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~Arcf NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WAIRTI'MEI REPORT ORIGINALLY ISSUED March 1946 as Advance Restricted Report L6AO5a CRITICAL COMBINATIONS aF IOCGITUDINAL AD TRAVERSE DIRECT STRESS FOR AN INFINIWT! LOFG FLAT PLATE WITH EDXES ELASTICALLY RESTRAINED AGAINST ROTATION By S. B. Batdarf, Manual Stein, and Charles Libove Langley Memorial Aeronautical Laboratary Langley Field, Va. WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre viously held under a security status but. are now unclassified. Some of these reports were not tech nically edited. All have been reproduced without change in order to expedite general distribution. DOCUMENTS DEPARIMENi L 49 zrILq Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation. V http://www.archive.org/details/criticalcombinat001ang TACA .RR iio. L6ADSa rT.'T CNAL . DVTS'FY C r.CM '.ITEZ K":? L.EFLCNA.'JTICS ADf,.I:CE PESTRICTED RETORT GRTTICAL C. INr,.TI. OF L, GITJDITit.. NE TRAi.ISVERSE DIRECT STFEEC FOR ;. I:FTNITELY L1 :G FL/.1 PL..TE I'iTH LDGES EL..STI A._LY ?EET'T..T iED .'.'I1ST POT..TT!'] By 3. B. Eatdorf, Yanuel Stein, and Cllhrles Libovc A tiheoret 'csl 1i,:est' .i t.r : wsz ma.d' of the buckling of an infinit; l: lor f l.'. laC1 w41 .h edges iltstically restralinc; r *in.t rntat ion uni..r 'rhin'iti ons if l1ngri. tudinal inc tran:.:vrs c d t t ress. Inter_ct on curves are ares m'n 2d th'jt "Lve L'h 'i ti3l corrmbir.stions of stress f:.r seve'jr l .lii ffrnt d 3re: :.f .l5st' ed'_e restraint, ir.cldii F rin. pi suu 'port n:rd ;.n 1'.tl; fixity. It ws f.junrd thit i. E'coorci sbl frL.ti n of the critical lons tu3di al stress .ay be a1;l.ied tL the plat.e vw;ithc.ut any redu.;ticn in the transverze comrressive stress required 'for buckling. I NTFP.'DUJTI ON Eecause the skin of an .'.rrl n jir flight is sub jected t_, ccmi.,r,atirni of stress, attention ha recently been ,iven to the problem of nlcate buIcklln.g whnn more than cne tre.s is .acting. The ore:nt opor is the tt.iird f series of papr;rs analyzing the elasticc buck ling, under the action of two sTr s: es, of an infinitely long flat plqte with erid. eiaily rrtrainrd a inst rotation and fully sunoorted'. The tv'o orevicus L&pers are reference 1, which deals with the int:rsction of shear and lonittudinal direct stress, and re:ferene 2, whion deals v.ith the interructionr of shear and transverse direct stress. 'h : rrcslnt oarer ,'.scribes the inter action of lc.n;itudincl snd tranzsverse direct stress. These three lDading combinations are illustrated in figure 1. ;.ACA ARR No. L6A.,a Interaction curves that give the critical direct stress coriibinations for several different degrees of elastic edge restraint, inclrdii simple su".:nrt and co iplete fixity, are presented for the case in which the * :.ltude of the restraint is independent of the buckle wave length. These curves are based on an exact solu tion of the ._fferential equation of equilibrium, the details of v'hich are :.ven in the appendix. S :DOL.LS 3 elastic modulus of plate material j Poissonts ratio for plate material t thickness of plate b width cf plate a lenr t.. of late (a > b) D fle.xural stiffness c' plate y'e unit l.th: x longitudinal coordinate y transverse coordinate w noianal displacement of a point on buckled plate fi'... its undeflected position X half v;ave ler.t> c buckle So rotational stifi..ess of restri:.ir :rediiu along ec:.es of plate, moment per quarter radian o.r unit ler., th dimensionless elastic edgerestra:lt constant ,SD ) NACA ARR No. L6AO5a 3 Cx applied uniform .l.ongitudinal cor'resi"e stress o appliedd unif.Cir transveise ccrcressLveG stress N = xt N = oyt ob2t k :, dl'mensionles: stress cofrficient k =  ; "V X 0 b t ky "D Rx _l.or.gitu i. nal i,; :etstress rPatio: rtio of ]3nituJi l .