The Ultimate Compressive Strength of Thin Sheet Metal Panels

The Ultimate Compressive Strength of'

Thin Sheet Metal Panels

Thesis by E.. E. Sech1er

In parti&l Fulfillment of the Requirements tor the Degree of

Doctor of Philosophy, CalifQ?'nia Institute of Technology~ Pasadena, California

on

:!?:art III ... ..-Flat Sheet .Panels Beyond t.'he 8ta.bilit.y

Approximate Theory fl'

-:Wirst

Par~ IV--•E.x.perimer1tal lt.pperatus ar:d Eesul·~s of First

r;x-Pa:rt VIJ: ... Application to C~.rrved; Sheet

VJ a

on

References

repor~

of

o~

ot

thin ehe~t metal panels a the gei1eral formulas for the

equations hs:ve been obtained

by

tb,eoreti

total load

UDsti:tfened 11 simply supported panel can be g.i~ven by

P=

c

JEc:rt,_

and ~ha't the c~nstant. C ill

b7

of

ted load&

lous

C

a

f

{A )

~

)

~

.{:

(II

t )

fr

~)

E • the Young's medulus of

t •

of

b

=

o:r

R llf rs41 us o'£

P~'"'"'''-:!·-~ tt bo tb

round to

:failure

ot

The author Wishee to his appreeiation of

the assistance !'urnlshed by members of the st.aff' of ·t.ne

Guggenheim Gre:dnate School of' Aeronautries at t.ne Calif'ornia

of Technology and also of'

by the

In parti eul

ar

he

c.

:B., llil1ika.n, and Donnell for their helpt'ul suggeetirinui

and criticiem&., Also itr. Raymond 0£ the Douglas Aircraft

co .. " who aid.ea by suggestions makir1g the work more a.ppli ...

cable to deaign~

G:rati tude is also

Institute

or

Technology, and in particuJ,a:r to Professor

Newell of that institutioD for giving access to their ex~

perimental and Qa•i.~ ..

Of the stu4enta aiding in weri:: ~ partieUlar

:mention due to ][essrs. Hutc1'J.ne 11

and Childers~ for their .,

The subject. of t.he stability thin pla:tes

:fi rat became of' praeti onl im.por tmu;,e in the :fi el de o:f

:naval architecture civil engineering. In

archi-tecture,. the designer was interested in failure of

side plates of a ship they were to

and shearing forces in the

ceuJe, "the problem watt l.ae:o eompli.catet!

In this

par ta of the by the additi co:! o:f t-he water pressure

acting normal to the plane of the plat~.. l engin..,

eer f"aeed a simi lsr pr()>blem when d~si gn:».ng l~~:rg.e plate

girders in which other dimensions o.:t t.he were very la.rge

in compari e:on to thickness of m;:lter1 used.. These

bea."Ils were .subjected to loads whi el1 ten.si on,. compres..,.

sion" and shear loads which in the pl.ane o:f the web" and it 1111"1!.~.s the deiii

tha. t buckl i of

¢ ~ pl'obl em ·to t::to eonstruct his beam

did

In both e>:f the above lure timS eo:nsi. clered

to have occurred when the plate suffered any appreciable

def'ormati 011 in a direettcn perpeudieulaI: to i.ts plane; since

any loading above this 'lllcri ti cal l.oatP' cauaed a, rapid. in-crease in the ae:for.wati 011 in a direct.i o:n parallel to the

plane of the sheet., This large defol"lnation, if accumulated ewer a large structure such as a ship or bridge t would

cause total displacements which eollld net be tolerated ir1

the desing of such a structure.,

Whe!l :flat and curved sheet; vms first used i'.'.l

aircraft construction. the tendency \"las to design the sheets

on the same basis as tbat used. previously.e i~e.,, on the

basie of the stability limit et:r the critical. buckliug load., 'fhi s 'buckl.ing load, however wae ''ITer,y low :fo:,r t.h.e ·thin

ma·t;.-erials that were used in aircraft construetiou and this

cri ti load induced stresses in al eh were

:far belaw ·the stresses that would cause tion in the sheet,.

It was also noticed that if

p&p.els were lQaded above the stabili

thin sheet metal

they continued

to carry a certain a."llount of increased losd

m

thout suffering

any perBuan~nt deformation., Consequently there arose the

second problem~ that ot: daetermiuing the ma:ximwn permi as:i bl e

load be:f ore the sheet suff'ered a perxna:uent de:fcrmationt or

bef Q"re the stresses reached the

is.l at any point.

e limit of the

mater-struetur·es,. such conditions as i.ndi above are quite

permissible., The normal flying or l.a:m:'ii.:ug loads ot1 an air0

plane are tisua.lly quite eompar loads ch ~Y

be applied under extreme conditious ods o:f

time.. Consequently it may be possible to design t.he strEtssed sheet. of such a structure so that i t is in the unbuckled

st~te for !lo-rm.al lead conditions and Cfin eerry considerable

ls.trger lo~ds :in the buckle-de s:t.~.te wne:;1 sub,j~'eted to extreme

eondi ticmsi;; In other ,, the

be based en t.he ultimate lead car1~yif1g

load should

caJpac:1ty of the

sheet rat.her tha..?J upon the load at \'iit'..ich. the $heet re.lches the sta.bi li ty limit ..

The det.ermi.uat.ion e:f the e)~af!.1'i c l.i ty limit of flat sheet members .has cally aud

theoretical. was by

:bis method. or ot.he:r methods giving

other~).

one

m.ethod

is given i:n II oi' this paper."

