An experimental investigation of the behaviour of stiffened plates in axial compression

ffCHn~lu"

VllEGTU;C;OOUW~:U"'DF. AN EXPERIMENT AL INVESTIGA

9

~~,iyt4!r\,\,<l() 10 . DH

fr

THE BEHAVIOUR OF STIFFENED PLATES

IN AXIAL COMPRESSION

by

R. C. TENNYSON

AN EXPERIMENT AL INVESTIGATION OF THE BEHAVIOUR OF STIFFENED PLATES

IN AXIAL COMPRESSION

by

R. C. TENNYSON

ACKNOWLEDGEMENT

The author wishes to thank Dr. G. N. Patterson, Director of the Institute of Aerophysics for providing the opportunity of carrying out this investigation. The guidance of his supervisor Prof. E. D. Poppleton is also gratefully acknowledged.

This work was made possible through the nnancial assistance of the Defence Research Board of Canada.

SUMMARY

An experimental study has been made of the buckling and post- buckling behaviour of a number of flat and curved panels.

The effect of the ratio of plate-to- stringer stiffness on the initial buckling stress was determined and the theory of Seide and Stein was found to predict the behaviour of flat panels accurately.

The ratio of the ultimate strength of a curved stiffened plate to that of a corresponding flat panel was found to increase with the ratio of panel width to radius of curvature. This increase was found for all values of stiffener-to-plate stiffness ratios. Further, as this stiffness ratio was reduced below a certain critical value, the ultimate strength of a stiffened curved plate, relative to a flat panel, increased rapidly.

TABLE OF CONTENTS ! " :

PAGE

NOTATION iii

J. INTRODUCTION 1

IJ. EXPERIMENTAL TECHNIQUE 2

..

2.1 Panel Construction 2 Test #1 2 Test #2 2 2. 2 Edge Supports 3 Tests #1,2 3

2.3 Auxillary Test Specimens 3

Test #1 3

2.4 Test Apparatus and Testing Procedure 4

Tests #1, 2 4

2.5 Experimental Measurements 4

Test #1 4

Test #2 5

lIL DISCUSSION OF EXPERIMENTAL RESULTS 6

3.1 Stress- Strain Curves 6

Test #1 6 3.2 Column Curves 6 Test #1 6 3.3 Initial Buckling 7 Test #1 7 3.4 Effective Widths 9 Test #1 9 3. 5 Ultimate Strength 10 Test #1 10 Test #2 12 IV. CONCLUSIONS 13

REFERENCES APPENDIX A APPENDIX B TABLES 1 to 9 FIGURES 1 to 21

TABLE OF CONTENTS (cont'd)

PAGE 14

15 16

A a b c D E GJ I I p L R s t

w

(iii) NOTATION cross- sectional area (in. 2) length of plate (in.)

panel width, measured between stringer centre-line (in.) torsional-bending constant

end-fixity coefficient

12(1 -J)" 2 )

(Ib. -in.) . plate stiffness,

₌

Young' s Modulus of elasticity (p. s. i. )

torsional rigidity of stringer cross-section, where E

G = 2

+

2 y (p. s. i. ) , J = torsion constant moment of inertia (in. 4)

polar moment of inertia (in. 4)

buckling instability coefficient,

=

(of mid bay)

12 (1 - Jl2)

(b

t

2 Çï

1r

2

R cr. actual measured length of column (in. )

L

effective length of column,

=rc

(in.) applied axial-compressive load (Ibs.) radius of curvature (in.)

distanee from stringer centroid to mid- surface of sheet (in.) sheet thickness (in.)

total plate width (in. ) effective width (in. )

initial value of amplitude of imperfection (in.) curvature ratio,

=

b2 (1 _ Jr 2) 1/2

(iv)

NOT ATION (cont 'd)

GREEK SYMBOLS

cr

p

{3

applied axial stress (p. s. i.)

radius of gyration of stringer cross-section (in.) axial strain (in. / in. )

screw diameter (in~ ) Poisson 's ratio SUBSCRIPTS

BT

yp cc p,m cr. st sh ult. tot. fp, cp (f, c) co. bending-torsion yield point column crushing plastic, metal critical buckling stringer sheet ultimate total

flat plale" curved plate

Lst, Rst E eff. c. g. se. (v) NOTATION (cont'd)

left, right stringer Euler

effe ctiv e

centre of gravity edge stress

1. INTRODUCTION

Available test data on the failure of stiffened curved plates is relatively scarce and results, generally, have not led to any consistent trends for the various geometrie parameters.

However. a recent investigation (Ref. 2) was successful in confirming a trend, which was implicit in an earlier work (Ref. 5), of increasing ultimate strength with increasing curvature. Results were pre-sented for panels having stable stringers and these showed a decrease in ultimate strength ratio (Pc /Pf) for increasing ratio of stringer-spacing to sheet thickness. As aresult, it was decided to study the effe cts of curva-ture on panels having unstable stringers and compare them with existing data.

An attempt was further made to establish the trend of the variation of ultimate strength ratios with stringer-to-plate stiffness ratio, while keeping the cross- sectional area ratio of plate-to- stringer a constant. References 6 and 7 were used as a theoretical guide.

Because of the two objectives set forth above, the present investigation was divided into two test programmes, the former being classed as Test #1, and the latter as Test #2.

Critical buckling loads along with effective widths were also determined during the course of the investigation.

(2) _:·1

Il. EXPERIMENTAL TECHNIQUE

2. 1 Panel Construction (Fig. 1, 2)

Test #1

Eight panels were constructed from clad aluminum alloy

sheet. having a specification QQA 362-T- 3 (24S-T3) and a thickness of

o.

040 inches~ The rolled channel stringer sections were of material

having the same specification but a sheet thickness of 0.075 inches. The

stringers. were first bonded to the curved sheet with a HYSOL .mixture

(10 parts of 2030, by weight, with 11 parts harden er AK), and allowed to

dry for at least twenty-four hours. Attachment was further secured·by the

addition of round head machine screws

(szS

= O. 138 in.) and nuts, at a pitch

of O. 6 inches. Such construction approximated a monolithic structure up

to failure. Since the area of attachment of the stringer's·flange was small

(approximately O. 23 ·in. wide) it was n?t considered necessary to form it

to the curvature of the panels.

To ensure a uniform distribution of compressive end:'load,

the .panel ends were cast in Wood's metal to a depth of three-quarters of an

inch. Holes were drilled in the plate and web of the stringers (approximately

O. 30 in. in diameter) through which the molten metal could flow to form a

type of rivet bond. The ends were then machined flat and parallel, normal

to the longitudinal axis of the string ers.

In designing the panel ends, the centroid of the Wood's metal

casting was arranged to coincide with the estimated centroid of the

stringer-sheet combination. The Wood's metal ends were cut about 0.70 inches

short of the panel's edges to allow the edge-support (Fig. 3,4) of the outer

bays to extend as far up the sides as possible.