*i'.ret stres:' p;r, sernt to critical str'c.s n rur r3 ILitud'.rcl e.'T.ression R trPnsvers? di ectstres rt; to; rtio of trans verse 2rc.rt tr.iess present to critical strees in pure transverse corrpresEs n SRES,il,' A!'D DTSCUTSIC.N The results of this investigation are given in the form of nnrdilenrsionE.l interaction curves in fij;ur,'? 2. Each o'int rn these c'rves reorPsants a critical combi natin of the stress coffi'cients kx and ky for a giver. elastic ed.erestraint constant E at. which an iinnite!l. lon, flat late will tuckle. The intraction curv. for plates with iirmoly su.orted ana cam"':d edges are given in fture 3 in tera of stress ratios rather than stress cceftici.ants. 'he calc'listed data uscJ to olot the intr.saotion curves are given in table 1. Acolicabilty of the interaction cur's. Critical comblin.ticns of 'lon5'tudrlnl arn: trsil,\rc direct stress for an infinitely lon flat ola :e vith edge.s either s.im)iy supported or 3lrcmod c bn b obtained fro.' the inteructin carves of figure 2. Critical con.binstions of dircrnt stress f',r rdlate witl. intern.edl.te elastic rest..:aint against ed. rottti.:n can ;lso be obtained from figure 2 for th,.c cares in v.hich th stiffness of the rcrtra'ning medium is independent of buckle wave length (E = a c:nt.t::.nt). Such edge restraint is provided 1TACA ARR No. L6A05a only by a medium in which rotation at one point does not influence rotation at another point. Edge conditions of this type are not ordinitrily encountered but .'.i ht occur when the restraint is fui.nislied by a row of discrete elements, such as coil spri',s or flexible clamps. Because of the creat variety of possible relationships between edge restraint and wave length, only the curves for edge restraint independent of wave lenth are shown. If critical stress combinations for a plate with con tinuous edge restraint (E dependent on wave length) are desired, they can be computed though somewhat labo riously by the method outlined in the appendix, pro vided the relationship between e3je restraint and wave length is known. This relationship is derived in refer ence 3 for the o.ecial case of a sturdy stifener, that is, a stiffener which t'.'ists without crosssectional distortion. The buckling stress for a finite plate can never be lower than that for an infinite plate haviv. the same ..'Ldth and thickness beca',se tLe finite plate is strength ened by 2..; ort alo:gz two additional edges. The use of figure 2 to estimate the critical direct stresses for a finite plate with edge restrsint independent of wave length, therefore, is in all cases conservative. Vertical portions of interaction curves. 7.:e vertical pot'tio:!. c.' t:e i;c.';ractic. c'.rv.?s (fig. 2) indicate that a cons'er:able amount of longitudinal com :pression may be p11ie to the plate without any reduc tion in the transverse coni'.rcssion required for buckli:L1. This result parallels the result of reference 2, in which it was found that a considerable a&_ount of shear stress could be applied to an infinitely long plate without any reduction in the transverse compression necess'.r to cause buckling. On the other hand, in ref erence 1 it was shown that the presence of shear always reduces the longitudinal compressive stress req'ir ed to pr1cduce buckles. This disparity in be:avior is probably attributable to the character of the buckle forms for the three types of stress. (See fig. 4.) The buckle form for shear alone (fig. 1.