Stresses around rectangular cut-outs in skin-stringer panels under axial loads - II

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Material Information

Title:
Stresses around rectangular cut-outs in skin-stringer panels under axial loads - II
Series Title:
NACA WR
Alternate Title:
NACA wartime reports
Physical Description:
46 p., 32 leaves : ill. ; 28 cm.
Language:
English
Creator:
Kuhn, Paul
Duberg, John E ( John Edward ), 1917-
Diskin, Simon H
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

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

Notes

Summary:
Summary: Cut-outs in wings or fuselages produce stress concentrations 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 around rectangular cut-outs in axially loaded panels. The formulas are derived by means of the substitute-stringer method of 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 two stringers. The present solution is based 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 solution as well as of the modified two-stringer solution.
Bibliography:
Includes bibliographic references (p. 44).
Statement of Responsibility:
by Paul Kuhn, John E. Duberg, and Simon H. Diskin.
General Note:
"Originally issued October 1943 as Advance Restricted Report 3J02."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

Record Information

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003806425
oclc - 124075771
System ID:
AA00009443:00001


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IV kc A L 3 4d




'ALADVSOR CO MITEE FORAERONAUTICS




TIE.REPORT
ORIGOA4ALY ISS"D
October 1943 as4
Advance Restricted Repor-t 3JO2

SU=AROMN RECTANGULAR =I-OUTS IN MRN-STRINME
PANMLS UNIER A=IA LOADS YI
ByPaul Kuhn, John 2. Duberg, and Simon 1.Diskin

langley. Memorial Aeronautical Laboratory
Langley Field,, Va.

UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE LIBRARY
PR0. BOX 117011
GAINESVILLE, FL 32611-7011(JM4






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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS



ADVANCE R ,' T RICTED- REPORT



STRESSES AROUND RECTANGULAR CUT-OUTS IN SKIN-STRIHGER

PANELS UNDER AXIAL LOADS II

By Paul Kuhn, John E. Duberg, and Simon H. Diskin

SUMMARY

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 around rectangular cut-outs in axially loaded

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

strinrger method of shear-lag analysis.

In a previous paper published under Lhte- same title as

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

structure containing only tiwo strinrers. The present

solution is based 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 Lhree-rtringer solution as well as

of the modified two-strinier solution.










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 caused by local

deformations taking place around the most highly loaded

rivets and relieving the maximum shear stresses.
INTRODUCTION

Cut-outs in wings or fuselages constitute one of the

most troublesome 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 behavior, under

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

finding the stresses in axially loaded panels without cut-

outs was given 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).










Skin-stringer panels, although simpler than complete

shells, are highly indeterminate structures. In order to

reduce the labor of analyzing such panels, simplifying

assumptions and special devices may be introduced. The

most important device of this nature used in references 1

and 2 is a reduction of the number of stringers, which is

effected by combining a number of stringers into a sub-

stitute-single stringer. In reference 2, this reduction

yas carried to the extreme of using only two substitute

stringers, one for the cut stringers and one for the uncut

stringers, to represent one quadrant of the panel with a

cut-out. The two-stringer structure can be analyzed very

rapidly but, being somewhat over-simplified, cannot give

an entirely satisfactory picture. In particular, the two-

stringer structure does not include the region of the net

section; and consequently this structure neither shows the

effect of length of cut-out nor gives a solution for the

maximum strihger stresses. These maximum stresses must

be obtained by separate assumptions. In addition, there

is no obvious relation between the shear stresses in the

actual structure and the shear stresses in the substitute

two-stringer structure as used in reference 2.

In order to obtain a more satisfactory basis of analysis

than that of reference 2, formulas were developed for a











skin-stringer structure containing three stringers. At the

same time, a new experimental investigation was made. con-

sisting 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

rapidity of the calculations.

The main body of the present paper describes the ap-

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

stringer method of analysis as well as a nodifiedgtwo-

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.

THEOR3ETICAL ANALYSIS OF CUT-OUTS IN AXIALLY

LOADED SKIN-STRINGER PANELS

General Principles and Assumptions

The general procedure of analysis is similar to the

procedure developed for structures without cut-outs











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

replaced by an idealized structure in which the sheet

carries only shear. The ability of the sheet to carry

normal stresses is taken into account by adding a suitable

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

stringer. The idealized structure is then simplified by

combining groups of stringers into single stringers, which

are termed "substitute stringers"; this substitution is

analogous to the use of "phantom members" in truss analysis.

The substitute stringers are assumed to be connected by a

sheet having the same properties as the actual sheet. The

stresses in the substitute sheet-stringer structure are

calculated by formulas obtained by solving the differential

equations governing the problem. (See appendix A.) Finally,

the stresses in the actual structure are calculated from

the stresses in the substitute structure.

It will be assumed that the panel is symmetrical about

both axes; the analysis 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 distances from the cut-out.











Symbols and Sign Conventions

Al effective cross-sectional area of all continuous

stringers, exclusive of main stringer bordering

cut-out, square inches (fig. 2)

A2 effective cross-sectional area of main continuous

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

A3 effective cross-sectional area of all discontinuous

stringers, square inches (fig. 2)

Arib cross-sectional area of rib at edge of cut-out,

square inches (fig. 2)

I KL 2 + K-2 2 + 2K

1B1 + K12 + 2K -n
K-2

C stress-concentration factor (fig. 7) *

CO stress-excess factor for cut-out of zero length

(Qf"g. 3)

D = I + '\ = + 2+ 2


E Young's nodulus of elasticity, kips per square inch

G shear modulus, kips per square inch

-K2 t 2 1 1
72 + X-3


1 + A
l -b1 (T'1 A2)










Gt2


Ot2
K3 EblA2

Gt1


K = -
4 Eb2 2



K= JKl2K22 -_ K3K4

L half-length of cut-out, inches (fig. 2)
K12 (1 22)
1 i2 -


p K3K4
P2 =
1 12 X22)