:1) and

r~Eu~r1 ts, are

!he treatment ~f e beyond

r,Jta,bilit;y limit is quit.e renent~ lt been o1rliained, ex""

periJ&entalJ.y a nurrtber at times but was first .eon&idex"ld

seriously in a publication· of Newell~s (Re:fe:rence 6) and i:u IL.A • C.,A,, Report llo. 356 (Re:f'ereuce 7) on l'irOrk done at

These no tempt

ga.v e the experimental data the hope that these results

would be a guide ~o

on s 001 "1frl!iS

m~.de by von aud it is this and its .ez; ...

pc.nsion >dlieh f orma the b~eic pnrt of this p~.pe:r.., _A semi-the:ore ti cal

edge ·compression it! the: :rJ.ane of ·the sheet, aud this theory

has been checked by a large nt!!nber of GXp~riir.ents,, Tr.ii s

has t.hen beG"n

m

th and. witbeut stiff en.ers*' the

personally by tb e o:f it

and part of it uas performed at c .. I,. T ~ by gradtu~,te stude11ts

in Aeronautics.., 1.£"ention of authors ru~d publictd.ione wtll

be made as their work is discussed.:1

II. The Stability Limit

atabi li ty l:bd.

'the method is

quite general ,

on othe1·

and to

x

s

stress o-

dis=-Y •

ot'

y ii:· b., See Fig$ l,.. All edges of

to

simply supported"' • if the

is

the general dirreren•~lal equation

(l)

mld, since er\\ -= 7""°"J:. o _{cr1 -:::..} -er

t

1.E

J'l-r

6 tJ. i! =er

-n_(,~"L) .J'1'\.

(2)

which gives the differential ecruation for this particular

(3)

the ca.se of simply suppor·ted edges" can

be given by an equation of the !orm.:

where

and are integers,. EqtW.tiO!l {4}

es the tlons

;l:.-:O

,

~ _{a.. f- 'Jl=-01} c._

J kl,.::;;: 0

put ti the values

into equ:ation (3) we get, after dividing through by

the on

t'E

f

C':)\

In"(~~:)~

(:rr-rJ-

er(

¥)~=-o

i"Z.(l-;...").) ~

(5)

wllich ean be solved for er , yi ding

l.

+'2..

l

J?.

.)l.

r '(

Vi

)~

l

~)"t.11..

0-=:. ll.b-,.;-•·-'·) ~

{.!

G.... b

J

(6}

{ '1)

(8)

.A plot of' K agah1st the b/a ratio is given in

Fig., 2~ Thi a family of curves indi that the critical.

the sheet has a length

this i a the eune of -_{,, a.iUl1} . . _ns t _{t:.ne .,} . . bl _~

for

b/a.

that1 l.$10 it

suffi eiently is

accu:::rat e to oonsi.der thgt

Thie method

buckli

to give r1cHial liwe&

o ...

~--~--1~~..,.._~_..~~..1,,,..~...J..~~..i...~-'-o I

very a~.tisfacto:ry for plates Qf the th:tckn;Jse used in naval

and. civil engineering as it. was poseil•l.e to otit&.in stresses reasc;mahle close to the ela.etic liri:ii t of t-be material t':ith

steel-

!n

thick could be used up to 30~ w.i.de and of any length thout having the buekling stress tl1e elaatie

limit (30,000 lbs

stress falls of£ proportional ·to the of the thickness

'E"f:S appli

design of aircraft of

materi 120

to a

a compressive get the buckli :ug stresei "Lrp

for any b/a ratio equal to, or greater thau l~O,. This ob""

vi ously means that :m.:unerou.;; stif:fene:rs must be used hJ order

to break up the sheet into a large nU!nber of very narrow

panels., This leads to an difficult

construe-ti on whi cb would not be used merely :rrom an eeonom:l c:

staua-If a panel of the same material h1 made 4* wide,

the buckling stress falls off to

ch is too low for ecorrnmi

cm this pauel is increased after buckl

average stress nf t.1vc1"" 7" can rea.ched

this last streee is still far belc;;: the cla.sti,.c limit o'f the material,, the increase is· impert.ant enough ·to be giv£n serious cousiderati.011 for thit1 sheet :met.el pDnels... Part III will consider thi .s increase in allor,;:;;b.Le stres$ ~boV~! the

&t.;;.bili ty li:a1i t i i i detail"*

III,,. Sheet Stability

pression

(Re:ferenoes 6

and etriking

obtained gave

buckli;ug

nearly

independent of the width of the

:::=s:

.

..

l

/~.

it- . '../

I

((l)

load ~ra-s

carrie~ by that part o~ the

was the

(C) the buckled state" In other

the

ila:r to the dth design

I

w

t-b

:first of

assumes th~t in th:J,s regi.on the plate

are

8}

of the she~t

by VOil

ene

-tl.E

'1..l l?L l..)

and eqi11a1~1 on

or

er:::

To

This gives~

:from wb.i

it

1tl.E+71...

l~

\1 .. ( '?~.<l,,."11..)

J..r

find

--'-

_{4w"L -}

..1

_JP-

=

o

J!

~ l.w

1T l. E 't-a. ~i1t.·c.a. l :::- \1..(l-j4-\)W11..

solving tbi s !'or effective

(9)

and the total load c-1:1rried by the sheet; 1 ftq'.:ia,l to

(

(9)

(3)

(ll)

{12)

'PToT-=

2.wto"-:::

~n.~:_.,.>

JEcr -t1..

(l.3)

- r

JE

_j,,.l.... - "--.f 'j a-~ ~I!.

f'om:"tl1 side :free (Fi.g" 4c)~ mis

ried out by Doiinell (Re:ferem:ie S) and

total 1oad of the same

nell's equation is "D ,

\ lr?J\~':it

:ro11owea. since they

def

om-(

)

with

t.he-an equation

as (15) 0 Don•

(16)

the

as a function of the ela.st-ic propert1es of the material" the

sqttare of the

of

the lo&d was i.nde:penrlent o:f the Width of the

sidering the value .of

r

for normal {metal)

to he equal to o.3$1 equations (15) and {16) give us

tp-a"->t

=

'LG I

i

Ecr

1

t

"t.

' . . r.:::"".:"

t

l...