(a)) can be transformed con tinuously into that for lojgitudinal +oJirGssion al..:e (fig. 4(b)) by a grsU.ual addition of compression and subtraction of shear. Neither of these buckle forms, however, can be continuously transformed into the buckle form for transverse c:.mpression alone (fi.j. 1(c)). The vertical port:.ons of t.e interaction curv s e:.:te:i. ndefinitely intc the t asicn 'e 1ji.n f1 k,. (For c n.venie;ce, in fig. 2 the curves a. scoped a t a s all ne.ative vclue of kx. ThiLs rorerLy cf Lh c.trves i.diCiat6s CtE t hCe presence of lonr itudinal tensicr :ac Inc efca t uLpoIn tl'e trans.vers e stresss necessary to :ro ..ce buc *l inc. SUI.L.iARY; <. .TUli^ ..bS Interactin curves are cresertied frro:.l hich critical combinations of ionTitudinal and trnsv;rse direct stress for an:L Inf'i:itely ion: flat plate .ith' edtes either sir.ply stpportcd or ,la' ,ed can be obtained. Critical cc.bi nr.tions cf d.ic str3s ' r interi.!edia5te elas _T',. restr&iLnt against c .ed rotat'no c.a also 0 obtained from tle ..nier&co .n c. yves i'cr those ca.ss La lich t:e sL2' 'ness 'i thn rc str':,i : 'd. .t .i : in e:endeit of blc:.:le wav.a .en tl.. 'or cases in hi t tifl ne s o. ti0e m.st;rain .C..1.L' ..c e..s upon lic buckle '.rave length a.d thie rel;tio. s' 'e 1;'.een t.1a t.o s13 : .'n, tie citical conr.iin..tinrn of' '.:ect stress c. be dPt r: .ned t.hou so .: .ct laorio.c ,^ b . :.. d t:.od cim lnr to that us:d .nr obtD inin ,, t. :.tti .ctijn c'.rvjS.. cons:3 ier.b'le a ,;:'rnt .'t lo.gi tudinal cc:';,;ressi n ini be i. .~. i:.cc; to n c ..m i..:.I .L lvi, :n3 .fl.. : l..te befo:'? there is an;. i.. d ,,ct ion :,.n tfi:. tr,:'_sv rze ca 1r 33$ ssion necessary.. to ,r ..l'Cea bckline:. The poresanc o' lori tudinal te:n.j.n .i :no, effect .uponL tlie tra:ive.rse stress :,ecessay to .prooc;.'c e b..cl:ling. The use c: tl:c inter:.?c c rvc toi e ter:.. ne ls.e critical stiuses ;'io a f icmi e i ?l Lte wjth. :' e r j''stra..t ind.epeede.t cf w avfE cngti ~ LU c seL ..Oe:'vative Lari ley ;o :moi'ial neronautical Labrrsator; Nattional Advisory Co.,niittee for Aeronzautic3 Langley Fpied, Va. TA.CA. ARI No. L6AO5a APPENDIX ,TT T, iK .' I." LOCI' P..T. UNTTDR 7.,0: D7'TCT STEESF ifr7r'til.. e'.cC_ *r nauZ.i..riun. The critical combination, of 1&'n tudinal and transverse direct stress that wi11 caue buckling in a, infinitely long flat plate with c..'s elatical.y restrained against rotation can be obtained 'c; olvinfg the differential equation of equi libriur. This equation, adapted fr m r.ae 324 of refer ence 4, is D + 2 a4 + + 2. + T 2 O (Al) bx4 x2 y2 b:i 6x2 yl)2 where x and Iy are positive for c:";ression. (The coordinate ~tte.' usecd ts given in fig. 5.) Equation (Al) may be rewritten and used in the following form: S+ 2 + + k + k W 0 (A2) Ox4 6x2 2 oy4 x 2 6x2 Y b2 y2 where wT . T2D and k = _1 ,]r*!on 7f r*iffre,:t' e c uat'n. If the plate is infinitely I.r in the xJirection, all disp2laci ~ nts are .riodic in 7 and the buckled surface is assu 1bd to have the form w = Y cos (Az) where Y is a function of y only and X is the h'lf wave length of the buckle in thi. x,..trection. Substitution :.x\A