NASA NACA-TN-2289-1951 Elastic constants for corrugated-core sandwich plates《波状中心夹层板的弹性常数》.pdf

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1、C_0_ oNAC;_rN22o9/_,_c,9 7/2._ _7NATIONAL ADVISORY COMMITTEEFOR AERONAUTICSTECHNICAL NOTE 2289/ J .ELASTIC CONSTANTS FOR CORRUGATED-CORNSANDWICH PLATESBy Charles Libove and RM.ph E. I-IubkaLangtey Aeronautical LaboratoryLangley“ Field, Va.),Washingtonebruary i_-_ - REPRODUCED BY - “NATIONAL TECHNICA

2、L_, INFORMATION SERVICEU. S. DEPARTMENT OF COMMERCESPRINGFIELD, VA. 22161:jProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-N 0 T I C ET-IIS DOCUMENT HAS BEEN REPRODUCED FROM THEBFST COPY FURNISHED US BY THE SPONSORINGAGI,_,_CY, ALTHOUGH IT IS RECOGNI

3、ZED THAT CER-i/_I_ PORTIONS ARE ILLEGIBLE, IT IS BEING RE-, E J*,!.; “,D IN THE INTEREST OF MAKING AVAILABLE,_, ,ibCIt INFORMATION AS POSSIBLE,QIProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-rU.S. DEPARTMENT OF COMMERCENational Technical Informatio

4、n ServiceNACA-TN-2289ELASTIC CONSTANTS FOR CORRUGATED-CORESANDWICH PLATESLangley Aeronautical LaboratoryLangley Field, VAFeb 51Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONALADVISORYCOMMITTEEFORAERONAUTICSTECHNICALNOTE2289ELASTICCONSTANTSFOR

5、CORRUGATED-CORESANDWICHPLATESBy Charles Libove and Ralph E. HubkaSUMMARYThe sandwich plate consisting of corrugated sheet fastened betweentwo face sheets is considered. Application of existing theories to theanalysis of such a sandwich plate requires the knowledge of certainelastic constants. Formul

6、as and charts are presented for the evaluationof these constants. The formulas for three of these constants werechecked experimentally and found to give values in close agreementwiththe experimental values.INTRODUCTIONA type of sandwich plate for which practical use has recently beenfound in airplan

7、e-wing construction consists of a corrugated metal sheetfastened, at its crests and troughs, to two ordinary metal sheets (see_for example, fig. i). The main advantage of this t_e of sandwich isthat the corrugated-sheet core not only serves to separate the facesand, therby, to achieve high flexural

8、stiffness, but it also carries ashare of any compressive loading applied parallel to the corrugationsand any edgewise shear loading. This type of sandwich has been calledcardboard-box construction (reference i) and also double-skin construction.It is referred to herein as corrugated-core sandwich pl

9、ate.Plate theories applicable to the symmetrical type of corrugated-coresandwich, illustrated in figure l(a), have been developed in reference 2for flat plates and in reference 3 for curved plates, a These theoriesare essentially homogeneousorthotropic-plate theories extended toinclude deflections d

10、ue to transverse shear, which can be significantfor the corrugated-core sandwich plate because of the relatively flexiblecore.aThe precedent established in reference 4 of referring to sandwichplates of the type shown in figure l(a) as symmetrical is adhered toherein. The type of corrugation shown in

11、 this figure is also calledsymmetrical.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACA TN 2289Application of the general sandwich-plate theories of references 2and 3 to any particular type of sandwich requires a knowledge of certainelastic con

12、stants for that type of sandwich plate. These constantsdescribe the distortions associated with simple loadings. They includetwo transverse shear stiffnesses DQx and DQy, two bending stiffnessesDx and Dy_ a twisting stiffness Dxy _ two stretching moduli Exand Ey_ a shearing modulus Gxy3 two Poissons

13、 ratios _x and _yassociated with bending_ and two Poissons ratios _ and _x yassociated with stretching.The purpose of the present paper is to present formulas forevaluating these elastic constants for the corrugated-core type ofsandwich plate. For the sake of completeness, formulas are alsodeveloped

14、 for evaluating the additional elastic constants that would beneeded for a rigorous extension of the sandwich-plate theories to theunsymmetrical type of sandwich. These additional constants_ denotedby Cxx_ Cxy_ Cyx, Cyy, and T_ describe coupling - for example, thecurvatures produced by extensional f

15、orces. The derivation and formulasfor the transverse shear stiffness DQy are essentially the same asthose given in reference 4 for the case in which interference betweencorrugation flats and face sheets is neglected_ but are extended slightlyto include the effects of stretching of the corrugation (i

