Comparison of solution to FE. note: the distance from flange edge is x in these plots while it was y in the derivation!!!

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Brandon Cobb
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1 Comparison of solution to FE note: the distane from flange edge is in these plots while it was y in the derivation!!!

2 Comparison of solution to FE

3 Comparison of solution to FE??

4 More elaborate solutions to the skin-stiffener separation problem Cohen, D. and Hyer, M.W., Calulation of skin-stiffener interfae stresses in stiffened omposite panels NASA CR 18468, 1987

5 Skin-stiffener separation: Impliations for design w flange t1 t skin M Material properties: E=137.9 GPa Ey=11.03 GPa Gy=4.86 GPa ν y =0.9 tply=0.15 mm E=11.03 GPa G=4.86 GPa Gy=3.447 GPa ν=0.9 νy=0.4 M results apply to other load ases as well!

6 Effet of flange/skin thikness ratio skin and flange have same layup: [45/-45/-45/45]n σ σ / yma 0 5 t1/t= inreasing t1/t M Distane from flange edge (y/t1) y t1 t M stress dies out within, at most, 10 flange thiknesses! impliations for design? stresses die out within 10 (or less) flange thiknesses as flange thikness inreases, peak stress inreases (BUT see net Figure)

7 Effet of flange/skin thikness ratio on peak interlaminar stress peak stress σ /σ yma avoid thikness ratios orresponding to this region Flange/skin thikness ratio (t1/t) y t1 t Flange thikness should not be lose to skin thikness!!

8 Why should we have t 1 t? free-body diagram of struture t 1 the interlaminar normal stress σ gives rise to M-M skin the higher M-M skin the higher the fore ouple reated by σ and the higher the peak value of σ

9 Why should we have t 1 t? from beam theory, t t for t t t t M t t for t t t t t t M M M skin ( )

10 Why should we have t 1 t? plot M-M skin as a funtion of t 1 /t (M-Mskin)/M Flange/skin thikness ratio t1/t M-Mskin is maimied when t 1 t!!

11 Effet of layup on skin-stiffener separation stresses σ/ σyma 0 o flange on 90 o skin [45/-45/-4545] flange and skin Distane from flange edge (y/t1) y t 1 0 dir is parallel to y t t 1 =t 0 o flange on 90 o skin is worst mismath: peak stress is twie the value of [45/-45/-45/45] flange and skin the higher the peak stress the faster the rate of deay (why?)

12 Effet of stiffness mismath peak stress σ/σyma 0 o flange on 90 o skin [45/-45/-4545] flange and skin Flange/skin thikness ratio (t1/t) the stiffness mismath almost doubles the interlaminar stresses roughly equal flange and skin thiknesses must still be avoided

13 Brief disussion on stress singularity omparisons based on the peak stress an be misleading the eat value at the flange edge is unknown; a full 3-D anisotropi elastiity solution would predit the stress is singular there flange t w y t1 skin anisotropi elastiity sol n σ M M present appro. sol n distane from flange edge

14 Brief disussion on stress singularity however, the elastiity solution is based on assuming homogeneous plies (matri and fiber are not epliitly aounted for) the strength of the singularity is weak (logarithmi or less) implying that the singularity is signifiant over distanes from the flange edge that are of the order of a few fiber diameters where the homogeneity assumption breaks down anyway; so singularity is of no onsequene in design (but still need to deide what to do with peak stresses alulated) an ombine the peak stress or some average stress or stress at a distane with an onset of delamination riterion

15 Skin/stiffener separation Summary of findings two fators tend to make the interlaminar stresses worse: flange thikness; in general, the higher it is the higher the interlaminar stresses stiffness mismath; the higher it is the higher the interlaminar stresses interlaminar stresses die out within ~10 flange thiknesses (important for suessive or staggered plydrops): h 1 h 10h 1 10h (similar onditions are used for internal plydrops)

16 Skin/stiffener separation Summary of findings in general, the flange and skin thikness should not be lose to one another one final note: if one ply is dropped, 10(tply) is only 1- mm. Plaing/dropping with suh auray in prodution is diffiult (and epensive) 1- mm (TYP)

17 Revisiting the stiffener ross-setion we have been onsidering all along require that the flange and skin layups have similar stiffnesses and different thiknesses dropped thikness rule

18 Other options for delaying skin/stiffener separation portion of skin overs flange portion of skin overs flange epensive! epensive! fasteners pins epensive! epensive!

