1、I= +I=|-_._-FCDZNATIONAL ADVISORY COMMITTEEFOR AERONAUTICSTECHNICAL NOTE 2601COMPRESSIVE BUCKLING OF SIMPLY SUPPORTED CURVED PLATESAND CYLINDERS OF SANDWICH CONSTRUCTIONBy Manuel Stein and J. MayersLangley Ae ronautical Labo rato ryLangley Field, Va.Washington_anuary 1952It |Provided by IHSNot for R
2、esaleNo reproduction or networking permitted without license from IHS-,-,-i_LL-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-IK NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSTECHNICAL NOTE 2601COMPRESSIVE BUCKLING OF SIMPLY SUPPORTED CURVED PLATESAND
3、CYLINDERS OF SANDWICH CONSTRUCTIONBy Manuel Stein and J. MayersSUMMARYTheoretical solutions are presented for the buckling in uniformaxial compression of two types of simply supported curved sandwichplates: the corrugated-core type and the isotropic-core type. Thesolutions are obtained from a theo_:
4、 for orthotropic curved plates inwhich deflections due to shear are taken into account. Results aregiven in the form of equations and curves.INTRODUCTIONThe use of sandwich construction for compression-carrying compo-w nents of aircraft will often require the calculation of the compressivebuckling s
5、trength of curved sandwich plates.In the present paper, therefore, a theoretical solution is givenfor the elastic buckling load, in uniform axial compression, of simplysupported, cylindrically curved, rectangular plates and circular cylin-ders of two types of sandwich construction: the corrugated-co
6、re typeand the isotropic type (e.g. Metalite).The analysis is based on the small-deflection buckling theory ofreference 1 which differs from ordinary curved-plate theory principallyby the inclusion of the effects of deflections due to transverse shear.The curvature is assumed constant and the thickn
7、ess small compared withthe radius and axial and circumferential dimensions. The core modulusin the transverse direction is ass_ued to be infinite; thus, considera-tion of types of local buckling in which corresponding points on theupper and lower faces do not remain equidistant is eliminated. Thecor
8、rugated-core sandwich is assumed to be symmetrical, on the average,about the middle surface, so that the force distortion relations arerelatively simple (see reference 2), and is assumed to have infinitetransverse shear stiffness in planes parallel to the corrugations. Thecore of the isotropic sandw
9、ich (flexural properties identical in axialand circumferential directions) is assttmed to carry no face-parallelstresses.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACATN 2601The results of the solution are presented in nondimensional formthro
10、ugh equations and graphs. The details of the analysis are given intwo appendixes.The results of the present investigation are comparedwith previouswork on the subject of compressive buckling of curved sandwich platesand cylinders (references 3, 4, and 5). This previous work is confinedto sandwich co
11、nstruction of the isotropic type only. The present paperincludes results for the isotropic sandwich covering a larger curvaturerange and gives the same, or more conservative, results.SYMBOLSiACDSDxDyDxyDQx,DQyDQECESEx,Eycross-sectional area of corrugation per inch of width,inchescross-sectional area
12、 of faces per inch of width, inches(2ts)flexural stiffness of isotropic sandwich plate, inch-pounds (2(1 - _S2)beam flexural stiffnesses of orthotropic plate in axialand circumferential directions, respectively, inch-poundstwisting stiffness of orthotropic plate in xy-plane,inch-poundstransverse she
13、ar stiffnesses of orthotropic plate inaxial and circumferential directions, respectively,pounds per inch (DQx assumed infinite for corrugated-core sandwich)transverse shear stiffness of isotropic sandwich plate,pounds per inchYoung Ws modulus for corrugated-core material, psiYoungs modulus for face
14、material, psiextensional stiffnesses of orthotropic plate in axial andcircumferential directions, respectively, pounds perinchProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 2601 3GCICV2,vh,G-4Nx.RZa,Zbabshear modulus of core material for isot
15、ropic sandwichplate, psishear stiffness of orthotropic plate in xy-plane, poundsper inchmoment of inertia of corrugation cross section per inchof width, inches3moment of inertia of faces per inch of width about middleof surface of plate, inches 3 (tsh_)mathematical operatorsmiddle-surface compressiv
16、e force, pounds per inchtransverse shearing forces in yz- and xz-planes,respectively, pounds per inchconstant.radius of curvature of plate or cylinder, inches2tsb_curvature parameters a2 = Z- ; zb2 = _ forR IS R2_Ssandwich plat_ CZa2 = 2tsa4(1- _$2).corrugated-core 7TR2_S- hZb2 = R2S for isotropic s