P3 =K2 (K,2 12)
X\2(%12 2 2)


P4 = 34
X2 l2 22)



Q1 force Ala acting on stringer 1 at rib, kips

Q2 force A2a acting on stringer 2 at rib, kips
R stress-reduction factor to take care of change in


length of cut-out (fig. 4)






8


XR difference between actual force in A, (or A2) at
the rib and the force Q1 (or Q2), kips -
a width of net section, inches (fig. 6)

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

bi distance from A2 to centroid of Al (fig. 2)

b2 distance from A2 to centroid of A3 (fig. 2)
T2Rt2
rl= -"

T2pt2 TIRti
r2 = A2(2R o)

Go2R
P3 = Eb2T2R



tI thickness of continuous panel, inches (fig. 2)

t2 thickness of discontinuous panel, inches (fig. 2)
x spanwise distances, inches (For origins, see

figs. 2 and 6.)
y chordwise distances, inches (For origins, see fig. 2.)

K2 + K \2 (C12 + K22)2 4J2
21


S=j KI2 + 2 12 + =2 1 4 2
2 V 2










00 average stress in the gross section, kips per square

inch

11 stress in continuous substitute stringer, kips per

square inch

O2 stress in main continuous stringer, kips per square

inch

03 stress in discontinuous substitute stringer, kips

per square inch

rib stress in rib, kips per square inch

a average stress in net section, kips per square inch

T1 shear stress in continuous substitute panel, hips

per square inch

T2 shear stress in discontinuous substitute panel, kips

per square inch

Superscripts on stresses denote forces producing the

stresses. Subscript R den:'tes stress occurring at rib

station.

Tensile stresses in stringers are positive. If the

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

are produced b; a tensile force acting on A1.

Simplified Three-Stringer Method

A principle for the effective use of substitute

stringers was stated in reference 3 substantially as fol-

lows:












Leave the structure intact in the region of the stringer

about which the most important actions take place, and

replace the stringers away from this region by substitute

stringers. In a panel with a cut-out, the most important

action takes place around the main stringer bounding the

cut-out. In accordance with the foregoing principle, the

three-stringer method is based on retaining the main

stringer as an individual stringer in the substitute

structure; one substitute stringer replaces all the remaining

continuous stringers, and another substitute stringer re-

places all the discontinuous stringers. The three-stringer

substitute structure obtained by this procedure is shown in

figure 2, which summarizes graphically the salient features

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

substitute structure, and the distribution of. the stresses

in the actual structure.

The maxinum stringer stress as well as the maximum

shear stress occurs at the rib station. The formulas given

hereinafter for the stresses at the rib station and in the

net section are based on the exact solution of the differ-

ential equations presented in appendix A. The formulas

derived from this exact solution for the stresses in the

gross section are somewhat cumbersome and are therefore

replaced here by formulas that are based on mathematical










approximations of sufficient accuracy for design work

(appendix B). The use of these approximations is the

reason for calling this method the simplified three-

stringer method.

Stresses at the rib station in the substitute

structure.- The stringer stresses at the rib station are

RCoA2
0l = A-) (1)



02, = o1 + RCO) (2)


where the factor Co, for a cut-out of zero length, 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 nore 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












T 1Rt = 6RCoA2K1 tanh KIL (3)

The running shear in the discontinuous panel at the rib

station is


To 2t = oA 1+ RCo + (4)



in which the factor D may be obtained fron figure 5.

The stresses a2R and T2R are the maximum values of

CO and T2, respectively, and are the raxinurm stresses in

the panel. The stress ai reaches its maximum at the

center lIne of the cut-out. The stress T1 reaches its

maximum in the grors section at the station where

01 = 02 t Go0
Stressed in the net section of the substitute structure.-

The formulas for tie stresses in the net section are ob-

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

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

in the continuous stringers are

RCoA2 cosh Kx"1
BC A2 cash KIjL

Scosh x
o2 = ( 1 + PRCo h L


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










length of the net section increases, the magnitude of the

parameter KIL increases and the stresses C0 and 72

converge toward the average stress 0 in the net section.

The running shear in the net section is

cinh KIx
T>t = ORC ArK in1x
1 o 1 cosh K1L


and decreases rapidly to zero at the center line of the

cut-out.

Stresses in the gro"s section of the substitute

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

from the exact solution given in 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 larger- distances from the

cut-out. The approximate solution assur.Pes 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 initial rates

of decay of the exact solution.

The stress in the cut stringer by the approximate

solution is


03 = o(l- e-rlx) (5)










The stress in the main stringer is

02 = (o + (02R co)e-r2x (6)

The stress in the continuous stringer 1 follows from

statics and is
A Al
01 = o + (%co 02) + I (o o) ()

The running shears in the sheet panels are

TltI = -Rt e-rlx 2Rt2 t e-r2x (8)


T 2t2 =. 2t2 e-r3x (9)

Strepsei in the actual structure.- By the basic prin-

ciples of the substitute structure, the stressescin the main

continuous stringer of the actual structure are identical

with the stresses in stringer 2 of the substitute structure;

the total force in the remaining continuous stringers of
the actual structure is equal to the force in stringer 1 of

the substitute structure, and the total force in the cut
stringers of the actual structure is equal to the force in
stringer 3.

In the shear-lag theory for beams without cut-outs
(reference 1), the force acting on a substitute stringer
is distributed over the corresponding actual stringers on











the assumption that the chordwise distribution follows a

hyperbolic cosine law. Inspection of the test data for

panels with cut-outs indicated that neither this nor any

other simple assumption fitted the data on the average as

well as the assumption of uniform distribution. It is

therefore recommended, for the present, that the stresses

in all continuous stringers other than the nain stringer be

assumed to equal ai and that the stresses in all cut

stringers be assumed to equal 03. (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 T1 in the substitute structure equal

the shear stresses in the first continuous sheet panel

adjacent to the rain stringer. In order to be consistent

with the assumption that the chordwise distribution of the

stringer stresses is uniform, the chordwise distribution

of the shear stresses should be assumed to taper linearly

from Ti to zero at the 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 from 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 immediate vicinity of the cut-outs. The dis-

crepancy is of some practical importance because the maxi-

mum stress in the rib depends on the chordwise distribution

of the shear stress at the rib. By plotting 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 cubic 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 is sufficiently accurate to repre-

sent the spanwise variation within this distance (fig. 2).

