REG NACA-TN-1425-1948 NONLINEAR LARGE-DEFLECTION BOUNDARY-VALUE PROBLEMS OF RECTANGULAR PLATES.pdf

1、7, j. f/,_/._._/, v._ 2.25. ThereforeEh 4a panel is unsafe if its design is based upon the slde-thrust considerationsonly, and the study of combined loadi_ is of great significance.A great number of authors have studied the buckling problems, an_considerable experimental work has been carried out. A

2、s a result,design formulas are available and seem to be accurate for most practicalpurposes. The bending problems, however, have Been studied by only afew investigators, and test results (references 21 to 23) are far tooscarce to Justify any conclusions. The combined bending anA bucklingproblem has

3、been studied in onl_ one case (reference 19), and even inthis instance the results are incomplete.Among the solutions of the large-deflection problems of rectangularplates under bending or combined bending and compression, Levys solutionsare the only ones of a theoretically exact nature. His solutio

4、ns are,however, limited to a few boundary conditions and the numerical resultscan be obtained only after great labor.The purpose of the present investigation is to develop a simpleand yet sufficlentl_ accurate mstho_ for the solution of the bending andthe combined bending and buckling problems for e

5、ngineering purposes,end this is accomplished by means of the flnite-difference approximations.Solving the partial differential equations by finite-dlfferenceequations has been accomplished previously. Solving the resultingdifference equations, however, is still a problem. In the case oflinear differ

6、ence equations, solutions by successive approximationare always convergent and the work is only tedious. Besides, Southwellsrelaxation l_thod may be applied without too much trouble. But, inorder to solve the nonlinesr difference equations, the successive-approximation _thod cannot always be relied

7、on because it does notalways give a convergent solution. The re2_xationmethod, since it isnothing but intelligent guessing, can be applied in only a few casesand then with great difficulties (reference 16).A study of the finlte-difference expressions of the large-deflectiontheory reveals that a tech

8、nique can be developed by n_ans of which thesystem of nonlinear difference equations can be solved with rapidconvergence by successive approximation by us_ Crouts method ofsolving a system of l_near simultaneous equations (reference 24). Byway of illustration, a sq_re plate under un!formnormal press

9、ure wi_1bo_idary conditions approximating the riveted sheot-stringer panelis studied by this method. Nondiz_nsional doflections and stresses areProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN No. 1425 5given under various nor_ml pressures. The

10、 results are consistent withLevys approximate num_rlcal solution for ideal, simple supported plates(reference 19) and Ways approximate solution for ideal clamped edges(reference 15), an_ the center deflections check closely with the testresults by Head and Sechler (reference 23) for the ratio pa4/Eh

11、4 aslarge as 120. The deviation for the ratio pa4_h 4 larger than 120is probably due to the approximations employed in the derivation ofthe basic differential equation.The procedure is quite general it maybe applied to solve theproblems of rectangular plates of any length-width ratio with variousbou

12、ndary conditions under either normal pressure or combinednormalpressure and side thrust.The present investigation was originally carried out under thedirection of Professor Joseph S. Newell at the Daniel GuggenheimAeronautical Laboratory of the Massachusetts Institute of Technologyand was completed

13、at Brown University, under the sponsorship and withthe financial support of the National Advisory Committee for Aeronautics,where the author was participating in the program for AdvancedInstructionand Research in Mechanics. The author was particularly fortunate toreceive frequent advice while workin

14、g on this problem from ProfessorRichard yon Mises of Harvard University. The author is grateful to bothProfessor Newell and Professor yon Mises for their manyvaluablesuggest ions.SYMBOLSa, bhx, y_ zu, v_gPE,Dlength and width of plate, respectivelythickness of platecoordinates of a point in platehori

15、zontal displacements of points in middle surfacein x- and y-dlrections, respectively (nondimensionaforT_ are ua/h 2, va/h 2, respectively)deflection of middle surface from its initial plane(nondimensional form is w/h)normal load on plate per unit area (nondlmensionalform is pa4_ h4)Youngs modulus an

16、d Poissons ratio, respectivelyflexural rigidity of plate _12(1 -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 NACATN No. 1425v2 ;52 ,:32_x2 _2_4 _4_xJ ay_ _xyGx“, _y“, “rxy“Ex , _y_ YXTIf I! ttEx _ Ey _ YX_4+ m4,4membrane stresses in middle surfa

17、ce (nondimensional%a T ,a2/m,2respectively) xy ,extreme-fiber bending and shearing stresses(nondimensiozml forms are ex“a2/Eh2 , Cry“a2/Eh2,and Vxy“a2/Eh2 , respectively)membrane strains in middle surface (nondimensionalforms are _x a2/h2, _y a2/h 2, and _x_ a2/h2,respectlvel_ )extreme-fiber bending