1>

-.:;.~

.::

\.1. Ii v E<r

'1

!mental values

0

r:1 Sr. I R.o~

Molil i.,;

mental values of Cr~ ealcula:ted f'r@m

foutHi in Ref'erenoe 1 are shown plotted

$eeE. that nearly elcl (a~%) of the

Gf do lie within the range

order to more

cat.et! by

au exper-imental program ?nu; started

of iree:hnol ogy in the Glagges.meim

was felt faults

,fO(J

1, and it iB

val u.es !tn.,'

California lnst1 tute

a~ mt1ch e.a po~s;t bl., e in the tei::;tf ng !1.,.T*T.

(l?)

IV"' nperimental Appa.ra.tus of First

Experimental Tests,,

The testing machine is

F1G. G.

in Figs. B and 9, and Fig., 10 snows .~ photograpl'l of the ma.chine

in use,.

me

upper head of the t.e.st:ing machine cons.ists of a

1 ten inch I-heam fastened to two.

upright ten i:uch channels11 ami

the lower~ moving head ot" the machine is a simiJ,ar ten ineh

I-beam pivoted, on ball bearings.

with a a.rm o:f eeven feet.,

which might introduced from

and lower beams were usetl in

]'.1g,. 10

order to eldn1.nate any deflection o:f the testing heads under

load distribution over

the edges o:f the specimen.,, The ho:rizont.al motio11 of the

lower beamf' be:tug approximately 0.-01 inch for every inch of vertical motion, eo11ld be neglectedg foe the vertical

motions obtained in the experiments wer~ in general less

than a o:r an inch ..

load was appl;t ed by meaJl~ o:r a seven and one•

half' t.on hydrauJ.ic Jack a.ml tbe valu.e~ gJ: the J.gutl.:s were

Fig. l l

by meaus of

and a

low-range pressure gauge"

Fig., ll"' The

low-range gauge read :\'.'rom

:0-2000 pounds aud the

gauge

were

per-iod1cally cal.1brat.e<1 against. a standard 'type t.es;t:i:ng m.achJne.,

fbe surf'aees of the I-beams next to the specimens

were faced th a strip of

heavy cold-rolled steel

having a 1200 V0 groove

:t'\lmli:ng lengthwise the

center of the beam<!" rbese

twtl V-grooves could be ac•

curately al iSJ1ed ll' one above

the other., by means of

ad-_Justment$ on the testing 12

machine~ The sides of the spec.imens were al so supported i:n V•grooves, in thfs 'case 30°11' which could be adjusted for any

wit\ th of' sheet d~aired and Vihi ch could so

that they were of

A photograph o;f the edge s11pports first "sed 1 s 3.hown in

manner I!

lower. edges acting i:n the plane of the

'first series

which were all 12fll long and which

sheet, al:ominum, and brass. th a

O~Ol2 to 0"067 inchtHl., Since the Ka.rma:in (Part lll)

ii1dicated that the total load

material~ a wide range

or

to check the equation!!' even though som,e of the materials

st.:ructura.1 materials.., It; wa.s also expected 1~hat :for very

large and of' the o this

1n order ~o establish limits of

the material was ordered crated and shipped flat

When purchased and was ueed as reeei ved t :l:"ej ecting however•

any specimens whiqh showed ci1nature or

:f ects.,, The ends were sheared iH.J.uare and loaded edges of

the specimens were made -cinular as there was a te1uieucy

for !lat. edges to ride on o:ue si.de of the upper and iower

on tl1e

cattsing i t to fa:Ll prematurely" A number -Of scattered ene inch wide specirnens were taken eloee to the l.arger compressio11

specimens and inese were used as

determination of

Tb e el asti. o p:roperti es were

deter-ing mea.StU"ed with balanced HU:ggen.,,. berge);" tensiomete:rs.. See Fig,, 13.,

The compression tests

were run ·on each specimen and the

value of the ultimate load iu

compression was tabulated,, From

th1sac using equatio11 (15) .. the

value _{ef Cf} could be determined§ since E aud

~

were given by the

stre.ss-strain curve of

:ror the detern1inat.ion of the

definition of

considered :suitable to check

first set of e.xperimentt?.J. tests,.

It

'Eor the

Fig,. 13 be

of was

this time it ~J:l

that the

a further

analysis in o:rder to determi1ne whether or not this d.iffereuce

was due to experimental inaccuracies or wlJ.et.ber C4\' ;J. was some

function of' t.he experimental variables"' Consequently a closer

investigation of the probable stress distribution was earri.ed

out and s second approximation to the prob£d::ile ultimate load

carri

Second Approximate Theory

The first assumption made in determining the

ul-timate load that could be earried by a thit1 sheet pm1el in

compression was that the center of the sheet carried ~10 load

t.he edge regions supported the load

by the sheet.. However 1t an investigati.~n of' the action o! a

sheet during .loading shows that this ass:umpt.ior1 needs to be

mod.if'ied in order to.more closely approach the actual load

At first, the sheet remains in its original pla.ne

form ~s the load is applied. At a eertain load., the. critical

buckling load .. which ea11 be determined by the equations

rived in Part I!9' the sh-eet buckles in a which is

is continued" the e.Qges remain restrained and eontin:ue to

being in the buckled state,,. c~:n carry little or no increase

in load. however they still carry the stress corresponding to

the critical buckling .load on the panel., The second

approx-imation to the ultimate compressive load is Ga.sea on t.b.e above

assumption8 i.,e. that the center p.ortion of the sheet carries

a stress e:orrespot:uiing to the critical buckling stress o:r the

given panel and that the edges of the sheet carry a stress

o-

distributed over the effective width

Consider a tz:11in sheet panel of length I., width b,

and thickness

t.

made o:f material with a Young's modulus E, e>.: ~., .. ~Pi

..

,.,

v BQnpol' ;;:- f,~Q on a.ll i'ow~

and subjected to ~ eompressiou load on the .upper aud l.Oll'<·er

ons over et:f'eetive on each side

uniformly distributed stress

i

w ~ I

1-- I w

er

a11d the center portion of the

sheet~ of width (b • 2w), afll"ries

the critical buckling st.ri$as

b the sheet,

Ob •

Ua1ng

Ka,rman assun1ptionv equation (12}

as

(12}

and the buckliug formulair equr:rtion (?) gives ll-i.. E+t...