If the stress 72 is distributed according to cubic

law, the stress in the rib caused by the shear in the sheet
i
is


orib = r- ib1 J


(10)


Modified Two-Stringer Method

The two-stringer method of analysis given in reference

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

stringer nethod previously described. It was found, how-

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

corporating some features of the three-stringer method,

partly by other modifications.











The nain features of the modified two-strinTer method

are summarized in figure 6. The cut stringers are re-

placed by a single substitute stringer; and all the uncut

stringers, including the main one, are also replaced by a

single stringer. Contrary to the usual shear-lag method,

however, the stringer substituted for the continuous

stringers is located not at the centroid of these stringers

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

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

determines the maximum shear stress, the spanwisc rate of

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

of stringer stress. The maximum stringer stress nust 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 stress in these stringers. No solutions

are obtained by the two-stringer method for the shear

stresses in the continuous panels, either in the net

section .or in the gross section.

Stresses in the substitute structure.- Throughout the

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

is

02R = e 1- + 2R(C 1)1 (11)

where C is the stress-concentration factor derived in










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

which is reproduced from reference 2 for convenience. It

may be remarked here that reference 2 placed no explicit

restriction 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 long cut-outs, implies that-

the factor C by itself should be used only when the net

section is long.

In the gross section, the stress in the main stringer

decreases with increasing distance from the rib according

to the formula

02 = o + (2R o,)eK (12)


The stress in the discontinuous substitute stringer is

03 = co(l e'Kx (13)

The stress 01 may be obtained by formula (7) when 02 and

03 are known.

The running shear in the discontinuous panel is given

by

T2t2 = oAKe-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-stringer method.







19


EXPERIMENTAL VERIFICATION OF FORMULAS AND

COMPARISON OF METHODS

Test Specimens and Test Procedure

In order to obtain experimental verification for the

formulas developed, a large skin-stringer panel was built

and tested. The panel was similar to the one described in

reference 2, but the scope of the tests was extended in two

respects: Very short cut-outs were tested in addition to

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

stringer stresses were measured around all cut-outs.

The general test setup is shown in figure 8. A setup

of strain gages is shown in figure 9. The panel was made

of 24S-T aluminum alloy and was 144 inches long. The

cross section is shown in figure 10(a); figure 10(b) shows

for reference purposes the cross section of the panel tested

previously (reference 2). Strains were measured by

Tuckerman strain gages with a gage length of 2 inches.

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

Strains were measured at corresponding points in all four

quadrants. The final figures are drawn as for one quadrant;

each plotted point represents, therefore, the average of

four stations or eight gages.

The load was applied in three equal increments. If

the straight line through the three points on the load-stress










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

to pass through the origin; however, if the necessary shift

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

ings was taken.

An effective value of Young's modulus of 10.16 x 10- ips

per square inch was derived by measuring the strains in all

stringers at three stations along the span before the first

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

sidered as including corrections for the effects of rivet

holes, average gage calibration factor, and dynamometer

calibration factor. The individual gage factors were

known to be within 4, percent of the average.

The average strain 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. The maximum

deviation of an individual stringer strain from the

average was 5 percent; about 10 percent of the points

deviated from the average by more than 3 percent. A

survey was also made of longitudinal and transverse sheet

strains at one station near the center. The average

longitudinal sheet strain differed from the average

stringer strain by 0.05 percent. The average transverse

strain indicated a'Poisson's ratio of 0.323.










Discussion

The results of the tests are shown in figures 11 to 33.

Calculated curves are given both for the exact three-

stringer method and for the simplified three-stringer method.

It may be recalled that either method assumes that the

stresses in all continuous stringers except the main stringer

have the magnitude c1 and in all cut stringers, the magni-

-tude 03. Because the values of oi and 03 do not dif-

fer very much for the two methods, the curves for them com-

puted by the simplified method are drawn only once in each

figure.

A qualitative study of figures 11 to 32 indicates that

the stress distribution calculated by the theory agrees

sufficiently i.eil v.ith the exferir.ental distribution to be

acceptable for most sr;-ess-analysis purposes in particular,

the ma-Ii',.u: :t.rsess in each panel agree fairly well with

the calc.uiLarc? ov?es. The most consistent discrepancies

are chargeable to the 3implifying assumption that the

stringer stresses e.re identical in all the stringers repre-

sented by one Fubztitute 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











the cut stringers are lower than the actual stresses is of

little practical importance because these stringers would

probably be designed to carry the stress ac rather than

the actual stresses. On the uncut stringers, however,

it may be necessary to allow some extra margin in the

stringers near the nain one. Aside from the consistent

discrepancies just noted, figures 11 to 32 show that the

stresses in the main stringers sometimes decrease spanwise

more rapidly than the theory indicates. It is believed

that this discrepancy also will seldom be of any consequence

in practical analysis.

Of paramount interest to the analyst are the maximum

values of the stresses. The quantitative study of errors

in the maximum stresses is facilitated by table V* The

highest stresses occur theoretically at the rib station

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

some small distance from this line. The comparisons are

made for the actual gage locations. The calculated values

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

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

solution and 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 local stress variations that were

found experimentally to exist in the panel without cut-

Co out. Presumably these variations are caused largely by
failure of the rivets to enforce integral action of the

structure.

The errorsin the maximum shear stresses computed by

the three-stringer method are consistently positive. The

discrepancies are possibly caused by the sheet around the

most highly loaded rivets deforming and thereby relieving

the maximum shear stresses. The errors are higher 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 is decidedly less accurate

than the three-stringer method for computing maximum

stringer stresses but that there is little-difference

between the two methods as far as the computation of the

maximum 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











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

tion for shear stresses in the continuous panels.