18、 and shearing strains(nondimensional forms are ex“a2/h2, _y“a2/h 2,and _,xy“a2/h 2, respectivel_)stress function (nondlmensional form is F_h 2)first-, second-_ ._ to nth-orier differences,respectivelyfirst-order differences in x- and y-directions,respectivelyFUNDAMENTAL DIB_EEEENTIAL EQUATIONSThe th

19、ickness of the plate is assumed small c_pared with its otherdimensions. The middle plane of the plate is taken to coincide with thexy-plane of the coordinate system and to be a plane of elastic symmetry.Aftersendlng, the points of the middle plane are displaced and lieon some surface which is called

20、 the middle surface of the plate. Thedisplacement of a point of the middle plane in the direction ofthe z-axis w is called the deflection of the given point of the plate.Consider the case in which the deflections are large in comparisonwith the thicknsss of the plate but, at the same time, are small

21、 enoughto Justify the following assumptions:Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-7NACA TN No. 14251. Lines normal to the middle surface before defolmztion re_innormal to the r_ddle surface after deforn_tlon.2. The normal stress cz perpendi

22、cular to the faces of the plateis negligible in comparison with the other normal stresses.In order to investigate the state of strain in a bent plate, it,is supposed that the middle surface is actually deforn_d and thatthe deflections are no longer small in comparison with the thicknessof the plate

23、but are still small as compared with the other dimensions.UnAer these assumptions, the following fundameni_l partialdifferential equations governing the deformation of thin plates can bederived from the compatibility and equilibrium conditions:bx_. 8x 2 8 2Y _y4 _k_X _j_ _X2 _y2_4w + 2 ;54w ;54w P h

24、_-2F _2w _2F _2w 2_x_ _x_Jwhere DEh 312(l - _2)the median-fiber stresses are_2F(_y - 8x282FT ! -xy _xand the n_diarJ-fiber strains are,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACA TN No. 14257_ = _2(i+ _) _2F_. axa_The extreme-fiber bending

25、 and shearing stresses are“ “ 2(1 -“_2) + _Y 2(_- .2)k._+ _SThese expressions can be made nondlmensional by writingF = -_- xh2E = aVt Wh Y =yam4 , = f:a,_2P =- E e_k_JEh 4where a is the smaller side of the rectamgular plate.The differential equations then become,., a2_, a%,_x,2 _)y,2+ 2_,2 ohy,2 _x_

26、 _xProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN No. 1425 9If _2 = 0.i, which value is characteristic of aluminum alloys, andthe primes are dropped, the partial differential equations in nondimenslonalform are_4F _2 1 52w _2w+ _ = _2 _2_4w

27、_4w _w lo.%, + ,0.8 _-._2w(i)_2F _2w _2F _-_y_+ _2 _2 2_-_ (2)The nondimensional mediam-fiber stresses are_X _2, _2FTX_T -= .-.m-and the mondimensional median-fiber strains are_2F _2Fx _y2 #_x2_2F 82FCy = _z2 P_2(3)(4)“Yx:y = -2(1 + #) _)2FThe nondimensional extreme-fiber bendin_ and shearin_ stress

28、es areProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-i0 ,YACATN No. 1425_X stI“2(1 +FORMULATION OF BOUNDARY CONDITIONS(7)The governing differential equations are 2 fourth-order slmul-taneous partial differential equations in two variables. In order

29、toobtain a unique solution in the case of rectan_,_lar plates, there mustbe four given boundary conditions at each edge.Before proceeding to the actual case, two theoretical bolundaryconditions may be mentioned:i. Simply supported plates, that is, plates having edges that canrotate freely about the

30、supports and can move freely along the supports2. Clampedor built-ln plates, that is, plates having edges thatare clamped rigidly against rotation about the supports and at the sametime are prevented from .having any displacements along the supportsActually, it is to be expected that neither of thes

31、e conditions willbe fulfilled exactly in a structure.The bending problem will be considered next, in which the bottomplating of a seaplane is to be studied. The behavior of the sheetapproximates that of an infinite sheet supported on a homogeneouselastic netwrork with rectangular fields of the sam_o

32、 rigidity as thesupporting framework of the seaplane.Because of the syn_et_y of the rectangular fio!ds, the displacementin the plane of the sheet and the slope of the sheet relative to theplane of the network must be zero wherever the sheet passes over thecenter line of each supporting beam. Each re