A2 T'L. L6Ai)a of the expression for w given in equation (A3) in the differential equation (A2) yields the following equation: d4Y 2n2 d2, n4 y4 n2 d2y + k.Y 1 + y =Y k (A4) dy4 \2 dy2' b22 o2 Y dy, Equation (A4) must be satisfied by Y if the assiued deflection is to satisfy the differential equation (A2). The expression i my Y = eb (A5) will be a solutlor. of equation (A4) when m is a ro)t of the characteristic equation m4 + ky 2m2 + =04 b (A6) The roots of this equation are iy = / 1, 2 + b S 1 V L2 + 4 +kx k) S= n/ 2 + 4Q{x ko) (A7 ) n3 = n/2 /k2 + 4 x ky) m4 = k 2 + k) J 2 k I ) j /1 r j The complete solution cf equation (A4) is therefore iyly im2y im 7 im47 Y = Pe + +e b + Re O + Se b (A8) where F, R, ad 3 are constants to be determined from the boundary conditions. S H..CA ;? "0o. L6AO5a The solution of the differential equation (A2) can now be written Simply im2y im3y im4y> w = e b + Qe + Re b + Se b os X (A9) Stability criterion. The boundary conditions that must be satisfied by the solution of the differential equation of equilibrium are (Y)y=b = 0 2 (Y) = 0 y=b / (A10) bb D ( b = o 4Su o) b Db = 4Sob The first two conditions result from the requirement of zero deflection along the edges. The last two condi tions express the requirement that the curvature at any point along the edge of the plate be consistent with the transverse bending moment at the point. If the conditions given in equations (A10) are imposed upon equation (AS), four linear homogeneous equations in P, Q, R, and S result. These equations are N\.A AR? I: ". L6'A5a i1m fro1 Pe Pe iml nmlPe p im. rm2 + Qe  '4 t ..c"^" i + 9Re + Re im2 I:3 + m,2n e + m3Pe i i.. iml + 1 id 1Pe + m2Qe ml2Pe  r, .) +im + me Le ' iin.r + SE 2 r 2~ = 0 im,5 im4 o 2 S Se = 0 Sin + mrnSe i m;R , + m.Re 2 ,JI S"iml1 + i lF e 2 im2 + r2ie " im3 + na t + m4Se = e J where 4S.cb D In order for P, Q, R, and S to heve values other than zero, that is, in order for the pl2te tD tuc':le, the determinant formed by the coefficiets of P, ., R, and 3 in equations (All) must equal zero. The e ansion of this retrminant is iven or. pare 13 rf reference 5 f:r the case in which the roots of the characteristic equation are of the farm 2 (A 12) m. n i+ 2 Y i" im4 +2 4 , + m4e 2 = 0 i_. im4 ~+ Re 1 2e ' + R2 Se4" Re 4 (^11) HACA ARR I:. L6A,5a In the present problem, the roots (equations (A7)) of the characteristic equation have the form of equa tions(A12), where Y = 0 y=o b2) + 1/ /k 2 b 2 k p2 ^ f k2 (4 cx kY) (A13) kv b2 21 b2 S y )k2 + 4 ky) Substitution of Y = 0 in the stability criterion given as equation (A19) of reference 5 yields a stability criterion that is applicable to the present problem. This stability criterion is a2 + e2) + (a2 2 2 sinh 2a sin 23 2 2ap L (cosh 2a cos 2p 1) + a + 2 j cosh 2a sin 2p (a2 + jsinh 2a cos 2p 0 (Al4) where a and p are defined in equations (A13). Any combination of values kx, ky, b/k, and E that satisfies equation (A14) will cause the plate to be on the point of buckling. Interaction curves for restraint independent of wave lenrtn. Tnc rroceure for plotting interaction curves is as follows: For a rIven value of c, a value of ky is chosen. Substitution of these values of ky and c in equation (A14) yields an equation in terms of kx and b/. A plot of kx acnainst b/h is then made. Every point on this curve represents a combination of kx and b/\ that will maintain neutral equilibrium for the given value of c and the chosen value of ky. :r.CA .UVP