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4 .4 .

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A% NO. 3J02 ,;’--.“~~n ,J‘~”II l;~jlilll Illillllll

3 1176 00139 1466\___ _____ . --- . .

3$‘ ATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

-. ‘/

~~ WAI’TIRII1 IUUBOIU!ORIGINALLYISSUEDOctober1943 as

AdvanceRestrictedReport3J02

STRESSES AROUND RECTANGULAR CUT-OUTS IN SKIN-STRINGER

PANEISUNDERAXIALLOADS-11

By PaulKuhn,JohnE. Duberg,andSimonH. Diskin

LangleyMemorialAeronauticalLaboratoryLangleyField,Va.

N’AC’A“W-

ASHINGTON

NACA WARTLME REPORTS are reprints of papers originally issued to provide rapid distribution ofadvance 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- 368

,-..-..,,... .—— ..... .... -. , . .,. . .. . ..—.—

NATIONAL ADVISORY COMMITTEE

“ I “-4iIVANCE’flW?dTtiCTSD

FOR AERONAUTICS

.’

REPORT .

STRESSES ARC)UNDRECTANQUMR CIh!-OUTSIN SKIN-STRINGER “

PANELS UNIMR AXIAL LOADS = 11 ““

“~ Paul Kuhn, John E. Dtiberg,and Simon H. IMskin

. WJMMARY “

Cut-outs In wings or fuselages produce stress con-

centrations that present a serious problem to the stress

analyst. As a partial solution of the general problem,

this paper presents formulas for calculating the stress

distribution aro~d rectangular cut-outs in axially loaded

pane1s. The formulas are derived by means of the substitute-

strln~er method oi’shear-lag analysis.

In a previous paper published under the same title as

the present one, the analysis had been based on a substitute

structure containing only twc stringers. The present

solution is basad on a substitute structure containing three

stringers and is more complete as well as more accurate than

the previous one. It was found that the results could be

used to improve the accuracy of’the previous solution

without appreciably reducing the speed of calculation.

Details are given of the three-stringer

of the modified two-stringer solution.

solution as well as

I

,, mmnmmm--m —.. . . . . m,,, --------- . .. - .-. . — . .

In order to check the theory against experimental

results, stringer stresses and shear stresses were meas-

ured around a systematic series of cut-outs. In addi-

tion, the stringer stresses measured in the previous in-

vestigation were reanalyzed by the new formulas. “The

three-stringer method was found to give very good accuracy

in predicting the stringer stresses. The shear stresses

cannot be predicted with a comparable degree of accuracy;

the discrepancies are believed to be ~aused by loc&

deformations taktng place around the most highly loaded

rivets and relieving the maximum shear stresses.

INTRODUCTION

Cut-outs tn wings or fuselages constitute one of the

mo”sttroublesome problems confronting the aircraft designer.

Because the stress concentrations caused by cut-outs are

localized, a number of valuable partial solutions of the

problem can be obtained by analyzing the be~nvior, under.

load, of simple skin-stringer panels. A nethod for .

finding the stresses in axlal~~ loaded panels without cut-

outs was ~lven In reference 1, which also contained sug-

gestions for estimating the stresses around cut-outs. In

reference 2, these suggestions were put Into more definite

form as a set of formulas for analyzing an axially loaded

panel with a cut-out (fig. 1).

.-. . a“.,

skh.-8tr*eF.. . .“.... . .......... .ahelh, are *hly

*e2ss aIt@ugh 6$mp19r-.tlyin qdmp~ete. .----Zndetexylyate.striiciiuies. In Ordep tca., ”.,

reduoe t~, }abor of an@yzlng meh W!qelst :~Mf’Y!I’113 .s . .

assum@imst az@..pp.ecZaldev~ces may be i.ntmd,uoed~ ~~x “ “’.” ‘ ‘“- ““” ““ ‘“

most important,detiee qf thla natupe used m hefer,enoes1! .. ., .,...

aqd 2 1s. a reductipn of.thg n~~r, ~ atringems~ which is; . . ....effeot.edby.qomb$n~ a n~er of str:ngers into a s&-. .. . .,’

at~hqpir..

the extreme

$or,the cut

Zn iefa~enoe ?, thZs r?duotlon

of using only two substitute ●. .8trtige??#and one,for the uncut

stringers, to represent one quadrant of ths pAnel with a

Cut-outo we two-stringer struoture can be.analyzed very. .

~ldly butr being somewhat O?er-slmpllfled, G&mot give..

an entirely satlsl?actorypicture, In particular, the two-. .

stringer.structure does not ticlude the,region of the net.

BeetIon; aqd ~onqequently t~s struoture neither ~hms the. .

effect of length of cut-out nm gtyes a eoltitlonfor the

mtmb.zn strl.ngerstresses, .. These maxlmqm stresses must“.

be pbtained by separate assup@ions. In addltlona there1...

is no obvious relation between the phear stresses t.nthe..” ..actual..atructureaqd the shear atremes In the #ubotiZtu’W.. . .two-stringer s~cture 4s used ~, ~.fermee 2-,.

In order to obtain a.rnQ??esat.~sfqctorybasis of ana~~.. . . .than that of r@menoQ 2, fo.~laa were developed for IR

. .

,— — —-—I

4 ‘“

skin-stringer structure containing three stringers. At the

same the, a new experimental lnvesttgatlon was made c“on-

slsting of strain surveys around a systematic series of “

cut-outs. “ Stringer strains as well as shear strains in

the sheet were measured In these tests, whereas only

stringer strains had been measured In most of the tests of’

reference 2. A study of the three-stringer method and of

the new experimental results Indicated that the accuracy of

the two-stringer method could be improved by introducing

some modifications which have no appreciable effect on the

rapldlty of the calculations.

The main body of the present paper describes the ap-

plication to a panel with a cut-out of a stiplified three-

strhger method of analysis as well as a modified two-

stringer method, Comparisons are then shown between

calculated and experimental results of the new tests and

of the tests of reference 2. Appendixes A and B give

mathematical details of the exact and of the simplified

three-stringer methods, respectively. Appendix C gives

a numerical example solved by all methods.

THEOFU3TICALANALYSIS OF CUT-UUTS IH AXIALI/Y

LOADED SKIN-STRINGER PM?!ZS

General Principles and Assumptions

The general procedure of analysis is stillar to the

procedure developed for structures without cut-outs

.-

—. .- —

.

..(reference 1). The actual sheet-stringer structure Is

.. -w. .-replaced by an idealtzed structure in w~ch” the sheet

oarries only shear. “Theability of the sheet to-carry

nomal stresses is taken Into account”by adding a suitable

effective area of sheet to the cross-sectional area of each

stringer. The idealized structure I.sthen simplified by

coniblnhg groups of strl~ers’ into sin@e stringers, which

are termed ~tsubstltutestringersll;this substitution is

analogous to the use of “phantom members’tin truss analysls.

The substitute stringers are assumed to be connected by a

sheet having the same properties as the actual shest. The

stresses In the substitute sheet-stringer structure are

calculated by formulas obtained by solving the differential

equations governing

the stresses In the

the stresses In the

the problem (Seeappendix A.) Finally,

actual structure are calculated from

substitute structure.

It will be assumed that the panel is symmetrical about

both axes; the analysts can then be confined to one quadrant,

It is furthermore assumed that the cross-sectional areas of

the stringers and of the sheet do not vary spanwise, that

the panel is very long, and that the stringer stresses are

uniform at large spanwise dtstances from the cut-out.

—- -—-. -.

.

#

.-

..— .i

6

Al .

A2

As

‘rib

Conventions “ :

effective cress-seotlonal area of all continuous

stringers, exclusive of main stringer borderl~ .

cut-out, ~quare inches (fig. 9) .“

effective cross-sectional area of main 69ntinuoils

stringer bordering cut-out, sqaare inches (fig. 2)

efiective cpos8-sectional area of all discontinuous

stringers sqmre fnohes (fig. 2). .

cross-sectional area of rib at edge of c-it-out,

square inches (fi~. 2) “

[ K12 + lb2 +.2RB=

i

A &K3KA

xl2 i-K22 +2E-.*KIU

c “stress-concentrationfactor (fiG: 7)

co stress-excess factor for cut-out of’zeho length

(fig..3)

D= Al + he‘~ ~

E Youngts rmdulus of elastic~ty, kipg per square Inch

G shear modulus, kips per square inch

$=g~+’.+~) . “ .-

.lz.~(++~)

Gt~K4=—

~2A2

7

..