33、ctezqgular fJeld willtherefor_ behave as a rectangular plate clamped along its four edges onsupports that are rigid enough in t_le olana of the sheet to preventtheir displacezrmnt in that _lane. At tile sa_m time these supports musthave a rigidity nor_l to the plans of the sheet equal to that ofthe

34、actual supports in the flying-boat bottom.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN No. 1425 iiThe rigidity of the supports will lie s_newherebetween theunattainable extremes of zero rigidity and infinite rigidity. Theextreme of infinite

35、 rigidity normal to the plane of the sheet isone that maybe approximated in actual designs. It can be shownthatthe stress distribution in such a flxed-edge plate will, in most cases,be less favorable than the stress distribution in the elastic-edgeplate. The strength of plates obtained from the theo

36、ry will thereforebe on the safe side if applied in flying-boat design. Reference mightbe madein this connection to a paper by Mesnager (reference 25), inwhich it is shownthat a rectangular plate with elastic edges of certainflexibility will be less highly stressed than a clamped-edgeplate. Thisdiffe

37、rence in stress may also be clearly seen by conLoarlngthe extreme-fiber-stress calculations by Levy (reference 19) and Way (reference 15)for simply supported plates and clamped plates.The impact pressure on a flying-boat bottom in actual cases,however, is not even approximately uniform over a portio

38、n of the sheetcovering several rectangular fields. Usually one rectangular panel ofthe bottom plating would resist a higher impact pressure than thesurrounding panels, and the sheet is supported on beamsof torsionalstiffness insufficient to develop large momentsalong the edges. Thehigh bending stres

39、ses at the edges characteristic of rigidly clampedplates would then be absent. In order to approximate this condition,the plate maybe assumedto be simply supported so that it is free torotate about the supports. At the sameti_ the riveted Joints preventit from moving in the plane of the plate along

40、and perpendicular to thesupports. According to the sameconsiderations as in the case of rigidlyclamped edges, the result would be on the s_e side. This case hasnever before been discussed and the stud_vof such a problem se_mstobe of importance.For the combinedbending and buckling problems hhe sameco

41、nsider-atlons will hold. It is evident, however, that as soon as the sidethrust is applied, there are displacements perpendicular to thesupported edges in the plane of the plate. Gall (reference 26) hasfound that a stiffener attaclled to a flat sheet carrying a compressiveload contributed approximat

42、ely the sameelastic support to the sheetas was required to give a simply supported edge (see also reference 20,p. 327). In combined bending and compression problems, therefore, itseemsalso i_ortant to study the ideal simply sup_0ortedplates. Theanalytical expressions for these boundary conditions ar

43、e formulated inthe following discussion.Simply Supported EdgeIf the edge y = 0 of the plate is simply supported, the deflection walong this edge must be zero. At Ole s_ tJJao this edge can rotate freelywith res!_cL to the x-axis_ that is, there is no bending moment M_along this edge. In this case, t

44、he analytical foz_itt_ation of the_hyslcalboundary conditions isProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12Similarly, if the edgeboundary conditions are(W)y_o = 0 Iy=ONACA iSNo. 1425(6)x = 0 of the plate is simply supported, the(W)x=0 = 0Since

45、 w = 0 along Z = 0, _w/Sx and 82w/_x2 must be zeroalso. The boundary conditions can therefore be written as(W)y=O = 0Sy-y_Jy=0= 0Similarly, on the edge x = O,(7)(W)x=0 = 0: o,_2Sx_oIf the plate has ideal simply supported edges, it must be free tomove along the supported edges in the plane of the pla

46、tej that is, theshearing stress along the edges in the plans of the plate is zero.AnalyticallyjTXy)y=O = 0oxy x_OProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN No. 1429 13or=0One more boundary condition is required to solve the plate problem

47、suniquely, and this may be obtained by specifying either the normalstresses or the displacements along the edges.For a plate having zero-edge compression, the normal stresses alongthe edges are zero. That is,_Y)y=O = 0or=0rThe strain in the m_dian plane isx -_ +_/x;(8), _+_/aq2Provided by IHSNot for

48、 ResaleNo reproduction or networking permitted without license from IHS-,-,-14 NACA TN No. 1429Therefore= _X 1 /_“h2_W_/ 5_y_Y= ey -l(_)22and the displacement of the edges in the x-direction iso -=constant .- 2while the displacement of the edges in the y-direction is-Constant Y 2 _Sy_ dyThe addition of side thrust may be exlpressed in teens of thechange in displacement of the edges.If Ex and _y are expressed in terms of the stress function F,UVI82F 82FJ y=Constamt _ B_ 1-82F 8,-F 1 _2-1J x=Constant i_x-_- B - _ L_) j