Critical combinations of shear and transverse direct stress for an infinitely long flat plate with edges elastically res...

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Title:
Critical combinations of shear and transverse direct stress for an infinitely long flat plate with edges elastically restrained against rotation
Series Title:
NACA WR
Alternate Title:
NACA wartime reports
Physical Description:
22 p., 5 leaves : ill. ; 28 cm.
Language:
English
Creator:
Batdorf, S. B
Houbolt, John C
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

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

Notes

Summary:
Summary: An exact solution and a closely concurring approximate energy solution are given for the buckling of an infinitely long flat plate under combined shear and transverse direct stress with edges elastically restrained against rotation. It was found that an appreciable fraction of the critical stress in pure shear may be applied to the plate without any reduction in the transverse compressive stress necessary to produce buckling. An interaction formula in general use was shown to be decidedly conservative for the range in which it is supposed to apply.
Bibliography:
Includes bibliographic references (p. 21).
Statement of Responsibility:
by S.B. Batdorf and John C. Houbolt.
General Note:
"Originally issued January 1945 as Advance Restricted Report L4L14."
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
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oclc - 124095404
System ID:
AA00009365:00001


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Full Text
'Ack L-mI3


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 32611-7011 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 AC-AUIST 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 stressed-skin 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 stif-enerr 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 stiffen-rs, 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 shear-strets ratio. Stres-
ratior may similarly be defl-ne.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 shear-stress ratio and Rc is longitudinal
direct-stress ratio. The same formula is recommended in
reference 3 for general use for the buckling of any flat











NACA ARR No. L4L14


rectangular plate, recardle-s 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 dra-n that
interaction curves in rtresc-ratio form are practically
independent of the diimernions of the plate.

The present analysis, L-w'ever, indicates that the
buckling of an infinitely long plate loaded in shear
and Lransverr-e 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 one-third to rri-re than one-half, ,d-I^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 r-dges
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 i-utLc 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,

in-lb ----t"
12(l -

E elastic m-dulus 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
men-ber at ed: e of plate, lb/rediai

Tc work done by- cimpressove force iJ'r half wave
length, in-lb

T, work done by shear f.rce r er half wave length,
in-lb

V1 strain energy in plate per l.alf wave length,
in-]b

V2 strain energy in ed.,e, restraint, p.Er half wave
lergth, in-lb

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-.ear-stress 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
P-Y) 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!,c-le, in.)

PDiT on'_ ratio

direct trees ?

tran-sveire direct scrcss, Ip.

Hear stress psi

an-lec betvrecn buc'



  • stress ratios

    shear-Ftress rntiD

    lori-itudnael dlrect--treFs 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 strs-r at \l-l.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 T-T and
    dimension ]ES bTucklir, C c-tfficients
    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 y-d~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. t-e 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 (A-e) 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 (A-19) 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 (A-23) 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 in-radians.

    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 + :2-CL (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 anal-zed ,y a sIrilar method, but
    the aa r, l.-Is I., quitc c m:,l.icated because ter.n? of third
    ord--r ..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 corre-stonl.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 Eulc-r value, the critical shear-
    stress coefficient kg car. therefore be either C or
    the value given by equation (Bl2). The concl-sion 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 Th-e sn--r 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 len-th 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 e-uation (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, t-i--' 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 elastic-strain 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 expres-3? 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 equ-ations (Th), (04), (CE'), and (C-)
    arc- no sb..~Ot'.*:ted in tlie bu2 use t:, the ;eq.iatiDns

    r2D




    j b-
    rL




    eliminate c'x:r, and N~ T'he resulting ecuatior. gives
    the critical z'n-.bination Df stress-es 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? de-enS- 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 tli-s 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 E-lsticall:, Re-trai-ned Edics 'inder C.mtrbined
    Shear and Lir.ct Btree A2.. F. T... 3 Kl., 124 Z.

    2. Sh-anley, 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, P-7'

    3. Anon.: Streng;th of Aircraft events. AR.-5, A. ..-
    I'av"-Civil "m-unittee *on t-.icr-aft Le irn riteria.
    Rvsd e d., Dec. 1942.

    4. Crate, Far ol., 5-atcrf, 3., and Fai, 3erE .:
    1T e Lffect of Tr.ter.-m1il Pr'esu e oni the Ecr.klin.,
    Stre s 3f' Thi'n-rI a-'. lle-t 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 Pdr-ls 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 :.F-C"ba', J hr:b ;VCuL,
    192i S R. Cli'rE.tovLr. Lcb',ij-chen r.cd Es. t in), p.. 11.-
    125.

    7. Timoshenk:3, S.: 'ih-eory of Llastic Stab-ility1 i1zGraw-
    Hill En?' Co., Inc., 1 .i_1 p. '-5.

    8. Stoweli, Elbrid-1e 2.: Critical Shear Strees of an
    Infinit.ly L:.nr Flat Plate .with Eqiual Elastic
    Festraints acairn st r.t-ati.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


    c-i





    a .
    4-


    Ln
    In kI




    r0 4
    L 0






    U --
    I u



    o a-




    Us.

    O O0
    -C
    .





    u
    8^

    -CC


    4-c
    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
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