R = /%2%2-%%

L half-length of’cut-out, inches (fig. 2)

K3K4P’ =

Al (A12 - A#)

K12 (K12 - A12)p3 =

?iZ(LIZ - A22)

‘4=*Q1 force Al= acting on atrlnger 1 at rib, klps

~ force A26 acting on stringer 2 at rib, klps .

R strese-reduction factor to take care of change In

length of cut-out (fig. 4)

..—. ..—— .--, . ,, . , , ,,... .. ——.— .

XR difference between actual force in Al (or A2) at

the rib and the force Q1 (or Q2), .klps

a width of net section, inches (fig. 6)

b half-width of cut-out, inches [fig. 6)

bl distance from A2 to centroid of Al(fig. 2).b2 distance from A2 to cqntroid of A3 (fig. 2)

72Rt2rl=.—

A3CS0

‘2#2 - TIRtlrz=-

‘2(U2R - %)

W.‘R

r3=-— Zb@2R.

tl

tz

x

thickness of continuous panel, inches (fig. 2)

thickness of dlacontlnnous

spanwise dlste.nces,inches

figs. 2 and 6.)

panel, inches (fig. 2)

(E%r origins, see

Y chordwise distances, inches (For orl@m, see fl~. 2.)

K12 + K22 + d(%2 + % )22-4K2

‘1 = 2

9

‘o “ average stress In the gross section, kips per square, . . , . .. ---- . . - . . ---- . .

inch..-. . .. .. .. . .

03

T1

T2

stress In continuous substitute stringer, kipe per

square Inch

stress In

inch

stress in

main continuous stringer, ki~s per square,

diacontinuouq substitute strtnger, kips

per sq~re Inch

stress in rib, ktps per square inch

average stress In net section, kips per square inch

shear stress in continuous substitute panel, kips

per square inch

shear stress in discontinuous substitute panel, kips

per square inch

Superscripts on stresses denote forces producing the

stresses. Subscript R denotes stress occurring at rib.

statlonm .

Tensile stresses in stringers are positive. If the

center line of cut-out is fixed, posltlve shear stresses

are produced by a tensile force acting on Al.

Simplified Three-Stringer Method

A principle for the effective use of substitute

stringers was stated in reference 3 substantially as fol-

lows:

I

Leave the structure intact In the region of

about which

replace”the

9trhgers.

the most important actions take

strhgers away from this’region

In a panel with a cut-out, the

action *akes place around the main strin~er

cut-out* In accordance with the foregoing

three-stringer method is based on retainfn~

,.”

the stringer

place, and

by subatltute

most important

boundi~ the

principle, the

the main

strln~er as an individual stringer In the substitute

structure; one substitute strhger replaces all the remlrdng

continuous stringers, and another substitute strl~er re-

places all the discontinuous stringers. The thr~e-stringer

substitute structure obtained by this procedure is shown In

figure 2, which summarizes graphically the”~alient features

of the method. The figure shows the actual structure, the

substitute ~tructure, and the distribution of the stresses

in the actual structure.

The maximnn stringer stress as well as the maximum

shear stress occurs at the rlb station. The formulas given

hereinafter for the stresses at the rib station and in the

net section &e based on the exact solution of the differ-

ential equations presented in appendix A. The formulas

derived from %his exact solutlon for the stresses In the

gross section are somewlmt cunibersome&d are therefore

replaced here by formulas that are based on mathenatlcal

i

11

approximations of sufficient accuracy for design work.... .......W--- . .. r---(appendix B). The use of these approximat~ons’iS the

reason for calling this method the simplified three-

strhger method.

Stresses at the rib station in the substitute

structure.- The stringer stresses at the rlb station are

.,,=+*)

u2~ = F(1 + RCO)

(1)

(2)

where the factor Co, for a cut-out of zero len@h, is

obtained from figure 3 and the factor R, which corrects

co for length of cut-out, is obtained from figure 4.

For practical purposes, the parameter B appearing in

figure 4 may be assumed to equal unity. (See appendix A.)

The length factor R depends, therefore, chiefly on the

parameter KIL. This parameter Is roughly equal to L/a

for usual design proportions; in other words, the length

effect can be related more directly to the length-width

ratio L/a of the net section than to the proportions of

the cut-out itself.

The running shear in the continuous panel at the rib

station is

—.. . ——

12

(3)

The running shear in the discontinuous panel at themrib

station is. .

K4

( )

~12 mT2Rt2 = - FA2 ~ l+RCO+— . .--..(4)

x

in which the factor D may

The stresses 02R and

U2 and 72, respectively,

the panel. The stress o~

center Mne of the cut-out.

be obtained from figure 5.

72R are the maximum values of

and are the maximum stresses in

reaches its maximum at the

The stress ?~ reaches its. .

maxtium in the gross section at the station where

al = 02 # CJo.

Stresses in the net section of the substitute structure.-

“The formulas for the stresses in the not section are ob-

tained

tance

in the

from the exact solution (appendix A). At a dis-

x from the center line of the cut-out, the stresses

continuous stringers are

( .)RCOA2 cosh Klx%=6 1“- A1 cosh KIL

( )cosh KIX02=F 1 + Rco

cosh KIL

length of the cut-out - or, more precisely, the

13

length of the net section - Increases, the magnitude of the.—

‘parameter KIL Increases and the stresses ul and &2

oonverge toward the average stress ~ In the net section.%10A

The running shear in the net section 1s

Sinh I{lx‘1% =

. ~RCoA2KI Cosh KIL

and decreases rapidly to zero at the center line of the

cut-out.

Stresses in the gross section of the substitute

structure.- The stresses In the gross section can be obtained

from the exact solutlon filvenin appendix A, but the for-

mulas are too cumbersome for practical use. A simple ap-

proximate solution can, however, be derived (appendix B)

that gives good accuracy in the immediate vicinity of the .

cut-out and reasonable accuracy at lar~ar distances from the

cut-out* The approximate solution assumes the differ-

ences between the stresses at the rib station and the

average stresses In the gross section to decay exponentially

with rate-of-decay factors adjusted to give the Inltlal rates

of decay of the exact solution.

The stress in the cut stringer by tk.eapproximate

solution is

CY3”= Cfo(l-e-rlx) (5)

... . — — —.

●

The stress In the

The stress in the

statics and is

&aln stringer Is

continuous stringer

-r .xUo)e 2

1 follows from

The running shears in tb.esheet panels are

-rlx

\

-~ xTltl = T2Rt2 e - “2

‘2Rt2 - ‘lRtlje

T2t2 =-r3x

‘2j3t2e

(6) “

(7)

(8)

(9)

Stresses in the actual structure.- By the basic pri.n-

clples of the substitute structure, the stresses in the main

continuous stringer of the actual structure are Identlcnl

with the stresses in stringer 2 of the substitute structure;

the total force fn the remaining continuous stringers of

the actual structure is equal to the force In stringer 1 of

the substitute structure, and the total force $n the cut

stringers of the actual structure 1s equal to the force In

stringer 3.

In the shear-lag theory for beam without cut-outs

(reference 1), the force acti~ on a substitute stringer

is distributed over the corresponding actual stringers on

— .— ----

. 15 -

the assumption that the chordwlse distribution follows a-. F,.... .- ,., ..----.-

hyperbollc cosine law. Inspection of the test data for

panels with cut-outs indicated that ~elther this nor anyma other simple assmptlon fitted the data on the average asx-

well as the assumption of unlfozm distribution. It is .

therefore recommended, for the presents that the stresses

In all continuous stringers other than the main strin~er be

assumed to equal U1 and that the stresses In all cut

stringers be assumed to equal tJ~● (See fig. 2,) The

validity of these assumptions will be discussed in con-

nection with the study of the experimental data.