19 5.5 Sandwih Struture M V faesheet V M N adhesive ore t N V t f faesheet V See: Plantema, F.J., Sandwih Constrution, John Wiley & Sons, NY, 1966

Sandwih struture omponents faesheet: any load-arrying omposite material preferred to have the outer-most ply as fabri to minimie damage etent due to impat layup of eah faesheet does not have to be

20 Sandwih struture omponents faesheet: any load-arrying omposite material preferred to have the outer-most ply as fabri to minimie damage etent due to impat layup of eah faesheet does not have to be symmetri (even though it would be preferred) as long as the entire sandwih layup is symmetri ore: honeyomb, foam, pins (X-or, K-or ) ; required to have suffiient flatwise strength and stiffness and two transverse shear strengths and stiffnesses adhesive: film adhesive, at least 0.08 mm thik; in some ases (e.g. X-or ) it an be omitted X-or

21 Honeyomb Core properties of importane ell sie s transverse shear moduli, G, G y transverse shear strengths F, F y out-of-plane stiffness E t out-of-plane tension and ompression strengths F t, F y deformation y deformation

22 5.5.1 Sandwih Struture by moving material away from the neutral aes, ore inreases drastially the bending stiffness of the struture=> inrease in bukling load at the same time, for ores thiker than 6-7 mm, transverse shear effets beome signifiant sandwih bending stiffness: D ij ( D ij ) f ( A ij ) f t t subsript f denotes single faesheet f (5.5.1) of ourse, one an also get Dij by defining entire laminate with ore stiffness values=0

23 Effet of ore thikness on bending stiffness of sandwih eample we saw before; faesheet: (±45)/(0/90)/(±45) faesheet thikness: mm A N/mm A N/mm A N/mm A N/mm D Nmm D Nmm D Nmm D Nmm

24 Effet of ore thikness on bending stiffness of sandwih D11 sand / D11 fae ore thikness (mm) range typially used in design

25 Everything omes at a prie the huge inrease in bending stiffness and bukling load would make the sandwih the ideal strutural element, BUT more failure modes must be aounted for; every omponent (faesheet, adhesive, ore) an fail and in more than one ways transitioning to adjaent struture (rampdowns) is not easy attahments usually require inserts whih an be epensive suseptibility to moisture absorption and freeethaw yles

26 5.5. Standard pratie for analysis N M S V V faesheet adhesive ore t f faesheet V V S M N t all loads shown are taken by the faesheet eept transverse shear S whih is taken by the ore moments are resolved as fore ouples: F fae t M t (5.5..1) f

27 Eeptions to the rule the assumption that all loads (eept for transverse shear) are arried by the faesheet is valid as long as the ore is very soft, whih is typial of most appliations; if the ore used has stiffness omparable to that of the faesheet (e.g. solid aluminum or arbon ore) the assumption is not valid

ore shear failure usually after faesheet antisymmetri wrinkling) faesheet dimpling or intra-ellular bukling (faesheet

28 5.5.3 Failure analysis panel bukling (bukling of sandwih panel as a whole) faesheet strength failure (tension, ompression, shear) faesheet wrinkling (loal bukling of faesheet on elasti foundation) shear rimping (preipitated by ore shear failure usually after faesheet antisymmetri wrinkling) faesheet dimpling or intra-ellular bukling (faesheet bukling between ell boundaries) adhesive strength failure (tension, shear) ore strength failure (tension, ompression, shear)

