Welded Continuous Frames and Their Components

Progress Report X

Local Buc~ling of Wide-Flang§ Shapes

by

Geerhard Haaijer and Bruno ThUrlimann

(Not for Publication)

This 'ltlork has been carried out as a· ,part of an investigation sponsoredjointly by the Welding Research Counciland the Department of the.Navy withfunds furnished by the following:

American Institute of Steel ConstructionAmerican Iron and Steel InstituteInstitute of Research, Lehigh UniversityColumn Research Council (Advisory)Office of Naval Research (Contract No.39303)Bureau of ShipsBureau of Yards and. Docks

Fritz Engineering LaboratoryDepartment of Civil Engineering & Mechanics

LehighUn1versityBethlehem, Pennsylvania

December, 1954

Fritz Laboratory Report No o 205Eo 5

205E.5

£'.A.!lbE QF QONIENIe.

~ page

I. Introduction '. 0 • • 0 • 1

II. Stress-Strain Relationships 0 • 0 0 2

III. General Stress-8tra1n Relations Appliedto Buckling of Plates e • • 0 4

rv o Application of Energy Method toDifferent Cases of Buckling of Uniformly

6'Compressed Plates 0 0' '. 0 0

~ ...

Vo Determination of 'G at Beginning of 'Strain-Hardening f~om Results of AngleTests 0' 0 0 0 • 0 • 10

VI. Tests on Wide Flange Sections 0 • • 11

VII. Summary of Results 0 • • 0 • 13..

VIII. Tentative Recommendations for theGeometry of Wide-Flange Shapes • • • i4

lX. Future Work 0 • • 0 0 0 15

X. Acknowledgments. 0 0' 0 0 , 16

XI. References • 0 • • • • 18

Appendix

Tables

Figures

'-

I. Introduction

In order to solve the problem of buckling 'of a plate,

the relationships petween additional stress and strains due to

the deflecti~n 'of the plate out of its plane have to be kno~m.

In the elastic range the assumption that the material

is isotropic and homogeneous leads to solutions which are in

very good agreement with test-results.

Several investigators have given theoretical solutions

for buckling in the plastic range based on different stress­

strain relationships. The discrepancies between the different

theories are basically due to these stress-strain re.lat~onships,

which by some authors are directly assumed and by others are

derived for materials with idealized behavior.

It is the purpose of the local buckling project at

Lehigh University to arrive ata solution of this problem which

to specify the dimensions of rolled shapes in such a way that

they can sustain sufficiently large deformations without occurence.,

of local buckling.

In the following theoretical considerations the stress­

s~rain relationships have been kept quite general. Comparison

of test results with the solutions obtained in this way will

give some indications of the value of the different variables

involved.

As long as no other information from direct tests

with regard to the stress-strain relationships is available, thi.s

seems to be the only way to arrive at a practical solution of

-2

the problem.

II. Stress~StrainRelat1Qns~

Consider a plate made out of a material exhibiting a

stress-strain curve in simple tension as shown in Figure 1.

The notation subsequently used is given in this same figure.

Take the center plane of the plate as the x-y

coordinate plane. Compressing the plate in the x-direction into

the plastic range up to a stress ~ may affect all the deformationo

properties of the material. Hence the tangent moduli Etxand

Ety in the x- and y-direction respectively are possibly

different. The same may hold for the Poisson's ratios Vx and

~ in the x- and y~direction. The shearing modulus Gt may also

be affected.

Call

vy

~-;;- ,= - E-·'y ty " . (1)

The assumption is made that no unloading occurs. Then

the relationships between the increments of strains as a function

of the increments of the stresses can be v~itten as

, Vx

Id€= ----- dcr + ,- dO'

Y E x E Ytx ty

Id'Y =-G d'T

xY t xy

(2 )

205E.5 -3

In words these relationships state thRt, ~Then uni­

axially compressed into the plastic range, the originally

isotropic material becomes orthogonallyan1sctropic.

For the c~se of unloading the material would again

behave elastically, but it is now generally agreed, that under

test conditions no unloading occurs ',rhen a plate starts to buckle.

In the Appendix a short review is given of the

derivations and assumptions with regard to the stress-strain

relations used in different theories of plate buckling. The

results are summarized in Table 1 giving the assumed or derived

Due to the heterogeneous yielding process of steel this

~. material presents an additional problem. A tynical stress-strain

.~,

curve is sho~m in Figure 2.

As already emphasized in Progress Report T(l) yielding

occurs in so-called slip bands and the strain II jumps" from Ey

to Est'. Therefore, as long as the average strain 1s in between

Ey

and Est: the material isnohhomogeneous. Tests have shown

that this yielding process is very erratic(l).

For the rotation of a plastic hinge it is in general

desirable that the average strain in the flanges of a VW shape

reaches the strain hardeniIigrange (E average ~ Es~).· Then the

material 1s again more or less homogeneous and only this case

\"ill be cons.idere'd in .the folJ,.o,:,ing deriva·t;tons.