Again, by the principles of the substitute structure,

the shear stresses T~ in the substitute structure equal

the shear stresses tn the first continuous sheet panel

adjacent to the r~in stringer. In order to be consistent

with the assunptlon that the chordwise distribution of the

stringer stresses is uniform, the chordwise distribution

of the shear stresses should be assumed to taper llnearly

from 71 to zero at tpe edge of the panel (fig. 2).

Similarly, the chordwise distribution of the shear

stresses in the cut sheet panels should be assumed to

vary linearly frma T2 adjacent to the main stringer to

zero at the center line of the panel. Inspection of the

test data Indicated that this assumption does not hold very

..- . .- — —— . .

—

*16

well in the

crepancy is

immediate vicfni.tyof the cut-outs. The dis-

of some practtcal importance because the maxi-

mum stress In the rib depends on the chordwise distribution

of the shear stress at the rib. By plottlng experimental

values, it was found that the law of chordwise distribution “

of the shear stress T2 at the rib station could be approxi-

mated quite well by a cubio parabola. The effect of this

local variation may be assumed-to end at a spanwise distance

from the rib equal to one-fourth the full width of the cut-

out ● A straight line 1S sufficiently accurate to”repre-

sent the spanwise varlatlon wlthln thfs distance (fig. 2).

If the stress T2 is distributed according to cubic

law, the stress In the rib caused by the shear In the sheet

1s

T2Rt2b

— [1 - Wl‘rib = 4&ib (lo)

Modified Two-Stringer Method

The two-stringer method of analysis given in reference2

Is more rapid than, but not so accurate as, the three-

stringer method previously descrtbed. It was found, how-

ever, that some Improvements could be made, partly by in-

corporating some f’eaturesof the three-strl~er method,

partly by other modifications.

. . .-

17

The main features of the modified two-stringer method.-

are summarized In figure 69 The cut stringers are re-

placed by a single “substitutestringer; and all the uncut

stringers,“inoludlng.the -in one, are also replaced by a .

single stringer, Contrary to thg usual shear-lag method,

however, the stringer substituted for the continuous

stringers Is located not at the centpold of these stringers

but along the edge of the cut-out. The substitute structure

is used to establish the shear-lag parameter K, which

determines the maximum shear stress, the spanwlse rate of

decay of the shear stress, and the spanwise rate of change

of stringer stress. The maximum stringer stress must be

obtained by an Independent assumption, because a single

stringer that Is substituted for all continuous stringers

obviously cannot give any indication of the chordwise

distribution of

are obtained by

stresses In the

stress in these strlnEers. No solutions

the two-stringer method for the shear

continuous panels, either in the net

section .orIn the gross section.

Stresses In the substitute structure.- Throughout the

length of the net section, the stress in the main stringer

Is

a2R = ~ rl + 2R(C - 1)3 (11)

where C is the stress-concentrationfactor derived in

.-—- - -

18

reference 2. Values of C may be obtained from figure 7,

whloh Is reproduced from reference 2 for convenience. It

may be remarked here that reference 2 placed no explicit

restrtctlon on the use of the factor C; whereas the use in

formula (11) of the correction factor 2R, which varies from

2 for short cut-outs to.1 for 10n.g cut-outs, implies trot.

the factor C by

section Is long.

In the gross

itselt should be used only when the net

section, the stress In the main stringer

decreases with

to the formula

The

The

03

by

increasing distance from the rib according

( ) -ILKU2 =cfo + 02 - 00 eR

(12)

stress in the discontinuous subst!.tutestringer is

03 = O.(l - e-=) (13)

stress cl may be obtained by formula (7) when 02 and

are known.

The running shear in the discontinuous panel is given

T2t2 = - ooA3Ke-Kx (14)

Stresses in the actual structure.- The stresses in the

actual structure are obtained from the stresses in the sub-

stitute structure under the same assumptions as in the

three-stringermethod.

,.,

19 “

EXPERIMENTAL.WRII?ICATI(M!J(YFFORMULAS AND

..W9kiPARXSbit03?Bt@iH@S”””: --” ““

Teat Sp@alWIs and TeaII~Procedure .zAAA

In order to obtain experimental verification for.the

Comxmlas developed, a Mrgs akin-stringer panel wad built

and tested. The p#n&l”was a~llarto the one described In

reference 2, but thet$e@pe d“ths tedta was extended in two

respects: Very s-* ~ut~?lts were .testedin addition

cut-outs of average Ieugth, and shear stresses as well

to

as

stringer st.resms We&e measured around alilcut-outs,.

The general teat setup Is aho~ In figure 8.

of strain gages ts s?lownin figure 9. . The panel

of 249-T aluminum allay and was 144 incbLe8 long.

A setup

was made I

The

cross section Is shown in figure lC)(a);fi~ure 10(b) shows

for reference purposes the cross section of the panel tested

previously (reference 2). Straifiswere measured by

Tuckerman strain ~ages with a“gage length of 2 inches.

The gages were used In pairs on both sides of the test panel.

Stnains were

quadrants.

each plotted

measured at corresponding points in all faur

The final figures are drawn as for one quadrant;

point representip,therefore, the average of

four statj.o~sor eight gages. .

The load was applied $n three equal Mmements. If

the straight 11.nethrough the three points on the load-stress

..- -—— —---- -- . . —-

.

20

plot did not pass through the origin, the line was shifted

to pass throu& the origin; however, if the necessary shift

was more than 0.2 klp per square inch, a new set of read- -

ings was t*en. :

An effective value of Young~s modulus of X!.lb x IO? ~.ips

per.square Inch was derived by measuring the strains.1’n”all

stringers at three sta$$oqs along the span .beforethe first

cut-out was made. This effective modulus may be-con-

sidered as Includlng corrections for the ef’fectsof rivet

holes, average gapy calibration factor, and.dynamometer

calibration factor. The individual gage factors were

known to be.yithln ~~ercent of the average..

The average stra~n at any one of the three “stations

in the panel without.cut-out did not differ-by more than

0.05 percent from the final total average..

deviation of’qn Individual stringer strqtn

avera~q was 5 percent; about 10 percent pf

. The maximum

from the

the points- “~ .

deviated fr.pmthe average by mor8.than”3

survey was also made of longitudina~ and

strains at one station near the center.

percent. A.

transverse sheet

The average

longitudinal sheet strain dlf~ered from the average*

stringer strain by 0.05 percent,. , Tho average transverse. .

strain indicated a Poissonrs ratio .of 0.323.

#

——-. — -

.

al,.

Discussion. .

.-, -The-resul.ts:ofthe-testware .sbpwn-in-figures 11 to 33.

Calculated curves are given both for Me exact three- .

stringer method an”dfor the simplified three-stringer method.

It may be recalled that either method assumes l@t the

stresses in all continuous stringers except the main stringsr “

~ve the magnitude 01 and In all

.tude U3. Because the values of

fer very much for the two methods,

puted by the simplified method are

cut stringers, the magni-

al and 03 do not dif-

the curves for them com-

drawn only once in each

figure.

A qualitative stud~ of figures 11 to 32 indicates that

the stress distrlb~~tiancalculated by the theory agrees ,

sufficiently ~:ellv.iththe experimental distribution to be

acceptable for moat stress-analysis purposes - In particular,

the

the

are

msxhnm stresses In each panel agree fairly well with

calculated ones. The most consistent discrepancies

chargeable to the simplifying assumption that the

stringer stresses are identical In all the stringers repre-

sented by one substitute stringer. As a result of this

assumption, the calculated stresses tend to be too low for

the stringers close to the main stringer and too high for

the stringers near the center line and near the edge of the

panel. The fact that the calculated stresses for some of

—— .—

aa

the cut stringers are lower than the’actual stresses 1s of. . .

Ilttle practical importance because these strtigers would. .

probably be designed to carry the stress co rather than. .

the actual stresses. On the uncut stringers,“however, ‘

it may be necessary to allow some extra margin in the “

stringers near &he matn one. Aside from the consistent “’ ~

discrepancies just notecl,figures 11 to 32 show that the “ “

stresses in the main stringers sometimes decrease spanwlse

more rapidly than the theory indicates. It Is believed

thdt this”dlscrepancy glso will seldom be of any consequence

in practical analysis.