Comparisons of the maximum 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 chordwise variation of the observed and

computed rib stresses is shov.'n for three cut-outs. Because

the chordwise distribution of shear stress in each sheet

panel between two stringers is essentially constant, rib

stresses computed by formula (C10) will be too small when only

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 Is 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 the effective

area of the rib,



"Langley lMemorial Aeronautical Laboratory,
National dPvisory Committee for Aeronautics,
Lai.gJey Field, Va.










APPENDIX A

TDXACT SOLUTION OP THREE-STRINGER STRUCTURES

For a two-stringer panel constituting one half of a ..

symmetrical structure, the application of the basic shear- .

lag theory yields the differential equation

-- KT = 0 (A-1)


which is given in slightly different form in reference 4.

In the anal7rris of a skin-stringer panel with a cut-out,

'a three-stringer substitute structure is used. (See fig. 2.)

Application of the basic equations of reference 4 to a

three-stringer structure yields in place of the single equa-

tion (A-l) the simultaneous equations

d2T1 2
1.2 T 11 + K3 "T = 0

d2T2 2> (A-2)
-- K.2 T + KT4T 1 0
dx2 J



On the simplifying assumption that the panel is very long

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

far ends, the general solution of the equations (A-2) is


71 = c le x + c2e 2


(A-3)


1___.-- -_lX 1 K -2Xx
T2 3 cle +\ 12 72) c2e (A-4)

in whtch cl and c2 are arbitrary constants.











Because the structure is assumed to be symmetrical

about the longitudinal as well as the transverse axis,

a the analysis may be confined to one quadrant as shown in

figure 34(a). The analysis can be simplified somewhat by

severing the structure at the rib and considering separately

the net section and the gross section. The solutions for

the two-stringer structure representing the net section can

be obtained from reference 4. The solutions for the

three-stringer structure representing the gross section are

obtained conveniently by considering two separate cases of

loading. In the first case, stresses aC are assumed to

be applied at the far end, and the stresses at the rib

station are assumed to be uniform and equal to the average

stress C necessary to balance the stresses oa. The

forces at the rib station ,existing in the stringers are

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

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

to load the stringers 1 and 2 at the rib sr.ation. These

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

In the net section the boundary conditions are as

follows:

At x = 0,


TI = 0 (from symmetry)











At x = L,
XR
a -

XR
2 A.


The substitution of these conditions in the solution of

equation (A-l) yields

X XRK1 sinh Kix
T\ coTh KIlT (A-5)



l-- d cOsh Y'Ix
c"- (A-6)



.. R acoh ]Kx
a..= 0--1 (A-7)
Ag ccsjn 1iL *

The superscript X indicates that the stresses are those

caused tby the action of the X-forces. In order to obtain

the total stresses, the avr-rage stress a must be added

to o,- or oA. The she-or stress TX is the total

stress br caisce the uniform.: stress 0 is not accompanied by

any L-hpar stre's.
V.-n ? he i.1-for-e c3 -uarf Ppplied to the gross section,

the bc'-.ni-ary c:.!diions at x = 0 are

01 = -- C 02 =- r2= a 3 = 0
A1 A 2-









Applying these conditions to equations (A-3) and (A-4) gives
the following solutions for stresses:

2 (P e-klx e-A2x)
I P2I P4e(A-8)

(1 x12 1 4K2 -22 )
2 = e- e (A-9)


Y 2 p2 -Xlx 4 -k (A 10
SQ 1 e2 e / (A-10)


(2 2K12x12 -t \x 4 2 K%_ 2 x
02 t-+ 1-je tL- -t1jee
0o A72 1L e71 t] ]1


2 2 l2 1') -"lx 4 2, -2x
3 o + A7tI I K3 c 72 K3


The superscript Q indicates that the 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 = 0,
XR X
W1 = 2 = AQ 3 = 0

and the corresponding equations for stress are
R -Kx -k2x1
TX= L(PI + P2)e P3- + P4e j (A-lI)











r XXRFP, + P2) (c12 q12) -X (3 + P4K1l21 22) .-2x
=2 e -T K3


XR I' + P2 1x + P4) -.X2x
S_ 2


:2x ( + p) ) t2 P1 2 X_- 1,2) -X1x

0(P 2 -1 1- e)13-Xe
i-



AIL ) r 4 22 (xi






S- P222)
If the shear strai-is In the net section and in the
rngr- section, vl>.c-' are determined front equations (A-5)
(".-", and (s-li), are equated at the rib, the following
rel1tii-)nrl'.ip ":rvween X-^ and Qz results:

x' + 7'- --" "-" P+ K1 tanh K (A-1)
2 + a q 1F3

FP)r ft- r.: ..cr f zer-z ecgt.h, L = C, equation (A-13)


*'X.-r,
: -. 7- 2 = C oQ2
V',.. .1 1. 5::,I/












Por any length of cut-out,

1
X- 1 ----- = QC R
NR = Q2C 1 + B tanh KIL =Q2CR


where

1/ 2 + K22 + .

SKI' + + 2K -


Values for Co can be obtained from figure 3. In

figure 4, the factor R is plotted for vari.ou. values of

K1L and B. The value of B nay be assun.ed equal to

unity with little loss in accuracy in the determination of

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

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

responding to this value used.












APPTEDIX B

SIMPLIFIED SOLUTION OF THREE-STRINGER STRUCTURES

The solutions for the stresses in the gross section

given in appendix A are too involved for practical use, and

a simplified method was developed. This method assumes

that the differences between the values of 02Ro and

T2R, obtained by the exact solution, and the corresponding

average stresses in the gross section decay exponentially

with rate-of-decay factors adjusted to give initial rates

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 0a and T1 are then derived from the solutions for

02 and 03.

