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ARR No. L4L14 i6 . i .: NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WAIRTIMEjI REPORT ORIGINALLY ISSUED January 1945 as Advance Restricted Report L4Llk CRITICAL COMBINATIONS OF SHEAR AND TRANSVERSE DIRECT STRESS FOR AN INFINITELY LONG FLAT PLATE WITH EDGES ELASTICALLY RESTRAINED AGAINST ROTATION By S. B. Batdorf and John C. Houbolt Langley Memorial Aeronautical Laboratory Langley Field, Va. UNIVERSITY OF FLORIDA DOCUMENTS DER RTMENT 20 MARSTON SCIENCE LIBRARY FL 326117011 USAI 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. L 183 =1 12 4 L(O( vcF'&ur? NACA ARR 11. L4L14 ITAT'IONAL ADVISORY COVIIITFEE FOR AERONAUTICS ADVANCE RESTRICTED REPORT CRITICAL CDj;BI3TATIONS OF SHEAR AND TRtaNSVr.FEE DIRECT STrfS; FOR A:l I!F II'IT'ELY I:ON. FLAT PLTE VvITY EDOGS: ELhSTI C.'LI.Y Fr.STRAil'.D ACAUIST ROTA'TI'?I1 Fy S. B. Batdorf and John C. H.ubolt An exact solution and a closely concurring approxi mate energy solutionn are civen for the buckling of an infinitely long flat plate under combined shear and transverse direct stress vith edges elastically restrained against rotation. It WvEs found thi. an appreciable frac tion of the critical stress in pure she.r may be applied to the plate without any. reduction in the transverse compressive stress nrce.ssaryi t:o pr.r.duce buckling an interaction formula in general use. .as shown to be decidedly cnnc rvatlve f.r the range in which it is sup posed to apply. I: TR CLU'7T I UI In the design of stressedskin structures, considera tion must sometimes be riven to the critical stresses for a sheet under a combination of shear and direct stress. The upper surface of a wing in normal flight, for example, is subjected to combined shear and compressive stress, and the lower surface is subjected to combined shear and tensile stress. The upper surface may then buckle at a lower compressive or shear stress than if either stress were acting alone. The critical shear stress for the lower surface may be increased by the presence of the tensile stress. NACA ARR No. L4L14 Tf the wing has cloFel sFpaced chordwvie stiffeners, the skin betveen two adjacent stifenerr may be regarded as a long sheet olichtly curved in the longitudinal or chordwise direction ani straight in the transveroe or spanwis7e direction loaced in shear and transverse direct stress. A conservative preliminary estimate of the critical stresses may be obtained if the sheet is con sidered to be flat and infinritel; long. In the present paper, the critical stresses are compouted for an infin itel,7 on,7 flat plete loaded as indicated in figure l(a). The corresncpndin legalization c.f the case of a wing with spanwvise stiffenrs, sh:'n in figure l(b), was treaLnt in refererce 1. < [ I I I (a) TjTe .1 ? ac' r.n prcble. solved in present paper. LI, I (b) Type )f loading problem solved in reference 1. FIgure 1. Buckling of an inifinicely long plate under c3robnined lads. NACA ARR No. L4L14 3 CCNVENITIOQIAL INTERtACTION! FOrVTJLAS For buckling of structures under combined loading conditions, no general theory has been developed that is applicable to all cases. Stress ratios (reference 2), however, provide a convenient mletlod of representing such conditions. For example, the ratio o' the shear stress actually present in a structure to the critical shear stress of the structure when no other stresses are present may be called the shearstrets ratio. Stres ratior may similarly be deflne.iel for each type of stress occurring in the structure. It is generally assuinmd that equations of the type 1lP + Rq + Rr + ... = 1 (1) may be used to express the bucf.linr conditions in the case of combined loading (reference Z, p. 1 18). In equation (1), R1 ,R.T, and R7 are tresz ratios and p, q, and r are exponents chosen to fit the known results. (All symbols are defined in appendix A.) Such a formula rives the correct results when only one type cf loading is present and has the further advantage