hJ^Cf\L'H^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED March 19^6 as Advance Restricted Eeport L6A05a CRITICAL COMBIHATIOKS OF LOHGITDDINAL AND TRABBVERSE DIRECT STRESS FOR AN IKFIirETELY LOBG FLAT PLATE WITH EDGES ELASTICALLY RESERAIKED AGAIHST ROTATIOU By S. B. Batdorf, Manuel Stein, and Charlee Llbove Langley Memorial Aeronautical Latoratory 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. L - U9 DOCUMENTS DEPARTMtNf Digitized by tine Internet Arcliive in 2011 witli funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation. http://www.archive.org/details/criticalcombinatOOIang FACA /'RR No. L6A05a NATIONAL >iDVTSOFY CGMyHTTEE FCR AERONAUTICS /vD¥;jCCE EESTRICZEH REPORT CaiTICAL COMBINATIONS OF LOKGITJDINAL AND TRANSVERSE DIRECT STRESS FOR i.N INFINITELY LONG FLAT PLATE '^ITE EDGES ELASTICALLY RESTRAINED AGAINST ROTATION By 3. B. Eatdorf, J/anuel Stein, and Cliarlss Libove STJI.TMARY A theoretical lnvestip;ation v;as made of the buckling of an infinitely long flat plate v/ith edges olastically restrained against rotation \and_er combinations of longi- tudinal and transverse direct stress. Intersotion curves are presented that give the critical corcbir.ations of stress for several different degrees c-f elastic edge restraint. Including siniple suyport and ccrnplote fixity. It W2S fo'jnd that an appreciable fraction of the critical longitudinal stress may be applied tc tVie plate without any reduction in the transverse compressive stress required for buckling. INTRODUCTION Because the skin of an airplane in flight is sub- jected to combinations of stress, attention has recently been given to the nroblem of plate buckling when more than one stress is acting. The present paper is the tiiird of a seri?s of papers analyzing the elastic buck- ling, under the action of two stresses, of an infinitely long flat plate with edges equally restrained against rotation and fully supported. The two previous papers are reference 1, which deals with the intcrsction of shear and longitudinal direct stress, and reference 2, which deals v/ith the interaction of shear and transverse direct stress. Tho present paper describes the inter- action of longitudinal and transverse direct stress. These three loading combinations are illustrated in figure 1. rl AC A /PtR No. L6A0"^a Interaction curves that give the critical direct- 3tres3 conibinatlons for sev3ral different degrees of elastic edge restraint, including simple support and CGJ.iplete fixity, are presented for the case in v/hich the nagriitude of the restraint is independent of the buckle wave lej-.gth. These curves are based on an exact solu- tion of the differential equation of equilibriu:ri, the details of v;hich are given in the appendix. SYMBOLS t b a D elastic nodulus of plate naterial Poisson's ratio for plate material thichness of plate width of plate length of plate (a > h) flexr:ral stiffness of plate per un' Et^ .ength i£(i - /) longitudinal coordinate "crans verse cooramate w normal displacement of a point en buckled plate fror,i its undef lected. "oosition half v/ave lengtli ci: biiclcle rotational stiffness of restraining nediuin along edges of plate, iiioi,ient per quarter radian per unit length dimensionloss elastic edge-restraint constant (¥) NAG A ARR No. L6A05a cr^ applied uniform longitudinal compressive stress o_ applied unixor-ir transverse ccrr.pressivo stress y ^ / o^^b^t k_, k,^ dimenslonlesr stress coef f icientr; ^ k^ = — ^ — ; x» y ^ ^ -y X ^^D ' a/o^t k = --^-.