16、n addition tobending) and the prevention of anticlastic curvature in the elements ofthe sandwich plate. The former effect can be important when the sandwichcross section approaches a truss_ the latter_ because the length of thesandwich plate parallel to the corrugation axis is several times thecorru

17、gation pitch. The results obtained for the bending and twistingstlffnesses Dx, Dy, and Dxy for the symmetrical sandwich correspondto the slightly less precise formulas of reference 5. (Transverse shearstiffness was not evaluated in this reference. A slight difference indefinition of the symbols Dx a

18、nd Dy exists between reference 5 andthe present paper.)Because the formulas developed are generally rather involved_ chartsare presented for one of them_ the transverse shear stiffness DQy, andapproximations are given for several of the others_ together with theresults of numerical investigations of

19、 the accuracy of these approxi-mations. In calculating the charts and in investigating the accuracy ofapproximate formulas_ a family of corrugation shapes consisting ofstraight lines and circular arcs was considered. The bend radii of thecorrugation_ measured to the center line, were generally taken

20、 as 0.18times the corrugation depth hc_ but departures from this value werealso considered_ as were departures from symmetry.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 2289 3As a check on the formulas, bending and twisting tests were run

21、onsamples of a corrugated-core sandwich plate. Experimental values ofbending stiffness Dy, transverse shear stiffness DQy_and twistingstiffness Dxy were obtained and comparedwith the theoretical values.The function of the elastic constants in a sandwich-plate theoryis first briefly described. A sect

22、ion follows in which the formulasfor the elastic constants for the corrugated-core sandwich are summarized.The tests and comparison between theory and experiment are then described_a discussion section follows, and a section of concluding remarks endsthe body of the paper. The symbols used in the bo

23、dy of the paper arelisted and defined in appendix A. A numberof them are also defined inthe text where they first appear. Appendixes B to E contain thetheoretical derivations.THEFUNCTIONOFTHEELASTICCONSTANTSIN SANDWICH-PIATETHEORYThe sandwich-plate theories of references 2 and 3 are based on astruct

24、ural idealization of the sandwich as a plate of continuous con-struction with material which is orthotropic with respect to the mutuallyperpendicular x-, y-, and z-directions. The modulus of elasticity inthe z, or thickness, direction is assumedto be infinite_ that is,local buckling of the faces is

25、not considered and the over-all thicknessis assumedto remain constant. Straight material lines normal to themiddle surface are assumedto remain straight, but not necessarilynormal to the middle surface, during distortion of the plate.This idealized structure can adequately represent a corrugated-cor

26、esandwich plate of either the symmetrical or unsymmetrical type for manypractical purposes, provided the core has sufficient stiffness to keepthe over-all thickness of the plate essentially constant and providedthe plate width (perpendicular to the corrugation axis is manytimesthe corrugation pitch.

27、 If the symmetrical type of sandwich (fig. iCa)is to be represented, then the elastic properties of the idealized-platematerial maybe regarded as varying symmetrically about the middlesurface through the thickness. In order to represent the behavior ofthe unsymmetrical type of sandwich (fig. l(b), t

28、he elastic propertiesof the idealized-plate material must be thought of as varying nonsym-metrically with respect to the middle surface.The behavior of a differential element of the idealized sandwichplate under load can be described by a set of force-distortion relation-ships. For an element of the

29、 symmetrical type of idealized sandwichProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 NACATN 2289(fig. l(a), subjected to forces and momentsas shownin figure 2(a),these relationships, as developed in references 2 and 3, are_2w Mx _! 1 _Qx- + My+ (

30、1)_x2 Dx _ _ x_-_2w _x _ i-_ (2)_y2 - Dx Mx - Dy + DQy _yNx _y_x - Ny (3)Ex F7_x Nx + Ny (4)_Y = Ex Ey1 1 _Qx 1 1_2w -_+ +- (5)_x _y Dxy 2 DQx _)y 2 DQy _x- _ (6)7_ Gxy82w 82w 82wwhere 2 -_3 and are the curvatures and twist of the middle_x _y _x _ysurface and _x, ey, and Fxy are the strains of the m

31、iddle surface. Thequantities Dx, Dy, _x, and so on which appear in the coefficients ofthe loading terms are the elastic constants. Each constant describes adistortion produced by a simple loading. For example, if all loadingsare zero except Mx, then, according to equation (1), _ _1 is theDxamount of

32、 curvature in the x-direction produced per unit of Mx-The behavior of the unsymmetrical type of sandwich (fig. l(b) ismore complex than that of the symmetrical type. In particular, a certainamount of coupling among the distortions may be erpected; for example,extensional forces may in general produc

33、e curvatures as well as extensions.The same type of coupling can be expected in a symmetrical sandwichsubjected to unsymmetrical loading. In setting up force-distortionrelationships for an element of the unsymmetrical type of sandwich, theloading on the element will be generalized as shown in figure