Except for panel #4, all panels were equipped with only three

strain gauges on each side (viz. one on each stringer and one on the

centre-line of the central bay). Panel #4 had an additional gauge on each of the centre-lines ofthe outer bays.

Test #2

As in Test #1, each panel was cut from a clad aluminum

alloy sheet having the same specification and thickness. Five flat panels

were made and another five were rolled to a radius of curvature averaging

thirty-two inches. The dimensions of all panels were maintained constant at:

total developed width

=

16 in. total length

=

18 in.

centre-bay width

=

.6 in.

The stringers werè cut from a sheet of HOMALITE (CR-39)

photoelastic plastic, having a thickness of O. 1875 inches. The

cross-sectional area of all plastic stringers was maintained constant at 0.24 in. 2, but the shape was altered to give a range of values of the parameter ~. These shapes are illustrated in Table #1. Attachment of the stringers to the metal sheet was made with an epoxy resin reflective-type cement. Conse-quently, a completely monolithic structure was maintained throughout the loading of each panel up to failure. The same cement was used to attach various widths of flange to produce 'tee' and 'cruciform' cross- sections for

the stringers. It is to be noted that at no time did the cement crack or cause premature failure of tne stringers .

To ensure a uniform distribution of compressive end load, the ends of the panels were machined flat and parallel. However, because of the extreme sensitivity to heat of the photoelastic material, the panel ends could not be cast in Wood's metal or some other material and con-sequently, they were machined bare ended. Since the stringer

cross-sections did not extend far from the attached sheet, the milling of the ends was not difficult, and did not cause chipping of the plastic.

2.2 Edge Supports (Fig. 3, 4) Tests #1, 2

To simulate simple support at the unloaded sheet edge, two stiff angle sections were fitted around it as shown in Figure 4. A strip of

sheet, having a thickness of that of the plate (t

=

0.040 in.), was inserted between the angle sections which were held together by a number of equally

spaced screws. By tightening the screws only enough to allow the edge

supports to stay clamped on to the plate edge (i. e. they could be easily slid up or down the edge) the possibility of their taking any compressive end load was eliminated. The inserted strip of sheet satisfied one of the two conditions of simple support; viz. it maintained the plate edge straight. To satisfy the second condition of simple support (viz. at each edge there must exist a zero bending moment) the supports were allowed to rotate freely at

the plate edges. Finally, the support was extended up the free edge to a distance of one- quarter of an inch from the loaded ends. This ensured small deflections of the edge normal to the plane of the plate.

2. 3 Auxillary Test Specimens Test #1

A pack specimen consisting of four channel columns, four

inches in length, was tested in compression (Fig. 6). Thus a compressive stress- strain curve was obtained (Fig. 7) for the stringer material, in which buckling deformations did not influence the strain readings.

( 4)

A plate specimen of sheet material was tested in tension, using both strain gauges and mechanical tensometers to measure deform-ation. A stress- strain curve for the bare sheet (the core of the alclad sheet) was obtained and it is shown plotted in Figure 8.

Eight channel columns ranging from four inches to eighteen inches in length, having their ends cast in Wood's metal, were tested flat-ended in compression, (Fig. 5, 6). The right flange of each stringer (as seen in Fig. 6) was provided with an edge support roughly simulating that of the attached sheet so that compression tests on the specimens yielded a column curve which would be applicable to the actual panels. The variation in the stringer length was sufficient to cover only the short column range, (see Table #2).

The aspect-ratio of the centre bay remained constant at

a

=

3 and the two edge bays had an aspect-ratio of 3. 6. Two stiffeners were Rsed on each panel at a distance of six inches apart, and five inches from the unloaded edges. A summary of the panel specifications and parameters is given in Table #3. Table #1 summarizes the panel parameters for Test

#2, (see also Table #4).

2. 4 Test Apparatus and Testing Procedure Tests #1, 2

All column and panel specimens were tested in a standard 60,000 pound Tinius-Olson, four screw, tension-compression machine (Fig. 2).

At the beginning of the test on each panel, load increments of two hundred poimds were applied up to some point well below the calcu-lated buckling load. Testing continued only when a fairly uniform strain distribution was recorded by the strain gauges across the panel. This state was obtained reasonably weU af ter successive machining of the panel ends. Mter buckling, the load increments were increased to five hundred poimds, and testing continued till failure occurred.

The stringer strains were obtained during Test #2 from

observations of the fringe patterns on the plastic flanges, using conventional calculation methods (Ref. 9). However, the mid-bay centre-line strain was determined by a strain gauge.

2. 5 Experimental Measurements Test #1

Mter rolling each sheet, the radius of curvature was measur-ed every inch in the longitudinal direction for three stations across the

panel (viz. at the mid~bay centre-line and at each stringer centre-line). The edge bay's curvature was obtained at the bay's centre-line. From these readings, it was found that the mid-bay's curvature varied very.little (viz. as little as + 1/2 in.), whereas the edge bay's radius varied as much as

+

1 inch. Averaging these readings yielded a mean radius of curvature for

each bay. ,

The straightness of the panels in the stringer direction was estiniated by means ,of a steel straight- edge and 'feeIer gauges'. 1t was found that the deviations were allless than 0.005 inches and it was consid-ered that these surface irregularities were acceptable.

The panel and stringer strains were measured by SR-4 (type

A-3) and NRC strain gauges, having gauge factors of 2.06 and 2.08

respect-ively. It was found convenient to use the longer NRC gauges for the mid-bay centre-line strain, and the SR-4 gauges on the stringers. In order to obtain the mean compressive stringer strain not affected by stress concentrations around the screw holes, four screws were omitted. The gauges were then attached, one to the outer flange and the second diametrically opposite on the sheet. A monolithic structure was maintained almost up to failure because of the bonding. The strain gauges were cemented to the sheet material with Eastman 910 cement and then connected in a four-arm bridge circuit. This provided a temperature- compensated mean compressive strain proportional to the sum of the strains in the two active gauges at any one station on the panel. An SR-4 strain indicator (type N) completed the circuit.Since lead wires were short, no corrections were made for lead-wire resistance.

Huggenburger tensometers were mounted on both of the edge bays (two tensometers at each edge) to measure the strain at the edge support. They were removed when the strains encountered after buckling were such às to deflect the tensometer's indicator off-scale.

Ame's dial gauges were arranged to measure stringer twist-ing and bendtwist-ing deflections durtwist-ing the compression tests.