I:>. L A'ca J Since the plate will buckle at the lowest value of kx that will maintain neutral Ieqlilibrium, orly the minimum m value of k, i :21.een firon ,r:ie plot or lk, arai..t ,/\. This process Is re ,eaLed for other as..iled values of 1:,, and each timn a minimum alue of k1y is deterrrined. Finally, the int . action cur",e of 1:., against the nminium value of k. can be Plotte. for the .vn value of . ?zir tn ecal E~ of a :laPe zlith si:;rply supported edes ( = .), t :n (A.4) i.? imilfi d to such an extent .act th.e m:. r.. tion of ':,, with respect to b/k can readily be dn.c analytically.. The equator. of the inter action curve fcr r = O can then he given explicitly as / , 1 2 k 2 1 + (Al) The' plotting procedure just li:'u;sed rnd the analytical solution f)r the case .of simply supported edpes equationn (Ail)) 1ive only the curved portion of the interaction car'es. The conclusion that the vertical portions also represent critical itre:s combinations and are therefore Froperly a cart of the interaction curves depends upon an aru'L'ent analoEgoa to that at the end of appendix E of reference 2. This arpgu:ment is based on the fact that the end point of the curved portion of each curv1e can be shown to rerpreent a comnbinat ion of stresses for which the buckle wave length is infinite. When the wave length is infinite, the lonritudiinal stress can do no v;or': during, buc':lin. Accordingly, the transverse stress required to produce bucklin' is the same as it would be in tlhe absence of lonp'itudinal compre'ion. Inasmuch cs a reduction' in longitudinal ccmpreEsiin tends to increase rather than to diriini'h the wave length, the sane arFgument applies to all points on a vertical line below the end point of the curved portion of each curve. For a Fiven value of c, conequently, thoe critical combinations of stress for which the buckle wave length is infinite are defined by a straight line of contant ky thOat starts at the end point of the curved portion and extends indefinitely into the tension region of kx. This value of k, is the value corresoondinc to Euler strip buckling and is relateJ to e by the equation (adapted from equation (A21) rf rftrence 6) tan  2 NACA ARR ii. L6AO5a In reference 7, the problem of buckling of finite plates under combined longitudinal and transverse direct stresses is investigated. The, results given in refer ence 7 further substantiate the existence of the vertical portions of the interaction curves, inasmuch as the finiteplate interaction curves are seen to have portions that approach vertical lines as the lengthwidth ratio of the plate increases. In figure 6 the interaction curves for infinitely long plates with simply supported and clamped edges are compared with the curves, based on the results of reference 7, for similarly upported plates with a lengthwidth ratio of 4. Interaction curves for restraint dependent on wave length. Interaction curves for a plate with edge restraint dependent on the wave length of the buckles can be obtained by a slight modification of the method outlined in the preceding section. This modification consists in computing a new value of E to be used with each new assumed value of b/X;.no other change is required. This method can be applied only when the relationship between e and b/A is known. For the special case of a sturdy stiffener, the relationship of E and b/A is derived in refer ence 3. ZLCA AR il:. Lc. a ".' REFERELCES 1. Stox.ell, Elbridge Z., and Schvartz, Edward E.' Critical Stress for a. Infinitely Lonr Flat Plate with Elastlcally Restrained Edce under Combned Shear 'ann Direct Stresc. .!A.