0: ... k

-b IZ(I -.,µ.'!!.) b

"-(7)

(l, 7)

(l2a)

(7a)

depending upi:m the L/b ratio given

Tbe total l ends is given by:

aud the total load carried by the center f!ac·t1on 1s :

,

1'~(b-lw)tcrb= ~l.:aEt}(\- ~

1

\~~)

(.19)

By adding the two. ve get the total load by the

combinatio11 as::

'° -

P-t

'P~ ~ecr

t\

~;r

[1..,.

~

[±..,.

~Et.?.

l

l°'Tn - - "Z.ot

VU

b oC2 _{0- b'l.-_j}

Letting

b:Y sueh a

l -

Ir

t

ii - ~~ b

val·tte ( 20a) we

"O _ ~

+l.

:z.ifr.

~ .\!\ _ ~t ",,_] -=

r

JEO:"

t'l.

r ~" -

"£11",

,...

~

r.

+ l« ") al.?. .fl~ ._,.. ~

in which

{21)

(20a)

(22)

tnis

(20b)

vl.i"t.h the value

o.so

and theu leaves 6f as a f.unotiou of

the L/b o is greater u:ui ty

and that. i t is conservative to U:f.le this value for K f'o:r al.l

values of L/n,,. Taking theu" 0 eh

give.s a •slue o:f« a 3.,29, we set for cf~

cf

=

[1.£11

-t

3."r

i- "·

G

'°A

j

c2•>

hol,d up to when the two e1feetive widtha would

Join, or when (b - 2w) s 0.., From this point on, the failure

would be purely a yield point fe,ilure and the total load carried wottl.d be equal, to

'f'::.

b

t<r :::

CJ.'

-V E<T

f''"

(25)

place,.

of

and~ putting it into the ~c:rm

we f'or

C ''-

k: 1r1.

fE

t. -

K

Ti~ ~-=-

3,,

1.). (27)

.f -

olt.

{~

b - cl

:for the region i11 whieh :K :r: 4.,.0 an<:1 OC.

=

3"'29., This gives a straight line as shoml in Curve III in Fig,. 15. '?he

clii"'-after buckling takes place.

In a recent paper

by

Cox of England (Refere:uce 9)

e, ~lightly d.if:f erent method 7nii user.'l tc calculate the allow=

able compressive strength. of' :flat sh~etll' using very nearly the sa!l'ie .iui tial assumption as used in this seeend a,:pprox-i1mat1 on and a formula watil obt.a.ined which CQUld be ptit i11to the f'orm

where

and where

l

is defined as in the above derivation,; This curve is show--.a as Curve IV in Fig_. J.5.,, Also, for :referenceii

the two values of

Cr

:foUlid from the first approximate theoxy

Fig,., 16 shows the exper:bneatal value-a ot' Cf' :frmri

F't.G. lb

Table I plotted agaiust

A

4 Th1 s r.lot ehOl'!S- that there is a

defini~e rela.tionahip between

Cr

antt

A

even though the

ch i& .taken a~ the :mean curve through the

plotted poiut.sll' does not agree with any of the theoretical

Tnis first set of

truat could be au.;pported by a flat panel could be given by

the equation

and. S'eeond p tha-t the e~f:fieiant

Cf

wa.e a function

or

th(~

parameter

for a wide range o:f materials 11rlth different elastic proper•

tigated. the affect of' the length of the specimen on the

deter.minati o?l of the exact shape of' the C:r vs ..

l

curve was

made on flat sh~et panels., Wlith the fact established that

the value

or

Cf eould be plotted as a. :funct,ion of

l

> a, much

larger series of panels were designed in order to better

determine this d~penda:nce.

In order to obtain a number of points over the

whole range of' possible values of

1

a 11.:u:aries of specimens

were choseu O., 064" az:ui 0"' OS21t l 7ST Dural

0 am!t Oji\ th wi

:from

l

that

the value of Cf w~..s more doubtful £or am.all values of

A

the

ccncu:!ntre"tion o:f e~perimeutal points v-ras made greater as

A

decreased. This lead t.o speeimens ranging from 0,,8f'

For the dete:rmi.:tia ti on

or

the dependence of Cf on

the length of' the panel, seve:u differeut lengths o:r sjflecimens

to tt in _{This gave}

from 3 to 21 inches

those used in the first group

being

of thix1

to

leaves of these blocks was negli.gible compared to the

The experimental resulta of these tests are given

i:n Table Il plotted in. enn be seen from

.l.O that there ii;, no

c:r

L/b ratios

C'f ·a& L/b decreases as show by the dotted curves~ This is

as would he expected as tl'le stress carried by the center

:r·egions of the panel i:nereases as the L/b ratio decreaaes

from 1.0.., lfowever !t for design purposes'!' it is :felt that the

iu;ie of the soliG. curve in Pig,. 18 :for all values or J,.,""b will

In fable II the valtie of was tak.en as the,t point

on the curve of the material

eorre5pc:n-ded to a 0,,.2% el onga ti on o:f the ., Thi$

was donejc :first. to agree with the d:e~ini ti on of the yield

point given in the design :requirementst and second

because use of this value seemingly gave less &eat ter

to the experimental point$. fhis scatter V;ms particularly

noticeable in materials with a .stress-strain curve !!hieh

departed slowly :from a straight linef failure seemingly bei:ng

retarded until a st:rl!s.s had been reaehed which was

consider-ably beyond the proportional limit of the material,. A

s-euasion of the sfil.a,pa of the

of

the equations fer flat sheet~ equat.ions {l6} (21)., This

0

L-~~...;._~~-l..~~~'--~~-'--~~--L.:-~~~~~~'.'""-~~-:-~~~~~~~-0 .1 .z .3 ·"' .s It .e. ., .a .9 1.0

't6

was particularly :fortunate since the whole theoretical

dis-cussi on has been based on ~he assumpt.i ou that the small f!,e ...

fleetio:n theories hold beyond the stabili.ty limit.II' This

assum.ptio11 has made in the derivation of all of the curves i:u Fig. 15,. Obviously, after the sheet has gone i:nto

flections in w~ich the stresseB

introduced.

eonai d.ered as as tl:i.e stresses

due to direct load111 This: treat ...

but as yet is not in a form t;hat

F1'1.