Once'a satisfactory solution for the strain-hardening

range is obtained, solutions for the intermedia.te plastic range'

20%1.5 -4

.~

..

(E y <. Eav < €s~) can be derived in a 1:1ay outlined in Progress

Report T. (1)

It should be emphasized that for the above presented

general incremental stress-strain relationships, Etx , Ety , lIx ,

~y and Gt possibly depend on the loading path. Theoretically

the plate remains plane until buckling occurs. Therefore the

lo'ading path would be~ O'j varying from 0 to 0'0 ,,,ith O'y ="'xy = 0 ,;

However, ~ue to unavoidable initial imperfections, for the actual

loading path~ .C' =i: 0 and.,. :j: o. Direct tests on, for instance.,y x y

tube specimens under different combinations of loads (tension,

internal pressure, torque) could give the answer to this im­

portant prob1emo

III. General Str§§s-8t.rain .l\elations ApnlielL to Buckling

of Plates

When a plate is loaded by forces acting in its center

plane a state of stress may be reached such that besides the

plane position also a bent eoui1ibrium position becomes possible:

the plate buckles.

If the transition from the plane to the bent position

would occur under constant loads, unloading on the convex side of

the plate would 'be inevitable. However, under test conditions

btick1ing occurs 'with an increase of load such that no strain

reversal takes place.

Shanley proved th~t the same happens in the case of

axially'loaded columns. Agreement between his theory applied to

rectangular steel columns, which buckled in the strain-hardening

range,andtestresultswas shown in Progr~ss Report 8(2) •

205K.5 -5

Mx y

Consider again, a plate uniformly compressed in the x­

direction up to a stress ~ causing orthogonal anisotropy.

Calling the deflection perpendicular to the center plane w

the expres~ions for the bending and twisting moments become

M' = _ ~.~L ('d2W +v 'd~~_.)

x. I-v V 'dx 2y 'dy2x Y

~Jj = _ Ety J (~:~- +v ~~.. \)y I-v }I- \ 'dy 2 x 'dx 2

x y. I

'd 2w- - 2 Gt T ax'dy-

with I =t~/12

t = thiclmess of plate

The condition that the bent position is an equilibrium

position can be expressed by the following differential equation

(4)

..'

2H = v 0 + v 0 + 4G t 1y x x y

The derivation of these equations may be found in the

pertinent literature (3) • Only if M~" = Ox Dy ,an assurnpt':l:an made,

by Bleich(4), solutions of this differential equation can be

easily obtained.

The condition that both the plane and the bent position

are equilibrium positions can also be expressed in terms of work o

fhe additional work done by the external forces due to bending

of the plate must equal the change in strain energy of the plate •

205E.5 -6

This renders the following integral equation

: (dW)2 ',.It- (d2W')2 ('02W\2(7 t, Jf - dxdy = JJ .0, \ -2' + 0 ~2 JI +,0 dX ! x\ OX I Y dy .

l_ '

, '. (5)I )f 02w\!?J 2w\ (-a 2wy'l

+ ,(. 11 0 + 11 0 \ --, ( --)+ 4 G T', -- ! 1dxdy. Y x x Y \ CX 2/\dy 2/ t oxoy;!\ ',. , ,. _1

When external restraints are provided to the plate the

right-hand member of equation 5' has to be supplemented by

additional,terms expres sing the vork done by these restraints.

By assuming an appropriate deflection surface equation

5 gi,ves an approximate solution. The degree ,of approximation

depends on the correctness of the assumed' deflection surface. In

any case the result will be on the upper side.

rl. Annlication 9f Energy Ne~hoQ~.iQ_DifferentCas~s of :B}1cJ5:1ing

of Unifor~ly Compressed Plates

I. Rectangular plate with the loaded edges x =0 and x =~

hinged, the unloaded ~dge y =0 restrained against rotation

and the unloaded edge y =b!2free (Figure 3).

In their paper on buckling of outstanding flanges

Lundquist and Stowell(5) assumed the following deflection surface,

[

2y ~(2Y\ 2. ,- ~ - + B --- ,jW - t-, b \b /

Which is known to be good in the elastic range.

(2 Y\) '2y\ 'I /2y' 5Jl

+ a, b) + a2 (-b-) + a~(b) ,

at =. -1.C076

a 2 =+0.5076

a)= -0. 1023

.' nXSin Z~ (6)

205E.5 -7

The ratio B/A. depends:on the amount of restraint. In

the case ,of elastic restraint with .1/; = moment per unit length

required for a unit rotation

Substituting w +n the energy equation and performing the in­

tegrations gives

/2·· .. ('7,)2+ E 1,-) -'

tY'.b 7T

(2 )2 .pC .. +p2 (,- (vy Etx + v. E t y 1 \b~ 1-73+-Fc~+p2c ~.