Test #2

Using a circular reflection polariscope (Ref. 9) (i. e.: a circular polariscope in conjunction with specimens having a reflective sur-face on their back sur-face, thus causing the incident polarized light to be reflected back to the pola'riscope)" and calibration data for CR- 39 photo-elastic plastic, the strain distribution in the stringers could be determined throughout the' complete loading range of the panels. Two dimensional plane stress equations were employed to calculate the actual st~ain in the loading direction. The panel centre-line strain was observed by means of

a strain gauge, set up in the same manner as outlined in Test #1.

( 6)

IIl. DISCpSSION OF EXPERIMENTAL RESULTS 3.1 Stress":"StraIn Curves

Test #1

The compressive stress- strain curve for the stringer mater-ial, using a pack specimen, is shown in Fi~ure. 7. The mean value of E for the stringers was found to be "10.4 x 10 p. s. i.

The stress- strain curve for one of the eight columns tested in compression is also shown in Figure 7. It is of interest to note that because of the large bending deformations of the legs of the channel section, a rton-linear relationship was obtained. Thus an apparent value of E as low as 10 6 p. s. i. was observed. However, a 2% offset in strain, drawn parallel to the best straight line through the test points, produced a yield point stress in close agreement with that given by the pack specimen, viz.

a-

yp ;= 40,000 p. s. i. compared to 38,000 p. s. i. given by the 4" stringer. Atension stress- strain curve for the sheet material, using strain gauge data, is plotted in Figure 8. In computing the nominal buck-ling stress of the alclad sheet, the value of E for the core material was used. Thus no cladding reduction factor was required to modify the nominal buckling stress. The mean value of E yielded by tests was 10.4 x 10 6 p. s. i.

A summary of the mean compressive Young's modulus of elasticity obtained from the panel tests is given in Table #5. Emean was derived from the average values of E from the stress-strain curves of both stringers and the panel centre-line, (eg. Figure 9). Stress-strain curves for two of the panels are plotted in Figures 9 and 10. From Figure 9, the constant displacement of the centre-line strain from the stringer strain indicated that the centre of the panel was lower than the panel edges. Until the divergence of these stress- strain curves was small, in comparison with the buckling strain, the panel ends were not accepted as being suffici-ently parallel, and further machining was neceS:sitated.

3. 2 Column Curves Test #1

Table #2 summarizes the failing loads and column parameters for those tested in compression. The choice of slenderness ratios (L/

(J)

.

was such that they feU in the short column range (Figure 11). The longer stringers failed by torsional instability (see Figure 6), and a calculation of this failure load for the 18" column, having its right flange laterally

supported against buckling, using the equation

<iëQ-

()[n=l. r&;T

+~

C )

.

I~

L~ ~

(Ref. 1), yielded a value of 7650 Ibs. This value was based on the as~ump-tion that the stringer rotated about the juncas~ump-tion of the attached flange and

(7)

, ,

the w~b. The measured value of the corresponding column was 7500 Ibs.

Also shown in Figure 11, is a semi- empirical curve given

bythe above equation for the range 0~crco~<::ryp'/2. For <ïyp/2!S~0~<ryp' the equation

~o

=

<::I

_{yp -}

cryp~

14

Q"""BT is employeä. Very good agreement with experimental results is obtained. However, it was necessary to deter-mine an end-fixity coefficient from one of the test results.

çr-cc2

A Johnston's parabola q-co

= <J

_{cc - - -} was calculated

4'lE2

for the shor:t column range, first assuming the column was restricted to fail about an ax·is passing through the point of attachment parallel to the web, having a crushing strength equal to that calculated by the methods of Referencè 1. A second J ohnston I s parabola was then calculated, assuming the column would fail about its least axis of inertia, and further assuming that the crushing strength was equal to the yield point stress of the column' s material. Both parabolas were based on the test result at Lip = 23. 4 (~o)' which allowed an end-fixity coefficient to be evaluated. The former para-bola had C equal to 2.6. lt was the second parabola which produced the best agreement with experiment. Consequently, a simple "rule of thumb" in calculating the failure load for torsionally weak short columns appears to be to use the latter parabola based on a test specimen having the appro-priate boundary conditions. Such a calculation is much easier to perform than a torsional instability analysis.

3. 3 Initial Buckling Test #1

Buckling was always detected by a sudden drop in the panel centre-line strain, accompanied by a sudden change in the stringer strains (Figures 9, 10). Up to buckling, deformation of all panels varied linearly with the applied load. Consequently, any change of slope indicated buckling.

A summary of initial buckling coefficients and critical strains of the central bay of the panels is given in Table #6. The instability coeffi-cient was computed for the mid-bay, along with the curvature parameter, and both are shown plotted in Figure 12. Linear theory (Ref. 4) along with Wenzek's empirical equation were also used to derive values of Kc. A correction term taking into account the effect of initial imperfections (Ref. 8) was used to modify the computed results of Kc and these are also plotted in Figure 12. The value of the imperfection is given by Wo = 0.005" (based on the largest deviation from flatness observed). From Figure 12, it is seen that Wenzekts equation predicted initial buckling very closely for the panels of Test #1 which behaved like clamped edge support as indicated below. Further, it is observed that for ZB:::»25, Kc vs. ZB tends toward

(8) ,~\

! \ /

From the results of flat panel tests, it was found that

Kc

=

7.98 (ZB

=

0) for the mid~bay. This correspond~ to a condition of

clamped edge support all around (Ref. 1). The tensometer data of the flat

plate edge~ bays (giving strains at the unloaded edges) indicated that buckling

occurred almost simultaneously with the mid~bay (see also the curved panel

data in Fig. 9). Assuming therefore, that

e.

cr for the edge~bay was the

same as that for the mid~bay, Kc was calculated and found to be 5.7. This

corresponds to a condition of simple support at the unloaded edges (Ref. 1).

Buckling was detected by a sudden change in stringer strain

(see Fig. 10), using photoelastic techniques to determine strain. Table #7

summarizes the observed buckling loads for both flat and curved panels

(for the mid-bay only). Having determined the ratio

of(~b)

eff. for each

panel, (summarized in TableJi~) the buckling instability coefficients were

computed and plotted versus(- o ) in Figure 13.

bD eff.

Using the data of References 6 and 7, for flat panels stiffened longitudinally with stringers possessing very little torsional stiffness, and

providing edge support to the mid-obay below the simple support condition,

good agreement with experiment is obtainefu (Fig. 13). Further, it confirms theory that an optimum value of ( bp) eff exists beyond which no

significant increase in buckling strength wil result. That is, for the flat

panels tested, having Ast

,.Ë...,

and

~

constant, the flexural stiffness

btsh t b °

(EI)

reaches an optimum value at the simple support condition which occurs at

(EI )

~

43. 5 as compared to 35 predicted by theory. The discrepancy arises from bD eff. the fact that at -

(EI)

~ 67. 5, the torsional stiffness of the

bD eff.

stringers was not negligible. Consequently this provided a buckling coeffi-cient Kc=- 4, whereas theory, which assumes a stringer support having

zero torsional stiffness, predicts Kc

=

4 beyond the optimum(

EI)

bD eff.