17

ca-n:fi:rmation o:f'

J;n investigation o:r the exr-erimeutsl curve crf F,ig,,. lS

it is the theor~ticel curves o.:f 16

:for nearly all valu~s of

l .

the loading over the ef'feetive width cannot be trn:i ?orm

as

the stress mu.st be a ma~x.i.mlillm ·~he

in Fi

would

tion... The position of' the experimental curve shows this t.o be correct exeept ?ery small Tal ue.e of ) ....

:r·he increase in the value o:f Cf at sm&J..J. val~ ues oi·

"lralues of

l

huUcate

that either t must be If b

the

value of the stress carried by the center.portion of the sheet

proportionally,,, there is a tendency

to increase

for decreasing

values of L/b is shown

by

the

dotted curves

th.at !'or de-flign purposee i t is· felt that the use of t"'le solid

curve"' neglecting the of L/b j,s advisable""

There is another e:r:reet, bowever,f which WE!S first

natieed when testing de specimens of' thin brass., Ot"ling

easily noticed and the bnelcli.ng action was_ visible thrtru&hout the loading_.. 'fak:h1g,. fo.r e~ a sheet 12" long" 14" .wid$ ~

and 0,,016* thick!>· corresponding to a value of ).

=-

0,,025l> the

·buckling f'iret started at a very 1-ow load and was iu the !orm

of one halt-wave~ as would be expected from the buckling

equation~

i:mately

the edges

Thi!:! conti.nued tL'tltil a load. erqual to appl"Olt=

of the ~ailure was and then

t,he sheet bu.ekled in of five

o·istt tb.e center 01.'

finally a number of small ~7a:Ves (

Fig., shows clea.rly the mult:lple wave fg_rm whieh thin.a

wi.de sheets go int.o,.. This ple wave eat! also

:oot.!eed in ·a, cf the Scpecimens shOtnl in

Fig<!' 19~

The :e.bove action was observed :f o.r all. of the

panels tested l?.nd the e:ffeet seemed to decrease 8.S the sh:eet~1

becanle narrowerq This wattl.d seem ~o 11.Hiieate toot the

wider sheets the distribution o.f et.:ress on the lottdec! edges

could not be accurately eXJ:iressed ev·t;m by ·the iaeeoru'1 met,hod

waves probably gives a sti, f:feulng ef':feet which is shown by i:u.,.

crea~e 111 the value e:f a.s the V<::-~lue of }( decreases.,. A

more caref'ul investigation

ot

the actual wave torn .of the

sheet . under load is being plam1ed in order to se-e i. f it will

not give a better ide~ as to the aetual stress distribution

acro;;;;s the loaded edges~

Fig.. 19 shows a number of photograp11.s of test

panels taken after :failnre-t i'""n which the type of' wave

ean clearly seen., Al so$' more rigid edge aupporta for

the edge$ of the sheet C.?..:ll be seen ill these

r

'

l

't

[-,·r r

<I

r-),

1:1

,J

-"" ___ _!

i

!

_I

VlI., on to Pa:nels

method of Part V"' wil~l ntiw be expanded to cover

a simply

R

\

T

.load in t.he plane

a- -0f the sheet and

parallel to the

s the

cylin-dri

l.

l

tnodu1 us of the

and the yield

l>oi:ut to be <r~ ... Since the Si•"ies of the sheet are simply

supported~ they will undergo a stress condition sim:i.l.ar to

that found for the

at the very edge

at. sheet,. i~elll' at :failure the stres:ia

be

a;

al~.di if.hi"' atreco wil.l. decrease

as the center of the sheet approached~ However~ for

eimpl.if'ica.tiou, it will be asa~ed that the region very close

to the edge will act as a flat sneet a:ud will strpport a

stress <.r over an effective dth

~rom the e-quation for flat sheet1

(29)

(30}

,. Appendix B.

total load carried by the tuo aide regions

wil.1 then be

"P-==-

z

w+cr-=

L+

Je:.(J"'

t

~

(31)

specimens could be cl.osely approxin1at.ed by the

r;:-- - A,,!'>. 1'-

.::t

{

32 }

UC. - u•;JJ.._ R

an<l a recent paper by Redsl:uaw i:u (Reference ll) J:ias theoretically that 5a.in€ critical buckling

as :for a complete eireru.lar cylinder., value oi the

cie.nt in eqtw.tion (32) also agrees with the

ax:i:.erimental d.ata gi:ven in an N .. A,,,C"'A .. Report by Lundquis·t

(Ref~renee 12) ..

'llllill U'1t Ci;.I'ry onal loe..d

·t.he _the

aides whicn will enabl~ the side regions to tiomu

stress stbove t.he critical stress V&;lue for the center regiottS9

as-that the st:rez.s distribution over v simply suppor·~ed

curved sheet panel is as f'ollo1fie:

a)OVer e region o! dth ¢Cw~ en each siclel> there is aeting a.

a- .,,

b}OVer the cent!'al region of" i1r:i.dth (b - h) t,here j,s

by Fig<:r 20.,

load carried by the side regions is given by

e01t:ta1;1ou (Sl} the J.oad cal"ried by the

Pc-=

{b-lw}cr-"'t: bo-c.+- 2.wcrc.t

{33)

( )

which!' if put into the :rorm used in fla.t sheet caleu:lati ons • becomes

(35)

where

(36)

which can he put irito the form

where

1=

~+

_{<5 I?}

uo the equ.i:;ttion eau be ll\!Titteu aa a t·u:uetio:i:i of

A

a:ne

1

alone,

as given below

e-_ E

+

+-~

b

Jia..

c: - -

er

-

= - -

$ - -

=AV\

R b 6 R. ·a- \ (38)

(39}

o.f' C plotted V) is given in Figia

plate~ "\ t £.. ~ O,. a1ui C •

Cf:'"

which gives ths tI·l.lee.x:-µ~~r:i1J;riii,:111;:~J..