If it is a.si-med that the stresses in the cut stringer

can be expressed by

a = o(l e-rx)

then
do3 -rIx
dx- = 0orle

but, from the basic shear-lag theory,

do3 T2 -rlx (B-
dx 3 = orle (B-1)

Therefore, at x = 0, TR
So 2
rl-*A'oo











The stress in the main continuous' stringer can be

approximated by
02 = ao + (a2R ao)e r2x

which yields

= -,2R o)r2er2x (B-2)


but, from the shear-lag theory,

do2 T1t1 T2t2 -r2x
-= + A -(02 co) re

Therefore, at x = 0,

r2 M t2 R 1t)
A2(02R cq)

The value of a, can be obtained by statics from 02
and a3 and is
A2 A
1 = + (o 02) + o 03) (B-3)


If the value of T2 is assumed to decay exponentially,

then
T2 T 2e-r3x

and

d2 -rex
-dx T2R r3e


but, from the shear-lag theory,

dT2 G "-3x
S= Eb 2 0 = -T2Rr3e











Therefore, at x = 0,
Go002R
G Y R
r3 Eb2 T2R

The shear stresses in the continuous panel can be

determined from the rate of change of 0o. From the shear-

lag theory,
Sdo- Tltl (B-4)
dx A1
Differentiation of formula (B-3) yields

do1 A3 do3 A2 do2
(B-5)
dx A! dx AI dx

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

obtained in (5-5) gives
do1 A2 r2x A3 rlx
S= 2 o)re x ~A orle (B-6)

Finally, substitution of (3-G) in (B-4) yields

Tltl = 2R to2e' (T2t2 TlRtl)e -r2x











APPENDIX C

NUMERICAL EXAMPLE
Analysis by the Exact Three-Stringer Method

The structure chosen for the numerical example is the
16-stringer panel tested as part of this investigation.

The particular case chosen is the panel with eight stringers

cut and with a length of cut-out equal to 30 inches. This

cut-out is the one shown in figure 8. The cross section

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

AI, sq in. . . 0.703
A2, sq in. . . 0.212
A3, sq in. a a . 1.045
tl, in . .. . 0.0331
t2, in. . . .. 0.0331
bl, in . . .. 5.96
b2, in . . 7.56
L, in . . 15.0

These data yield the following values:

KI2= 0.01295
K22= 0.00944
K3 = 0.00995
K4 = 0.00785
K = 0.00664

Prom these parameters follow the factors for the rate of

decay, which are

/12 + 22 + + 2)2 4K2
xl = V 2 = 0.1421


1:1 + K22 VQK2 + K2)2 4K2
X2V : -2 = 0.0467












The computations of stress may more easily be made in

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

of which are


K 9(KI 2 2 2) 0.01295(0.01295 0.00218) = 0.0545
1 l 2 X 22 0.1421(0.02021 0.00210)


13 4
P2 =
1i 12" x22)


0.00995 x 0.00785
0.1421(0.02021 0.00210)


= 0.0305


12 1 -2) 0.01205(O.C1 5 0.0202 1)' 1
3 \(x12 -_ 2) 0.0467(0.02021 0.00218)


P4 = 2 13 4
- -


0.00995 x 0.00785
0.0467(0.02021 0.00218)


= 0.0927


The reduced stress-excess factor is


RCO =


P4 P2
P1 + P2 P3 4 + Ki tanh KIL


0 00.0927 0.C305 =0.296
0.0545 + 0.0305 + 0.1117 G.0927 + 0.1065


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

O = 1.90 = 3-.82 kips/sq in.










37

and

0 = .5 8.21 kips/sq in.

Therefore,

-R = RCA.. = 0.296 x 8.21 x 0.212 = 0.514 kip

3 .n 3 _.r: t' -.- ,-'tjon.- The shear stress in the

substitute cai." c r. -, reP -, :on is found by equation (A-5)

,", : nh i:E1x
1 ; il K, L



0-5 '.124 x 0.11Z sinh 0.1138x
-j .0,21 cosh 1.707


= 0.620 sinh 0.1138x

At the rib station, x = 15.0 and

T1R = -0.620 sinh 1.707 = -1.65 kips/sq in.

The stringer stresses are found by substituting in

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

Scaosh K'x =0.514 cosh 0.1138x
oI = = E.21 -
A1 cosh K1L 0.703 cosh 1.707

= 8.21 0.257 cosh 0.1138x



R- cosh Elx 0.514 cosh 0.1138x
2 A2 cosh K1L 8~ 0.212 cosh 1.707


= 8.21 + 0.850 cosh 0.1139x












The maximum stringer stress occurs in the main stringer at

the rib, x = 15. The nearest gage location was at

x = 13.5, where


02 = 8.21 + 0.850 cosh 1.536 = 8.21 + 2.05 = 10.26 kips/sqin.


Stresses in the gross section.- The stresses in the

gross section are obtained by adding the solutions for the

stresses due to the X- and Q-forces.. The shear stress in

the continuous panel is obtained by adding equations (A-8)

and (A-11). The final solution thus obtained is
= -0.142lx -0.0467x
T e 4.57e


At the rib station, x = 0 and


T1 = 2.92 4.57 = -1.65 kips/sq in.


This value of TI1 checks the one previously obtained

for this sane station in the net section.

Substituting the constants in equations (A-9) and (A-10)

and cornhininf give

T- = -2.13e-0-1421x 4.95e-0.046?x


At x = 1.0O, the point of maximum observed shear stress,


To = (-2.15)(0.809) (4.95)(0.031) = -6.33 kips/sq in.