Curved panel data is also included in Figure 13. An

opti-mum value of stringer support is reached at

(EI\

~ 35. However, it is

bDI eff.

to be noted that roughly the same buckling strength is obtained if the

string-ers are removed. This occurs because of the increase in buckling strength

with increase in bay width (see Fig. 12). The values of Kc in Figure 13 for

(

bD eff

EI)

.

=

0 were deduced for an aspect ratio of three from experimental

results on panels without stringers having an aspect ratio of one, using the formula

12 (1 - Y 2)

Ci:

(b)2

=

2 c~

Th'at is, when the' stringers reached their Euler buckling load, their bending ene'E§Y was sufficient to cause premature buckling of the curved sheet. At (_),

=

0, the large value of Kc for the curved panel is thought

bD eff. '

to have arisen because of the absence of a stringer causing premature '

buckling of the attached sheet. Beyond the optimum value of ( EI ) ,no

" bD eff.

increase in buckling strength wiU arise as long as the ratio of

Ast remains constant (0.03 in the present test). However, as the stringer btsh

torsional rigidity is increased, the sheet will then be altered in its buck-ling strength because of the change in the edge constraint. Figure 12 shows

a definite incrèase in }Çc for an increase in stringer edge s~pport. The results With( EI)

~

67 for Test #2 are shown in Figure 12 for

compari-bD eff.

son with those of Test #1. The torsional rigidity of the stringers in the former test was small (almost zero), and since(EI ) was greater than

bD eff

the critical value (i. e. : 35), it is considered that the results approximate the simpie' support condition. It is of interest to note that at ZB~ 190, the panel tested had no stringers, but did have an edge support described in Section 2.2 which was shown to be simple support. '

The data shown indicate the effect of edge support on buck-ling and the results are consistent with a curve drawn parallel to Wenzek's but lowered by an amount A Kc ~ 5.

3.4 Effective Widths Test #1

Since only three strains were measured on the mid- bay, (viz. two stringer strains and the panel centre-line strain) it was necess-ary to make some assumption concerning the strain distribution across the panel in the post-buckling regime. Sechier and Dunn (Ref. 1) suggest a cosine variation of strain across the panel af ter it has buckled, and the results of Reference 2 tend to support this (Fig. 14). The strain di stri-bution for panel #4 is shown in Figure 16, in which it is assuméd that the strain at the unloaded edges is the same as that of the nearest stringer. Figure 9 shows that this is areasonabie assumption, The overall validity of the cosine variation was confirmed by integrating' across 'the panel (Fig. 16) and comparing the computed load with the actualload (see Table #8) .

Effective width curves are plotted assuming the above co-sine variation, in Figure 15. Comparison is made with the calculations of Sechier and Dunn and Wenzek, both of whbm assume an edgesupport lying somewhere between the simple and clamped condition.

(10)

In plqtting Figure 15, the following assumptions were made: (1) The ,edge· strain (at the unloaded' edg~s) is closely approximated by the stringer strain.

(2) Af ter buckling, some portion of the sheet enters the yield point region. Hence the effective width curves must be corrected for inelastic behaviour (Ref. 1).

The effective widthformula given by Reference 1 is: We

b

.

(f.c.r.

)

(E.c.r-,)

11 (

z's"t.)

==

0·25 I

+ __

-

I fof'

1'1

==

0' 3'7 ~

~5t ~ st. { c ; ; . ' J P ·

References 4 and 5 propose an effective width equation formulated by Wenzek:

1"or

f.'Cf.c~.

(5

0·3

~2 )(~r

'

2.

~e

=

1

2.'4e.

Rt

(S-+O.3

b2.)_

b

f

5+0.'3

:t2.]

b - R"t

R.

1-nr\d

~

'("

ê..>

&.f"

-

2-~

0

(~ ~~

) 1/2.

~ ~2.

It was found that the experimentally derived effective widths followed the predicted trend of SechIer and Dunn and Wenzek. However, since the edge support of the mid- bay approached the clamped condition, it was to be expected that the predicted values were rather conservative.

At any given stringer strain, the effective width for a curved panel was larger than that for a corresponding flat panel. For example, taking

S

t equal to 800 in.

lin.,

the following effective widths were

"measur:d": R

=

45 in. R

=

32. 1 in. R

=

18. 5 in. We

=

2.08 in. We

=

2.20 in. We

=

3.0 in.

The increased effective widths were caused by the increase in buckling strength of the more highly curved panels. In turn, the column properties of the stiffener:s were altered according to the panel' s curvature. This effect caused a 20% net gain in ultimate strength for the curved panel (R '" 13 in.) over a corresponding flat panel (see Section. 3. 5).

No effective widths were measured in Test #2. 3. 5 Ultimate Strength .

Test #1

"

At failure, the maximum load indicator on the test machine remained stationary at the last load which the panel absorbed with increas-ing strain.

( 11)

Failure of all panels occurred by torsional instability of the attached stringers, aided somewhat by premature tearing of the stringer flange from the sheet at the position where. the screws were omitted. Although tearing was present in all panels, it occurred at different stages of loading, depending on the number of sorews missing (see Table #3). That is, for panels number 4, 6, 7 and 8, tearing occurred at two to three thousand puimds prior to failure, whereas for panels 3 and 5, tearing was almost co-incident with failure. It is concluded now that had smaller strairi gauges been used with only two screws missing, tearing would have been'" delayed until af ter torsional instability.

The load decrement due to tearing may be approximated by the difference in ultimate loads of panels 5 and 6 (TabIe #3). Thus, as a first approximation,· an additional load of 3500 pounds may be assumed acting on panels 4, 6, 7 and 8 at failure. As aresult, the predicted ulti-mate loads must necessarily be conservative.

The Ame I s dial gauges which were arranged to measure .any

twisting and bending of the stringers, showed that negligible deflection

occurred prior to failure. However, at failure, twisting was sudden and catastrophic.

Figure 17' presents a plot of (Pcurved ) versus the

. Pflat ult.

curvature ratio ZB for the panels tested, along with data taken from

Reference 2'. At constant

bit

ratio, the ultimate strength of a curved panel was found to increase with respect to a flat panel, with increasing curva-ture. The data of Test #1 was such

tha~

PC) varied parabolically with

ZB' " \Pf ult.

The data of Reference 2 when collected with the results of Tests #1 and 2. were plotted in Figure 18 as(Pc ) versus b/R for

(

EI) Pf ult.

constant bD eff. values. It was found that for panels having very stiff stringers, the sheet thickness did not effect failure to any degree,' as shown by FMure 18. "It can be seen that the two curves for b~ 142 ,70

having an ( - ) ratio of 250 collapsed to' a single curve Jhen plotted bD eff.

against b/R. " " .