34、 2(b).The forces Nx, Ny, and Nxy are no longer assumed to be applied in “the middle plane; each has an arbitrary plane of application, denotedProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 2289 5by I, II_ and Ill_ respectively. The strains _

35、xJ _y_ and 7xy aremeasured in these same respective planes. The force-dlstortion relation-ships for the element are then given by the following generalization ofequations (i) to (6):_2w Mx _Y My + ICxxNx + , I _Ov_x-7= -o-_+ Dy , c_: + DQx- (i)_2w _x My+, 1 _y2 = D-_Mx- Dy ,%xNx + C_y, + D% _y (2),

36、I Nx _Y Ny_x : i-CxxMx - Cyx_, + - -Ex Ey(3)% :,-Cx_x-c_: -_xNx+_z (4)i I Ex %- + TNxy + +- (5)8x 8y Dxy , , 2 DQx ()y 2 DQy 8x, Nxy7xy = 2TMxy ,+, , Gxy(6)The boxed terms are the terms that have been added to express thecoupling behavior. The coefficients Cxx _ Cxy _ and so on in the boxedterms are

37、 the coupling elastic constants. The presence of eachcoupling elastic constant in two equations is a consequence of thereciprocity theorem for elastic structures. (Further consequences of the_x _y _ _Ylreciprocity theorem are that - = and _ = _.Dx Dy Ex Ey /Through a proper choice of locations for p

38、lanes I, II, and IIl_some uncoupling may be effected for any given sandwich. Plane I may bechosen so that Cxx or Cyx is zero_ plane II so that Cxy or Cyyis zero_ and plane III so that T is zero. Thus_ in general_ three ofthe coupling elastic constants may be made equal to zero. In specialProvided by

39、 IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 NACATN 2289cases, proper choice of locations of planes I, II, and III will resultin still further uncoupling. For the symmetrical sandwich_ of course,choosing these planes to coincide at the middle surface of th

40、e platecauses all the coupling constants to vanish.THEORETICALRESULTSElastic Constants for Symmetrical SandwichIn appendixes B to E, derivations are madeof formulas for theelastic constants for the general corrugated-core sandwich plate. Theformulas obtained are nowgiven in reduced form for use in c

41、onjunctionwith the force-distortion equations (1) to (6) for the symmetricalsandwich plate. Generally, the subscript C denotes the core, and thesubscripts 1 and 2 denote the lower and upper faces, respectively. Inthis section, however, only symmetrical sandwiches are considered andthe subscript 1 is

42、 used for both faces. It should be kept in mindjtherefore, that the definitions of manyof the terms appearing in thefollowing formulas for the elastic constants apply only to the symmetricaltype of sandwich.Bending stiffnesses.- The formulas obtained in appendix B for thebending stiffnesses Dx and D

43、y arewherei_x = EC_C + _ Eltlh21E-Iy = _ Eltl h2Dx = EYx (7)1 - 2 1 - 7xE1ECPoissons ratio of face sheet materialmodulus of elasticity of face sheet material, psimodulus of elasticity of core material, psiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,

44、-,-_CA TN 2289 7IICtIhmoment of inertia, per unit width, of corrugation cross-sectional area about middle plane, inches3thickness of each face sheet, inchesdistance between middle surfaces of face sheets, inchesFor practical sandwiches, the moment of inertia _C contributed by thecore is often small

45、compared with the moment of inertia which the facescontribute to cross sections perpendicular to the corrugations. Insuch cases, _ is very nearly unity_ and the following approximationElxto equation (8) may be made(8,)This approximation implies a neglect of the restraining effect of thecorrugation o

46、n the Poisson expansion or contraction of the face sheets.Results of a numerical survey of the accuracy of this apDroximatio_ aregiven in table I for the symmetrical sandwich = 1.0 of the commontype shown at the top of the table. The table gives the ratio of theapproximate value of Dy, as computed f

47、rom equation (8), to the exactvalue of Dy, as computed from equation (8). The error in the approxi-mate value is seen to be small over a large part of the range of con-figurations considered and, in extreme cases, no more than 6 percent.Poissons ratios associated with bending- The formulas obtainedf

48、or the Poissons ratios associated with bending _x and _y are(see appendix B)_x = _i (9): (lO)DxExtensional stiffnesses.- The formulas obtained in appendix B fortne extensional stiffnesses Ex and Ey, reduced to the symmetricalcase, areEx = EAx (ll)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACA TN 2289%Ey =i 12(l1(12)whereEA-x : EcA C + 2Elt IE-Ay = 2Elt IB area, per unit width, of corrugation cross sectionperpendicular to corrugation axis, inchesIf, once again, the restraining e