val.ue for the :flat plate; since iu equation (39) and Fig .. B-2

Cf i.s taken from the value o:f Cf as a :n.metion ef'

A

as de.,,.

te:rrnined e~perimentally., Thua11 in Fi :B-2 , the curve f 01'

1

=

tJ (!!orresponds to sl"leet.s wi t11 in:fini t.e radius er flat sheet specimens.

curves for the value of C are so limited

by their intersection with the o.urve giving the yield point

aud point& to the right. of thio curve ind.ioate that th'9

stress in the center of the sheet ail- by (32)

is higher than the yield point stress cf t.he material aud

t:.t.w.t the pa.nel would f:.dl as a yield point failµre at a value

and Gale ou the streugth of' cu:L"V&d

sheet panels series of

in

C :from the experimental value of the total load by the equation

(

~

'Pfitf (40)

VEa-.,

t'l.

1n liihi ch 1 t has been assumed that E • 107 a.nd o-~ • 36, 000 lbs./sq.in11 The resul·ta1 of this calculation are give:u in Table III.,

~rom 'fable JII it. c~n be seen tha.t the above method

o~ calculation gives a very good check on the values of the

curved sheet constant as determii1ed by experiment"' and could:

be used ttireetly .for the calculation of the maximum compresei.on

load that could be resisted by a curved sheet panel withou.t

sti:f:feners.. IIowever 11 a cl.oser inspection of' the e.xperimental

a.nd theoretical values seems t() indicate th.at there is a

consistent varia.t.ion of the value of C with the length of

the speci.me11, the shorter speeime11s g1vit1g higher, and longer

specimens giving lower!!' va1ues than are obtained by equation

(39)~ Taking the average ratio between the theoretical value

of C and that obtained by experiment for each length, we

obtain

iation

For L

=

6"

C(exp) • lit.l C(t.heory)

C(exp) •

i.o

C(theory)

C(exp) ~ 099 C(theory}

There are two possible explru1ations o:f this

var-th length .. the first being that the stress<»c. is a

'function of the length of the specimen .. &i,nce the last aeries

ot:

tests 011 flat sheet pro-ved t.hat there was 110 varia~iou

of importance irt Cf with variations in length. This source

of error., howeve:rt does not seem importaut as the value of

.::Tc. has been checked for a large range of leugths of specimens

The second possibil.i ty

ot:

error lies in the method of edge support used for the paneis4 The ai del:l of the aheet. were supported in V-grooves with in1 angle of 45° which is

too large. These grooves initially hold the sheet in a

straight line, but if any buckling takes place in the center

of the sheete the edges are free to move out of the grooves~

This motion will allow the edge of the sheet to deflect

perpem:Ucular 'to the plane et' tb.e plate aml to a.et as an

Euler co1mn with a restricted range of' buckling,. Thi;;; would teud to decre.as.e the load carried by the edge region and this

bu.ck.ling load wou1d very likely be a function of the length

of the coltmu:i. An investigation of the sti:f:fened sheet

panels,, ct.1sctu:H:u;~d lat.er~ seems to 1ud1 cate that t.he &econd

source of error is the most probableo

On the ba&is of the above discussion, i:u1d on the results of check with stiffei1ed. curved _sheet panelsie it is felt. that the value of C obtained from equation (39) or

Tl.II~ Sheet Panels th Stiffeners ..

Sheet pane.ls are :never useii in the construction

of an airplane simply supported as those o:f the experime11tal.

tests we have just been discussing, but always have

stif-:fener shapes forming the bocu11da.ries of the panel. a.re

psrt.1 cularly interested tllen in the load that

can be carried by a thin sheet panel to which are attached

~tif'feners rtmni11g in t.he direction of t.lle applied eompressio:r1

for this determination a method suggested by

Lut.u1qu1st (Ke:rere.uee 14) .ts it:~ed¢r The e!'recti ve widtll 0£ tbe sheet is to act with the stiffener section as

l l fail u:nder ,a,

certain eh will a ftmcti on of

the type

ot

sti;ffener~ the e:ffecti·ve width o:f the sbeet.

the eoltUn.- For the determination of the effective widt.h11.

tha equation for f'lat sheets is used

w~

C+

If

T

2.

ye;-Cf i.s a function of ).. where

l~ ~t:

It can readily be seen that. the determination of the true

ations. fhis calculation is earried

out

in detail iu Appen ...

dix A and consists in starting with some arbitrary value of

cr;t-

e.g. the value of. the :failing stress of the stiffener with nQ sheet attached, calculatiu.g the effective si d:th

combined sheet and etif:fene;r section, and trom. this obtaini:ng

a. second value of

oSt ..

This procedure 1a .carried out un1~il

erst and a.re ecmpa:tible, which usually invol-res not more

than three approximations., Then, the total. load carried by

a sti£fened .sheet panel (fl.at) is g:iVEm by the sum of the carried by these sheet•pl ua"!'stifi"ener eolt.mltlt1:L. This

gives

where .a₁ 1a the area o"f the st:iffetier, ~ ii' -the area of

the sheet attached to the stiffener'" and

u;.

_{7 is the final}

(41)

In sheet panels th s.tif'feners,

the pan.els carry;, i:n a.ddi tion. tc the above load carried by

pai-t of the sheet lltlxt to the · ati f':f eners, a, load in the

center of the panel given by the equation

~

I :.

~-

(

b-2.

W)

t

(

42)

load that can be carried by a stif:fened curved sheet.

is given by the equation

~cT~

Ltr;t(

~,-t ll~)

-t

I_(b-2.w)t

(43}

where the first term indicates ·the contribution o.f the edges

of the sheet with the attached sti ff.e:ner sections and the

seeoud tei'm indicates the co:ut.ribution of the ce11ter portion

of the sheet, or the part due to the curvature of the panel ..