It is not so obvious, however, that the failing load should be independent of the plate thickness when the ratio (EI) is·less than

bD eff.

the critical value which ensures simple support. To verify this result, three panels were constructed (see Table #6, 7, 8) having an( EI) := 18

bD eff. and a

bit =

72. Except for the radius of curvature, all three panels were identical in geomtry and physical properties. The curve joining these three values of

(~)

, passed through b/R := O. 18 at a point which was

Pf ult.

close to the data of Test #2 appropriate to b/R = 0.18 and

bit

:= 150.

( 12)

From Figure 18, an envelope can be formed passing through (

EI) = 0 and(EI) = 00 (viz. at p. c = 1). It should be possible to

bD eff. bD eff. Pf

draw a universal set of curves for any(EI ) lying between the above bD eff. (R)

boundary curves from which ultimate load ratios ~ may be calculated. Pf ult.

Since the flat panel ultimate load can be calculated rather accurately (Ref. 1) this permits an easy estimation of the curved panel' s ultimate strength. However, more test data is required before such a calculation could be performed with confidence.

Calculations have been made of the ultimate strength of both flat and curved panels, based on the approach outlined in SechIer and Dunn (Ref. 1), but using the experimental column curve to determine the string-er strength. A summary of the computed loads and the observed loads is contained in Table #9. From these results, the calculated loads are

consistently higher by 30/0, Since the actual ultimate loads were a result of stringer instability and tearing, it is clear that had no tearing occurred, the failure loads would have been larger.

Test #2

From colour photographs (Fig. 19) taken of the photoelastic patterns present in the stringers at failure, it is clear that the mode of failure was by Euler-type bending. The effective length of the stringers was approximately measured by noting the two points of contra-flexure on each stringer which co-incided closely to the regions of zero stress. It was observed that LO~ L/2, or the end-fixity co-efficient of the stringer

ends was roughly equal to 4 (clamped-end condition). ,

c71'2 EI

The Euler failing load, given by P E

=

L2 ' is plotted versus Pult. in Figure 20. As the stringer stiffness becomes sufficiently large to produce a condition of simple support along the edges of the mid-bay, the failure of the panel is dictated by the column failure of the string-ers. That is, the mode of failure is primarily bending of the sheet about some flexural axis fixed in the stringer cross- section, and any change in P ult. is due to the stringer. The intercept on the PuIt. axis of Figure 20

may be interpreted as the load supported by the buckled sheet. On the other hand, for low EI where the stringer- stiffness is small compared to the plate's stiffness, it appears that the stringers tend to bend about some axis in the plate' smiddIe surface, and the stringers behave less like isolated columns.

This is further illustrated in Figure 21 where P c/Pf is plotted against(EI ) for b/R

=

0.18. There is a sharp change in slope

(

bD eff.

near EI)

=

67 where the mode of failure changes. An analysis of this bD eff.

(13)

Table #7 summarizes the buckling and failing loads of Test #2. A short note contained in Appendix A explains the significance of

(:~)

eff. .

IV. CONCLUSIONS

(1) The theory of S-eide and Stein predicts the effect of the ratio of plate-to- stringer stiffness on the initial buckling stress for flat panels very

accurately. In particular, the existence of an

oPtimum(~)

is bD eff.

confirmed. That is, increasing the stringer flexural st ffness beyond the optimum results in no significant buckling strength benefits.

(2) The experimentally derived effective widths followed the predicted trend of Sechler-Dunn and Wenzek. However, the predicted values were very conservative, as was also found in Reference 2.

(3) Two semiempirical formulae were found to represent the failure of a channel section stringer supported along one flange, (see Section 3.2). The use of these formulae in conjunction with the method of Sechler-Dunn allow-ed a reasonable estimate of the failure load of a curvallow-ed panel to be made.

(4) The ratio of the ultimate strength of a curved panel to that of a flat panel was found to increase with the ratio of panel width to radius of curva-ture, the function being independent of bit. This increase was found for

all values of stiffener to plate stiffness ratios. At constant b/R, the

ulti-mate load ratio decreases towards unity as the ratio of ( EI ) increases bD eff.

1. SechIer, E. E. Dunn, L. G. 2. Soderquist, A. B. T. 3. Gerard, G. 4. Gerard, G. Becker, H. 5. Ramberg, W. Levy, S. Fienup, K. L. 6. Seide, P. Stein, M. 7. Seide, P. 8. Peterson, J. P. Whitley, R. O. 9. Frocht (14) REFERENCES

Airplane Structural Analysis and Design, 1942

Experimental Investigation of Stability and Post-buckling Behaviour of Stiffened Curved Plates, UTIA TN 41, 1960

Journalof the Aeronautical Sciences, Vol. 13, Oct. 1946

Handbook of Structural Stability Part I, Buckling of Flat Plates Part 111, Buckling of Curved Plates and Shells, NACA TN 3781, 3783, 1957

Effect of Curvature on the Strength ofAxially Loaded Sheet Stringer Panels, NACA TN 944

Compressive Buckling of Simply Supported Plates with Longitudinal Stiffeners, N ACA TN 1825, 1949

The Effect of Longitudinal Stiffeners Located on One Side of a Plate on the Compressive Buckling Stress of the Plate Stiffener Combina-tion, NACA TN 2873, 1953

Local Buckling of Longitudinally Stiffened Curved Plates, NACA TN D-750

( 15)

APPENDIX A

A Note_ on (EI) (Ref. 7) bD eff.

In solutions presented prior to Reference 7, the buckling instability phenomena was treated only for stiffened plates for which the centre of gravity of each stringer cross- section lay in the middle surface of the plate.

However, if in practice the stiffeners are very large com-pared to the plate, (viz. for large

:~),

the attached plate has little effect on the bending of each stiffener about its own centroidal axes. On the other hand, for smal! stringers attached to strong plates (viz. for small EI) it is the plate which governs the bending action at buckling. As a

resu~P,

the effective moment of inertia of a stringer varies between that moment of inertia taken about the centre of gravity of the stringer cross- section, and that moment of inertia taken about the plate'smiddle surface.

Tables 1 and 4 summarize the calculations necessary to cornpute(

:~)

eff.' which were originally outlined in References 6 and 7.

(16)

APPENDIX B

A Simple Analysis of the Ratio ( Pc) Pf ult.

Assume two panels, one curved and one flat, are identical in all respects, viz. geometrically similar, and having the same material properties.

Flat Panel Analysis

Assume the ultimate load of a flat panel is given by

n

m

~~

Puit,

==

2

<rS~

Ast

+

L.

t

S

\k.

a.x.

I I _ bIl.

