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[

t I '

. s •__

• t

BIFURCATION AND COLLAPSE ANALYSIS OF STRINGER ANDRING-STRINGER STIFFENED CYLINDRICAL SHELLS WITH CUTOUTS

1^1f^1J1 ;by

A. N. Palazotto N ((Associate Professor (Structures)

Mechanics 6 Engineering Systems Department ^^'a1^ fMClllrtAir Force Institute of Technology, WPAFB ,e Dayton, OH 45433, ,^ • _^^

& ABSTRACT

This paper presents resultsfor cylindric,l shell configurations

z~ using the STAGS computer program. Discontinuities have been imposed

upon the shell's skin by incorporating symmetrical cutout openings.e^1 -

In addition, the surface is stiffened with both stringer and ring-

-w stringer arrangements.W M0 -w E4

0

CO .aN

'.The cutout problem has been shown to be highly nonlinear for i

P4 ^ --tawHwV

L) z v u smooth surface shells, but the author has found that bifurcation andH T.

QxeE-4Hw collapse loads are close when one is considering stiffened skin con- f

0 z t figurations. In order to arrive at this conclusion, it was necessaryHE-404

H .a

" to evaluate the following:U A S ^*2 x

"C

En V

Hcomparison between smeared and discrete stiffener theory for

H a,.aIx

`p a'u 4.4

linear solutionsx cc -

^.r-HHA

H N numerical finite difference convergence as directed toward

w H buckling determination.-a0W w

0N collapse load results with the various skin stiffeners.

V H x 0I4 to

:04;utit w

N'This paper also includes a linear bifurcation study relating to

EniC

6.44 H •r4Z4 tn ^ stiffening effects around cutout areas present within stringer and

siag-stringer shell surfaces. Comparisons have been made between a

vititty of geometric positions considering cutout frame and thickened

akfn additions. Th y° investigation points toward an optimum positioning.

,11 PAGE laQUALMOF ppOR

i

1

t

kNOMENCLATURE

2a - cutout dimension

A1 ,A2 - cross sectional area of stringer and ring

bl,b2 - spacing of stringer and rings

E - modulus of elasticity

GJ torsional stiffness

h,t - shell thickness

Iy ,I, - moment of inertia of stringer,

L - length of shell

Nxy - resultant shear force per unit length

n - number of nodes in finite difference scheme

P =cr bifurcation load

P . bifurcation load for shell with no cutout0

R = radius of shell

u,v,w - displacement in x, y and radial direction

x,y,8 - coordinates

Z - Batdorf shell parameter (1 - v2)h(L/R)2(R/h)

B - slope of deflected surface at boundary

v = Poisson's ratio

INTRODUCTION

It is obvious the cylindrical shell is an important structural

configuration within the aerospace industry. One finds an unlimited

amount of research papers directed toward this common shell surface.

Yet, the practical consideration of skin cutouts and their stiffening

has, for the most part, presented certain amounts of analytical

difficulties. Until-recently, the stiffening of these cutouts became

an experimental trial and error procedure [1]. It is now possible to

2

obtain a computational evaluation of cutout effects and reinforcements

using an all purpose shell program developed by Lockheed. This program

has been named STAGS (Structural Analysis of General Shells). It con-

tains the overall capabilities of either nonlinear or linear shell

analysis leading to bifurcation or collapse loads for several different

shell geometries, in particular cylindrical shells [2, 3, 4, 53.

Studies have shown that a shell buckling resistance is increased

with the addition of stringers and rings [6, 7]. As mentioned previously,

the positioning of cutouts as geometric discontinuities creates a struc-

ture which is extremely practical. A literature search indicated very

little work, either experimental or analytical, in the area of stiffened

cylindrical shells.with cutouts. Thus, the author has carried out such

a study with the assistance of the STAGS program. The remaining portion

of this paper will discuss the

linear bifurcation solutions for both ring and ring-stringer

stiffened cylindrical shells with particular interest given to conver-

gence of results due to finite difference refinements.

• collapse nonlinear solutions for the above mentioned shell

structures. Comparison has been made between linear and nonlinear

solutions.

• comparison between smeared and discrete stiffener results.

• geometry and stiffener effect on buckling.

• buckling load increase due to stiffening in the area of the

cutout.