The calcula,tit>u of' the predicted load ou the basis

of equations (41) and {43) bas been made f.or a number of

sheet and stiffener combinations tested at I .. 11\, and the

results are given in 'fables IV s..ud Vo ~able IV gives the

ca,lculatione for flat panels and Table V gives the ealenlated loads for curved panels.

In Table V i~ will noticed

two values are given for the predicted l.oad,. These are di&•

.cussed detail Appendix A it is Cfn1y t10<

mention l1ere that the speoimene giving these valties lie

out-side of the U'5ual range 0£ monocoque deeign,

inas-much as they have a very small radit.is ~f curvature coupled

with a. large sheet thickness. Di.sregardiug these values,.

the remainder of the tests show good agreement between the

experimen~al· values of the load and. the load: predicted by t.he met.llo4 Just ou~l-lned.,

In ~he case of stiffened pe.?1els z there arises

anot(.her variable which as yet has not been discussed. This

is tbe ty:pe of attachment between sheet and stiffener and

the apa.oing of the attachment points. On the ba,sis of some

recentf< Q11published work at C"l .. T~ on the .relative stren~rth

of stif:fener and sheet a:ttacllment m&thotis (Reference lo) it has been :found that spot-welding, Jtf well done. gives the

highest failure loads, rivets a.re from 5-16% lower and

bolt• will give failure loads which are consid.erably below·

the loads ee.rried. by the riveted specimen.Se Also_, as would

be expected" the failure loads drop off as the spacing of

the attaehment. points is increased.,

l t will bo no'tice-d in 'r~bl.e V that. there do~s not

seem. to be any consistent. variation in the load carried by

the panels for d.i:ff ere11t lengths of specimens, other than

that expected :from a shortening of the column.. ln other words,

the discrepaucies between the predie:ted aud actual loads are

not a function of the length of t-he specimen.,

the fact that values for

taken throughout., owing to la.ck cf' more complete d~ta by the

The a..2swnptio11 • 107 lbs~/sq~in. i~ probably

coarrect to :t however 4 the a:1un11mption ol: G'"'': 36 ,000 lbs./

sq .. 1%1. is much more doubtful, since this value varies

tlomparatively wide limits for dii'ferent sheets o! the same

shee~s of different thickuees.ese It ie expected tha~ the

experimental scatter could be appreciably decreased if' the

aett~al values of' E and ~ were to be ueed for each separate

panel tested. The values assumed,. however., are good. aiea11

values" anfl tbe use o:t t.11.e true va1uee for eaQ"h panel woulu

probably appreciably alter the average correlation betll:een

and theory,.

Another possible souriC:.\2

or

eca.tter for the stiffened

panels lies the teating method. used. were

made up. the e11ds were maehi:ned parallel and were then tested

h1 a parallel bead testing as !lat ended columns. This tenes to give a "fiery doubtful value of the e:u.d fixity a.s the e:f:t'ective erui fixity will have a tendency

\d th di:f:fere:r..t thicknesses of sheet~ the accuracy of the

machining operation"' the type o:f

J?or the above reasons,. i .. e. that the actual

pbysiea.l prope:eties of the materia.ls were not availableir and

"that tbe eud :fixity of the panel3 was aomerr.hat dollbt!ul :r i t

experimental and theoretical va.lues is not excessive, and that

the method used to calculate the predicted load on stirrened

Con el usi en:

i t thought tb.at the

treatm:e:nt as a special problem in ~all defleetions and tl1e

pbysi eal a,11alysis of' a large fHlr!lber cf experim.ental tests

haa: l to are the equations

rp -:

C

J

E

ci

t "'·

c:

=

f-().,'1)-=

4f~

t \

Jr

':a)

for C as a fuuetion cf

A

and

1\

whi are appli.eable to

correspond to necessary ·~o

go to a more elaborate m1alysis o:r the problem!' using the

de:fl ons.

The applieati on of the above analyai~ to flat and

of an effective

Y1idth of sheet acting with the stiffener sectio-n is

jueti-fied by the agreement between ex:perimenttl tests a.nd

cal.cu-there should be e011sidered the type a.nd spacing of the

attachment

Although the testis on curved sheet have all bee:n

on " it is felt that the method ttsed will

apply equally well to other materials., 'l'he strength of the

Sl.lpportee sides the has been to given by

the above equatious very sati s:if'ae:torily for any material and

the allowable stress in the center section is taken from

range of materials.

iippendix A gives in detail the method of

ea.lcu-latittg Iii, stiffetH~d panel. ~.nd. Appendix :B contains the curvea

o:f c and Cf as wel.l as several curves which simplify the

calculations o'f: values needed in using this methoelo

J..,...,G.,.HeBryaa=•Prec.,.. London Math. Soc~ vol<!> * 1891

: Part II'4'

of El as ti city ..

6.w a Age-·Nov.. and .Dec. l.Sl30

7 .... L-11Sehuman and G~Baclt••W'I' Report No.

s.,...,Th.,

von Karman, Sechler 11. L"H,,Dounell ... -Applied Mech .•

Trms.,. of t.he A,.S.,Jlit,.E ... ru:ne 1932.

1554.,

Report No ..

l.3·~J' .s.)fe11iell "'Gele••"•"A Report on Aircrai"t nte::i..~ial&

l4·•E,,::Ei..,.Lundqu:ist--]t,.J\.,e .. A .. fech't' note no.,. 455..,

15--C"'B":Erutehi.n;s--Tbe Relative Stre:ngth of Stiffener aud

Appendix A

The method

or

ealculgting the ultimate load tha:t

the failing stress of the combined eol'Umnt- which is a

:fun-ction ef the radius of gyratienir t.he leugthll and the end

In ~rder to f aeili ealeulations. a eul"'Ve showing

th c v a.r i on of the gyration th wi Of

a very ca1culation ...

of I ,/i\ arrived at an e-quation

Since the feil.ing stress of a column is diree~l.y prop~rtional

to the squar~ uf' the radius of gy.;ration,, the equa:tiou is

w.ritten in the :ferro

where

~

.,-

-

radius oi'

1-a-

Lr-r·( tYJ

·t

( t-u·

1tr-gyration of th1e stiffeuer

( )

alone-f

-

~ radius oi' g:-1r a ti o.n of the stif.fener plus the at·tached

sheet.,

Ao- e ~rea of stifiener alone

s •

distance from the center of the Bheet to tb.e neutral

axis of the &ti£fener~

t • t:r1i ckness of the sheet

~ • total width of sheet acting with the stiffener

iuto curve f"or.m ru1d is given in .Fig"" .B-3.