\,. A7C io. ZKl13, 1945. 2. Batdorf, S. E., and iloubolt, John C.: Critical Combinatimns of Shear and Transverse DOrect Stress for an Infinitely Long Flat Fate with Edges Elastically Restrained against Rotation. InCA .4RR '0o. L4L14, 1945. 3. Lundquist, Eugene E., and Sto'.vell, Zbridge Z.: Restraint Provided a Flat Rectangular Plate by a Sturdy Stiffener alonr an Edge of the Plate. !T1CA Rep. 1io. 735, 1942. 4. Timoshenko, Q.: Theory of Elastic Stability. icGraw Hill Book Co., Inc., 1936. 5. Stowell, r:ltridre Z.: Critical Shear Strer of an Tifin'tely Long Pat Plate with Equal Elatic Restraints against Rotation along the Faraliel Edges. 'ACA ARR ';o. 3K12, 1943. 6. Lundquist, Eucene E., Rosman, Carl A., and HMubclt, John C.: A Nethod for Determrinin the Coliu.r Curve from Tests ,'f Columns x'iti: Equal Restraints afcinst Rotation on the Ends. ::ACA T': Io. 9C3, 1943. 7. Libove, Charler, and Stein, fanuel: Chairts fr Critical Combinations of Lonritudinal and Trans verse Direct Stress for Flat Rectanular Plates. "IACA .'::i: . L '. t, 1; 6. T C A ARR I:o. L6A05a  2C 0 10 0, rI Co 0) O . .0 0 CO 5 O) to0 0 Q'0v 02,10 t) 0 0 0 .0 0 0 * V 0 CO tQ (X O H NO K O 0 O 0 (0 f H i 0 S) 0) tO H t C) O * 0 1V II O0 tO CO t1 0C 0 CO S o0 L H v > CD to C0 0 02 C l H .0 02 10 O H o 0 0 N t0 tO I S O H t 0 > t0 0 I i0 CJ 0 O NP ; C I L1 0 0 0 H H 0L G I ) C O O C LO I OQ CN C H H ( 0C C H C tO tO C to S O o to (DO Co C0 (0 N I M I (0 C02 0 0 0 1 '1 10 II I 5 > CO Y) H t C C ) I 0 10 0I O H V 0o IC S* I H H H H H z0 I I I I ii S0 0O Co 0 O 0 0 0 S, 9 S 0 04 0 0 i r 0 H No r(O C E 0 0 O rr? 0 CO 4t H NACA ARR No. L6AO5a (a) Loading combination treated in reference 1. I, t't t tt ttt (b) Loading combination treated in reference 2. 4, 4, 4, t t t t t t t t t (c) Loading combination treated In present paper. Figure 1. Buckling of an infinitely long flat plate under combined loads. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS Fig. lac Fig. 2 NACA ARR No. L6AO5a __ I a 0 17 N N .0 0o cO S/0' 0" .. O xL  o '+ U S*a c* a a . 8 /  _o ' 4 . SE  .0 0" OC 0 C Z; C0L 0 E e o ICn Y, No + a, C 2 ~ ~ ~ ~ ~ 1 as u (+ b c Oc E x o oa NACA ARR No. L6A05a Fig. 3 E //c  O 0. ,as S 0 Lo  A 3 u 10 IZ' 0 0 w9 rq c 4 C 0 01 o co C I4 Im ,1 . S O 0 C C E c I . /I~h S  ^ ; , //c Or I //r ^ I / ^ ' NACA ARR No. L6A05a A A Secton AA (a) Buckle form for shear alone. Lodes'.s 1 Section BB (b) Buckle form for longitudinal compression alone. I 1 1 1 \ \ \ I. t t t t t tt t Seton CC (c) Buckle form for transverse compression alone. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS Figure 4. Buckle forms for a simply supported,infinitely long flat plate. Fig. 4ac NACA ARR No. L6A05a NATIONAL ADVISORY COMMITTEE F01 AERONAUTICS Figure 5. Coordinate system used in the present paper for an infinitely long flat plate. Fig. 5 NACA ARR No. L6A05a 101 0 S b a b 8 Clamped ay edges r k ,"F Eit% 6 tr = i  4 _____________ z l4 4 0 1 0 I 2 3 4 5 k, NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS Figure 6. Comparison of interaction curves for infinitely long plates and interaction curves for plates having a lengthwidth ratio of 4. Fig. 6 UNIVERSITY OF FLORIDA lIIl IrI IU I II II II II 3 1262 07761 917 8 UNIVERSITY OF FLORIDA OOCUMiENTS DEPARTMENT 120 f tAARSTON SCIENCE LIBRARY L:DX 117011 , ..;.,,LLE FL 326117011 USA 