of being nondilmensional. Equation (1) implies, more over, that the presence of any positive fraction of the critical stres of one type reduces the amount of another type of strosE required to produce buckles; this implication appears reasonable and ias been proved true in some cases (references 1, 2, and 4). In reference 2 the following interaction formula is given for an infinitely lonr plate with clamped edges loaded in shear, and longitu6inal compression: R,1.5 + Re = 1 (2) where R, is shearstress ratio and Rc is longitudinal directstress ratio. The same formula is recommended in reference 3 for general use for the buckling of any flat NACA ARR No. L4L14 rectangular plate, recardles of the direction of com pression and the degree of edge restraint. Later theoretical work (reference 1) shows that, to a high degree of accuracy, for an infinitely long plate with any degree of edpg restraint loaded in shear and longitudinal comp:ression R2 + c = 1 (3) The sare formula was found in rcfrrence 5 to be appli cable to simply supported rectangular plates of aspect ratios 0.5, 1, and 2; and the conclusion was dran that interaction curves in rtrescratio form are practically independent of the diimernions of the plate. The present analysis, Lw'ever, indicates that the buckling of an infinitely long plate loaded in shear and Lransverre compression (fig. l(a)) is not adequately represented either by equation (2) or (2) or by any formula of the type of equation (1). Two independent theoretical solutions to this buckling problem are given in appendixes 5 and C. Appe.di.: E contains the exact solution of the differer.tial equation of equilibrium, and appendix C contains an ener r' solution leading directly to an interaction formula. This energy solu tion, which Elves approxi.nate values only, was made to obtain an initial quick survey of the problem and to provide a check on the results of the exact solution. Approxinate interaction formulas in substantial agreement with these results were givcn for the cases of simply supported and clar:ped edges in reference 6. RESULTS A:iD DISCUSSION In figure 2, curves are given that indicate the critical combinations of shear and transverse direct stress for an infinitely long plate with edges elasti cally restrained against rotation. These curves are computed from the exact solution presented in appendix B. The degree of edgF restraint is denoted by c, which is defined in appendix B in such a way that zero edge restraint corresponds to simply supported edges and NACA ARR No. L4L14 5 Infinite edge restraint indicate cl.arped edges. A similar set of curve. is given in terms of stress ratios in figure Z. The n'u.ieriica] values used to plot figures 2 and 3, together v:ith the valeies found by the energy solution, are given in table I. Ihe most strlkinr feature of these result is that an appreciable fraction of the criticls stress in pure shear can evidrr.tly LF a.plied to tii plate without any reduction in ch;ee conprsve tre nrecu C ary to produce buckling. (eec fig. Z.) Thi. fraction ivrie, from about onethird to rrire than onehalf, ,dI^Fndinr on the degree of restraint. At hear stresses higher than those correspondirr to this fraction, the cornPoessive stress required to produce buckl!er is reduced 07 the ;resence of shear. The result that the com.,reFsive buckling stress is entirely unaffected by thle pro ence of a con siderable amount of irhear is robabily peculiar to infin itely loni plates. It is to o~ ex'ncted, however, that this result will b. closely ,pr.ached in the case of long finite plates. In fi ure 4 a coTpari son is miade between the exact solutions and the interaction 'rT.ulaF o:f equations (2) and () Equation I2'), hic s tn' "nteractior formula in general use, is .se.n to be decid',dly conservative. C ::C U 3S T O'S The exact solution. of the differential equation for the buckling of en infinitely long flat plate under com bined shear and tra'nsverse direct sLres vith rdges elastically restrained against rotation indicates the following: 1. An infinitely long flat plate Tray be loaded with an appreciable fraction of its critical stress in pure shear without causing any reduction in the transverse coimpressive stress necessary to produce buckling. 