17、andwich plaaxial length of plate or c_linder, inchescircumference of cylinder or circumferential width ofplate, inchesdepth of sandwich plate measured between middle surfacesof faces, inchesProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4kxa, kxbm,n
18、2pra,rbNACA TN 2601_Xa = Nxa_-_2 ;compressive load coefficients ES_SW2Nxb 2kxb ESTS_2 for corrugated-core sandwich plate_Xa= Nxa2. Nxb2-, kxb =DS,2 DS.2hfor isotropic sandwich plate)Ynumber of half-waves into which plate or cylinder bucklesin axial and circumferential directions, respectivelypitch o
19、f corrugation, inchesdeveloped length of one corrugation leg, inchestransverse shear stiffness parameters ES_Sn2a = -_;DQ?rb for corrugated-core sandwich plate_a DS“2“ DSn2= DQa-_, rb - DQb-_ for isotropic sandwich plate_ZtctsWx,y_C#Sthickness of corrugation material, inchesthickness of face materia
20、l, inchesradial displacement of point on middle surface of plateor cylinder, inchesaxial and circumferential coordinates, respectivelyPoissonTs ratio fo# corrugated-core materialPoissong ratio for face materialJJProvided by IHSNot for ResaleNo reproduction or networking permitted without license fro
21、m IHS-,-,-NACA TN 2601 5_x,_y Poissons ratios for orthotropic plate, defined in termsof curvaturesPoissons ratios for orthotropic plate, defined in termsof middle-surface strainsRESULTSCorrugated-Core SandwichThe theoretical compressive buckling load for a curved rectangularcorrugated-core sandwich
22、plate of axial length a and circumferentialwidth b can be obtained from the following equation:EI 82 1kxb = la_2 + l/a_2 n2 +_,D Z_,_j 1 - _s2 _i+ m2 (b)2% 2(1+ _s)_Zb 2 m2b /(1)whereNxb2kxb- EYs_2m,n number of half-waves into which plate buckles in axial andcircumferential directionsProvided by IHS
23、Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 NACA TN 2601rbEgsW2D_ is the transverse shear stiffness, obtainable from reference 2, andim_l =I b 2hEcAGi+ - SJET_sEC_C1 + _EsA Sl+b SEC_ CI +The details in the derivation of equation (i) are presented in appendix
24、A.In using equation (1), in general, different combinations of integralvalues of m and n must be substituted until a minimum value of kxbProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 2601 7is obtained for given values of the other parameter
25、s. This minimumvalue of kxb determines the buckling load.For the special case of an infinitely long curved plate (a,m-_),kxb must be minimized with respect to the axial wave lengt_ a/m andintegral values of n.For the special case of a cylinder, the width b is equal to 2_R.Thus, R becomes involved in
26、 the parameters kxb , rb, Zb, and a/b.Equation (1), therefore, is not well-adapted to studying the effect ofchanges in R on the buckling strength of the cylinder. It is moreconvenient, for the special case of a cylinder, to consider equation (1)rewritten in terms of slightly different parameters as
27、follows:i - _S12 +n2(_) 2_i+ m2ra 2(i + _S)+Za2 1_h m2n2a(2).whereNxa2kxa E#S.2ES_S .2ra =2Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACA TN 2601Za2 - 2tsahR2IsThis equation, unlike equation (i), can yield the limiting results foran infinitel
28、y wide flat plate (R_) compressed in the short direction.For cylinders, different combinations of integral values of m andeven integral values of n must be substituted until a minimum value forkxa is obtained. The values considered for n for the range of practicaldimensions are: n = 0 for short cyli
29、nders (axisymmetrical buckling) andn = 4, 6, 8 . . . for medium or long cylinders. The case n = 2, whichis not considered here, corresponds to column buckling. For medium orlong cylinders, the same results or slightly conservative results areobtained if, instead of minimizing kxa with respect to eve
30、n integralvalues of n, kxa is minimized with respect to the circumferential wavelength b/n. The latter procedure is used in this paper.Because of the large number of elastic and geometric parametersappearing in equations (1) and (2), the critical load coefficients canbe readily obtained only for ind
31、ividual corrugated-core sandwich sections.For purposes of illustrating the application of equations (1) and (2) toa particular corrugated-core sandwich, the section shown in figure 1 hasbeen selected. The required physical constants calculated for this sec-tion are also shown in figure 1. The value
32、of D_ is computed from theformulas and charts presented in reference 2.The process of minimization is carried out for this particular sec-tion and buckling loads are obtained for infinitely long curved platesand cylinders of arbitrary over-all width and length, respectively, andof arbitrary radius.