Of paramount interest to

“value8 of the stresses. The

the analyst are the maximum

quantitative study of errors

in the maxtium stresses is facilitated by table 1. The

highest stresses occur theoretically at the rib station

but, for practical reasons, measurements had to be made at

“some”small distance from th$s line. The comphktsons are “

made for the actual gage locations. The calculated values

for the three-string6r method are based on the exact solu-

tion but; in the region of these gage locations, the exact

“solutionand the.simplified solution agree within a fraction

of 1 percent. “-

The errors shown by table 1 for the maximum stringer

stresses computed by the three-stringer method are but

_ ...

little larger than the 100al stress variations that were

found experimentally to exist in the panel-without cut-

OUt● Presumably these variations are caused largely bysy failme of the rivets to enforce integral action of the4

structure.

The errozwin the maximum

the three-stringer method are

shear streises c&puted

consistently posltlveo

discrepancies are possibly causedby the sheet around

by

The.

the

most highly loaded rivets deforming and thereby relieving

the maximum shear stresses. The errors are hdgher than

those on the stringer stresses and corrections to the

theory appear desirable In some cases. The criterion that

determines the accuracy of the theory cannot be definitely .

established from the tests. A rough rule appears to be

that the error increases as the ratio of width of cut-out

to width of panel decreases.

The errors given in table 1 for the.two-stringer

method show that this method 1s decidedly less accurate

than the three-stringermethod for computing maximum

stringer stresses but that there is llttle difference

between the two methods as far as the computation of the

maxfmum shear stresses Is concerned. A general study

of the two theories indicates that this conclusion drawn

from the tests Is probably generally valid. It may be

24.

recalled here that the two-stringer method gives no solu-

tion for shear stresses in the continuous panels. “

Comparisons of the maximm observed rib stresses and

the computed stresses are given In table 2. Two values of

computed stress are shown. The smaller value was.obtained

on the assumption that the filler strips between the ribs

-and the sheet were effective In resisting the load applied

to the ribs; whereas the larger’value was obtained on the

assumption that the filler strips were entirely ineffective.

In figure 33, the chordwlse variation of the observed and

computed rib stresses Is shown for three cut-outs. Because

the chordwlse distribution of shear stress In each sheet.

panel between two stringers is essentially constant, rib

stresses computed by formula (10) will be too small when cmly

a few stringers are cut. The computed values of rib stress

were therefore determined by calculating the shear stress

at the center of each panel according to the cubic law and

assuming this shear stress to act in the whole panel.

The agreement between calculated and”observed.rib

stresses 1s not all that could be desired. The discrepancy

may be attributed to the approximation used for determining

25

the shear stresses and the uncertainty of tha effectivew. ...,. ,..... ... ........... .

area of the rib.,.........-... ...

8?

s’ Langley M&orial Aeronautical Laboratory,National Advisory Committee for Aeronautics,

Langley Field, Va.

. .

----- .

26

APPENDIX A

EXACT SOLUTION OF THTWE-STRINGIX STRUCTURR3

For a two-stringer panel constituting one half of a

symmetrical structure, the application of the

lag theory yields the differential equation

d2T-K2T=0

G

basic shear-

(A-1)

which Is given in sltghtly different form in reference 4.

In the analypis of a skin-stringer panel with a cut-out,

a three-stringer substitut~ structure is used. (See fig.2.)

Application of the basic equations of reference 4 to a

three-stringer structure Vields in place of the single eq~-

tion (A-1) the sinultaneoun equations

On the simplifying assumption that the panel Is very long

and that it is uniformly loaded by a stress 00 at the

far ends, the general solution of the equat~ons (A-2) is

+ +xT1 = cle + cae (A-3)

in which c1 and C2 are arbitrary constants.

..

. a?

—

.-,Because -tha”struotureIs assmed to-be symmetrical

about the longitud~.nal-as well as the transverse axis,

the ~i~~is may be confined to one quadrant as shown Inz

xfigure 34(a). Ths analysis can be simplified somewhat by

severing the strac%ure at the rib and considering separately

the net section and the gross .sectlon. The solutions for

the two-stringer structure representing the net section can

be obtained frm reference 4. The solutions for the

t.hree-strlr~erstructure representing the flrosssection are

obtained conveniently by considering two separate cases of

loadln~. In the first case, stresses U. are assumed to

be applied at the far end, and the stresses at tlie rlb

station are assumed to be uniform and equal to the average

streaa z necesaury to balance the stresses O.. The

forces at the rib station%xisthg in the strln~ers are

called the Q-forces (fig. 34(b)). In the second loading

case, a croup of two equal and opposite forces is assumed

to load the strl~ers 1 and 2 at the rlb station. These

forces are called X-forces (fig. 34(c)).

In the net section the boundary conditions are as “

follows:

At x=O,

T1 = O (from symmetry}

—-

, .-. , .- ,. .....-—.. -. .. . --- . . -------- —

28

At” X= L,

The substitution of these conditions in the solutlon of’

equation (A-1) yields “

x xR~l SiIlh Klx

‘1 ‘-~ cosh KTL (A-5)

The superscript X Indicates that the stresses are those

caused by the action of

the total stresses, the

to alx or U2X. The

the X-forces. In order to obtain

average stress ~ must be added

shear stress 71X is the total

stress because the uniform stress ~ is not accompanied by

any shear stress.

When the Q-forces are epplied to the crass section,

the boundary conditions at x = O are

..-.

-.

a9

Applying these conditions to equations (A-3.)..and-.(4)4)@ves.. . .

the followliigsaihtlons for.stresses: ..

Q2.( +x@ = ~ W .- p4e )

-%. .

..●

[

% P2(K12 - ~12) e-?i~x P4(K12- ’22) 42XT2Q = yK3 K3

e1

UIQ = Cro

02 Q %‘“o+~

{[

Pa

q 1-

e-klx

~-?qx r‘4 ~.—.?b2

The superscript Q lndlcates that the

(A-9)

(A-1O)

}

-%e

stresses are those

caused by the action of the Q-forces.

The boundary conditions due to the application of the

X-forces are, at x =

and the oorrespondlng

T1x ‘R[=.q (P1 +

‘3=0

equations for stress are

+xP~ e 1-(P3 + p~ea2x (A-n)

— ...

,, .,-,,,...

,.

., .. .

,. . . . . . . . . . .

... . .. .

.,,- .,

,..

. .

31

. . . .. ”...,

For any iength of’cut=out, ,.

= Q2COR?KR=%col+B~ati KIL .

Values for co can be obtained from figure 3. In

figure 4, the factor R is plotted for various values of

KIL and B. The value of B may be assumed equal to

unity wfth llttle loss in accuracy In the determination of

stress; but, if a more exact solution is desired, the actual

value of B may be computed and the curve in figure 4 cor-

responding to this value used.

— —-. — .- ~-—_.—-—. — —

. . .. . . . ..— .. . . — —. ...I

32. . .

SIMPLIFIED

%e solutions

APP~!DIX B

SOLUTION OF THREE-STRINGER STRUCTU~S

for the stresses In the gross section

given In apFendlx A are too Involved for practical use, and

a simplified method wa”sdeveloped. This method assumes

that the differences between the values of U2RD U3RJ and

T2R, obtained by the exact solution, and the corresponding

average stresyes In the grozs section decay exponentially

with rate-of-decay factors adjusted to give Initi”alrates

of decay equal to those of the exact solution. These..

rates can be written simply in terms of the stresses at

the rib and the properties of the panel. The solutions

for al and 71 are then derived’from the solutions for

U2 and IY3.

If it is assumed that the stresses in the cut stringer

can be expressed by

u= = uo~l - e-rlau

thenda3 -r x

= aorle sF

but, from the bas~c shear-lag theory,

d03 T2 t2-rlx

x’-~=oorle(B-1)

Therefore, at x = O,.72Rt2

rl=-cA300

1-----

. -. - - -n. .,..- ,,..

The stress

approxhated by

33

r- . . . . . . . . .. . . . . . ,+. . ., .,-- . . .

In the main continuous Btringer can i% .