29 Sandwih panel bukling - ompression angle 90 o unless the ore is very thin, transverse shear effets are important! Plane setions remain plane but are no longer perpendiular to the mid-plane treat the sandwih as a wide olumn; then from [1], the bukling load (per unit width) for an isotropi beam is given by N rit N Erit kn 1 t G Erit 1. Timoshenko, SP, and Gere, JM, Theory of Elasti Stability, MGraw-Hill, NY, 1961, p 351 N Erit =Bukling load without transv shear effets k=shear orretion fator G =(ore) shear modulus (in dir of loading) t =(ore) thikness ( )

30 Note on shear orretion fator k inonsisteny between derived and assumed throughthe thikness strain distributions derived: 3 Q h 1 h / quadrati in ζ => strain γ is also quadrati in ζ ζ τ

31 Note on shear orretion fator k assumed in first order shear deformation theory: γ w Ψ P ζ w deformed u u w w independent of ζ! undeformed P ζ

32 Note on shear orretion fator k reonile inonsisteny by making sure the work done is the same for both approahes h γ deformed In general, work done: W h/ h/ d 1 st order theory h/ onst W d h/ Kirhoff quadrati distribution 3 Q h 1 shear fore Q h / 3 Q 1 h h/ 6 Q W d G 5 hg h/ h/

33 Note on shear orretion fator k 1 st order theory Kirhoff quadrati distribution W W Q 6 5 Q hg shear orretion fator k 5 5 Q Gh h 6 6 instead of Q h

34 Sandwih panel bukling - ompression for a sandwih, the through-the-thikness shear distribution is (very nearly) uniform and k 1; then, rearranging eq. ( ): why? N rit t N tg G Erit 1 N Erit for uni-aial ompression was found before in eq. (5..3.1): N Erit D 11 m 4 ( D 1 D a 66 m ) m ( AR) D ( AR) 4 ( ) ( ) with a the panel length (load // a) and Dij given by (5.5.1)

35 Sandwih panel bukling - Eample (±45)/(0/90)/ (±45) faesheet with Nome HRH-10 1/8-3.0 ore D Nmm D Nmm D Nmm D Nmm A N/mm A N/mm A 891 N/mm A N/mm faesheet properties t Core shear stiffness G LT =4.1 N/mm b a=b=508 mm a

36 Transverse shear effet on sandwih bukling load Panel bukling load (N/mm) a b no transv. shear effets 1000 with transv. shear effets Core thikness (mm) at t =3 mm the differene in bukling loads with and without shear effets is already 1%

37 Sandwih panel bukling under N y shear y N y N y N y N yrit G G G N G y yer y t t ( ) where N yer is the bukling load under shear for simply supported plate without shear orretion

38 Sandwih panel under shear for N yer an use the epression derived at the very beginning of the ourse: N yer yrit or (see Advaned Composites Design Guide, DoD/NASA, 1983) N yer D1 D 11 D D 11 D3 81D 4 b 3 a D1 D1D D1D3 D b a a D11 ( D1 D66 ) D 3 4 3a b b 81D D 4 a b 4 a b 4 a b D 1 18 D 18 D D D D a b a b a b 0.79 for 0.5 a/b<1 fator introdued to orret the epression derived by eigenvalue problem

39 Sandwih panel under shear for 0 a/b<0.5, interpolate between the value for a/b=0 given by N D D if yer D 3 1/ D1 D a D11D D D 11 D D D D D N yer D D D a D D D D if D D D D and the value for a/b=0.5 given in the previous page

40 Sandwih panel bukling under ombined loads use the interation urves we had before but orret the individual bukling loads for for transverse shear effets

41 Wrinkling (1) wrinkling is a loal bukling phenomenon where the faesheet bukles over a harateristi half-wave length l unrelated to the panel length or width a l (1) Hoff, N.J.,Mautner, S.E., The Bukling of Sandwih-Type Panels, J Aeronautial Sienes, July 1945, pp 85-97