LINEAR AND NONLINEAR RESULTS

The primary concern of this section is to establish, for the shell

^UTG^ IB-""; ^3

i

I

1

geometries investigated, a finite difference mesh arrangement which

can be used in collapse load solutions. Thus, one is looking for a

grid system that reduces convergence difficulties and also handles any

large displacement gradient which may arise in the nonlinear solution.

In order to approach the optimum finite difference approximation

economically, one should first do a great deal of parametric study

using the simpler linear bifurcation theory.

Figure 1 shows the shell properties incorporated into the conver-

gence and nonlinear investigations. One should notice the clamped

boundary conditions assumed plus the rectangular shaped internal

stringer. Since the loading conditions are taken to be axially sym-

metric, the analysis has been carried out over a one-eighth shell sec-

tion imposing the prebuckling and buckling boundary conditions for a

linear solution.

As previously mentioned, a collapse shell load value can only be

confidently evaluated, using a numerical finite difference technique,

if convergence characteristics are established. This is usually done

through first considering linear bifurcation results. Figure 2 indicates

per values found for a shell with a 24" x 24" cutout using various nodal

arrangements. One notices the increase in the number of nodes within

a finite difference mesh arrangement. Figure 3 shows that, if a dis-

crete stiffener approach is incorporated into the s ..udy, convergence

becomes erratic; i.e., the buckling load is not monotonically increas-

ing or decreasing with respect to the number of nodes. Yet, smeared

theory results show a definite convergence trend from above. In both

sets of results, a scatter of solutions amounts to no more than 3

percent. Consequently, it is possible to obtain good results, again

4

1qAtj,

IEi

I

for the governing conditions, using approximately 600 nodal points

spaced such that a circumferential mesh llne is present at and between

stringers and an axial mesh line is separated as far apart as the

stringers spacing.;

In order to further convince the reader that a detailed refine-

ment of mesh size is unnecessary even with a 24" x 24" cutout, one

may observe the next sequence of diagrams. Figurer 4 through 8 present

data representing stress flow and buckling modes in the:area of the

cutout. Discrete and smeared theory are evaluated making use of two

different mesh refinements. It becomes obvious that not only do the

smeared and discrete theories give close results, but a mesh arrange-

ment of one line between stringers gives almost the same stress values

as an arrangement with three mesh lines. This insensitivity is

apparently due to the stiffening effect produced by the stringers.

The author believes that it is now possible to evaluate a collapse

load for the structure considered using a mesh arrangement consisting

of 602 node points with one mesh line at and between stringers. Figures

9 and 10 give values that are both within reason and satisfy good

convergence characteristics. Figure 9 shows the closeness between

collapse and bifurcation loads. The figure depicts a load displacement

curve for that point displacing the greatest quantity using a nonlinear

technique. From Figure 10, it may be observed that a collapse wave,

along the bottom boundary, contains no large displacement gradients.

Furthermore, it is interesting to note that the pre-bifurcation curve

is much different than the collapse displacement. Yet, the buckling

loads are relatively close. This is due primarily to the increased

stiffness present in a linear analysis which produces a less sensitive

moment effect.

1

'^ I

h ^^

The work discussed up to this point relates to stringer stiffened

cylindrical shells. A like study has been carried out for a shell

stiffened by both internal stringers and rings with a 27" x 24" cutout.

A smeared theory approach was only used in the convergence

investigation for this particular skin configuration. Again, as

Fig. 11 indicates, the smeared theory gives convergence from above.

One obtains a critical load value, for 714 node points with mesh lines

between stringers and at the ring locations, which is within 4percent

of the value obtained using 2262 nodes. Thus, mesh refinement does

not appear to be necessary. Figures 12 and 13 present the collapse

load and mechanism for the shell. There is a larger collapse load

compared to the bifurcation value, but the difference amounts to only

ll . percent. Furthermore, it is shown in Fig. 13 that the displace-

meat collapse wave does not change drastically and therefore can be

considered well behaved with at least five nodes per half wave. Thus,

it appears the mesh arrangement is adequate.

STIFFENING AROUND CUTOUTS

A limited investigation has been carried out into discrete stiff-

ever effect upon shell cutouts. This section discusses certain find-

ings for such a study.