The stress in the continuous substitute stringer is

found by combining equations (A-10) and (A-12). The

final result is


01= 3 82- O.07e-0.1421x + 4.63e-0.0467x


Similarly, the stresses in the main stringer and in

the cut stringers are found by-adding- the proper values of

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


02 = 3.32 + 5.55e-0.1421x + 1.26e-0.467


arnd, in the cut strinaeyS,


03 = 3.32 .46e .14 3.6e


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 Miethod

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

stringer method. Compute
21 2
K1 0.01295 x 0.00944
So *= 1.565
'.3fKl4 0.009.5 x 0.00785


0.00664 0.704
K22 0.00944

Prom figure 3,


Co = 0.600











From figure 4 for KiL = 1.707 and the exact value of
B = 1.10, there is obtained R = 0.492.
The stresses in the continuous stringers at the rib
are,by formulas (1) and (2),

0R = 8.21 1 (0.492)(0.600)(0.O' = 7.48 kips/sq in.

02R = 8.211[ + (0.492)(0.600)] = 10.63 kips/sq in.


The running shear in the continuous panel at the rib
is, by formula (3),
TIR t = -8.21 x 0.492 x 0.600 x 0.212 x 0.1138 tanh 1.707
= -0.0547 kip/:..
The maximum running shear in the cut panel is qonputed by

formula (4). The value of D is obtained from figure 5;
with K = 0.00664 and K12 + Kg2 = 0.02239, D = 0.189
and

0.00785 0.01295
T2 t2 = -8.21 x 0.212 x 0.015 + (0.492)(0.600)+ 0.0664
= -0.234 kip/in.

S2 0.234 = -7.08 kips/sq in.
R 0.0331


The stresses in the net section are computed as for
the exact solution.










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

section can now be computed

T2 t2 -0.234
r .2 4 = 0.0587
AR30 --= 3.82 x 1.045


T2Rt2 T1Rt1
r' 2- -p _-
A2 2 %o,)


-0.234 + 0.055
0.212(10.63 3.82)


= 0.1236


G r2R
r3 2- _2-
-b 2T2


0.380 x 10.3 0.07
7.56 x -7.05


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

03 = Z.82 e e-0057x)


and in the nain stringer by formula (6) is

02 = 3.82 + 6.93e-0.1236x

The stress in the continuous stringer can be found by

formula (7).

The running hears are, by formulas (8) and (9),

Tlt1 = -0.234e- 00587x + 0.179e-0.1218X


T,,t = -0.234e-00755


At x = 1.50, the point of maximum observed shear stress,
T2t2 = -0.234 x 0.893
= -0.209 kip/in.











and
0.209
'2 0..0331

= -6.31 kips/sq in.

Analysis by the Two-Stringer Solution

The basic data rernain as before. Compute

a 9.38
b ~ 14.06

= 0.667


aA3 1.045 x 9.38
b(A1 + A2) 14.06 x 0.915

= 0.764


Then from figure 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)J
= 9.G5 kips/sq in.


By statics,

i, -- 7.70 kips/sq in.

The rate-of-decay factor is computed from
-2 0.380 x 0.0331 (1 1 \
-- 7.56 0.9715 1.045)

= 0.00342


K = 0.0585











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

-0.0585x
02 = 3.82 + 6.03e


and,by formula (13), the stress in the discontinuous

stringers is

03 = 3.82(1 e-0.0585x

The stress in the continuous stringers may be found by

using formula (7).

The running shear in the cut panel is,by formula (14),


T2t2 = -3.82 x 1.045 X 0.0585e-0.0585:x

= -0.234e-0.0585x

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


T2t2 = -0.0234 x 0.916

= -0.214 kip/in.


and
T_ 0.214
2 -- 0.0331

= -6.47 kips/sq in.










44


REFERENCES

1. Kuhn, Paul, and Chiarito, Patrick T.: Shear Lag in Box
Beams Methods of Analysis and Experimental Investi-
rations. Rep. No. 739, IACA, 1942.

2. Kuhn, Paul, and Moggio, Edwin M.: Stresses around
Rectangular Cut-Outs in Skin-Stringer Panels under
Axial Loads. MNACA A.R.R., June 1942.

3. Kuhn, Paul: Approxinate Stress Analysis of Multi-
stringer Beans with Shear Deformation of the Flanges.
Rep. Io. 636, NACA, 1938.

4. Kuhn, Paul: Stress Analysis of Beams with Shear Defor-
nation of the Flanges. Rep. Ho. 608, NACA, 1937.





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46


COMPARISON


TABLE 2

OF OBSERVED ANID CALCULATED !IAXIMUMII RIB STRESSES


[Load on panel, 15 kips]
Calculated
Number of stringers Half-length Observed Csulated
cut-out stresses tri/s se
Total Cut (in.) (kips/sq in.) (b)in.

16 2 1.5 1.57 2.51 1.48
16 4 1.5 2.20 3.29 1.94
16 6 1.5 2.73 4.42 2.60
16 8 1.5 2.89 5.73 3.58
16 8 8.0 4.30 3.32 2.91.
16 8 15.0 4.77 3.28 2.88
16 10 15.0 5.49 4.48 3.94
16 12 15.0 6.75 6.49 5.70

aFiller strips ineffective.
Filler strips effective.







Slt I t 1 t


Figure 1. Axially l panei Ith cut-out.

Figure I.- Axially luadal panel with cut-out.


NACA


Fi_. 1





NACA Y---' Fig. 2


SH -Main5tringer

I 1I : 7

4 Arib


















tt
------/ ---. .



,../ /'/ 5ubshtute structure
I, /9"/







/ / ,,,?


i *1 !I/





Figure i_.-5tress distribution around cut-out by three-stringer method.





NACA


__ __/ __ __ C o


'-4




0











-'-It
02 /




o CCC
O/















7/ / .
/i/ ,


Fig. 3


C%o
c'j ^
'% gl






















a) 1
r-4


aS



LO


H













Ad
0
.,4




"-4








-4O
-4I I











I------0--


NACA


Fig. 4








Fig. 5


+* C
L
O
"4"-








L
I b-
-.
._.'


NACA





NACA


y--,
./'

.
-- -A--4. cut-out


nj 1W 1W 1W 1W 1W -V 1W
~ Lcd


&- - -


,


'i


Stringer stresses














Substitute structure







b









hear stresses


Figure 6.-5tress distribution around cut-out by modified two-stringer method.