.. .

but, from the shear-lag

du2

==

Therefore, at

The value

Tltl.x

x= o,

r2 =

of 01

+

theory,

T2t2— = -(,CT2-A2

-.O.) re-r2x

(B-2)

‘2 (.U2R - ‘O,)

can “beobtained by statics from 02

and 03 and is

‘2 ‘3‘1 = ‘o + q (.OO- (B-3)%2) + ~ & - ~3)

If the value of T2 Is assumed to decay exponentially,

then

~ \ ‘2=72R=r=

and

dTz -r xx= ‘T2Rr3e 3

but, from the shear-lag theory,

dTz

x=&v72 -*d=3 “m--r x

-T2Rr3e.

——-—

.. . —. -—.

34. ... ..

Therefore, at x = .0,

The shear stresses in the continuous panel can be

detemnined from the rate of change of (S1. From the shear-

laC theory,dul Tltl~=~ . .

Dlfferentlation of fomnzla (B-3) ~ields

(?3-4)

(3-5)

Substitution of the derivatives (B-1) and (B-2) already

obtained in (E-5) ~lves

dul A2 A3—=.—

(- ao)re-r2x -

-rlxdx ‘2R , aorle (B-6)

Al q

Finally, substitution of (B-6) in (B-4) yields

.

.-.

35

. . IL ..-. --

A~kT!NDIX ()” - - ‘“””

NUMIIRICALE2UlllP~

1 Analysis by the Exact Three-Stringer MethodY\ The structure chosen for the numerical example Is the1

16-strfnger panel tested as part of this Inves”tigatlon.

The particular

cut and with a

cut-out Is the

case chosen is the panel with eight stringers

length of cut-out equal to 30 Inches. This

one shown In figure 8. The cross section

of the panel is shown tn figure 10(a). The basic data are:

Al, sqin. . . . . . . . . . . . . . . . . . . . . . 0.703A2,,sqin. . . . . . . . . . . . . . . . . . . . . . 0.212A~, sqin. . ● i . . . . . . . . . . . . . . . . . . 1.045tl,ln. . . . . . . . . . . . . . . . . . . . . . .0.0331t2,in. . . . . . . . . . . . . . . . .’. . . . . .0.0331bl,in. . . . . . . . . . ...”. . . . . . . . . . 5.96b2,in. . . . . . . . . . . . . . . . . . . . . . . 7.56”L,in. . . . . . . . . . . . . . . . . . . . . . . . 15.0

These data yield the following values:

K12 = 0.01295K22= 0.009441:3= 0.00995

% = 0.00785x = 0.00664

From these parameters follow the factors for the rate of

decay, which are

K12 + %2 +Al = (%

2 + %2)2 - 4j&

2

Kf + K2 )22

AZ =- 4K2

2

= 0.1421

= O●0467

36

The

terms of

of which

PI.=

P2 =

P3 =

p4 =

With

computations of stress may more eastly be made in

the constants PI, P2, P3, and P4, the values

are

= 0.01295(0.01295 - 0.00218)0.1421(0.02021 - 0.00218)

“ ‘3K4 0.00995 x 0.00785=“0.0305

AI(A1 2. L22) = 0.1421(0.02W!1 - 0.00218) .

. . . .. .

., X3-4 . . . ‘otioo995.x b:oo785‘.O.C467(0.02021 - 0.0021G) .

= 0.0927A2(?12 : X22)

The reduced stress-excess factor Is

Rco =

=

r~ - l?~z

p~+P2-P3- P4 + K1 tanh KIL

0.0927 - 0,03050.0545 + 0.0305 + 0.111’7- G.0927 + 0.1065

=0.296

.“.. ..

a force of 7.5 kips acting on the half’panel,

klps~sq in.

. .

I

——.

37

and “- -.”.’ .. ., . .?..,.-, ----,.

7.50 ““ “ “5=—

0.915, = 8.21 kips/sq In.. .

Therefore,. .

XR = RC05A2 = 0.296 x 8.21 x 0.212 = 0.514 klp

Stresses In the nqt section.- The shear stress in.the

sfistitute panel of the net section 1s fomd by equation (A-Q

X#l sh.h Klx ~‘1 = - ~ cosh KIL

0.514 x 0.1138 Sinh 0.1138X=-

=- 0.620 S~* 0.1138x

At the rlb station, x = 15.0 and

TIR = -0.620 Shh 1.707 = -1.65 klps/sq in.

The stringer stresses are foundby .substltutingin

equations (A-6) and (A-7) and adding the average stress

XR cosh Klxul=L- 0.514 cosh 0.1138x

Al cosh KIL = 8.21 -“—–0.703 cosh 1,707

= 8.21 - 00257 cosh 0al138x

XR”cosh K1x”=E+— 0.514 cosh 0.1138x

62 A2 cosh KIL = 8-21 + 0.212 cosh 1.707. .

= 8.21 + 0.850 cosh 0.1138x

-..—— .

38

The nwdmum stringer stress occurs In the main stringer at

the rib, x = 15. The nearest gage

x= 13.5, where

02 = 8.21 + 0.850 cosh 1.536 = 8.21

Stresses in the Rross section.-

gross section are obtained by adding

stresses due to the X- and Q-forces.

location was at

+ 2.05 = 10.26 kips\sqti

The stresses in the

the solutions for the

The shear stress In

the continuous panel is obtalnedby adding equations (A-8)

and (A-n). The final solution thus obtained 1s

T1 = 2.92e-@.l~~lx - *057e-0.0467x

At the rib station, x = O and

‘lq = 2.92 - 4.57 = -1.65 kips\sq in..

This value of T~R

checks the one previously obtained

for this sane station in the net section.

Substituting the constants In equations (A-9) and (A-1o)

“and combini~ gives

-o.1421x - ~095e-o.0467x‘2 = -2.13e

At X = 1.50, the point of maximum observed shear stress,

T2 = (-2.13)(0.809) - (4.95)(0.931)= -6.33 kips/sq In.

I

The stress In,.. ..

found by combfning

final result is%

“ 39

the continuous substitute stringer is,.. - ‘r,

equations (A-1O)

= 3.82-O.97e -0.1421x +‘1

Stillarly, the s,tresaesin the

.. ..-

and (A-12).

~ ~3e-0.0467x●

main stringer

The

and in

the cut stringers are found by adding the proper values of

the X- and Q-stresses. In the main stringer,

o~ = 3.82 + 5055e-0~142h + 1.26e-0.0467x

and, In the cut strlng~a?a,

-0-1421-x - ~ ~6e-0.0467x‘3 = 3082 - 0.4Ge ●

Plots of the computed stresses In the panel for this

cut-out are shown In figures 22 and 30.

Analysis by the Approximate Three-Stringer Method

The basic data are the same as for the exact three-

stringer method. Compute

K12K~2 0.01295 x 0.00944 = ~ 565K3K4 = 0.00995 x 0.00785 “

It = 0.00664 = ~ 704q— 0.00944 “

From figure 3,

co = 0.600

40

From figure 4 for KIL = 1.707 and the exact value of

B = 1.10, there is obtained R = 0.492. “

me stresses in the continuous stringers at the rib

are,by formulas (1) and (2),

[= 8.21 1 -

)](0.492) (00600) &;;~ .%R = 7.48 klps/sq In.

~2R = 8.21rl + (0.492)(0.600)]= 10.63 kips/sq in.

The running shear ic the continuous panel at the rib

is,by formula (3),

TIRtl = -8.21 X 0.492 X 0.600 x 0.212 X 0.1138 tank 1.707

= -0.0547 kip/lfl.

The maximum runni~ shear fn the cut panel is computed by

formula (4). The value of I) is obtained from figure 5;

with ~ = 0.00664 and K12 + K22 = 0.02239, D = 0.189

and

0.00785T2Rt2 = -8021 x ‘“212 x Tzm [ 1~+ (o,492)(o.600)+ :”::::: “

.

= -0.234 klp/in.

0.234To=-—‘R

= -7.08 kips/sq in.O.osa

The stresses In the net section are computed as for

the exact solution.

.— I

41

The rate-of-decay factors for the stresses in the gross---- ,, ..-.,-

section can now be conputed!-

‘i?#2rl=-— A300

T2Rt2 - TIRtlr2=-

‘2~02R - ‘O)

‘1a2Rr3=-—

Eb2T~R

-0.234-=-3a82 X Imo45 = om0587

. .