It is conceivable to use the STAGS program as a tool analogous

to physical experimentation. As stated in the Introduction, most

cutout stiffener problems are handled through a trial and error solu-

tion. A good illustration of this experimentation approach can be

obtained in research performed by McDonnell Douglas El]. In this work

a thin stiffening frame was placed adjacent to a cutout within the

nE 6 .

.I

5

interior surface of a Thor Delta Interstage cylinder. The effects of

a cutout and its reinforcement on buckling behavior were explored ex-

perimentally by physically varying the stiffener position to overcome

any skin rippling. It is now possible to explore the same problem by

Incorporating STAGS into the solution without resorting to extensive

experimentation.

Figure 14 indicates cutout stiffening results for the stringer

stiffened shell geometry (27" x 24" cutout) di:zussed previously using

the 602 node finite difference scheme shown in Fig. 2. Two types of

stiffened arrangements have been studied. The first is a discrete

rectangular frame (Area - 2.18 sq. in.) placed internally around the

opening. A plot is shown for each frame position relating a ratio of

stiffener weight to cutout weight removed against the critical load

normalized to a shell with no openings. It can be observed that as

the discrete frame is increased in volume, through a change in posi-

tion, the critical buckling value approaches results close to those

possible without any shell opening.- An optimum frame position is

possible which gives a higher buckling value compared to other positions

(see position 4). 'The second cutout stiffener studied is developed by

increasing the shell thickness along its internal surface. The cross

section and a plot of the weight to critical load ratio can also be

found in Fig. 14. 'Two stiffener widths have been evaluated producing

points A and B in the plot. It is apparent that as the size of cross

section increases the critical buckling ratio approaches the value for

a shell with no cutout.

The reader can observe in Fig. 15 bifurcation comparisons for a

ring and stringer stiffened shell with cutout reinforcements similar

I:

7

ORIGINAL PAGE ISOF POOR QUALITY

II

to the stringer stiffened shell just discussed. It is shown that as

the discrete frame volume increases with location, the bifurcation ratio

P/Po decreases. Yet, as the thickened reinforcement increases in volume,

a bifurcation ratio almost equal to one occurs.

STRESS SURROUNDING THE STIFFENED CUTOUT

M attempt has been made to determine the overall effect due to

cutout stiffening by investigating stress redistribution and eigen-

vector comparison at bifurcations. Figure 16 presents the stress field,

at the interior stringer stiffened shell surfa a, surrounding a cutout

in which a stiffening frame is positioned adjacent to the opening

similar to Fig. 14. In order to compare values to non-stiffened cutouts,

stress quantities have been normalized. The reader can observe that

the stress values decrease in areas adjacent to the discrete stiffener.

Yet,"immediately above the frame one can find highly concentrated ax

contours. Figure 17 indicates, for the same frame positioning shown

in Fig. 16, the complete reorientation of bifurcation waves along the

cutout edge as well as other frame axial coordinates. Points of maxi-

mum buckling wave amplitudes are depicted for specific angular measure-

ments. One can appreciate that not only are the half wave locations

different between shells without cutout frame stiffeners, but the

circumferential wave number is also affected. Figure 18 is a presen-

tation of the same'stress and buckling wave effect discussed in the

above two mentioned figures, but in this case cutout stiffening is

created by a thickened shell skin. Results indicate a similar total

effect.

EFFECT OF SHELL JLND STIFFENER GEOMETRY

Investigators of . problems related to cutouts in smooth surface

cylindrical shells [8] have determined a generalized cutout parameter.

ORIGINAI; PAGE 1S e

OF POOR QUALM

This parameter is the a /(Rt)^ ratio in which 2a equals the cutout width;

R and t are the shell's radius and thickness respectively. The writer

tried to produce a similar parameter for cutouts in stiffened shells.

This section described some of the work performed.

Figure 19 shows a plot of cutout parameters used in reference [8]

for thn present problems. It becomes obvious that a parameter similar

to smooth surfaced shells was not appropriate for stiffened shells

since a common curve is not obtained. The cutout effect turns out to

be very different compared to smooth shells because of the stiffening

produced in cutout regions. Therefore, a study of possible stiffener

geometry effects on cutouts was pursued. Figures 20 and 21 indicate

a greater shell sensitivity to variations in a stringer's axial rigidity

for given cutout sizes as compared to a stringer's flexural stiffness.