^ — , '^ rr'-D / Rj^ longitudinal di ■•^eot-stress ratio; ratio of longitudirsl dlrecjt stress present to critical stress in pure Icngitudin&l coirpression R transverse di reot-stras?; ratio; ratio of trans- verse direct stress present to critical stress in pure transverse compression RES'TLTS AND DTSCUcSIGK The results of this investigation are given in the form of nondlinenslonal interaction curves In figure 2. Each point on these curves represents a critical coifoi- nation of the stress coefficients k-^ and k„ for a given elastic edge -restraint constant e at which an infinitely long flat plate will buckle. The Interaction curver for plates with siwoly su'-forted and clamoed edges are given in figure 3 in tei'ms of stress ratios rather than stress coef t icisnts. The calculated data used to plot the interaction curves are given in table 1. Acplioabili ty of the interaction curves . - Critical combinations of "longitudinal and transverse direct stress for an infinitely long flat plate v;ith edges either simply supported or clamped can be obtained from the interaction curves of figure 2. Critical combinations of direct stress for a plate with interm^ediate elastic restraint against edge rotation can slso be obtained from, figure 2 for those cases in vdiich the stiffness of the rcp.tra'ning medium is independent of buckle wave length (£ = a constant). Such edge restraint is provided Ij. NACA ARR NO. L6A05a only by a medium in v/hich rotation at one point does not influence rotation at another point. Edge conditions of this type are not ordinarily encountered but might occur wlien the restraint is fiu-nished by a row of discrete elenents, such as coil nprings or flexible clarnps. Because of the great variety of possible relationships between edge restraint and v/ave length, only the curves for edge restraint independent of wave length are shown. If critical stress cor^ibinatlons for a plate with con- tinuous edge restraint (£ dependent on wave length) are desired, they can he conputed - though somewhat labo- riously - b^^" the method outlined in the appendix, pro- vided the relationship between edge restraint and v/ave length is knov/n. This relationship is derived in refer- ence 3 ^'o^ the special case of a sturdy stiffener, that is, a stiffener v/hich twists without cross-sectional distortion. The buckling stress for a finite plate can never be lower than that for an infinite plate having the same v/idth and thickness beca'ise the finite plate is strength- ened by support along t^'o additionS-1 edges. The use of figure 2 to esti:ic^te tiic critical direct stresses for a finite i^late with edge rsstraint independent of \.'ave length, thereforo, is in all cases conservative. Vertical portions of interacti on curves,- The vertical portion^ of the interaction curves (fig. 2) indicate that a considerable amount of longitudinal com- pression m.ay be applied to the plate without any reduc- tion in the transversa compression required for buckling. This result parallels the result of reference 2, in which it was found that a considerable amount of sheai-> stress could be applied to an infinitely long plate without any reduction in the transverse com.pression necessary to cao.se buckling. On the other hand, in ref- erence 1 it was shown that the presence of shear always reduces the longitudinal compressive stress required to produce buckles. This disparity in behavioz'' is probably attributable to the character of the buckle form.s foi' the tliree types of stress. (See fig. Ij.. ) The buckle form for shear alone (fig. l|-(a)) can be transformed con- tinuously into that for longitudinal compression alone (fig. li(b)) by a gradual addition of com.pression and subtraction of shear. Neitl^er of these buckle forms, however, can be continuously transformed into the buckle form for transverse com.pression alone (fig. I|.(c)). ITACA Ar:: No. LoAG^a The vertical portions of the inter-action curves e:.;tend indefinitely into the tension region of k-^. (?or c convenience, in fig. 2 the curves are stopped at a snail negative value of k-^« ) This property of the curves indicates that the presence of lonsitudinal tension has no effect upon the transverse stress necessary to produce buckling. SUIvHuARY C? RESULTS Interaction curves are presented froi'i which critical combinations of longitudinal and transverse direct stress for an infinitely long flat plate v;ith edges either simply supported or clarr-'ped can be obtained. Critical cc;nbi- nations of direct stress for