Fiq. 6







Fig. 7


(1)










0J


*NACA


(-'I black = o10/Z5") -







NACA


Fig. 8





Ln
r-4








+3
II








4--





V)




4








'-4


CO














0
4-1














UJ
(0
I-,
1-1

















:(1

t2.







NACA


Fig.9




















w
0
4--





O




0
1-4
p.

0

-I



IC





NACA


(a) 6- stringer panel.
(b) 15- stringer panel.


Figure 10.- Cross sections of test panels.


-- Exact solution


---- Simplified solution

(by three-5tringer method)


2 -

04

2
0--


-o----o-~


10
Distance


15 20
from center line


25 30
of cut-out, in.


Figure 11 .-Stringer stresses in 15-stringer ponel with 1 stringer cut ond L =8.3 inches.


L



61

4





40 -------------------------------


I 40
35 40


Figs. 10,11





--ACA -Exact solution F
Simphf ied 5 olu ion by three-stringer method





4



0 ... ...
2-



4 -






4 -

0 2


-n i
b 66



0 ---- ----- -- -- -






42 2


4 0 ----- -

04 _..


2

0 4
0 0l l 15 I5 3I-' ,5 4
Distarce from center line of :jt-.ut, in.


Figure 12.-Sinnger stre5se5 in l-strirer panr,.- itf 3 sIlrig-rs cut ond L=8.3 inches.





BACA Exact solution ,. 1.
Simplified solution by three-strirnger rrethod



0- ___- _----__-___
4





0

6








4



if)
b "------ -----








4

2
S4 -- -- ---


40


04-
o4- ---------



0 5 10 15 a0 5 30 35 40
Distance from center line of cut-)xt, in.


Fij re 13.-S'ringer stres.e5 in 15-stringer panel with 5 ringers cut and L=8.3 inches.






Exact solution "
impliedd ltir by three-stringer method
- -- Simpif'ied solution


2 -

0 8 -


40





6 -

4
e ~ ~ o---------------------
.. ---' -- .


64 "" ------




2-
,o)


04-

2-

4 0 -

2 -

D 4L

2

06---


0 15 20 25 3
Distance from center lirne c ;ut- cjt,irn.


0 3t 40


Figure 14.-Stringer stresses in 15-stringer panel with 7 stringers c .t ond L=8.31nches.


8

6


F'I 14


- 2
0
'-.

to"

Ln
cu


AI


5





NACA Exact solution Fig.
Simplified solution by three-stringer method
----- Simplified solution

8
6 o -
4-


0 101

8






0 00



4 60



0}0





0 4J- a- ~ ^------------


4 0


0 4 _

C, 0

0 5 10 15 20 25 30 35 40
Distance from center line of cut-out, in.


Figure 15. Stringer stresses in 15-stringer panel with 9 stringers cut and L=8.3 inches.






Exact solution
Exc solution by three-stringer method
----- Simplified solution


6

4 0









0
B-

4


2
04

40
08 -


04
4 -



4 0- ""---------
2



04


0 5 10 5 0 5 30 35 40


Distance from center line of cut-out, in.

Figure 16-Stringer stresses in 15-stringer ponel with 9 stringers cut and
Heavy main stringers.


L=-.3 inches.


IACA


Fig. 16





NACA 6 Fig. 17


L'- Exact solution

o ---6 5implified solution by three- stringer method

4



6 o

4 -

2

0 6


6-
I





------------




2r

01
I







L
6-




4 |---o u u------------

0


I o^


0 5 10 15 20 25 30 35 40
Distance from center line of cut-out, in.

Figure 17.- Stringer stresses in 16-stringer panel with 2 stringers cut and L=l.5irnches.





NACA


6



2F



0 O----


C

0 8-

6

4


0


C.
0---------------0-- -----------------------


1 r-

10



6

4


0~.

___________-------


0
-~ I I ~


Fig.ire 18.- stringer 5tr-sses i, 16-5trinqer panel vwith 4 stringers cut and L=I.l5 inches.


6 F Fii
4
SExact solution
S--- mplified solution by three-stringer method
--- implified solutioni


S 18


5 10 l2 20 25 30
Distance from center Iinm of cut-out, in.


35 40


^





EACA


Fig. 19


2

80


4


0 8



4


12 0

10

8


- -:00


a
0
I I I -


35 40


Figure 19.- Stringer stresses in 16-stringer panel with 6 stringers cut and L =1.5 inches.


6

4 0-
Exact solution
2 ----- Sified solution by three-stringer method


4 0


0 4


2-.
0


0 5 10 15 20 E5 30
Distance from center line of cut-out, in.


I-


U 0





NACA


Fig. 20


mULi oU ogIUI1
5nimplifled solutionI by three-stringer method


0 --

U



a



a

- ~ I I I


35 40


Figure 20.-5trinqer stresses in Ib-stringer panel with 8 stringers cut and L=l 1.5cne..


8

6 0

4
St lJ4,-


2 -


08



4


2


6

4



40'

2

0 4:





2
4 0



04


0-


0 5 !0 15 20 85 3-,
Distance from center line of cut-out, in.





Fi. 21


NACn 8

6

4

2

0


5 10 12 '0 25 3
Diktonce frOn center line ;+ c-tcit, in.


35 4,C


Figure ?.-Stringer stresscs inr- I6-.rr.ger Danel vvih 8 strinqer cut arind L=. inches


E E ,ct solution ,
- implified SOt'on by three-stringr meth.,j


0


2
12-


Lfl




Vi',


2










C.,





2'


-.- ----


II --





NACA 8 Fg.

6


4 -



080

6 -

4"

2-


C,


8 ____


cL 0
- 12-


b 8
L 6

+- 4
a) I


4 0


4- Exact three-stringer method
0 -4 5imifed three-strinnger method

2 -- Modified two -strger metho

4 0-


0 4-


0 -


_-* I -----
_i / C,


0 D 10 15 20 25 30
Distance from center line of cut-out, in.


35 40


Figure 2.-5tfringer stresses in 16-strirnger porel with 8 .,tringers cut and L=i. inches.