Consequently, the author explored various axial stiffness parameter

combinations by including or excluding the shells axial rigidity. The

success in obtaining an ideal function can be found in Figs. 22 to 24.

Observation reveals that simple geometric stiffener area is not enough

to establish a basic parameter. Yet, one may note an important property

coming from this particular set of figures. It can be stated that non-

linear shell effects become apparent using small stiffener area; e.g.,

0.05 sq. in., since the bifurcation curve reaches a minimum and

increases for an increase in a /(Rt)^ ratios. This has also been ob-

served in reference C8]. A reason for this inconsistency may be attrib-

V. uted to the elimination of moment effects in linear bifurcation theory

and thus, as cutout size increases a tension field occurs using linear

analysis which would normally be overcome if nonlinear relationships

were incorporated.

}^^:aItiAL GSU AL1'^^OF POOP.4 9

Ii

f

6F

An additional study wa4 pursued, this time in relation to the

stiffener and shell flexural rigidity. It was previously stated that

a cutout is less sensitive to a flexural parameter. Results verified

this conclusion as shown in Figs. 25 and 26.

As a final investigation phase, the author attempted to combine

r,dal and flexural rigidity. Figure 27 indicates some success for

rrasonable shell thicknesses.

' CONCLUSIONS

It is now possible to make the following .,conclusions:

1). The STAGS program has been used in studying buckling charac-

teriritics for a stringer stiffened, in addition to a ring and stringer

stiffened, shells. The bifurcation load is within 11 percent of the

collapse force for each configuration, and thus it is possible to

study buckling loads using linear analysis for configurations similar

to those investigated herein.

2) . Previously, cutout adm•fozrsaeat A+ detemol ed by ,e, trial

and error experimental approach. STAGS makes it possible to,investi-

gate this problem and determine the most effective reinforcement

kposition.

and 3). The study reported upon reveals that the parameter a/(Rt)

used in isotropic shells considering cutouts is not appropriate for

stiffened shells.

10

r

4

REFERENCES

E R. F. Snell, Experimental Evaluation of the Effects of Cutouts

on the Stress State and Buckling Stability of the Isogrid

Structures of the Thor Delta Interstage and Fairing, MDC

Report 62873, April, 1972.

(2) D. Bushnell, B. 0. Almroth and F. Brogan, Finite Difference

Energy Method for Nonlinear Shell Analysis, Computers and

Structures, Vol. 1, pp. 361-387.

(3) F. Brogan and B. 0. Almroth, Buckling of ,cylinders with Cut-

outs, AIAAJ., Vol. 8, No. 2, Feb. 1970, pp. 236-241.

(4) B. 0. Almroth and A .M.C. Holmes, Buckling of Shells with Cutouts,

Experiment and Analysis, Int. J. Solids and Structures, Vol. 8,

pp. 1057-1071, 1972.

(5) B. 0. Almroth and F. A. Brogan, Bifurcation Buckling as an

Approximation of the Collaps± Load for General Shells, AIAA J.

Vol. 10, No. 4, April 1972, pp. 463-467.

(6) J. Singer, Buckling of Integrally Stiffened Cylindrical Shells -

A Review of Experiment and Theory pressnt ±d at Van Der Neut

Symposium, 1972.

(7) D. L. Block, Influence of Discrete Ring Stiffeners and Pre-

buckling Deformations on The Buckling of Eccentrically Stiffened

Orthotropic Cylinders, NASA TN D-4283, January 1968.

(8) J. H. Starnes, The Effect of a Circular Hole on the Bu ,-kling of

Cylindrical Shells, Ph.D. thesis, California Inst. of Tech.,

1970.

big 11a33 11

U^

I

FIGURE CAPTIONS

Fig. 1 Shell Geometric Properties -=

Fig. 2 Mesh Arrangement

Fig. 3 Number of Nodes vs. Pcr (Stringer Stiffened)

Fig. 4 axAlong x 29 in. Mesa Line

Fig. 5 c Along x 33.5 in. Mesh Linex

Fig. 6 Comparing Various Mesh Arrangements for 8XY

Fig. 7 Nxy Near Cutout

Fig. 8 Eigenvectors

Fig. 9 N vs w (Stringer Stiffened)X

Fig. 10 Displacement at x 47 inches

Fig. 11 Number of Nodes vs. P cr (Ring and Stringer Stiffened)