2. An interaction formula in eneral use for rec tangular plates in combined shear and compression is 6 NACA ARR No. L4L14 decidedly corservetive when applied to an Infinitely long plate ir. shear and trar.nverse co.iprecsion. Larnley "e orial Aero iutLc t Laboratory M'ational Advisory Cr,Littee for Aeronautics Langley Field, Va. NACA ARR No. L4L14 APPE'LIX A SY: E C LS C1,C2 functions of edre restraint coefficient c ,iv\en in apnecndix C D flexural zstffness of plate per unit length, inlb t" 12(l  E elastic mdulus of materiall ps Ii comorespive forc6 per unit length, Ib/in. Nxy shear n.' force per unit length of plate, lb/in. S rotqtionil r.tiffness p.er inch of restraining menber at ed: e of plate, lb/rediai Tc work done by cimpressove force iJ'r half wave length, inlb T, work done by shear f.rce r er half wave length, inlb V1 strain energy in plate per l.alf wave length, in]b V2 strain energy in ed.,e, restraint, p.Er half wave lergth, inlb Y function of y associated v.ith c.eflectimn of plate during buckling b width of plate, in. b wvidth of plate in oblique coordinate system: of reference 1, in, c function of a, '., and \ kc,,k critical compressive and t.earstress coeffi cients, respectively NACA ARR No. L4L14 m flt f, f t root of a characteriC? c equation of appendix B functions Df restraint coefficient e given in rhpreronfix C thickness of plate, in. nb u = 2? w, y a E Y C 0 07 T e = tan R F: ,RZ Rs RC cr displacemaeent f" bucicled late from original PY) S1 1 )IL arnplitvde of atLrumed rave form of buckle lornitudinal coordinate of plate tra~i ver.e coardi nate of pIste functiIcns of \, y, and kC nrnr.irenr.?cnal coeff'jcient: of Edge reFtraint ore f t:j .paramIntFrs deteFrr.milnln buckle form nr. of tw) rararamter. determining buckle form (i.elf ..p.ve length of b!,cle, in.) PDiT on'_ ratio direct trees ? transveire direct scrcss, Ip. Hear stress psi anlec betvrecn buc' stress ratios shearFtress rntiD loriitudnael dlrecttreFs ratio critical (used as subscript) NACA ARR No. L4L14 APPEIDI': P SOLUTIOC1N Y DIFF'ERETIT i AL EtUAT ION Etateir.ent of pr:,blerm. The exact solution for the critical strsr at \ll.ich biick:lir:sg cctur irn a flat rectangular pl.te sub :'ected t? coarir.ed .hear and com pression in its own plane may be obtained by solving the differential equatioi that eroresses the equilibrium of the buckled plate. Tne plate is assumed to be infin itely long, and equal elasti3 'retraints again t rotation are assurned to be present. alone, the tv.. e . s of tine plate. Differential eq,,lati.:n. 7'gure e shE .s the. c: dinLate system used. 'the differential equation for equillbr Lun of a flat plate under .er he ar an. tr.a:.sve'r.e direct stress is (from reference 7) 4w w 64,h w + 2 +  + 2rt Dx 4 2 2 t "') It is convenient to write TT and dimension ]ES bTucklir, C ctfficients means of the relations y  k rr D T b't bht Substitution of the excrecsians from equations (E2) in equation (31) o w + Oa.t 0 : (5' T in term of the kc and k, by > (P22 / for a d a.d T gives 1) ) 10 NACA ARR No. L4L14 ,4w 56 8 w 4 22: 2 A2k 2 2 "1 ;2 + 12:c 62w + c + + =0 (Bz) 6x4 xy2 gy4 b2 6Xy 2 672 '3Sr ti r. of C ffrreritiaj. equation. If the plate is ir.finnit(. li .I' ".7 the yd~rectic.or', all di placements must be periodic in x and the deflection Furface may be taken in the form IT w = ':e (64) where Y ic a fur'ctinn Df  nly and X is the half wave lern th of tl' oud;!les ii. te y:.ir:ectifn. ST't i tirnt i o) f C: e ex.;ressior fnxr w from equi b'L n (4) in tie Liffe.r i c ~,rt equati ;i (; ) rives. th.e f':'llm ir:g a the Equaticr, th t d t.er:nlr F ': U 2kc + 'a 2d ik d  +   = 0 ( BE) :1 \b d..r b dy A sJlution of equLticn (E5) is v Y = e wl.re m is a root