(1)

where

n

=

the number of stringers,

m

=

the number of bays. Case A:

The strin'gers are assumed to be negligible load carriers but do in fact provide some edge support to the panels.

bil.

Pulb

~

ft

5

(f" ..

dx

(2)

-~h

Assuming ~is given by the following relationship (Ref. 1)

~c =

1[

(~e

-\-<k(') -

(~e

-

~(.(")

'-OS

2:X

]

then equation 2 becomes,

where

bt

<O$e (

2

2.

<r~e. \Ne

t

( 3)

Since the stringers are assurne d to be negligible load carriers, then the sheet material's yield point stress is the maximum stress allowable on the effective widths at failure.

4'0

\f

~ ~

(q"''f p )

5~ee± ~t

el"'I c:,J o • 0

=-

AI

<!<.f'"

+

BI

I

B

=

\f

_yp

bt...

or equation 3 may be re-written in the form

f>

vit. . Çf'IP'

t (

4

We.

where

t

,CI;f' ~d.S" are constants and

We

is some effective width which is a function of ~c.t""

o-yp

.

where

A

B

• 40

Puit

~

A

We

+

B

=

=

4

t

<Jyp. '"

constant '2. t

<::ryp'

~constant

for the unloaded- edge support condition remaining fixed. Curved Panel Analysis:

Assume the curved panel' sextra load carrying ability is due solely to its curvature . . (This ignores any support of the " el astic foundation" type.)

If it is further assumed that the variation in effective

widths ('JIJe) at failure of the curved panel and the flat panel is smal! (Ref. 1 and also Fig. 13), then the extra load must be carried by the buckled

curved portion of the panel. That is,

(b-

2.\Net)

must carry a stress

or

( 18)

Consequently, for a given panel geometr.y ,<k:~c:. is a ~onstant.

•

0 0 Puit.

~ \fyp.

t (

4

We

+

2.

Wease)

+

(b -

l.

'We ) t

<Îo--.

+

2.

tG'êtl (

be~\).e-Co

c

~

We&Je

'

-

We )

,.

Pvl"l l3r

c..

\"-Ie

+

D

where

C

= .

4-1:

:

( <:S""

p, -:

<kc

c ) constant 'V constant

D

=

~yp(l.tWe~)-

(J""<'I'ct(2.

~e

-

b

Consequently the ratio of. - 2.

beJ<ke)

Pult. (curved)

Pult. (flat)

CWe

+

D

,

AVJe+B

for panels having very small weak stringers. Case B:

Panels having very strong stringers which are able to support a great proportlon of the ultimate load may now be analyzed very easily.

" Pf ult.

'. ',.

T~e

main effectto the ratio of {PC ) is that of adding a very large constant to both the numerator an denominator

i. e.

(Pc)

~

C

VJe.+

D

+

K

_J<

_~

_Gyr

_As+

Y\

P.ç. \)

L. T,"'-

_A

_B

4-1<.

'<.te.

+

where

n

=

number of stringers

LiMat

.l+x.

.1..

C. f. ~

X ... CO

W\+~

where

i

and

rn

are large constants.

Thus it is concluded that as the stringers are increased in strength, th ere is a limiting value of support given to both a curved and flat panel, and as aresult, the effect of curvature in increasing Pc above Pf is dirtünished. Thus a limiting case of P c~ P f is reached for panels having very strong stringers. ~ the other hand, for panels posses sing very weak stringers, the ratio (_C) approaches its largest value when

Example: Case A

.

, , Panel Set nos. 1- 4 Test #2

A = 6720

C

=

6280

B

=

1630 D = 3240 where

Y-.k

and ~ .... were calculated from data of Ref. 1.

<:.

Panel Set

We. (in. )

(

(~)ultlalc.

No. (Ref. 1) 1 0.474 1. 29 2 0.456 1. 30 3 0.426 1. 31( 5) 4 0.360 1. 35( 5) Example: Case B

(~

)measured

1. 19 1. 30 1. 28 1. 32

A panel set was taken from Ref. 2 having a b/t~142. The stringer was rectangular in shape, i.e.; 3/4" x 3/8". Whence,

A

=

1830 C

=

1405

B •

1330 D = 2940 Again,

We..

and <iè~ were calculated from Ref.

We

=

1. 41 ",

W~e.

c

=

1. 45" and (q-c )

~

1. The results yielded 1. 10 compared with the measured value of 1. 13; .

q-f

ult.

"

" ,

..

Tl\BLE 1

P-ANEL DETAILS-DF'TEST #2

Panel Stringer a' b' Ast Est 2st

Znq

EIst

S-et Shape (in. ) (in. ) _{(bt) sh E sh} (in. ) _bDsh

No. 1 tee 0.7 0.6 0.0303 0.531 1.164 11. 2 2 cruciform 11/16 0.8

_"

0.373

_"

'4.63 3 tee 0.25 1. 05

_"

0.330

_"

4.60 4 tee 0.10 1. 20

_"

0.210

_"

3.02 5 none 0 0 0 0

_"

0 6 tee 0.20 0.50 0.0348 0.259

_"

2.7

NOTE: Refer to References 6 and 7 for theory of calculations.

NOTE: T:EE SECTION I

...-b---1

T

I Q

1

3 " Plastic thickness

=

16

STRINGER GEOMETRY TEST #2

CRUClFORM SECTION I

t-r-

b

- - I

TI

CL

,

1

,(: ;) eff. (p.s.i.) EpI.

x 10- 5 -, 67.6 3.1 32.4

_"

26.3

_"

11. 3

_"

0

_"

18.0 11 CENTROlDAL AXES

T

Zst

k, __

1/ 0·010

-

I:\:) o

(21)

TABLE 2

COLUMN PARAMETERS

Length Cross- Iyy _Lip . Sheet Failing Failing

L (in.) Section (in. 4) Thickness Load Stress

Ast. (in 2) t(in. ) (Ibs. ) (p. s. i)

4.0 0.239 0.1375 5.37 0.071 10840 44400 5.9 0.238 0.1375 7.82 0.071 10340 43400 6. 6 0.1095 0.0·247 14.0 0.050 4200 38300 7.9 0.239 0.1375 10.39 0.071 10060 42200 9. 6 0.1095 0.0247 20.1 0.050 3800 34700 11. 9 0.238 0.1375 15.6 0.071 8830 37100 12.6 0.1095 0.0247 26.2 0.050 3100 28300 17.8 , 0.238 0.1375 23.4 0.071 7530 31600 5._{9(1 0.238 0.0247 7.82 0.071 10140 42600}

Plate R L No. (in. ) (in. )

1 (1) _{en 18} 2(2) _{en 18·} 3( 3) en 18 4(4) _{45 18} 5(3) _{32. 1 18} 6(4 ) _{30 18} 7( 4) _{18. 5 18} 8( 4) _{13.5 17.7} Ast

=

0.238 in. 2

~ot

=

1. 12 in. 2 .Y'

=

0.03 (22) TABLE 3

PANEL PARAMETERS OF TEST #1

b t Rit bit

z=

_Per

(in. ) (in. )

_b~

(Ibs. )

Rt x 10-3 6 0.04 en 150 0 4 6 0.04 en 150 0 4 6 0.04 en 150 0 4 6 0.04 1125 150 19. 1 4.5 6 0.04 . 803 150 26.8 6 6 0.04 750 150 28.7 9 6 0.04 463 150 46.5 12 6 0.04 337 150 63. 6 14.3

(1) - No bonding; all serews present.

e,

erG x 10 (in. lin.)

