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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 [email protected]~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 = [email protected]~~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
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