Critical combinations of longitudinal and transverse direct stress for an infinitely long flat plate with edges elastica...

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Title:
Critical combinations of longitudinal and transverse direct stress for an infinitely long flat plate with edges elastically restrained against rotation
Alternate Title:
NACA wartime reports
Physical Description:
14, 6 p. : ; 28 cm.
Language:
English
Creator:
Batdorf, S. B
Stein, Manuel
Libove, Charles
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Airplanes -- Design and construction   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: A theoretical investigation was made of the buckling of an infinitely long flat plate with edges elastically restrained against rotation under combinations of longitudinal and transverse direct stress. Interaction curves are presented that give the critical combinations of stress for several different degrees of elastic edge restraint, including simple support and complete fixity. It was found that an appreciable fraction of the critical longitudinal stress may be applied to the plate without any reduction in the transverse compressive stress required for buckling.
Bibliography:
Includes bibliographic references (p. 13).
Statement of Responsibility:
S.B. Batdorf, Manuel Stein, and Charles Libove.
General Note:
"Report no. L-49."
General Note:
"Originally issued March 1946 as Advance Restricted Report L6A05a."
General Note:
"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 previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003638646
oclc - 71843726
sobekcm - AA00006238_00001
System ID:
AA00006238:00001

Full Text

~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


zr-ILq






































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









T-ACA .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, GIT-JDITit.. 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. Eat-dorf, 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:.-:v-rs 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 ffr-nt d 3re:- :.f .l5st' ed'_e
restraint, ir.cldii F rin. pi su-u 'port n:rd ;.n 1'.t-l; fixity.
It ws f.junrd th-it 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 trans-verze comrressive stress
required 'for buckling.


I NTFP.'DUJTI ON


Eecause the skin of an .'.rr-l n jir flight is sub-
jected t_, ccmi.,r,atirni of stress, attention ha- rec-ently
been ,iven to the problem of nlcate buIcklln.g whnn more
than cne tre.-s is .acting. The ore:-nt o--por 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-. e-iaily r-rtr-ainrd a -inst
rotation and fully sunoorted'. The tv'o orevicus L&pers
are reference 1, which deals with the int:r-sction of
shear and lonittudinal direct stress, and re:feren-e 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, inclrdi-i 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 le-r.-t> c buckle

So rotational stifi.-.e-ss of restr-i:.ir :rediiu along
ec:.es of plate, moment per quarter radian o.r
unit ler., th

dimensionless elastic edge-restra-:lt constant


,SD )








NACA ARR No. L6AO5a 3


Cx applied uniform .l.ongitudinal cor're-si"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,; :et-stress rPatio: r-tio of
]3nituJi l .*i'.ret stres:' p;r,- ser-nt to critical
str'c-.s n rur r3 ILitud'.rcl e.'T.ression

R trPnsvers? di ect-stres rt; to; rtio of tra-ns-
verse 2rc.rt tr.iess present to critical
strees in pur-e transverse corrpresEs n


SRES-,il,' A!'D DTSCUTSIC.N


The results of this investigation are given in the
form of nnr-dilenrsionE-.l interaction curves in fij;ur,'? 2.
Each o'int rn these c'-rve-s reorPsants a critical combi-
nati-n of the stress coffi'cients kx and ky for a
giver. elastic ed.-e-restraint constant E at. which an
iinnite!l.- lon, flat --late will tuckle. The int-raction
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 int-r.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
interucti-n 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 stif-ener, that
is, a stiffener which t'.'ists without cross-sectional
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 restrs-int 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 -p-11-ie 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 in-tc 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.a-tive vclue of kx. ThiLs rorerLy cf Lh c.trves
-i.diCiat6s Ct-E t hCe presence of lonr itudinal tens-icr :ac
Inc ef-ca 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 trn-sv;-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 tl-e ..nier-&c-o .n c. yves i'cr those ca.ss La lich t:e
sL2' 'ness 'i thn rc str':-,-i : 'd. .t .i : in e:-ende-it 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 c--itical
conr.iin..tinrn of' '.:ect stress c-. be dPt r: .ned t-.hou
so .:- .c-t laorio.c- ,^ b -. :..- d t:.od cim lnr to th-at 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 l-vi, :n3 .fl-.. :- l..te befo:'?
there is an;-. i.-. d -,,ct ion :,.n t-f-i:. 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
:,ecessa-y to .prooc;.'c e b-..cl:ling.

The use c: tl:c inter:.?c c rvc toi e ter:.. ne ls.e
critical sti-uses -;'io a f ic-mi e i ?l Lte wjth. :-' e r j''stra..t
ind.epeede-.t cf w avfE cngti ~ LU c seL ..Oe:'vative


Lar-i 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 STEES-F


ifr7-r'ti-l.. e-'.c-C_- *r nauZ.i..riun.- The critical
combination, of 1&'n- tudinal and transverse direct stress
that wi11 cau-e 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.a-e 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 x-Jirection, all disp2laci ~ nts
are .-riodic in 7 and the buc-kled 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. L6A-i)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)
d-y4 \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
nml-Pe 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 coefficie-ts of P, ., R,
and 3 in equations (All) must equal zero. The e- ansion
of this ret-rminant 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
p-2 ^ -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 lenr-tn.- Tnc rroce-ure 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 p-lot or lk, arai..t ,/\.
This process Is re ,eaLed for other as-..iled values of 1:,,
and each timn- a minimu-m alue of k1y is deterrrined.
Finall-y, the int .- action cur",e of 1:., against the nm-inium
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 rerpre-ent 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, con-equently, tho-e critical
combinations of stress for which the buckle wave length
is infinite are defined by a straight line of con-tant
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 r-ftrence 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
finite-plate interaction curves are seen to have portions
that approach vertical lines as the length-width 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 length-width 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 Sch-vartz, Edward E.'
Critical Stress for a. Infinitely Lonr Flat Plate
with Elastlcally Restrained Edce- under Co-mb-ned
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 Fla-t 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 Stre-r of an
Tifin'tely Long Pat Plate with Equal Ela-tic
Restraints against Rotation along the Faraliel
Edges. 'ACA ARR ';o. 3K12, 1943.

6. Lundquist, Eucene E., Ros-man, 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. Lib-ove, 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, r-I 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


N-O K O 0 O 0 (0
f H i 0 S) 0) tO H t C) O
*

0 1-V

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





4-t
H






NACA ARR No. L6AO5a


(a) Loading combination treated in reference 1.


I,


-t-'t t tt t-t-t
(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. la-c







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;-

C0-L 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 w-9





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 A-A

(a) Buckle form for shear alone.



Lodes'.s 1


Section B-B
(b) Buckle form for longitudinal compression alone.


I 1 1 1 \ \ \ I.




t t t t t tt t Seton C-C

(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. 4a-c







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 length-width ratio of 4.


Fig. 6







UNIVERSITY OF FLORIDA
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