Fig. 12 N vs. w (Ming and Stringer Stiffened)

Fig. 13 Displacement at Collapse Along x - 47 inches

Fig. 14 Cutout Weight Ratio (Stringer Stiffened)

Fig. 15 Cutout Weight Ratio (Ring and Stringer Stiffened)

Fig. 16 ox at Bifurcation With Discrete Stiffener at Cutout

Fig. 17 Effect on Bifurcation Wave Due to Cutout Stiffeners

Fig.. 18 Bifurcation Wave and Stresses for,Thickened Skin

Fig. 19 Linear Buckling for Cylindrical Stiffened Shell

Fig. 20 Effect of Stringer Area on Pcr

Fig. 21 Effect of Stringer Inertia on Pcr

Fig. 22 Effect of Stringer Area and Opening on Pcr

Fig. 23 Effect of Stringer and Shell Extensional Area on Pcr

(Al 0.28 sq. in.)

Fig. 24 Effect of Stringer and Shell Extensional Area on Pcr

(A1 - 0.08 sq. in.)^

O'EXEDING PAGE ^3 v^ LG1I ^43;'fi`1

4F ^

Fig. 25 Effect of Stringer's Moment of Inertia on Pcr (A 0.08

Fig. 26 Effect of Shell and Stringer's Moment of Inertia on P cr

(A 0.08 sq. in.)

?IS. 27 Effect of Combined Perimeters on P cr

. .= day ^^

it

sT-

or T tKh0,35"

A = area of stringer

0_$„ I Y = 0.01493 in.4Y

GJ = 30302"in.2

t

b

E. =10,000 ksi;

A : = area of ringi

Clampedpre—buckling buckling

Q ; v4 -0

Q ; u 0 same

®v/3 =o

Q ; w=r /,;s 0 w-r=uU. 0.09 in. =/j=0

•--09, Y

O

Q Q)x

1i

' t 24 at 1.5" 1 18 at 3" 24 at 1.5 01at 3"

4 3t 4 at7 -{

5"5„

9^d 18 at3 14x43 1.5" 23x43= .T

j a n 602 0=989

Pcr =629,171Pcr =618,930#

12 at36 at 1". 18 at 3" . 1,36 at..Q.S:_.- 1.S" 18 at 3" ..... .

• 4 at Oat5„ 4„

13 at32 x 55= 1 " 54 x 67 a

-

27 at n =1769 n=3618 1

36 att0.511

Pcr =631,562Pcr = 627,757#

n number of wales

R = 51.3" L= 94 91 1 = 0.28 " b i = 3"

Cutout =24" x 24" Discrete Stringer Approach Used

^RIP^^ ^ ^

1

^

^t z

•

GO; gym"..

•N m Cghs

•. v N SM

v

b y yS ^^

tj toN Nd

'• y0

44 m

j' N

VS Hjf:. y O t

tics i •

S'i . • _ '

4 OMO

ND

s

4c; -

w .

x =29"

R =57.3"

L= 94"

r

1.5" grid discrete

-- — ---1.5" grid smeared

•--x---x 0,5" grid discrete

yPon

x

22

x =29" , I

20 11"

12"

18 — — __ x 41"

h16

8'.

114C Nv b

0 12

bx.10

•^^ 8i

6 1

.^

'21

2 44

6 8 10 11 14 16 18 10 22 14

= 2. y (degrees)

4,QIJ ,I7I

so ^

24 X=33.511i

I.! X s 4110

Iaid discrete

— --1.511 aid smeared

-)(-X-0.5" aid discrete

1

8 / 10 12 14 16 18 20 22 24

/ y (degrees)i

-X 4

1

Y

i^

im

1000

-^- x n 33.5"2'

^ 12" J400-

x = 47.. - — -

200

z0

2 8 10.t

-10 `y(degrees)

-4

-60

-800

x w 33.5"

/00-f 16 18 10 22 14

1.5" grid Discrete

---1.5" grid Smeared

-X--X-0.5" grid Discrete

C

V •^

Y-

600-

• goo ^ ^ '

x- t9'^t

400 x = 33.5"

_ 12"A 200 x- 47"—* —

x = 29"r.3dc 0

4 4 6 8 10 12 16 18 20 22 24

—200 y(degrees)

using 1.5" grid discrete—400

i

—600x - 33.5"

—800 Shear Distribution

OP PGO Q PAGE

QUA

• 9

c

M N rl O ..r N 91 ^ W! wO O O O O O' Co O' O' If

0 .05 .10 .15 .20 .25 .30 .35 .40-NAL PAGE ZS w (displacement) in.