of the characteristic equation n4 +2 + m T2( m +( )4 = (s) B:cept for th.e s'ib tituTion of 2(b )2 2kr fir 2 equation (56) is i.de:.tical with equa tion (Ae) o' appendix A of reference 8, in which NAdA ARR No. L4L14 11 equation (B2) of this appendix was solved with kc = 0. With this change, all the results obtained in appendix A of reference 8 are applicable here. The stability criterion for combined compression and shear is therefore the same in form as that for shear alone, given by equa tion (A19) of reference 8, which is 2a (Y2 +(cosh 2a co. 2, cos 4y) 4,2p2 2) 2 + 2) (42 + a2)l sinh 2a sin 23 + Ea(4y 2 + a2 2) cosh 2c sin 22 + 3(4Y2 2) sinh Za cos 2: 4apY sin 4 = 0 (B7) The relation between k. and a, p, and y is also the same in f1rm as that in equation (A23) of reference 8, namely, k = ( + 2) (8p) S2 rrb In the present report, however, a and p have the following values: a c + \ v(,2 + c)2 _ (S)4 S(B ) S= \c + . c)2 , where c l/Y2 + 2 2 k c = y 4V + "c NACA ARR No. L4L14 As in reference 8, the restraint coefficient c is defined herein by the relation Sb D where S is the ratio of a sinusoidally applied moment to the resulting sinusoidally distributed rotation of the restraining element measured inradians. Evaluation of kg corresponding to a selected value of kc. The procedure for evaluating k,, after values of kc and E have been chosen, is as follows: A value of bA is selected; a series of values of y are assumed until one is found that, together with the" corresponding values of and n computed from equa tion (.9), satisfies equation (B7 ; kg is then com puted from equation (B8). Another value of b/A is seleQted; a new set of values of y, a, and p is found that satisfie.s the stability criterion; and a new value of ks is computed. The entire process is repeated until the minimum value of k. can be found from a plot of ks against b/k. When e is a func tion of b/k, E must be reevaluated each time a different value of b/, is selected. The minimum value of ke and the chosen value of kc, when inserted in equations (B2), give a critical combination of shear and direct stress. Evaluation of ks when kc..has value corresponding to buckl.ini as E'uler column. One critical combination of shear and compression is simply ks = 0 and kc equals the value corresponding to buckling as an Euler colCrn. The curves giving critical stress combinations, however, did not appear to be approaching this point as their construction progressed. It was therefore neces sary to determine whether values of kS other than zero are critical when the Euler compressive stress is reached. The determination of k., when kc reaches the value at which the plate buckles as an Euler column requires special treatment, because k4, given by equation (E8), becomes indeterminate when the wave length becomes infinite as suggested by the energy NACA ARR No. L4L14 13 solution. The result that \ become infinite when kc takes its Euler value is readily checked from equa tion (Cll) for the special case of E = 0; for this 1 case p = q = r = 7 and (kc)cr = 1. From equation (B8) it is clear that, if kQ is to remain finite when the wave length approaches infinity, either Y tO (a) or a2 + :2CL (t) For case (a), when c = O, it fIllows from equa tions (59) that, to small quantities of the second order, a = 1 2 + 2u2 i  ,.( 10) where nb U=  2X If the value? of a and p from equations (B10) are substituted in equation (B7) and the resulting equation is expanded, with only the lowest powers of u and y retained, =2 4 ~u (B4) 64 6n2 Substitution of the values of a and p from equa tions (B10) and (B11) in equation (B8) gives as the final result for c = 0 2n 1C = \6 4 62 14 FACA AER No. L4L14 Case (b) cen be analzed ,y a sIrilar method, but the aa r, l.Is I., quitc c m:,l.icated because ter.n? of third ordr ..rut '=v retained. For. E = 0 and = nm, case (a) and case (b) were four to lead to eyactl:y the same re.ul t for k.. A :,al,'.e of : other than zero when kc takes its Euler value may