1、REPORT No. 899A GENERAL SMALL-DEFLECTION THEORY FOR FLAT SANDWICH PLATES“ByCHARLESLIBOVEand S. B. B4TDOEFSUMMARY.4 smalldejkction theory h dew.?.opedfor the eilm$ic behatiorof orthotropic jla=t plutes in which de$ectiom due to hear aretaken into account. In tkk theory, which corers afl types oft san
2、dtih con.Nruction, a plate is characterized by serenphytical coants (jce #Q7nesses and two Pcrieson ratios) ofwhich six are independent. Both the enerqy exprewion. andthe dierenttil equaki.m.e are dewloped. Boundary condMonscorregpomiing to .m”mply suppotied, clamped, and ela z measumd normal toplan
3、e of plate and x anj D, -Din . .- -.DQZ,Dogk) PyP7.? TVha, bvv, “-”-Ku, 0intensity of middle-plane tensile force pmallel to*z-plane, pounds per inch .-.intensity of middle-plane tensile force.parallel toyz-plane, pounds pm inchintensity of middh+pkme shearing force pmallolto yz-plane and m-plane, po
4、unds per inchflexural stifl%es of plate with anticlaeticbending unrestrained, nch-pou moment M= dy on the opposite facewould be shown acting clockwise). The twisting momentand middIe-pIane shearing force acting on any cross sectionare known, from equilibrium considerations, to be equal tothe twistin
5、g moment and middle-plane shearing force actingon a cross section at right angles. The symbols M=Vand N=vtherefore appear in both of the faces shown in figure 1.For convenience, in this report the z-direction is sometimesreferred to as the vertical direction and pkmes paraIleI tothe -plane are samet
6、imes referred to as horizontal planes.PHYSICALCONSTANTSThe physicaI properties of the plate are described by meansof seven constants: the fle.xuralstiffnessesD=and DU,the twist-ing stiffness D=Y, the transverse shear stiffncmes Q= andQr, and the Poison ratios p. and P“. Definitions of theseconstants
7、 are obtained by considering the distortions of theditlerentialelement of figure 1under simpleloading conditions.Let alforces and moments acting on the element be zero,except for the moments M. acting on two opposite faces. Theefbct of M= is to produce a primary curvature z in themiddle surface of t
8、he element and also a secondary curvaturea% which is a Poisson eflect. Then D= is defind as thenegative of the ratio of moment to primary curvature orD.= (1)T2when only M. is acting, and =is defined as the negative ofthe ratio of Poisson curvature to primary curvature orVwz=x2FW (2)5?when only ilf.
9、is acting. hro other distortions are assumedbut and only Mr is acting.(4)If, now, all of the forces and moments are equaI h zeroexcept M.p acting on all four faces, the onIy distortionproduced is a twist %, and D=, is defied as the ratio ofaxtwisting moment to twist orD.”=+ (5)axwhen only MW is acti
10、ng.The transve shear stiffnessDQ=is defied by lettingoglythe shears Q. act on opposite faces of the element-(except foran infinitesimal moment of magnitude Q, dy dx required forequill%rium). The distortion is assumed for the momentto be essentially a sliding of one face of the element withrespect to
11、 the opposite face, both faces remaining plane, Asa result of this sIiding, the two”faces parallel to the z-planeare distorted from their rectangular shape into paralIel they can be evaluated theoretically if theproperties of the component parts of the sandwich are knownand if the pateis of simple c
12、onstruction. In any event., theProvided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-142 REPORT NO. 899NATIONAL ADVISORY COMMITTEE FOR AERONAUTICScoustants can be determined experimentally by means ofbending tests and twisting tests on beams and panels of
13、 thesame sandwich construction as the pate. A description ofthe testsrequired is given in nppe.ndixA.AIthough seven physical comstantshave been. discussed,they need not.all be independently determined for if any threeof the four constants D., DV,P., and tiare know-nthe fourthcan be evaluated from th
14、e relationship/.L=DU= = “ (8)This relationship, based on a genendization of llaxu7c11s.reciprocal law, is derived in appendix B.The shear stiffnesses DQ, and D, merit some additionaldiscussion. HE distortion due to shear was assumed to bea sliding of tlm cross.sections over each other, the cross see
15、-.tions remaining plane and the shear strains remaining con-stmt for the entire thickness of the plate and equal to. thoshear angle 7* or 7r. Actwdly, if the plate is continuousenough for cross sections to exist at aII,under shear the crosssections generally tend to warp out of their plane condition
16、(p. 170 of reference 8); this warping makes the shear angle,as defined for equations (6) and (7), meaningls. The shearstrain varies with depth and an average shear strain wiUhaveto be used as the effective shear angle 7= or YVfor purposmof defining effective shear stiflness Daz or DaV. If the exper-
17、imental method is used. (sew appendix A), this difhdty isnot encountered because, instead of a shear angle, curvaturesare measured, and the stiffmssw. obtained are autcmmticallythe effective stifl%essw.Daspite the general tendency of crow sections under shearto warp, the assumption that they remain
18、plane (though notnormal to the middle surface) can bo. shown to. be almostcorrect for those sandwich= in which the stiffness of thecore is very small compared with ths .stiflness of. the faces(for example, Metalite, honeycomb). For such sandwichesthe shear stdlnesses Du= and Dcau.be readily calculat
19、ecl,because the faces may be resumed to take alI the direct bend-ing stress and the vertical shear may thmefore be assumeduniformly distributed in the core. The shear angles = and7Wwill then be constant throughout the core.For those sandwiched in which cross sections under shearmay not ba wmmed to r
20、emain plane, the tendency of thesecross sections to warp introducw a further complicationwhich can, however, be resolved by means of a justifiablesimplifying assumption. This compljcation is due to thefact that if the cross-sectional warping is partially or com-pletdy prevented the effect will be to
21、 increase. the shearstiffness DQZor DQV. The shear stiffnesses, thus, depend notonly on the properties of the plate materials but also on the.degree of restraint against cross-sectional warping. For thepurpose of the present theory the shear stiffnesses Dz andD, are assumed to be constant throughout
22、 the. plate anclhave the values tlmy would have if cross sections were alIowedto warp freely. The error caused by this assumption willbe mainly local in character, being most pronounced. in theregion of a concentrated Iateral load, where a sudden changein the shear tends to produce a sudden change i
23、n the degreeof warping which is prevented by continuity of the platv.The error wiIIprobably bo mgligiblo in the cuse of distribululkinds, for which there are only gradual changm in the shww.A discussion of this erm.rin connection with bcama is con-tained in pagss 173=174 o.reference. 8 amin rcfercn.
