c s~o)tan-l(kltano)--( k~sin~ok~- c s2 ) tan -1 (k2 tan ) --sino coso log(lq- \*2tan= )} r~ 1 + k~ = tan z -= ~ Ho(O), say. (2) 8

10 The stresses are related to the stress function by the relations ~2 ~ -- r 2r + ~~ a' 2a (21) i.e., (k~ ~ cos 2 -- sin 2 ) c7~ = k~ tan -1 (k~ tan ) (k~'~ cos2 --sin a ) -- k2 tan -1 (k2 tan O) sin2 (1 + kl'~ tan~ ) + 2 log l+ka atan so = Hi(O), say, (22) (k~ ~ sin" -- cos a ) ~o = k~ tan -~ (k~ tan ) _ ) -- k2 - tan -~ (ha tan ) sin2 (1 + k~'tana ) 2 log 1 q-k~ atan ~"~? ---= Ha(O), say, (23) ~r hi /tan -l(kltan) k2 /tan -l (ka tan ) } cos2 (1 + kl"tana ) + 2 log 1 +ka atan ~' = H~(O), say. (24) A complete set of stress resultants which are independent of r is therefore ~ = a~ + a2 sin 2 + a~ cos 2 + a4h~(o), ~o = a~ -- a~ sin 2 -- a3 cos 2 + a~h~(o), ~o = a2 cos 2 -- a~ sin 2 + a4h~(o), where the a's are arbitrary constants. (25) Tables of these H functions follow. 9

11 TABLE 1 H.functions for X =.25 o Hi(o) ~o(o) ~3(o) (deg) o "7452 1" J " q-o

12 TABLE 2 H functions for X = O. 5 o Hi(o) Ho(O) Ha(O) (deg) O O O" O " O" t. O474 t" 3682 t.533 t.5295 t"

13 TABLE 3 H functions for X =.75 o Hi(O) H~(o) H~(o) (deg) "24813 "48581 "7433 " "6264 1"1982 1"35 1" " " " "4863~ 1" " " "3353 1"3593 1" " " " " " " " ' "685 12

14 TABLE 4 H functions for X = 1. o Hi(o) H2(o) H~(o) (deg) " ' I " "63 --1" "288 -t-" t" t" t" "

15 TABLE 5 H functions for X = 1.5 o ~1(o) H2(o) gdo) (deg) " "9963 1"9473 1" " " " " "6797 1" " " " " " " , " " t to t" " t"29822 to ,

16 ~ \ ~ o o o ~ ~ \ \~oo oo --2 '2,4- -~ 6 1. Fro. 6. Stress distribution in the neighbourhood of a 45-deg corner (constant-stress edge member along = deg; free edge along = 45 deg). 6 -"4 -'~.2 '4 O'G.8 1, FIG. 7. Stress distribution in the neighbourhood of a 6-deg corner (constant-stress edge member along ---= deg; free edge along = 6 deg). 15

17 ~ ~o i O -,6 -" ,4 OB OB l,o FIG. 8. Stress distribution in the neighbourhood of a 9-deg corner (constant-stress edge member along = deg; free edge along = 9 deg) Y i 2 FIG. 9. Stress distribution ill the neighbourhood of a 12-deg corner (constant-stress edge member along = deg; free edge along = 12 deg). 16

18 lt~ FIG. IO. -~,-4-3 -~ -I o ~ Stress distribution in the neighbourhood of a 135-deg corner (constant-stress edge member along = deg; free edge along = 135 deg). 17

19 2- ~= 1.5\ 1.5 g = -75, X = -5. X=-2~ X=O,, / ~ 5 6" 7" 8 ~ I" 1t" 12 IBO Fro. 11. Variation of ~=o,,,,/~... with c~ and X. I, " I / A 3.~.,-'/ = t 5 "1: 3o 4 8o 6o. 7o. 8 9 ~( 1" tto 12 ~ ~ 15 -t FIG. 12. Variation of 6... /z=o,,, with g and X. 18

20 45" ' -O * de -~ "8 1. FIG. 13. Stress distribution in the neighbourhood of a 45-deg corner (constant-stress edge member along = deg; stiff edge member along = 45 deg) '8 O-Z "4.6 ~8 1. Fro. 14. Stress distribution in the neighbourhood of a 6-deg corner (constant-stress edge member along ---- deg; stiff edge member along deg). 19 (618:) A*~*

21 9# ~,.," " o O-~t 4 O.G Fm. 15. Stress distribution in the neighbourhood of a 9-deg corner {unit shear applied along edge = deg; stiff edge member along = 9 deg). -o4 -o.~ o o.2.4 o~ o.~ 1. FIG. 16. Stress distribution in the neighbourhood of a 12-deg corner (constant-stress edge member along = deg; stiff edge member along = 12 deg). 2

22 o 9 FIG. 17. O4 OZ O v'~ u'a Stress distribution in the neighbourhood of a 135-deg corner (constant-stress edge member along = deg; stiff edge member along deg). m I.O.o.J / j ) )1.O-I- O. FIG. 18. Contours of constant z~y/~,,,~ for a rectangular panel, shox~_ng the approximate range of validity of the -distributions. 21 (618) Wt. 17/68 I{,9 9]54 Hw. PRINTED IN GREAT BRITAIN

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