be fou cd in the same manner f:.r oth,:r valu.cs :,f' edge restraint. FDr any value of the restraint coefficipnt E, IcC j n i ke + 2 f k + in T\k L\ r c 2+ 2 C s "/ "c . (cos n 1) r; 3 s '. (s12)  kc k ,2En .jk. + .n  sin r T 2\.. + " k, + E cos n K ( 2 \ 2 + 4 + 8 + ?c2 C+ cos T 1 where k, has the vwlui correstonl.rLr to Eulrr column buc:klit,, at this vqlue pf c. The relationsh:ip between this E:ulr value of kc and the c rrespon.dinr c LI givrin by tre eqiati :n (from refer,ce 9) n \f kc t r NACA ARR No. 14L14 'Vhen kc reaches the Eulcr value, the critical shear stress coefficient kg car. therefore be either C or the value given by equation (Bl2). The conclsion that ks can also have any vala bes'.:'een tlese lijmitL is plausible on t:he baSis of the follo'wiprg rhr:ical con sideretion The snr stress does nc. w.'r: dluri.n buckling when tl.e stress condition is such that the plate buckles with an, intinit. ,'a.e Jlencth. The effect of shear, furthermore, i" to reduce the ws.ve l~ig.th to a value of the order of thE width Df thn ;:plte. The wave lenth at the tir.:e I'ickling, ocars is Iinfinte, however, when the pdate i" eiSher in pure compression or at the value of k. Fati sfyirin euation (E12). 'T:) fact means that, for values of ks bet.vecn 0 and that given b. equation (B12), the shear strec is not great enough to force bu.c Clin: in s.rt waves and there forc does not REs'i i.n .,rodC,. ri. r bucl.les. I, ti' range of shear stress the c:oipreer'ive 3t.ess' nece' .arv to prdu.ce buckling is, cr3nsequerntly, the Zuler tress. NACA ARR No. L4L14 APPENDIX C SOLUTION BY ENERGY METHOD FOR EDGE RESTRAINT INDEPENDETT OF WAVE LENGTH The critical stress is determined on the basis of the principle that the elasticstrain energy stored in a structure during buckling is equal to the work done by the applied loads during buckling. If the structure under consideration is an infinitely long plate under combined shear and edge compression with edges elasti cally restrained against rotation, this equality may be written c +Ts = V + V2 (Cl) In reference 1 an energy solution was given for the type of loading shown in figure l(b). The deflection function used in reference 1 is also suitable for appli cation to the solution of the type of loading shown in figure l(a), which is the loading considered in the present paper. The values for Ts, V1, and V2 may accordingly be taken directly from reference 1, but Tc must be recomputed to apply to the case of transverse compressive stress. The following substitutions are used to transform the energy expressions from the oblique coordinates of reference 1 to the rectangular coordinates used in the present paper (fig. 6): Reference 1 Present paper y/b1 = y/b (C2) bl cos Q = b For brevity the following notation is also adopted tan p NACA ARR No. L4L14 By us of equations (C'2), the expres3? ons from reference 1 that are used in th.e present paper may be rewritten as follows: bG TV = w 2 2 f Vl "2 )2f 1 D 22)(2 + @2 23 Vr, z c W' W %"2? (C5) (C 4) (2 }) where f (7 4 \ 1  4 \ 1 n 2 1 fg = C + f7 fe ^ and c is the restraint coeff .crent defined in appendix B. The work done by the c=z rr..e"i.ve force cer half wave length may be written ("C) b \ Tc = iy )2 Jx dx ,. 2 As in reference 1, the assumed deflection function is 1 : 1+  C 18 NACA ARR No. L4L14 w = w, 4' ) ( + 4)2s3 CO 'x + Sy) (C7) t " _ TWhen the erprpezrion for w from equation (07) is substituted in (Ce) ard ith indicated operations are perf.rmrr.d, j"TT2t (c2 Tc = wo 4 f. + f (C) Valurs from equations (Th), (04), (CE'), and (C) arc no sb..~Ot'.