42 Wrinkling two different failure modes: symmetri (1) antisymmetri mied modes are also possible (1) symmetry refers to the loal half-wave and is about an ais perpendiular to the plane of the panel

43 Symmetri Wrinkling N N N N A l sandwih is infinite in y diretion faesheet bukles loally over a half-wavelength l with simply supported BC s defletions vary linearly with over a portion of the ore

44 failure loalied over a short distane; broken fibers (brooming), delaminations, adhesive and ore failure Symmetri Wrinkling

45 Symmetri Wrinkling satisfies the & BC s on w assume defletions of faesheet and ative portion of the ore are given by l w A sin ( ) determine the wrinkling load by energy minimiation total energy (=potential-work done) per unit width: U f U W A ( ) where U f is energy stored in eah faesheet, U is energy stored in the ore, and W is work done by applied fore N per faesheet

46 Symmetri Wrinkling assume that deformations in the plane of the faesheet (u,v) are negligible (=> ε o =ε yo =γ yo =0) strains and stresses in the faesheet are given by ( ) where E f is the faesheet (membrane) Young s modulus appropriately alulated then, per unit width, d w EI d w I E dd w E U f f f f ( ) where I=t f3 /1 and (EI) f (D 11 ) f w E f f E w

47 Symmetri Wrinkling strains and stresses in the ore are given by u w w u 0 by previous assumption G E w G w E ( ) where E and G are ore aial and shear moduli respetively then, per unit width, 0 1 dd w G w E U ( )

48 Symmetri Wrinkling N N A l work done per faesheet (per unit width): W N 0 d dw d N N d dw ds dw 1 ds dw ds dw d 1 for 1 small d dw ds w ds for dw 1 ds small dw ds W N 0 w d = ( )

49 Symmetri Wrinkling use ( ) to alulate neessary derivatives: A w A w A w os 1 os 1 1 os ( ) substitute in ( ) and arry out the integrations ) ( 3 4 A N A G E A EI f ( )

50 Symmetri Wrinkling to determine the wrinkling load N wr, minimie the energy (differentiate Π with respet to A and set the result equal to ero): N A wr 4 0 A ( EI) Euler olumn bukling load f 3 ( EI) E f 1 E G G ontribution from beam on elasti foundation; ompare with result from setion 5.3. N K mm 0 ( ) elasti foundation ontribution when it onsists of torsional springs 4 EI kl m 4 L EI m same when m=1 and k=e /

51 Symmetri wrinkling the epression for the wrinkling load, eq. ( ) is still in terms of two unknown onstants, l and sine we are looking for the lowest bukling load, determine the value of l that minimies N wr N wr ( EI 0 E ) f 1/ 4 substituting in ( ): ( ) N wr E ( EI ) f G 3 ( )

52 Symmetri wrinkling in a similar manner, the ative portion of the ore, an be determined, N wr 0 3 / 3 E ( EI) f G 1/ 3 and substituting for I f =t f3 /1 / 3 3 t 1 f EE G f 1/ t f EE G f 1/ 3 ( ) use this epression to substitute in ( ) to get l 3 1 1/ 6 1/ 3 t f E E f G 1/ t f E E f G 1/ 3 ( )

53 Symmetri wrinkling use ( ) and ( ) to substitute in ( ) to get N wr E E G 1/ t f f per faesheet! ( ) this epression has been derived by many people making different assumptions and using different methods; typially, the only thing that hanges as the approah and assumptions hange is the oeffiient; for a good review of most of the methods, see: Ley, R.P., Lin, W., and Mbanefo, U., Faesheet Wrinkling in Sandwih Strutures, NASA/CR , January 1999

Lecture 11 Buckling of Plates and Sections

Lecture 11 Buckling of Plates and Sections Leture Bukling of lates and Setions rolem -: A simpl-supported retangular plate is sujeted to a uniaxial ompressive load N, as shown in the sketh elow. a 6 N N a) Calulate and ompare ukling oeffiients