I:u the above equ&J.ti<m

J.

is taken as the total

w ,----w

"I"

w ,...

i

u

u

;

Li

b

Fu~. A· I

wiClth sf' sheet scti:P_g with the sti:ff'ener (See Fig .. A-1) ..

In the case of a contiuuoue structure t.be value of

.R

woUld obviously t)e twiitce the e::fTecrtt1ve wid:tl'l.p or

ending ~. stiff er1e:r section then

1-::.

w+d

Far the sheilt (J'l(erhang beyond the stiffener

aitach-ment. point, the sheet is considered bo be acting as a panel

aimply supported on three sides with the :fourth side free.

The buckling stress of such a panel is given by atl equation

ch ean be put into a form giviug the effective width of

Whenever

aa found above is less t:1w1 the actual overhang

of is grea.ter than the overha:ng value., it is obvious that

Example l.--Cousider :first the :flat sheet p~t-iel Ii'

No o ll ₁ 'l'a1ile IV. 1.1he known value:;:; are

-1::-= ~01 ~'I

Ao:::

.oSt.'3 .., .• -... •

0 -:::: .1«000"

,o

~

:::') = . &S"lb-+

1:/L"' •

4t,S.f

b-=S·"'U"'

equati~n1s or the curves bx Appendix B ~

Repeat

t/Ao=

,·!>1lS'1 S/~

0

: f.o;;..\

A=

.DC't.«i Cf

Cf-= &.rz.

..

{) = ,(..,.3.6

,..

cr/o-o

= • 'i ~

, (j;"..1 •

=

. 0-1)

' • . (Al(l$t'1, II'\

and the total area under thi$ stress is

T"nen the load carried by end stiffeuere is Pe

= .

,311 i l'l.~oo -::= :i.o<oo"'*

dth of acting

stif.feners ia

The- saaond approximatic:n gives

~'=.01'li

C.t-== M~ 2..

w

=

,J(,C/"

)l ::. .Tl~"

O'bo-='

41.l

.·.Cf'J.c::::•<j>'lt: Zit11..001tt-ta•1

The total area thi"s ~t1~ese

the load carried by the center stiffener is

'P IC--:::: • 010.'5 :!( i.'l...fl..O Q

-=

f..firoel ~

rhis gives a total load ed

For a stiffened curved

as

ie the smae except that there must added the load ce.rri ed

is subjecteti to the stre.s~ ~ '!' This lo£:~d is

?

(b-"lw)Tuc

(!. 9'\ le lj" :::

which is carried in addition to the loe.d c~.:::rried by the

st-iffener plus the attached, sheet., This j_g the method that

has been followed in calc:nuatiug the values gf Table V ..

In Table V there will be noticed a number of

specimens for which two ealeulated values l1ave 1:;eeu shovm.,

The :first value listed is that obtained by the method

ou:t-liue4 above i in 'ti"hieh the stiffener~ take a ~tresecr. ... si:ul

) i i

CJstz.. respectively~ and ce11ter of' the sheet t,akea a stress

cr- .. Consider!! :for example, t.he l2ff long specimen with

three stiffeners" made of 0"'052° sheet. e:ud ben~ to a radius

'le.

=

31200 lbs;!)' /sq,,,

Calculated in the ordi:uary manner" the predicted

which iliil

Since, in the above case, the failing stress of

all of the stt:ffene:rs is far beloc; the all.owable compressive

stress of the curved sl'l~:Hst" the stiffeuers very likely buckle

at their failing stress and precipitate :failure of curved sheet bet.iveen them .• lone; bf!'f'ore the sh.~et has reached its maximum allowable stress. For this reason, it is suggested

that !o:r conservative deeign for this case~ the whole panel be assumed to fail as soon as stiff'euer wi

value ofc.iSt reaches its failing stress11 a.ud. that the sheet

between st,iffe:uers ~an undergo no higher stresses than this

value of C\+(""'"-<!U:)., If this method is usedt!c the:n in the above

exwnple·t1

and the total load carried by the i e give11 by

which is 18% below the val:ue obtained. by test.,

It is obvious from an inspect:iJJn of the results in Table V that the higher orit.i Tsilues of the

sheet tend to retard failure the st1

the a.mount of this. effect is largely it seems

ferable to ignore i t altogether!' aud design as indicated. above

to the rigbt of the limiting linew we are a.ssurning that

TABLE.

I

t-\o _"'T"C-fa. b'' t" p• E a"»io -·t a: ¢ Jl" "E.a;

c

)._

l '1 fiY (~) ('I) IS) «•:l 17) \BJ ('ii lioJ

1 'DuRAI..

,,

. (l(,lo'i J.J

:r<-o

I· 023 :~7,330 5".l.t;ll ~OS 2.a"IS'

101.d.-2. 'i .ot.i.o 3,0o j.Ol.3 _'l11,)?»0 S". l.'i " ... 1 • .:..1<:. J J G.O

.3 .., 1& . 01ot.." '1,

•uo

j.031 "I01'i00 l.·'"11 I.HI ' ()~-q 7

"1 0 • OC..11 "i1'*oo j.031 ·~o, 'to.a

t..·n ;\ ..

I .Si I ' 115"'1

S"

,..

2 , o.i..12- 3,880 1.031 ·"IO,"fOO t.·"l1 •' 1,H:.7 ₅ 3-ieo i::J _" I 80 .ol '"!·00 l.·'1 ,11.""" j. 2-b'i . Slloo

l S;e.f..L. 3 'Ii .0113 Mo .v .. 6'1 .11.'t, .s:s-o ~" 13 i\ •• 1. ;"i'i

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