NACA 12 Fig. aZ.











8

6
Exact solution
4 rimpliie .t by three-stringer method







4


12 0
10 I


g8.
C) 6 "

S 4






4 0


20 4
4 0 ,



0 4 "'

2 --


0-
0 4 J- -

0 ID 15 20 25 30 35 40
Distance from center line of cut-out, in.

Figure 23.-5tringer stresses in 16-stringer panel with 10 stringers cut and L= 15.0 inches.




NACA 18

16 c

14 -



10 -


4


0 12 ----


,-c-


4-
--- Exct sution
SI- -51mphfied scuionJ by three -strirer method


60

4
4 -

04



4 0-







40



O -


0


0 '-, IO 5 ._-', 2 ., I,
Diar;ce tr r-rr :ertEr in, o,'f ,-Cut in


Figure 4.-Stnnger 5reses in 16 r r.]er c ro i- 1 strl '-r d Lct :. *
_.'r ,. ,. J L O -...


FiE. ::A


c

U)-


tr)
b"
u"'"
u";
al>
L
G5


35 40





FI C. 6


p
In
LI)
CL,
I
'ft


Figure .5.-5, e r stresses in 16-stringer pinre 2 vit nrir s t rid L=1.5inches.




--- C1 5-lutiun
-- -- S[li fie i solt'ln

(b. ti ee-s -trnger metl id)


-o -

6




I


0


r ---
__ I I20 J 30 35 40
Dist n,: f-rr n r ter i.t- in.


Figure 6.-5her .-. -':n 16-str'nLur' p.' -I,' 4 f, ...r '..ut arJ Lj l.i ncres.


-2
0 -C

-6 .






Di: Jr. fr :.rr' :erter lir, -, cut-our, in


Ul)


NACA






NACA Fig. 27


Exact solution
---Simplified solution by three-sringer method
- S im p lifie d so u n


2 -



-2
,p


- U
2'0


c


a -4 -


0 -8

L


-6
4 -



-4 0-

-4 -

0-4

-2 K

0


Figure27-Shear stresses in 16-stringer parel with 6 stringers cut and L=1.5 inches.






HACA


Exact solution.
xac--- f solution by three-stringer method
Simplified solution


-2-

0





-2

00

2-



i -4





-6r

-4

-2





-4 01-



0-4

-2 L-

o0 5 10 i5 20 25 30 35 40
Distance from center line of cut-out, in.


Figure 28.-5hear stresses in 16-stringer panel with 8 stringers cut and L= .5 inches.


Fig. 28






NACA


0 0
o -- ^r----------------------


-8









-6
-4 -







-4 --^------






0-4
-4 0 --------------------





-2 -
0-4

0 5 10 15 20 25 30 35 40


Distance from center line of cut-out. in.

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


Fig. 29




Exact solution 1
xc. sid solution by three- stringer method
----- Simplified solutions


ri






Fig. 30


---Exact three-stringer method

-----Simplified three-stringer method

---Modified two-stringer method




p Li U


_Z?


u

2


0



0


-2

0

2





Cn


-4-


In


-4
-4 h-





0-4



-4 0




-2



Distance from center line of cut-out, in.


Figure 30-Shear stresses in 16-stringer panel with 8 stringers cut and L=15.0 inches.


NACA






Fig. 30


--Exact three-stringer method
-----Simplified three-stringer method

-Modified two-stringer method






0
2-



-2




0 5 .. 0...o 3 5 40
2 -
.d
















-4 0
S-4-





-2

-2-

0--
0 5 10 "1 20 25 30 35 40
Distance from center line of cut-out, in.


Figure 30-Shear stresses in 16-stringer panel with 8 stringers cut and L=15.0 inches.


IACA






Fig. 31


Exact solution
5implif led solution by/three-stringer method




0


-2

0

2-



-4-


-8a-

-6 -





i i
-4






-4






--4

0-4-



0 ,10 15
D'istanc- f'rym


"---- ________ 0


20 25 30
center line jf cut-out, in.


35i| 4


Figure31.-Shear stresses i,.- 16-stringer panel An !C tringers cut ana L=5.0 inches.


NACA


35 40





NA- Exact solution \
Simplified solution by three-stringer method
-4 o o
0 0
-,







01

-2




S-2
S-4


S-4-

-6


-4













05 0 20- 2'5 30 35 40
-2-


-4







Distance from center line of cut-out, in.


Figure 32-Shear stresses in 16-stringer panel with 12 stringers cut and L=15.0inches.








NIACA


I f1
| ,I I
"I I

B .'- l C. I
U A)
| U i i --. I
L fl. t.- LO
01U F-I -., .^
-- .- .q I- -"- --- .. -. -. -
*
... .....- --''-I

,i ,--^I ,; I I /
r-i r r-4
,,-i ..: i f !i
___ ,1C I / L _


iI i I r


[.. .. J-.. .. .
I I








I / I !



co i c --s J N
*uL be /S9dJ *ss5J'4g


7 p

I ~~L-- I
I----
I ,~------~---
j.~p: ~1hItZ -_






I~--T-. TLT~-- ___
I ~ I -~
I I I 2.


?ig. 33


0 1
*-





r.4







(-.


1-4


wI In iw
G) ) ()











4 ..1 L. t
.-1 i:- --1





,4 --
II II II
-'1 I-








0 J .1
- .11


Sr-I r*-

Q lm t


l. L.LI U)
c .; c,



.,.--, .,--. u.-























U _______ "~- -
I.'
~ N


I'
x I
___ I __

I----. --


NIACA


Fig. '4


F'-'
.1*
0


1)

~Ij




C\.




'-I

'C-.,









~1//
.16


UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE LIBRARY
P.O. BOX 11f7011
GAINESVILLE, FL 32611-7011 USA


*, ,
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44 11

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