interLiedlate elastic restraint against edge rotation can also be obtained from the interaction curves for those cases in v/hich the stiffness of the restraining mediuir. is independent of buckle v;av3 length. ?or cases in which the stiffness of the r-jstraining medium depends upon the buckle wave length and the relationship between the tv/o is known, the critical combinations of direct stress can be deteri.iined - though somewhat laboriously - by a method similar to that used in obtaining the interaction curves. A considerable amount of longitudinal compression miay be applied to an ir^ijiitely long fld^t plate before there is any redu.ction in the transverse compression necessary to produce buckling. The presence of longi- tudinal tension has no effect upon the transverse stress necessary to produce buckling. The use of the interaction curves to determine the critical sti'esses for a finite plate with edge restraint independent cf wave length !:•■ in -.11 cases conservative. Langley i\;emorial Aeronautical Laboratory National Advisory COirinlttee for Aeronautic: Langley pield, Va. ilACA ARH No. L6An5a APPSNEIX BUCKLING 0? 1N?T"ITSLY LOITC PLATES UNDLR TWO DIRECr STRZS'??:: Differerti?Ll eq^iation ^f pquil?.briun. - The critical combination? o;^ longitudinal and transverpe direct jrtresn that v;ill cause tuckling in an ^Infinitely long flat plate v'ith ecre.3 ela?tically restrained against rotation can be oLi:ained by rolvir.f: the differential equation of eq-ai- libriurs. Thi3 equation, adapted from page 324 of refer- ence 4, is rhere IT^ and IT are posltlT-e for conprecsion. (The coordinate sy.vte/n upcd is given in fig. 5.) Equation (Al) raa77 be rewi'itten and used in the follo^vin.'? form: A 4 ^,4 .4 2 ^Z 2 ^2 O W . o - ■■■' , O \V , , . IT f + 2 -^-rr— ^ + -^-^ + k,, ■^.- -^-^ + k^ IL;. ^ = (A2) ox^ 6x26v2 5^4 x ^2 ^^2 - J ^2 s 2 v/here .-b' A - — TT^D and I\L.b''' k = -J- Solution of differential equation .- If the plate is infinitely lonsr in the :N--Jirpcti.c;n, all displacements are periodic in >• and the buckled curface 1.3 a??ur.ed to have the form w = Y C03 — (A3) A. Vi'he^'e Y i? a function of y only anr? \ is the half wave lenp;th of the buckle in th''-^ >c-c*reci;ion. Substitution ■AC A AR?. Lio. L6A05a of the expression for w given in equation (A3) in the differential equation (A2) yields the following equation: d% 2Tr2 d^Y r it4 Y - TT- TT d2Y k^Y + ^ 1-:.. ^ = (A4) dyi \2 dy2 • x4 ^ b2\2 "^" ■ b2 ^'^y dy2 Equation (A4) must be satisfied by Y if the assumed deflection is to satisfy the differential equation (A2) . The expression l^:iy Y = e ^ (A5) will be a solution of equation (A4) when m is a root of the characteristic equation -^^-.,).3...„.gg.. n4 + 1 2 i ;he roots of this equation are "^ m IT:- 1 = y ^ - r^ * i/i^y' " " %^-y- - h) Ay hS 1 ■"3 = Y^' - ^ - ipy- + 4 >-^(k3t - ky) ^""^ 1 The complete solution of equation (A4) is therefore im]_y im2y im^y lm4y Y - Pe ° + 46 ^ + Re + Se (A6) > (A7) (A3) where F, Q,, R, and S are constants to be determined from the boundary conditions. o NAG A ;iRR Jlo. L6AC5a The sclution of the differential equation (A2) can now be written w = VFe + Qe b + Re b + 3e im4y cos TTX X (A9) Stability criterion .- The boundarjr conditions that irust be satisfied by the solution of the differential equation of equilibrium are n (Y) ^3 y=— ^ 2 (Y) b y=- = -4S 'd^Y \ ^dy2 J /■\ y= b "2 4S o ^"'h y (AlO) 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 (AlO) are imposed upon equation (A8) , four linear homo.^eneous equations in P, ^i, R, and 3 result. These equations are NACA ARR llo. L6A05a im-|_ imo irr.j iin4 2 '? Pe + e -^ + Re ^ + Se = -ir:i]_ ~i^2 -im3 -im4 Pe ^ + •;ie ^ + ■"^ 2 Re -^ + Se 2 im2 + TTip'^'ie ^ + ni_j'-Re im.^ lTn4 ^ + m,2se 2 'x / l!ni in-^ ims lm4 ^ -lehTi^^jr-e " + m2Qe ^ + m-^Re ^ + m^^e '^ J - Q •irriT - mg •im^, m3_ Pe ^ + m2'^Qe "" + n^^Re -^ -iran -im4 -42.86 '^ + icln-iPe \ 1' + nio.