-UOR

C!

310

0 0 0 0 01

l'u !) „M„ N011031A30

2 17x42 =n -714

i

10 at1 2 at 0.15" .-1 -1 .5" 15 at3_ ,GS'^

4 at4.5of

4 atLAW

39 x 58 —30 at o - 22620.75"

1211

13.5"

Pa - 953,505

26 at 1.511 1 _15 at 3.68".

26 at 1.5" 15 at 30FI

4 at4.5"

4 at1.625"

24x4215 at A • 1009

0.75"

Pcr -965,024

n - number of nodesR-60" L-94A ) -0.28 " b l- 3"

A - 0.079 " b - 4.5"2 2

cutout 27" x 24"smeared approachRing—Stringer

cr - 985,527'

990

980

970

g 960

a.^ 950

1000 1500 2000

2500

number of nodes (n)

3000r

.U4 Mf .14 J's .L4 du .34 .3y.44 .44

w (displacement) in.INTERNAL RINGS & STRINGERS

C!

N

ftZ

dA

qt M N -i O NO O O O O O

1 1Vui) A, N011031J3a

CL

v.i

N

s m

C7 ^o

d

faCD

s,V A19 M1

tj

^ NN ^IN N

N v

of IS- VAZI P^0 QII 818

rr

INCREASED SHELLTHICKNESS ATCUTOUT

301

P/Po

DISCRETE FRAMEPOSITIONS

^+^2.25'2.25"

(3) 2.25"(2)

4.511

(1) 12..

3" 1.51.51.5

27 9qx24" CUTOUT

r

0.9Z0.76" 0.1 (B) (4)

o..A = 2.18

0.90 (2) (3)(1) 0.8,.

0.88(5) 2.725"

(A)

0.86W= wt. of added stiffener

0.84wo removal wt.P a PCr

0,82 Po Per no cutout

=8.30,822*

0.80—T

1.0 1.5 2.0 2.5 3.0 3.5

P/Po (no reinf) = 0.78w/w0

(Strin;er Stiffened Cylindrical Shell)

fJ,%-Wx

i

VARIATION IN 279#x24" CUTOUTTHICKNESS AT DISCRETECUTOUT (4) FRAME

(3) POSITIONS(2)(1) 12' ►W;

4 A 1.5 1.5"B 3" I 4.5"C 4.5"

0.9" 0.1"g..

(C)0.98 ' (B)

0.96

0.94 (A)FRAME'S X AREA= 2.180"

0.920.8"

0.90 2.725"

P/p` 0.88-f r(2)

0.86 (3)A = 0.28 0"

0.84 A 0.07911"2

0.82 w-wt. of added stiffenerwon removed wt. at cutout

PAGE

cr0.80 P = Pcr

oRI^Il3tTAI1 P" P no cutout

OF POOR Q0.18 -1,420,7134

fie,)0.76 '--j-

0.5 1.0 1.5 2.0 2.5 3.0

P/Po (no cutout) w./wo

- 0.68 (Ring—Stringer Cylindrical Shell)

1

1

I

i-S; r^:.R.. • ^ fly `T• • I• ..

or

mash 1 ,135

flo

1,34 133 1:05 . 1.,.6.:..:.:.

t -t

2.25" Stiff. Frare (—^iJ) ; (—iG) `z

1.24 ':.