-0.234 + 0.055=-0.212(10.63 - 3.82)

= 0.1236

00380 X 10=63 s o 0755=.

7.56 X -7.08 “

The stress in the cut stringers by formula (5) is

U3 = 3.82@ - e-0”0587x~

and in the main strinfjerby formula (6) is

CJ2 = 3.82 +

The stress In the continuous

formula (7).

The running

Tltl =

shears are$

6.936-0.1236x

stringer can be fo~d by “ .

by formulas-(8) and (9), .

-0 ~34e-0.0587x+ 0.179e

-0.1218x●

‘2t2= -0.234e400755x

...

.,

At X= 1.50, the point of maximti”observed shear stress,

T2t~ = -0.234 % 0.893

= -0.209 kip/in.

I --- —— .. . .- —..-

——— .——(

42

and . . . .

‘0.209T2 = - “ml ..

= -6.31 kips/sq in.

Analysis by the Two-Stri&ger”S’olution

Yhe basic data remain as before, Compute

. ..

9.38:’ 14.06

. .. . .

= 0.667

. .

hJL3 :“1.045 x 9;38

b~Al + A2j 14.06 X 0.915

= 0.764..

Then from fl&ure 7 is obtained . ~ ~

c = 1.195

The maximum stringer stress can then be computed by

formula (11) . . .

‘2R = 8.21[1 + 2(0.492 )(0.195~ “

= 9“.G5 klpg~sq in. “ -‘“‘ “

By atatlcs, - “’ ““,“

.-

%R = 7.70 k~ps/sq in.. .

The rate-of-decay factor is computed from

IF= 0.380 X 0.0331. . 7.56 ( )&“+&

= 0.00342 .,--

Ii= 0 ,,0q85. .

In the net section the stringer stresses are assumed to

be constant and equal to the.stresses at the rib. For the

gross section, by formula (12), the stress in the main

stringer is .

= 3.82 + 6.03e-0m0585x‘2 .

an~by formula (13),the stress in the discontinuous

stringers is

‘3 =

The stress in the

using formula (7).

The running shear

3.82 (1 - e-0”0585xj

continuous

In the cut

T2t2 = -3.82 x 1.045 x

= -0.234e-0”0585x

stringers may be found by

panel 1s,by formula (14),

0.0585e-0.0585%

At x = 1.50, the point of maximum observed shear stress,

T2t2 = -0.0234 x 0s916

= -0.214 kip/in.

and

-r2=- 0.214m

= -6.47 kips/sq In.

..--— —.-— . —.— —-—-

I

1.

2.

3.

4.

“ REFEBENCE9

Kuhn, “Paul,and Chiarito, PatrickBeams - Methods of Analysis and

TS: Shear Lag in BoxExperimental Investi-

gations. Rep. No. 7398 NACA, 1942.

Kuhn, Paul, and Maggie, Edwin M.: Stresses aroundRectangular Cut-Outs In Skin-Stringer Panels underAxial Loads. l~ACAA.R.R., June 1942.

Kuhn, Paul: Approxlnate Stress Analysls of Multf-strl~en Bems with Shear Deformation of the Flanges.Rep. KO. 636, NACA, 1938.

Kuhn, Paul: Stress Analysis of Beams with Shear Defor-mation of the Flanges. Rep- No. 608, NACA, 1937.

. ..

;MAXIMUMSTRESSES >COMPARISONOF OE-SERVED

——Stringerstresses

had Obeerved I ‘1’lwee-strlnfier

I Shear stressesI

bNumberofstringers

TotalCut

Two-strlnSersolution

Observedmaximumstreaa

~r-

(kip~/~qIn-) (klPS/eqtn.)(percent)’(kipe/sqin.) (Pe~~t

Half-lengthcut-out(in.)?“““mMw (kips/~q‘n”! (k~ps/sqin.)(perc~t)

Calculated Error(klps/sqIn.) (pe~;nt)

1 1 1 I16-stringertestpanel,length= 144.0in.

2.91.’3-?.1-6.6-3.4-1.1-1.2-1.4

7.’768.529.5010.9010.009.8512.8019.43

13.8-2.0-11.3-17.4-9.1-5.0-3.9-.9

-2.97-4.19-5.34-6.55-6.00-5.90-7.20-9.41

-3.91-5.03-5.90-6.68-6.40-6.33-7,62-9.77

31.720.110.51*96.67.45.93.9

-3.85-4.71-5.47-6.47-6.47-6.47-7.75-9.80

29.612.4

-:::7.89.77.44.1

1.51.51*51.58.015.015.015.0

15.015.015.015.015.015.015.015.0

6.828.6910.7113.2011.0010.3713.3019.61

7.1)28.8610.4912●3310.6310.2613*1419.34

I 15-gtringertestpanel,length= 141.6in.I -—8.38.38.3Q*38.38.3

3.2 4.60 29.5 ---.- ----- ---- ----- ----5.19 9.5 ----- ----- ---- ----- ----

-2:: 5.72 -1.7 ----- ---.- --*- ----- ----

-4.1 6.70 -5.4 ----- ----- ----- ----- ----

-7.3 8.23 -8.,8 ----- --.-- ---- ----- ----

1.5 6.31 8.2 ----- ----- --..- ----- ----

8.94 3.568.94 4.’748.94 5.1328.94 7.008.94 9.0210.85 5.83

3.674.765.716.798.365.92

7- end 8-stringertestpanels,length= 62.5 in..

7 1 5.0 20.0 18.85 18.39 -2.4 21.40 13.5 19.20 19.70 2.6 19.40 1.07 5.0 20.0 29.40 27●95 -5.3 29.00 -1.4 12.40 12.80 3●28 ;

13.50 8.95.0 20.0 20.25 19.21 -5.4 21.s0 5.2 16.80 19.70 17*3 19.80 17.8

. . 27.91 -8.8 29.60 8.60 10.60 23.3 10.20 15.7

Averageof absoluteerrorsfor all tests

:+ 4 200 3060 3-4 :; . . 11*2

9.8

a.Errora are

Calculated- fhamvedObserved x 100.

b iiArea of mah stringer Inoreaeed.

-.

8

46 ,TABIiB2 :

COMPARISON (IF’OBSERVED AND CMXUIATED MAXIMUM HIB STRIMES i[Load on panel, 15 kips~

Humber of strhger~

=16 2

4i: 616 816 816 816 1016 12

Half-lengthcut-out(in.)

Observedetressee

(kips/sq in.)

Calculatedstresses

(f:X1/fq&)

1.51.51.51.5

I 8.015.0

I 15.015.0

I

1.572.2029732.894.304.775.496.75

2.513.294.425.733.323.284.486.49 J

1.481.942.603.38 “2.91.2.883.94 .5.70

aFillor strips Ineffective.

bFiller strips effective.

● ✎

‘..

—

IU.4

.“..-, -.. .-

t

. .

1

1+

.-

— —

—

—

l?ig.1

—

., ---- ._= .

Figure 1.- Axiallyloadedpanelwithcut-out.

NACA Fiq. 2

rib

x

Stringer stresses

//

///’

///— ——

,/

$?-/zMaiwki~— Cut-alt

“7=7--47

‘Az ‘A3

—i-?AIJ

Shear stresses

Figure Z.-Stress distributim around cut-d by three-strimp method.

1.0 10 divisions on 1/20 EnRr.

c(’)

-====

‘1.() 1.2 1.4 1.6 1,8 2.0 2.2 2,4 2.6 2.8 3,0 302 3.4 3.6 3.8 4.0

Figure 3.-

K,2K22

K3 K4

Graph for cc).

‘“-’*

1.0

.9

.8

●7

R

.6

.5

.4.4 .

(1 block = 20 c

1 I

T.visione on 1 25 En

%=

=1.0

I

II 2.0 2

K,L

Fgure 4.- Graph for fwtor R.

%1

&.*

,06 (1 block = 20/25)1)-,

.05‘ \‘ ~

\ I

\‘-..