*:ted in tlie bu2 r2D j b rL eliminate c'x:r, and N~ T'he resulting ecuatior. gives the critical z'n.bination Df stresses and r.ay be written as k = L2 + 9 l + 2(1 + 30)f 2, IF% f 6f + kc. f2 + 6 f (C9) "1;? Equt ion TL s1' Ltat, for a <:lected value of kc, t'e crittcel ~ iFear stress? deenS ds rpon the wave length ard the annle of t!e buckle. Sirne a structure buckles at the e13 l t 'ties at v',hLcl irstablillty can occur, k i:s :rinirizeu '.ith respect to wave length and anrle NACA ARR No. L4L14 19 of buckle. The minimum value of ks with respect to value of wave length is determined from the condition 6ks 6 = o (c10) which gives (when e does not depend on wave length) Gy (ll) f + _ kf2 f 2 kc Substitution of this value of wave length in equa tion (C9) giveT 1 + 2 fl9 I :F j= + fl3 kcfo] + ~i21 + 3)f : 2f1 (C12) f l (1+ 38.> C The minimum value of kg with respect to angle of buckle is found from the condition = 0 (C l) which gives 1/'2 2 L Tl( + f3) kc f2 + f2 6 (014) 1( + f kc? 2 + ,f2 k 20 NACA ARR No. L4L14 Tf tlis value of 9 ir substituted in (312), the final result is the fjllwing interaction formula, in which .k is 7iven in term: of kc and the cdge restraint C: Lo2 = 4?i+ ^.(C2 kc) + (402 kc) J2( C2k) (c15) where f + c 01 = 1 I1 2 .  ~+ 1 \ : +" 1 , + 2 2 f., CU = f, 1 4 (+ +  + 1 .. 2\ 1.12 2 +' + + + STT2 \1J' and = r c IIACh mRR !Io. L4L14 21 1. Stovelil, Elrridge Z., and Schv.artz, Ed.ward 3.: Critical Strese for ar, Infinitely Loni Flat Flate with Elsticall:, Retrained Edics 'inder C.mtrbined Shear and Lir.ct Btree A2.. F. T... 3 Kl., 124 Z. 2. Shanley, r R., and' Ry.er, E. 1.: 'ftrFer at!tos. The rN.swer to th.e Colr.bin.?e L;adinE Prob. lem. Aviation, vol. 3S, n.. e, June 1.9?, pp. LC, 9, 43, P7' 3. Anon.: Streng;th of Aircraft events. AR.5, A. .. I'av"Civil "munittee *on t.icraft Le irn riteria. Rvsd e d., Dec. 1942. 4. Crate, Far ol., 5atcrf, 3., and Fai, 3erE .: 1T e Lffect of Tr.ter.m1il Pr'esu e oni the Ecr.klin., Stre s 3f' Thi'nrI a'. llet Ci1:ct.l ar C: li.i er? under ors ion. F:'CA :. .1 L4E 7, 1 . 5. Hopkins, H. 0., and Rao, :. V. 3. C.: The Initial Buckl:inz of F'lat FPectan,.u:r Pdrls under Combined Shear anid Co:mpre i *n. r. "'. S .E. J244, Pritish R.t.. ., a'arch 1I.4. 6. WVarner, Ferbert: Ther rin.truLkionS und Eertchnu,,rsfragen ,:Ces Te :.FC"ba', J hr:b ;VCuL, 192i S R. Cli'rE.tovLr. Lcb',ijchen r.cd Es. t in), p.. 11. 125. 7. Timoshenk:3, S.: 'iheory of Llastic Stability1 i1zGraw Hill En?' Co., Inc., 1 .i_1 p. '5. 8. Stoweli, Elbrid1e 2.: Critical Shear Strees of an Infinit.ly L:.nr Flat Plate .with Eqiual Elastic Festraints acairn st r.tati.rn, alone, the Faralle?. Edges. 'ACA .i'F i Z '.:12, ?. 9. Lundquiit, L.&eie L., Ro ra Crrr] A., a.id Hoi uolt, John C.: A "Method for DLter:rininru the Column Curvc from Tsts. of ColuP.ns with Bqual Eetraints agai nst Rotariorn .on the Lnds I:ACn T11 iI?. 9032 1 D .41. NACA ARF No. L4L14 TABLE I. VALUES OF ko A!D Rc WITH CORRESPONDItG COMPUTED VALUES *" Ks AND R. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS i/: NACA ARR No. L4L14 Fig. 2 // Sd I L I .0 u SL______ 0 _i 0 / . L q I ,! II, 0 I" CL 4 /I I 1 !lu u ; _/ L1 ) L 0 00 S U U > 54  b+ + In c c e 1/ o at ana  / 1  ? Bo0^^ i * 7e I 77 o <\J 1T>' NACA ARP No. L4L14 / c c 0 L. o oL 0 u * S0t ~t~~^\ ~ZlL~~ nt P4 o g 0 "1  O . e I  1 1 1 1  1 1  1  1i 4 I N~ U w 8 0n a oo ' C o 4 on Uo I. u o ci a . 4 Ln In kI r0 4 L 0 U  I u o a Us. O O0 C . u 8^ CC 4c L 0 V 0 tl_ c Mt lz Fig. 3a NACA ARR No. L4L14 (b) Shear with compression. Figure 3. Concluded. 1.0 0 NATIONAL ADVISORY ___MMITITE FOR RAUTICS Fig. 3b NACA ARR No. L4L14 0 .Z .4 .6 .8 1.0 R NATIONAL ADVISORY S COMMInEE FOR AERONAUTICS Figure 4. Comparison of correct interaction curves with a curve formerly proposed for infinitely long plates under combined shear and compression . Fig. 4 NACA ARR No. L4L14  x Figure 5. Coordinate system used in Appendix B for on infinitely long plate under combined shear and transverse compression. c NATIONAL ADVISORY COMMITIE FOR AERONAUTICS Figure 6. Coordinate system used and wave form assumed for energy solution in Appendix C; inclined lines indicate nodal lines of buckles. Figs. 5,6 *: 1 i i i ? r UNIVERflY OF FLORIDA UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 1 0 M~.ARSTON SCIENCE LIBRARY P.O. BOX 117011 GAINESVILLE, FL 326117011 USA "N wit 
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