-320 320 488 533 578 900 1140

(2) - Bonding only; tested past buekling, then unloaded, serews added, then tested to failure. p_{ult . is not reliable.}

(3) - Bonding, plus serews, three missing per stringer. (4) - Bonding, plus serews, four missing per stringer.

. P ult". (Ibs. ) x 10- 3 22.5 23. 3 23.75' 20.75 24.75 21. 3 22.75 . 24.75

. Panel Set No. 1 2 3 4 5 6(1) (23) TABLE 4

SUMMAAY OF PANEL PARAMETERS TEST # 2

-

.

(~~)

eff.

Radius of Curvature

. flat _curved flat curved

67.6 67.6 00 33 32.4 32.4 00 31 26.3 23.6 00 30 11. 3 13. 1 00 33 0 0 00 32 18 18 00 14. 3, 18.2

(24) - _

TABLE 5

EXPERIMENTALLY OBSERVED MODULUS OF '-ELASTTCITY OF PANEL'S CENTRE BAY

-.... (1)

-Panel Radius E

No. R (in. ) _{p.s.i. x} 10- 6

: .3 <X> 10.8 4 45.0 10.1 5 32.1 10;0 6 30.0 10.4 7 18.5 10.0 8 13.5 10.7

-

_{E c}\-.+ E Lst.+ ER~tl E

₌

(1) 3

--Emean

=

2.E

~ 10.3 x 10 6 p.s.i.

Plate No. 1 2 3 4 5 6 7 8 (25) TABLE 6

BUCKL1NG INSTABILITY COEFFICIENT

AND CRITICAL STRAlN SUMMARY

~=

t

cr x 10 6 b2

(l_J)4)'/2.

Exp't. (in./in. ) Rt . 0

-0 320 0 320 19. 1 488 26.8 533 28.7 578 46.5 900 63.6 1140 Kc=

~,"12

(l -

.v2~b)2

".~ E t

-7.98 7.98 12.2 13.3 14.4 22.4 28.4

Panel Set No. 1 2 3 4 5(2) 6(3) (26) TABLE 7

SUMMARY OF BUCKLING AND FAILING LOADS

FOR PANELS OF TEST #2

Pcr PuIt

flat curved flat curved

(lb,s. ) (lbs. ) 1300 2200 (4380)(5) 6580 5565 1000 2160 3850 4980 910 2000 3820 4900 460 700 2530 3860 150 2300 1000 2346 \

11~0

J

-{

38-1~

~

4790 13300

t

5060 \

(1) Based on( EI,) for curved panel.

bD effe

( :ftu

( 4) 1. 185 1. 295 1. 284 1. 32(1) 2.35 {1.-33

(2) Panel Aspect Ratio ~ 1. 1. and no stringers, all other pan~ls had

an aspect ratio of ~ 3.

(3) Three panels were constructed in this set. Each panel had a

total developed width of 7.45", and a length of 8.3".

(4)(

Pc} Pf ult.

was computed for a Q ~ O. 18

R

(5) The flat panel which failed at 4380 lbs. had a smaller end fixity

than the corresponding curved panel. Hence, another flat panel of set 1 was tested and restricted to fail with a higher end fixity.

ExampIe:

(27)

TABLE 8

COMPARIsaN OF CaMPUTED LOAD (FIG. 15) WITH ACTUAL LOAD APPLIED Ta PANEL

Applied Load Computed Load

(Ibs. ) (Ibs. ) 5000 5080 6000 6050 7000 7020 8000 8050 9000 9050 P _{app Ie} 1· d

=

6000 Ibs.

At

ot (under curve):::: 7537 in. 2 (Fig. 15)

8

L ~ 493.,M--in./in.

E.

R . ~ 687~. lin.

st. st.

Ptot~

10.4 x 10 6 (0.04 x 7537 + 0.238 x 1180 x 10-6)

=

6050 (Ibs.)

where Ast.

=

0.238 in. 2 t sb. = O. 04 in.

I·

Panel No. 4 6 7 8 Flat (28) TABLE 9 COMPARISON OF CALCULATED AND OBSERVED F AlLING LOADS

Puit.

(l)

observed (Ibs. ) 20750 21300 22750 24750 Pult. calc'd (Ibs. ) 21400 21930 23155 23900 20500(2)

(1) Failure was premature due to tearing of stringer from plate. (2) Based on extrapolated value using equation of Kanemitsu and

1" .t.. '4 T 3" T 4" oL

T

1 .... ---14~"---'1 .. CAST END'S CHA1;NEL COLUMN c:::::::::~--1 L)

4"

C_(WOOD'S META

1

1.94" ""if

·

.

.

#6

MACHINE 8,CREW • AT 0.60" PITCH

'r-

~

·

I I 1 I _I • _I I

.

0 •

•

.... 1

.

I~ • •

•

•

•

•

•

I ... ..,..-S 18" - - 0.04011 o • G7 5"

.t;!~=~~=e=====:::'--"'-:r

CREW

CENTRE-LINE-!} ...

~

0.161'" 0.546" ,sTRINGû: DETAILS I - - 5" _{6" - -...}

...--41"...,.

41

~"

... , , - - - 16" ---1~ EOOE SUPPORTS

STEEL ~~GLE SECTIONS

STRINGEh MATEl1IAL Q.!; .• A362

CURVED PANEL CONSTRUCTION SHEET MATERIAL 24S-T3 ALCLAD

ALUMINUM

TABLE OF PANEL PAf\AiwtETERS

Panel ti m n

No. (IN) ( IN) (IN)

1 1.55 ~' .54 2 1.55 2.54 3 1.55 2.?4 4 45 1 ')Ç) .cu 2.54 5 32.1 1.24 2.:;4 6 30 1.24 2.:.4 7 1~3.5 1.22 2.51; Q _{13.5 1.21 2.5 4} u FIGURE 1

.,.

S IFFENED CU VED PLATE IN COMPRESSION o.1ACHINl!.

" LOADI G PLATEN ChST END SUPPOrlT (WOOD'S ~TAL)

'

- . '

/

\.EIGHI "G HEAD

~

FIGURE 2

f

I

I I

0.70"

1.