.03t+

I !-. \ “%283. “J. J

n, tn2-

Figure 5.- Graph for factor D.

I

(n

=F=e%!?’\’:\’\T’\”L

Figure 5,- Graph for factor D.

NACAY ‘---7 F iq. 6

~–.

\

//

‘//I

/L’-––––// ‘“”’

/’ \,‘,\

///

l\// \

L——————–———. !1

Strirger stresses

7

5ubstitute structure

-b

7

5hecr stresses

Figure 6:3ress distribution arourd cut-out by modified two-stringer method.

I

I .W

.1

i3gure 7. –

.2 .3

Approximate

,4 .6 .8 1.0 ~

b

2 34 6 0 10

stress-concentration factor C for stringer stresses(from reference 2).

-J

Fig,ure [{.- Test set-up on lG-Stri~~er panel with eight stringers cut and L = 15 in. ~

Figure 9.<-Typical set-up of strain gages. b

lTACA Figm.10,11JM87

Lt-.o33l

L tJ-.m

A

r’ ‘[+1-

A

~ 703.125-21.8%‘ t-

IM

(4 IQxltlel

I.KnoL t=.cm6

I-E+’4 L L =

(0)16-stringerpnel.(b)15-stringer pnel.

Fw IO.- Crosssectmof test ponek.

— Exact solution ---- Simplified solution

4 L(by three-str-inkr method)

o

L—..—,—.—————

z

.~_ I I

00 5Dis&ce fr~m cent$ line o?cut-out:n.

35 40

F@re II .-Str.~ stresses in 15-stringer panel with 1stringer cut andL =8.3 inches.

!?. .

YACA

.

4

z

0

4

2

0

6

4

2

0

4

2

0

Exact Solutbrl

1—--- — 1%pliiitd*t*mbt~*-striqff mdhd

Fig. la

+.—.. ..-. . . .. —.- .-. ...- ....—-—-- .-.—— ..—. . .

(+i4k

.u. ..— - & ._____ ,, .2;-

0t-

—--— —-——---—-—.——-—..-

1-1

J .> ‘.

.—

k --—— ——---——......-—-.—.—...-.8,6

L4,“-~_._.____ -:_______.-..—

2

0: — —-. . ... .. . . .I

4,

I ------—” ---”..I

L!

2“ .~—3

001 I 1 I I

5Disbce f~ cent%”m of %-out, i;

3!3 40

Fin 12.-Stringer stresses m15-strirwr pad wiih 3 strkg.rs cutd L=8.3 idea

I -.

,

IAOA

6 r— hct Solutkxl

)‘---- sblymedsdutkm ~

rig. 13

tkee-5tringer ~ttiod

I2 .,-.,..- ., .-,.,> ,-,..

0I

o

6L

I4 L. . -+

w a

0: .—. -—. —. —... .- - --- -- -. .-—--- .

0 l—— - -—___ — —-... .

8-.

4“2~=,&.-_:____ ., ...––.

2 ~-

04’~ ‘- ‘—-—

2 ‘.

04-

- . -. --- —-------- “

#

~~

Fi~me13-S+r”qer stresses in I%tr”kqer panelwith 5 s!nngerscutod L=83imhes

-———-

Emt sdutii

1Simplified solutionby three-stringer method

rig.14

8

[6,!

t

4.0’ Q”--Qw o

m .,. . r

c1

21

00

6T

8

6

4

2

0

4

z

o

4

2

0

-a-- —-–—

i

KJ

8 I > A-6

t4,

2

0Li

t4-

2t

0c

Fwe M.-%ingw s!resses in 15-sh%ger pad W-M7stringers cut d L=83”tis

NACA

8

‘6

4

2

0

—————

r

Exwt splution

Simplified solution}by

Fig. 15

three-stringer method

1-0

10,

8“-00

6 ;

4t

o ——. — ————_0

tvl

.—

0

ci-——————

0 0 0 0

00

n 0

.)

0

c)0

0 0_o---zr

2

L

0 00

00I I I I I

5

D&nce fr~m cen%r line % cut-m~ in.35 40

Figure 15.-3-ringer stresses in 15-strimjer panelwith 9 stringers cut andL=8.3inches.

!? ‘-..’

6

4

2

0

8

6

4

2

0

4

z

0

4

2

“1

Exact solution

}

‘Fig.16

----- 5implified solutionby three-stringer method

l_

1

t

I

Zt

L-2’ 00 0

00 I I I I I

5!k$ance f;;m cen& line % cut-ou~, in.

35 40

Figure 16- Stri~er stresses in 15-stringer punelwith 9strhgers cut andL=8.3 inches.Heovy main stringers.

E!–J

6

4

2

0

6

4

2

0

6

4

2

0

Fig. 17

Exoct solution

---–– Simplified solution}by three-stringer method

6

2t

‘i

I6;

4+-

10

8

6

4

2

0

r

,-,

i- 0 ————— __________

1-

4 __ —.. ——.—— —..—__-— ~-- ——=i

2’

[ KI

/<---’/’

o-” I I I I I

o 5Dis&ce fr;m cent% line o?cut-ou~in.

35 40

r-.

i-igure 17.-5tringer stresses in 16-strings-pm-d with 2 stringers cut andL=l.5inches.

8..m.n-m-— -—- ,., ,.,, ,,., - , , ,. , ., , —. ——-——. —.. . ,.

NACA

.r-

,=

.b

6

4

2

0

6

4

2

0

8

6

4

2

0

4

2

0

Fig. 18

I Exact solution

r ----- Simplified solution}by three-stringer method

2t

Oi

8

6;() c1

4 -

2’r

() o 0—, ——— —— ____ ———— —— ____

L?

10’

8b6’ 0 ‘=,=

[i‘=_.

4-0 .=_____-— __ . —. —___ ___

o ..-——— ---—-—

—-...—..—.-—.—

2’(LJ

L~1 c1

o- I I I I I

0 5

Dis?ance f;gm cen~r line&f cut-o:!, in.35 40

I-igw-e 18.-5tringer str=sses in 16-stringer panel with 4 stringers cut and L=I,5 inches.

NACA 6

4

2

0

8

6

4

2

0

12

r0 0

w

Exact solution

)‘---– %7pMed solutionby three-stringw methd

8 ,, .. . ,-, ,.

2’r

c) o “U

r8

6A o———————_

4 -———— —.— ~__ _

2 ‘-

0

10r\ ~\

8t

z

04

2

40

z

04

—_—-o—— ———

0 ___ ——--— ——

5

Dis&ce frA?n centg; line oFZut-wt17n.35 40

i-igre 19.-Stringer stresses in 16-stringer PJnelwith6 stringers cut andL=I.5inches.

Pp,..,

NACA 8

6

4

2

0

8

6

4

2

0

4

2

0

4

z

,r Fig.20

t Exact solutionI

)bythree-stringer method‘---– Simplifiedsolution .-

8.,.

I21-

t--——_ _

I

6

4

12

v\,

10

t

\o\

8r

o

--—— _

o

2

0

4

—

ot1-

}04-

2 ‘-

o–0

—-— . . . .—.. .—...—— .—.—0

0

0

0.——

0

I I I I

Ficjv-e 20.-Stringer stresses in 16-stringer panel with 8 stringers cut and L=l.5inti::.

■.,m—.mmmm— . ,—.- !.-... . .-—-——. - .—

NACA 8

6

4

z

0

8

6

4

2

0

4

2

0

4

2

Fig. 21

[ Exact solution——._—

1Simplified solutionby three-strjngq- method

8.

6!-O,) ,> c1

41

21-

0-L&---------

t

.; “--- ____——

-. -—-—.——r —

Ot---.-

41-

2’

I0

04

2 ‘

o l-- _L_____

,.—-

1 -

/

‘7

o 5Di~;nce f~;m ten%- line ~f cut-o;:, in.

35 4C

Figure Z1.-Stringer stresses in 16-stringer m-d with 8 stringers cutandL=80 inches.

NACA 8

6

4

2.