1. 123 0155 0.95 1.054.50"- ..,;

1.119 0.55 0 t

1 0.53 0689

0 1 85 ^ -.: 0.51 0*. 85~ _.•._.._

IGINAL' FAGE IS

POOR QuA1,Tly1315 M .1

016 0, 43 I r 1:05

normalized vXto stress 0.86 1.1 0

field with no frame• r

o 15 a:43 095 1.05 ti•' • •!^

^. at 6ifurc^ition ( inner edge of shell) (ctringer stiffierad cyl. shell) ,•r:^ti

r^ 1 t I ^ ^. 1'^-^ 1

0

{Il

i'1 ,

2,25'

frame

Point of greatest ratio

2,9 6.75"

z - 33.5"

n= circumferential full value

0

max, ampl. at x = 33.5"frame 0°, 24°, 41°, 58°, 72 0 n=10

no frame 0 0 , 14°, 300 , 480 , 62 0 , 760 n =12

1 2.2

9"

ime

X=

U

max. ampl. at x = 33.5"

frame 0', 117°, 29°, 44°, 58°, 76° n=12

no frame 0°, 140 , 30°, 48°, 62°, 76° n X12

^*- 4.5 11 imax. ampl. at x = 33.5"

frame 160 , 21°, 44°, 580 , 720n=10

no frame 0°, 14°, 30 0 , 480 , 62°, 16°n-12

max. ampl. x 26.75"

frame 14°, 260 , 44°, 580 , 720n-10

no frame 0°, 16°, 300 , 480 , 62 0 , 76°n-12

Note: 2 .2 (multiple of stress atpoint with no frame)

max. ampl. at x=24.5" (frame) ampl.=amplitudeframe 190 , 21°, 44°, 58°, 76° n-10

no frame 0°, 16°, 30°, 48°, 62°, 76 0n=12

(stringer stiffened cylindrical shell)

r^q ^_

I I I I ^ ^

0 1"

0.16"

Sect. A—A

^ I' A 1.54 1.64 1.03

x 2^"0.2K0.36;M 0.98

= _(}1)0.47 0.73 1.0

4.5"x = 33.5 0.56 0.51

(-15)0.^5 0.88

(-23)I (ks0

normalized to

stress field with no

stiffener

max. ampl. at x = 29"—

frame 0", 240 , 410 , 58 0 , 72 0n-10

no fra me 01 , 140 , 30°, 480 , 62° 76° n- 12

max. ampl. at x = 33.5" ampl.= amplitude

frame 0°, 24°, 41°, 58°, 72 0n-10

no frame 0°, 14°, 300 , 480 , 620 , 76 0 n-12

n—full buckle wave

(Stringer Stiffened Cylindrical Shell)

! 4 n

ORIGINAL PAGE JSOF POOR QUALITY

ii

P/PoClamped Stringer Shell

Po-Buckling Load — No Cutout1.0

6t -0.05"

0.9

.R /t=1146

R/t =5130.8

24x24

0.1 R./t=286

0.6

t =0.2"0.5

0.4

5 10 15 20

25

2a./(RtY'

MGIN AL PAGE ISOF POOR QUALITY

aoas

0

o ..a

N

O1a m

i

c

AgiSM

t -

N

R On O u^

mi8 n rJ2

q HSc C2

m

DRIGINAD PAGE MOF POOR QUAL y.

d° r► '*CL. c a o c

t -

t

I

F,

0.8-`

0.1 t = 0.0

0.6

A,=

0.5

0.4

1

00

Pipo

1,0 ^

0.9

"

pF paoR ^V^-,r-^

5 10 15 20 25 30 35 40

aI(Rt) %/A , /bt

1 Z 3 4 5 6 1

a/(Rt) 11/1+ A, At

t

P/Pe

1.0

0,9

0.8

0.1

0.1

0.^

0,

PRO

.I

O

0

0,

0.

0.;

a/(Rt, "A + A I /bt

0 5 10 15 20 25 30 33

a/(Rt)'' /(I I /bt' )

L

:. PIPo

1.

PRO

1,0

1p

IS

0.8

0.1

0.1

0:

0.,

0.

* ^ i

1 z i + J

a/(RtP/(1+I I /bt 3)

r, Y r

0

i

1

P/Poto Sher thickness

LOA v-a (0.1811") area of iffener

^^ / t=0.05" (0.08)

(0.28)

t=0.05" (0.28) ^, it =.2

^`` rrt=0.1 (O.QB)t=0.2(0.28)

t- 0.1"(0.08)

r

0.]♦.

0.6

!rte

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

a/(Rt0/(1 + A I /bt) (1 ♦ I /bt 3 )

r_` ey ,,