^e where c = + in-^Re 4S.b + D -ims -l:n4 \ m43e ^ y -^(AU) J In order for P, %, R, and S to have lvalues other than zero, that is, in order for the plate to tuchle, the determinant formed "by the coefficients of P, Q, R, and 3 in equations (All) must equal zero. ^ The er.pansion of this determinant is .^iven on page 15 of reference 5 for the case in which the roots of the ch.aracteris tic ■ equation are of the form mi = Y + 3 ^ 2 m-r. Y - 8 + la *->, (AI2: rxi \t la J 10 I^ACA ARR No. L' ;A05a In the present problem, the roots (equations (A7)) of the characteristic equation have the forn of equa- tions. ( A12) , where P- a = (A13) Substitution of y = in the stability criterion given as equation (Al9) of reference 5 yields a stability criterion that is applicable to the present problem. This stability criterion is \2 ,2 i f' sinh 2a si:x 23 - 2ap -- (cosh 2a cos 2p - l) / 1 + cja'a^ + P^ ) cosh 2a sin 2p - pia^ + ;?^jsinh 2a cos 2f j = J (Alii) v/here a and p are defined in equations (A3.3) . Any combination of val\ies k^, ky, b A j ^^-^ ^ that satisfies equation (A14) will cause the plate to be on the point of buckling. y e y Inieraotjon curves for restraint independent cf v/ave lenrth .- The procedure for plotting interaction curves is as follows: For a ri'/en value of e, a value is chosen. Substitution of these values of k- in equation (A14) yields an equation in terms and b/X.. A plot of k^ against b/\ is then Every point on this curve represents a combination and b/X that will maintain neutral equilibrium or and of made . of k. for the criven value of e and the chosen value of ^y -iCA mR No. L6A05a Since the plate will 'buckle at the lov/est value of k^ that will maintain neutral equilibrium, only the minimum value of k.^ i? caken fi'om the plot of k^^- against h/x. This process 1? repeated for other assumed values of ky, and each time a minimum value of ky is determined. Finallv, the inte''-^action curve of ky against the minimum value of k,,- can be plottei for the given value of e. For the special ca^e of a plate with siriply supported edges ( - = 0) , equation (A14) is simiplifled to such an extent that the mrinimizatlon of k^ with respect to b/A. can readily be done analytically. The equation of the inter- action curve for r = can then be given e.cplicitly as / , ^ k^ =: 211 + \/l - I^yy (Al£) The plotting procedure just discussed and the analytical solution for the case of simply supported edges (equation (Al£)) ^ive only the curved portions of the interaction curves. The conclusion that the vertical portions also represent critical stress combinations and are therefore properly a part of the interaction curves depends upon an argument analogous to that at the end of appendix E of reference 2. This argument is based on the fact that the end point of the curved portion of each curve can be shown to represent a combination of stresses for which the buckle wave length is infinite. When the wave length is infinite, the longitudinal stress can do no wor!c during buckllnp. Accordingly, the transverse stress required to produce buckling is the sam.e as it would be in the absence of longitudinal com.pression. Inasmuch as a reduction in longitudinal compression tends to increase rather than to diminish the v/ave length, the same argviment applies to all points on a vertical line below the end point of the curved portion of each curve. For a given value of f, consequently, those critical com.blnatlons of stress for wxid ch the buckle v/ave length is infinite are defined by a straight line of constant k„ that starts at the end point of the curved portion and extends indefinitely into the tension region of k^. This value of ky is the value corresoondlng to Suler strip buckling and is related to e by the equation .(adapted from equation (A21) of reference 6) tan - 12 II AC A .'\RR No. LoAO^-a In reference 7, the problem of buckling of finite plates under combined lonf;itudlnal 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 finite-plate interaction curves are seen to have poi'tions that approach vertical lines as the length-width ratio of the relate increases. In figure G the interaction curves for infinitely long plates ^vith simply supported and clamped edges are compared with the curves, based on the ref..ults of refereiice 7, for similarly supported plates ?;?