08

6

Fig. 22

0.El

cl

o0 0

c1

OL108 L._. _.__. ~

o 06

4 !-1

7 11-

40 I

t

G ~-

2‘– Exactthr-e+tringernwthod o

04 ‘--- Smtifiedthree-stririgwmethd

I

UJa –-– Mofifi~ two-striiw methcd

4 0;

I

o

I I

o 5Dis’?ance f?? cen;~r line % cut-o:!, in.

35 40

Figure Z2:Stringer stresses in 16-stringer panelwith 8 stringers cut and L=15.Oinches.

NACA

&.—

b-

12

to

8

6

4

2

Fig. 23

0 c1o

[I ~~

}

Exact solution - ~.o

[‘---- 5implified mlution

by three-stringer method

z~

120 ~

0’

I e

8\\

\\\6’

I

c ‘.= -----—._ ––G4

2 ‘-

04I 3 u

2’_—— —_—— —

[

0

40

2’

04[

2’

[

0

40

0“0Di5Lce frgm cent: line o?cut-cu~in.

35 40

FigureZ3:Stringer stresses in 16-Arin~r panelwith 10stringers cut andL= 15.0inches.

I

NACA1)>1

8

6

4

2

0

14

12

10

8

6

4

2

Fig. 24

0 \\

UJ\

\

\\

c ‘.,‘-.

‘.----

‘._ -c-1.

————Exact solution

)- 5in~dified solutionby three-stringer method

2’

[04-”

z’‘-

40 12 ‘-

04 -

.,

I

2 ‘-

0 i I

——— .-~

2

.

--’-+

I I I I

o 5 35 40Dis&ce fr;% centerline of ~ut-out,;.

Figure 24:Stringer stresses in 16 stringer panel with 12 strinqers cut and L=15,0incines.

1-

NACA Figs.25,26

: -:+$. -, —..—,Q -2 -

--— -*———-,+..

k- 0 ~’ ——.-. .—Ui” –6’aG)

iii-4 [

–2t

Fkjurc ?5.-51 ear stresses in 16-stringer pmel with Z +-ingcrs cut md L=l.5inches.

————Exact 5Llutim7

---– Simpifle(j 5oldicn

(by thee-stringer metl~od)

----- -–’ –’~” -i———___— “--—-—————e_ –.-,. . .- __._—_——...—

\.

.\‘%

“%:< \ ..--- —-_ ———————___ ___ ____ ___—-.——— ———___ .==..

—I I I -T’ ,5 10 15 20 *5- 30 35 40

NACA Fig. 27

Exact solution 1‘----Simplified solutionby three-stri~er method

1–2 ,

I.

6 ‘2$- 0

I_

——_._--—o —c1

0 ————_—__----

—— -Q_——._

1- 8 ,~——

gm

–6 ;o

‘4 i

[ A

-2-— ———__——

-40,—-—. .—— — ——_

t ~-

-\-2 0 0

r)

L,-10 ~

-2 ----~

00I 1 1

5 10 !5 20 25 30 35 40

Distance from center line of cut-out, in.

Figure27.-Sheor stresses in 16-stringerpaneIwith 6 stringers cut andL=I.5 inches.

m. m■ ——., 9———...—,. M .! . . . ! I !! ..-.!. ! ! l-—-. -———--

i ‘=

Fig. 28

————.

Exact solution.

)Simplified solutionby three-stringer method

o .

1-2

0 0

-2 - ———— ——— __ -— .

-6

I-4 ,

—.2

/o-4 ~

-2

I-40,

-2

I0-4,

“z

t00

‘.

___ -—_

\.———— ———————. —— ___ _

o

I I 1 1 1 [ I I5

Dis~nce fro’; cente~ine of ;ut-out ,%.35 40

F~ure 28:5heor stresses in 16-stringerpanelwith 8 stringers cut and L=I.5 inctws.

NACA Fig. 29

—————

Exact solutim

Simplified solution1by three-stringer

-27

0 0

I-z ;

o 0

k

//- ~

Ja-—..–––– —— ———

F- ‘2 o —————_ .—

:“ –8 ()

cl)

-6h t

-4 ;= <-—._>

-2 !- ---- ——_ —--–————_ -.-——0-4,-

——. —___‘A

—

I-2 r o-4 f):.

-2

t0-4~

-2 -

0 —

I

(-jo&-_~.l I I I I I i I

5Dis?ance fi-;m centefiine of ~&-out.;,

35 40

Figure 29.- Shear stresses in 16-stringer panel with 8 stringers cutand L = 8.0 inches.

III —

Fig.30

bat three-stri~er mettd

-----Simplified three-stringer mettd

—-—Mcd&d two-stringer method

1-2,

0

1 0

2,I

1-2 ,

0

2;I

-4 ~

t–

0

-2_—— .2 —-3—. ——_ .—

0, I

-4

-z

-6

t—

t0-4,

-zt

.0

–40~

-2

tc1

o-4 ~

-2

L ~_~. I I

00 ‘51 I )

40

Diskce fr& cente?’line of?ut-wt,%.35

Figure 3Mheor stresses in 16-stringer punelwith 8 stringers cut andL=15.Oinches.

I

—.—,,, ,. ,.,-, ,. , .,, ,. ,,, .—---. !!—-— .

1-

NACA Fig. 31

k.—

–2

—————

Exact sclution,, . . ....5implif ied solutiit

by three-stringer

0

2,-

-4 .-0

0

–2__ ——— ——_.

0;

0-4 )

-2

1-4 0,

-2

to-4 y

–2t

‘>o

>———_ o

figure 31.-Shear stresses in 16-stringer pand with IC stringers cut and L=15,0imhes.

mmm

l—

NACA

I——————

-4

–z ;.— .,.

Exact mlution

}

Fig. 32

Sirnplifid solution by three-stringer method

o 00—..—.—. _ ——. —

o!.—

u u

-10

-8

–6

-4

–2

0

-8

-6L I

.—-4

~Q i.s2

-2,

Uj-–fjpt –4m

-2

9

-4

–2

o

-4

–2

0 -

.+-

\\

\.

c1

o

.OL ~-~—-5 1’

Dis~nce from center line of cut-out, in.

I I I I 1Zo 25 30 35 40

Figure 3ZShear stresses in 16-stringer panel with 12 stringers cut and L=15.Oinches.

“E”I

I

1,

0 8 16 0 9 16 0 8Distancefromcsnterline. of panel, in.

(a)16-stringer-panel; 8 stringerscut; L = 1.5 inches.(b) 16-stringsrIWM; 10 stringsrscat; Z = 15.0 i.xka.(c) 16-stringer~nel; 12 stringerscutt L = 15.0inchee.

Figure33.-Xxperimmtal andcalcuktedrib utre~aes. Panel load

.

(a)

q = al

(h)

1:*

III

I

i

I IiIIil1

J (c)

.

TITLE: Stresses Around Rectangular Cut-Outs In Shin -Stringer Panels Under Axial Loads n

AUTHOR|S):Kuhn, Paul; Duberg, John E.; and others ORIGINATING AGENCY: National Advisory Committee for Aeronautics, Washington, D. C. PUBLISHED BY: (Same)

'0- 9319 (None)

OIIO. kGBtcr MO.

nnuntHO ACfNCV KO.

Dan Oct '43 Unclass. U.S. Eng. 78

UUSTBAIIOM photos, tables, dlagrs. graphs

ABSTRACT:

Formulas are given for calculating stress distribution around rectangular cutouts in axial loaded panels. Present solution is based on a substitute structure containing three stringers and Is more complete as well as more accurate than the previous method based on a structure with only two stringers. Three-stringer method was found to give very good accuracy In predicting stringer stresses, but shear stresses cannot be predicted accurately. Discrepancies are believed to be caused by local deformation around highly loaded rivets and relieving maximum shear stresses.

DISTRIBUTION: Request copies of this report only from Originating Agency SUBJECT HEADINGS: Cut-outs, Structural Stress analysis (90800)

Theory (28235); DIVISION: Stress Analysis and Structures (7) SECTION: Structural Theory and Analysis

Methods (2) ATI SHEET NO.:R-7-2-U

At, Documents Division, InraUlgonco Ooportmont Air Malarial Command

AIR TECHNICAL INDEX Wrlahl-Pottonon Air Forco Sou Dayton, Ohio

ATI 209 03

Same as 9519