24、cc 0,. . DIFFERENTIALEQUATIONSFOlt PLATEDISTORTIONEQUATIONSNWEquiitious can be derived relating the curvatures andah VW“ andheJvistaxay at any point in thtiplate h the iuicr-nal shears nnd moments acting at that point.Equation for the curvature #.An expression can h!obtain where, from equations (3)
25、and (4),The work.done by the first group of foces 111=dy in aesocia-lbf,tion with the curvature KYT produced by the secondrgroup isif. dy (y, g d.r)orSimilarly, the workly dz in associationM=MY,dx dydone by the second group of forceswith the curvature p= produced byzthe first group is-“,dz(%dy)orEqu
26、ating the expressions for the two works and ehn.inatingthe common factor ilf.ilwdx dy givefrom which is obtained equation (8).Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-APPENDIX C,.DERIVATIONOF EQUILIBRHJMEQUATIONSAND GENERALBOUNDARYCOLHTIONSBY
27、A VAR1ATIONALMETHODIn the body of this paper only free, simply supported, and clamped edges were considered, These typos of boundaryconditions are characterized by the condition that the moments and verticaI forces at tho boundaries acquire no potcntiaIenergy as a result of the pIate.sdeflection. Th
28、is condition hokie by virtue of the fact that either the moments and forces at thebounclarieeare zero or the points of application of the nonzero boundary reactions do not move. A more general type of sup-port, in which neither of these conditions holds, is discussed in the following section,Potenti
29、al-energy expression.For simplicity a rectangular plat “d by to be wo at the boundarn x= a“Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.4 GENERAL S3L4LL-DIlFLECTIOF7THEORY FOR FLAT SANDWICH PLATES 155 _lf a pkde is elastically supported at the bo
30、undaries, the elastic support may sometimes be conveniently thought of asmade up of three rows of closely spaced discrete spr at each edge: a row of reflectional springs, a row of rotational springs,and a row of torsional springs, having the known stnesses per inch kl, k,and kg, which may vary along
31、 the edge. For thistype of support the vertical shear reaction at any point along thti edge is proportional to the vertical deflection at that point-.L-and the twisting and bending moment reactions am proportional to the corresponding rotations of an originally vertical line “”element in the edge. T
32、he boundary conditions for this type of support can be obtained from equationa (C3) and (C4) by set-ting at x=Oat x=aaty=OThe signs in the above boundary conditions follow as a result of the directions assumed for positive shears and moments.APPENDIX DDERIVATIONOF EQUATION(27) FOR THE POTENTIALENERG
33、YOF THE EXTERNALFORCESii rectangular plate the edges of which are z= O,a andy=O,ZI is considered (fig. 6). The boundary conditions as-sumed are the usual conditions corresponding to zero workby the reactions; that is, each edge is either free, simplysupported, or chtmped.The horizontal loads N=, iVr
34、, and N,r are assumed first tobe applied at the boundaries with no lateral load. As aresult the middle plane (and all horizontal planes) of the platestretches; thus, the constant stretching energy discuwwlpreciously in connection with the strain energy of the plateis produced, and alight shifts in t
35、he points of application ofthe edge forces i%, iVV, and N= are cau%d. These newpositions of the points of application are used as the arbitraryfixed reference points in any future measurements of thepotential energy of the horizontal edge forces.If the Iateral load q is now applied, the middle surfa
36、ceacquires the displacementa u A. E. H.: A Treatise on the Mathematical Theory of Elas-ticity. Fourth cd., Dover Publications, 1944, pp. 173-174.Woods, Frederick S.: Advanced Calculus. Ginn and Co., 1034,pp. 317-331.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