■ th a length-width ratio of 4. Interaction carves for restraint dependent on vave length .- Int'^;raction curves for a' plate viith edge restraint dependent on the wave length of the buckles can be obtained by a slight modification of the m.ethod outlined in the preceding section. This modification consists in computing a new value of f to be used v;ith each new assumed value of b/Xjno other change is required. This method can be applied only vv'hen the relationship betveen e and b/x, is known. For the rpeoial case of a sturdy stlffener, the relationship of e and b/\ is derived in refer- ence 3. NACA ARr. No. LoA05a 13 REFERE1\'CES 1. Stov.'ell, Slbrldge Z., and Schwartz, Edward B.? Critical Stress for an Infinitely Long Flat Plate with Elastically Restrained Edge-^ under Co^nbJned Shear ana Direct Stresc-. NAOA^ARK No. 3K13, 1943. 2. Batdorf, S. B., and Houbolt, John C.r Critical Combinations of Shear and Transverse Direct Stress for an Infinitely Long Flat Flate with Edges Elastically Restrained against Rotation. !IaCA kRR No. L4L14/ 1945. 3. Lundquist, Eugene E., and Stowell, Zlbridge Z.: Restraint Provided a Flat Rectangular Plate by a Sturdy Stiff ener alone- an Edge of the Flate. KACA Rep. No. 735, 1942^: 4. Timoshenko, S.: Theory of Elastic Stability. KcGraw- Hill Book Co., Inc., 1936. 5. Stowell, Elbridge Z.; Critical Shear Stress of an Infinitely Long Flat Plate with Equal Elastic Restraints arainst Rotation along the Parallel -es. FAGA^ARR No. 3K12, 1943. 6. Lundquist, Eurene E. , Rossrian, Carl A., and Houbolt, John C.: A Ilethod for Determining the Colui.m Curve from Tests of Columns with Equal Restraints against Rotation on the Ends. NACA TN No. 903, 1943. 7. Libove, Charles, and Stein, Manuel: Charts for Critical Combinations of Longitudinal and Trans- verse Direct Stress for Flat Rectangular Plates. NACA /JiR x^-. l6.v03, lw[;.6. Ik NAG A ARR No. L6A05a X J4 o 00 1-1 1^ l-l n o M O o Ph CO w o o td Eh M O (jO •<4 W m Eh !> w to lO to iH m c^ o •K '^f cn> CO o ' O C\J .. KJ %l^ "sf LO CO CD CO o — -v ■ rH 8 II O Lfi o o o o o O t>. O CO LO o o o o o w r^ • • • • • • • • ■st< w to to c\: H o 1 '0 1 \ £> o Oi o: o lO o o> CD M iH a> ■<\i c^ CO H CO c:^ O M 1 O W J CVl t^) c^.-) •* lO ti-,-; CD 00 O — • II , 1 in KJ a) to C\} O ^< CO ^ K'V 00 i;-— ♦^i rH ^ t> 00 CD rN • « • • • • • • C\J W CO C-J H o C-V! 1 1 1 c- rH CD CJ ^ C\j CO to N o C^ fH CO .>J r- w to V ^9^,-C^. Oi K5 to ■* ^ CD !> ' 1 to 11 w o o O r-\ ■vt' !> CO CD l>> ^0 C\} O X> rH LO CO LO ^ • • • • • • • • C\J w C\) rH rH C\5 1 1 1 1 C\J CM H CO CD CD CD J> K o (0 KD CD CO Cvl CD ■sH II ^.2-.,^> W to K.1 fi ■^ li'3 CO - CD ■vt< K5 0.1 rH vti CO t>5 CD LD Cv! O CO ^ to oa ^J ■ • • • • • • • • rH <-\ <-\ r^ r-l 1 to 1 O o ^- o fH ^0 o to CO N O ^ o o ■sfi r- Cj 0?. '^h ^ • ■ • • • • • • m o V O C\) ,. (M w to to t-O "^ ^'^ UT 11 o to m LO o LO o O t>5 o O^ CO rr- LO CO o o ^ • • • • • • • iH o rH 1 CO 1 o M Eh >H t3 « < O &; CO o H p:; > H < 1-1 O IS? O H H W ^ EH o o NACA ARR No. L6A05a Fig. la-c (a) Loading combination treated In reference 1. 'f.^ ■f^ y -< ^ ^ ^^ Y -^ T-^ T -'' T ^ ^Y (b) Loading combination treated In reference 2. i i i I 4 i I i i I t 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. 2 NACA ARR No. L6A05a U N II '-9 u N 3: I b'^ o I- ■g -•- c n c -•- ? i- -♦- E (U o o k. o M- >4- >^ n o — (> o o -«- o 'i- (0 o Q) o en c 1n (U ^ 1 -t- 0) o u CT> X -o ;/) ■6 a> (D -C (U in ^ s> i- -1 0) a > lU (D -^— r a o o Q- -»— -(— .»_ u o • n -D <*- v_ C C- o en o r iw a h— 1 — C) n o 1 c «_ -n >. OJ -) V -»- +- -4- o NACA ARR No. L6A05a Fig. 3 ■d c CD c « (D o cd -4J «2 iH in o a. V tH k. 4J -P a E CD CS > ^ J rH O (1> 0) -IJ Jh -h •M c ■a -H tM rH C a -H c •H G "d CO p« -IJ fn >. •rl O to tn a: c o rH t>5 K > m hr\ 01 C