I

T

0.60"

I

1

L

-

.

1.0

~

IJ-o

30"

I I

,

....

,

"

~ I I

,

I '-...,

'-I I I I METAL/ 1 WOOD'S

c;

_I I END CASTING

,

'::

I -SThlNGEh I

I

I~' _i~lAChI}.jE SCrtEW - I

tJ=

0.138"

I

..

,

_I ", I

I-5"

(EDGE) PANEL CONSTRUCrION FIGURE 3 WOOD'S î'JETAL PANEL

/

"

(

1'-,

'-", 0.50

.

J

c

70"

"

1

0 ,-( _\ I

t

''''

_I

.25

11 ,

'0

I I .P ANEL }!;DGE _I (BA~) I 1

0

1 I I 1

0

,

;---..

-1

A~GLE IftON EDGE

SUPPORTS

SnEE T H~ SEK T

t= 0.04"

~-A~~GLE IhON EpGE

WOOD' S rIllE TAL SUP~OnTS

fOUNDED ED(.ES

EUGE SUPPOrtT c.:O~S'I'hUc.:tION

3TIFr~NED CHANNEL COLUMN IN CO~WRE53ION MACHINE -~ .

,

_~..

"

_{t .} I~G PUTEN .' -,,:.-- CHANNEL COLU1>1N BUCKIZD F LAl-. GE

CAST END SUPPOrtT

F' A ... - (wOOD'S I-1ETAL)

•

•

CHANNEL

COLU~illS

TESTED

IN

AXIAL

00MPRESSION

ITH RIGHT FLANGE

LATERALLY

SUPPORTED

AGAINST

BUCKLING

~_ ....

, 'PACK I

bPECltv!EN CONSISTING OF

FÓUR

CHANNEL

00LUMNS

·

(L

=

4")

"'

'0 r-l x-• H

·

Cf.l • Çl., Cf.l Cf.l II-l ~ 8 ti: ~. ~ Cf.l rIl rx:1 ~ p.... ~ 0 0 H <I! H ~ .

COMPRESSIVE TEST OF CHANNEL STRINGER SECTIONS SION TEST ON 24S-T3 ALUMINUM CLAD C:iiCEir

AXI Est. Ast. = 0.238 in. A pack

=

4 Ast 0.952 PACK 4 IN". STRINGER 2

%

OFFSET 10.4 )(10 pslo X 10 X102 FIGURE 7 t=0.040 in. 2 A sh.

=

0 •. 09 in. 0 r-l X

·

H • Cf.l

·

p.... Cf.l Cf.l f:iI ~ 8 Cf.l f:iI :> H Cf.l Cf.l >rl ~ !l. :.E O· 0 ., ~~ . '-~ H

.

~

1IIIIIIIIIIIIIIJlIIIIIIIIIIIIII E core =1004 X100 psi.

AVERAGE STRAIN GAUGE READING IN/IN. X 104

... ' - ;

.

-o ~

3

lil > H Vl al

~

~

_o ·0 .-1 ~ _' H ·x I ~ O . i.it .H . .-1 Pi Pi ~

APPLIED LOAD

vs.

MEAN CO~WRESSIVE STRAIN

l' OR A STIF.r'ENED CURVED SHEET

CRITI~AL B CKLING LOAD

UNLOADED EDG E •

PANEL CENTHE LINE 0

R 18.5 (IN)

b 150

t

t .... 0.04 (IN)

A;:::1.12 (IN)2.

LEli'T AND RIGHT STRINGERSAV

MEAN AXIAL COMPRESSIVE STRAIN ( IN/IN»)C

\0

4 FIGURE 9

AF .PLIED LOAD va. 1::1 STHAIN FOR A CURVED SHEET

STIFFENED WITH PHOTOSTRESS STRINGERS

CRITICAL BUCKLING LOAD

R b t t A 33 (IN) 150 0.04 (IN) 0.65 (IN)

·I:!. STRAIN IN/IN)

><

102.

ct '!-H+t-t

rr--P"TF'P :-t!-H:j:j::pir'

m

i

f~-fr~

i

~

-"

IQ

i

---.

~

H dl , ~

-Jl Jl ~

,

-r:

8 Jl Ó ;Z; H ~ H ~ L<o Z ~ ~ ~ 0 ~ 1111111111111111:

YIELD POINT STRESS

<i"ee=(Jyp JOHNSTON PArtABOLA C

-=

2.6 SEMI-EMPIRICAL CURVE rTl ITl i JOHNSTON PArlABOLA

c

1.3 (jee eale.

EXPERIMENTAL TEb'fS RESULTS 0

COLUMN CURVE FOR CHANNEL SECTION

Length of Column

Radius of Gyration of Stringer

FIGURE 11 Ll... ' " LLLl..ll.LJ..L.ll.U..L.L t-t-H-+1-::r:t '-\ ·1-}-1 r::,:;:t

--t-

-m

t _-.

-

rt

_{tiJt - J -~}

_±!+::1:'

td

_{H~ ~':_ =t-H.+-_' _ -.}

~

~;: ::j-.' -' --:~l~

-- - --

::r

::j: - -nH-' _T -tJ-+ 1+ 4- -- - - - -- -- -~- :t- r

=m-I 10ft -- _.

-- ".!=- - ! _ + I I I - J

Ui: -- ]

~UCKLIN"G IN~'l'ABILITY COEH ICIENT VB •. CURVATURli: f<ATIO

-~~ -' j ---i REF. 4 -"REF. 8 w 2

.:..c..

.kt WEN'~K I

'

i-

j TEST

111

EXPERIMENT -- T_ b 150 TEST #2 t -CLAMPED _ ~IlvJ.PLE SUPPORT

CURVAT URE RATIO -b 2 _{(1 v2 )1/2} ZB Rt -' -_Ot --- -H-I+' - -- -- --- -FIGU RE 12 H- +

K -

<ïcr.

b2t c

_1("

_{2 D} ' j

!st

=0 bt Ast=O

bi

BUCKLING INSTABILITY COEFl'ICIENT VB •• (EI)

bD eff.

FOh FLAT AND CUHVED PLATES. (AR.~3)

__ • ~ •••• a-". SIMPLE SUPPORT THEORY -EXP'T. ~t bt 0.03 ~t=0.03 bt i+i b-150

(I~J

OUJ

opt.

FLAT PANEL ZB- 0

J

CURVED PANEL ZB=26 t THEORY (REF. 6,7)

(~~)

effe FIGURE 13 i LIH :-.-, H t

• ;2ó H

-3·0 1.5 1800 '&00

EFFECTIVE WIDTH

VS.

STRINGER STHAIN

WENZEK~

R=co'~

EXPERIMENT

o R= 00"

A R=~5"

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