33、Nondimensional curves presenting these results areshown in figures 2 and 3 giving the load coefficients kxa and kxb andtherefore the load as a function of the over-all dimensions. The dashedcurves give the results obtained if transverse shear deformations areneglected. The equations for this case ar
34、e presented in appendix A. Itshould be noted that the b/R and a/R curves are cut off where thedimensions of the plate or cylinder would no longer be consistent withthe requirements of small-deflection theory.Isotropic-Core SandwichThe theoretical compressive buckling load for a curved, rectangular,i
35、sotropic-core sandwich plate of axial length a and circumferentialwidth b can be obtained from the following equation:Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2K NACATN 2601 9kxb= +I a +r b n2a 2Zb 2 lfa_ 2(3)wherekxb DS_2DS_2rb DQb2andh2DQ =
36、OC h - tS (see reference 6)The details of the derivation of equation (3) are presented in appendix B.The values of m and n to be used in equation (3) are obtained in thesame manner as discussed in the previous section for the corrugated-coresandwich.For the special case of an infinitely long curved
37、plate (a,m-_),kxb can be minimized with respect to the axial wave length a/m andthe number of circumferential half-waves n. The results are plotted infigure h. The equations for the theoretical buckling load coefficientsof infinitely long curved plates and the ranges in which they hold areProvided b
38、y IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I0 NACA TN 2601kxb _ (z+ rb)2+ Zb2 l-rb (_)when_dl - rbZb(l + _o)2(5)when -. Equations (2) and (5) are not exact but are quite_2 = rbaccurate for the curvature-parameter ranges indicated.For the special case of a
39、 cylinder (b = 2.R), it is convenient torewrite equation (3) with parameters in terms of a rather than b asdone in the preceding section; the resulting equation is+ m _ ._ m2kxa = + (7)1 _ n)2(a)_ E n,2/a,2-1 2m-_ + ra + (m “_ + (m/_,_.JProvided by IHSNot for ResaleNo reproduction or networking perm
40、itted without license from IHS-,-,-NACA TN 2601 iiwherekxa= DS_2DS_2ra =D_ 2Za2 =2tsa4(l - _S2)R2 empirical “knockdown“ or reduction factorshave therefore been proposed for ordinary curved plates in compression(see, for example, reference 8). The sameshortcoming maybe expectedof the present buckling
41、 solution for curved sandwich plates; the reduc-tions required, however, will probably not be so severe as in the caseof homogeneouscylinders, partly because the greater thickness of thesandwich plate reduces, relatively, the importance of initial irregu-larities (see discussion in reference 3). Som
42、eexperimental data (refer-ence 5) seemto indicate that no reduction is required for the isotropicsandwich with a core of sufficiently low shear stiffness. Further inves-tigations are required, however, before this conclusion maybe taken asgenerally valid for all curved sandwich plates.CONCLUDINGREMA
43、RKSA theoretical solution is presented for the elastic buckling loadin uniform axial compression, of simply supported, cylindrically curved,rectangular plates and circular cylinders of two types of sandwich con-struction: the corrugated-core type and the isotropic type.Like the results for flat sand
44、wich plates, the results for curvedsandwich plates show, in general, that the effect of finite transverseshear stiffness is to lower the buckling load. For given cross-sectionaldimensions and properties, this effect diminishes as curvature increases.In the range of practical dimensions, however, the
45、 effect of transverseshear deformations on %hebuckling load is always important.For isotropic, sandwich, curved plates and cylinders of low trans-verse shear stiffness (weak cores), the critical compressive load NxProvided by IHSNot for ResaleNo reproduction or networking permitted without license f
46、rom IHS-,-,-NACATN 2601 15becomesindependent of curvature and equal to the transverse shearstiffness DQ, a result found also for flat sandwich plates of lowshear stiffness.Langley Aeronautical LaboratoryNational Advisory Committee for AeronauticsLangley Field, Va., September5, 1951PProvided by IHSNo
47、t for ResaleNo reproduction or networking permitted without license from IHS-,-,-16 NACA TN 2601APPENDIX AANALYSIS OF CORRUGATED-CORE SANDWICH PLATEA corrugated-core sandwich plate with the corrugations orientedparallel to the x-axis _my be considered to be an orthotropic platehaving infinite transv
48、erse shear stiffness in the axial direction; thatis, DQx-_. For such a plate loaded in axial compression (Nx posi-tive in compression), the general equations of equilibrium for ortho-tropic plates developed in reference 1 reduce toGxy 8_w 82wLDW + _ LE-1 _ + Nx _ _R2 28x28y3and(A2)where LD is the linear differential operator defined byLD= Dx 84 _x _x2_ 84 Dy 8_1- tI,x4Xy_x 4+ - _li,y + 2D:_d + ! - Ii,xl.i,y/Sx28y2 + i -“ “_1_Provided by IHSNot fo