+ 4 (. I-V.v IG/2

) '~+jiS·P'__:L\. y Ii> i /3+PC 1 +P2 (2 .

wi th

(8 )

(1 = 112+2/5 a1

+ 113 a 2 +217 a3 = 0.23E93

C = 115+1/3 a1 + 1/7 1a 2 + 23·1 + 114 1a 3 + a 1 a 2 1 + I 19 1a ~ + 2 a 1 a 3) +2 1 2

115 a 2 a3

+ 1111 0\32 0.04286=

C3

= 4 + 12 81

2 + 144/5 a/ + 400/7 a 2 + 12 a1

+ 16 a2

+3

20 a 3 + 36 a 1 a 2 + 48 a 1 a 3 + 80 a 2 a 3 0 . 567 I2

C.. =1+ 2 a 1 +3 3 2 +4 a 3 = 0.0984

C,=2/3+2 a1

+ 11516 a 12 + 148 2 '+113 (II a 3 +9 a1~2' +

1/7 126 a1

a3

+ 12 a22 1 + 4 a

2a

3+20/9 a/ = 0.0216

C6

=2(/+a 1 +a 2 +a3 ' = 0.7954

C7 = 4/3 +3 a 1 + 1'/5 19 a 12 + 16 a 2 '+ 113 112 a 1 a 2 + 10 a 3 ' +

1/7 (30 a1

a3

+ 16 a/+ 5 a 2 a 3 +25/9 a3

=0.17564

. A minimum value of 0'0 is obtained for 7, ~iven by. \ \

flO)

In the limiting cases "rhen the edge y = 0 is hinged or completely

fixed equation (8) reduces to

a. Edge y = 0 is hinged (p = 0) and ~ =L

(2tV G 7T

2Etx /b \2 ] .

CT - ----: "-----:\ - 1 + G •cr - b ' 12(1-1/ 1/ l\2Li . t

x Y .

For a long plate the first term can be neglected and

(11,)

(12)

b. Edge y = 0 is completely fixed (f:.! = (JJ )

The minimum value of (To is obtained vThen the half- .

wave length Z satisfiesrE--­

21 __ I 646 ,"I t x

b '~E~

CT c r

Then

= (2:)' r· 275 \~-:S- 0.506 (1/ Et

+ 1/ Et

I___~. ~ x _1.__ +12 (1-1/ 1/ I

x y.

1.371 (13)

2. Rectangular plate with the loaded edges hinged, the

(14)

For this case the following~ ~

i I Y2) 7TYJ 7TXW = j 87T1---2 -1/4 + IA + B.I cos- sin -::;-

L \d . d ~

unloaded edges having equal restraint against rotation (Figure 4)(6)

deflection surface is used

the ratio B/A depending on the amount of restraint. For

elas,tic restrCiints with ,¢::: moment per unit length required for

a unit rotation

, 1.j.,dP = BfA =­

20 y(15')

Then

1/4 + (C1 +3!._1T2!~~p2~2

1/ 4 + /1C 1 + f:/ C2 t

-9

with

1/4 +l,C 1 + p 2CIf

I 14 + I,C + I~, 2C +. fJ 1 f-' 2

(16)

C1

= 1/2 - 4/1T 2

C = 1T 2 /60 + 1/4 - 4/1T 22

2C3 =1/4 - 2/1T

C =5/12 - 4/1T2

'f .

In the limiting CEl.ses, 1,vhen the unloaded edges Y = ± d/2

are hinged or completely fixed the minimum values of CTo are

with

a.

CT c r

d

""---'fIE\. I t l(\!---\j Et Y

(18 )

-10

"tiith

b.

O"cr

Edges y= ± d /2 are completely fixed ('p= (JJ )

7T2(t)2 ~~554 \I~-:r';~-'+ 1.237 (vyE tx +vxE ty )

= ~ -; L=' '-I-V'xVy

+-,!

---'

• (20)

(21)

.1

v. Dete!ID1nation of Gt at the Beginning of StraiD-Har~inp'

from Results of Angle ~ests

When trying to apply any of the above derived ex­

pressions for buckling stresses, the immediate difficulty arises

that only the values of Etx and V x are knO'WIl. Etx can be obtained

from coupon tests and V x ::: 0.5 for the strain-hardening range

(incompressible material). Fortunately, it is possible to obtain

a value of Gt from the results of angle tests.

Figures 4 and 5 of Progress Report T(l) show that angle

specimens A-31, A-32 and A-33 failed due to torsional buckling

at about the point ,of strain-hardening.

The critical stress for torsional buckling of angles

0" = 35 ksi

1c r

b v ::: V ::: 0.5 Gt s = 2,580 ksix y

- = 8.82t

r v =0.5,v y ::: 1.0 Gts =2,510 ks i2l j

x

- =2.74b

205E.'

b. As-delivered material (Test A-33)

,..11

(J" :: 45 ksic r

b 11 = ,l-'y = 0.5 G = 3,270 ksit s

:: 8.7 x-2t

LL::11 :: 0.5,1I y 1.0 G, :: 3,210 ks i

2.6b x t.

b

From the general expressions summarized in Table I

numerical values of Etx, Ety ' Gt, 11 x and 1Iyare computed for the

beginning of the strain-hardening range taking

E= 900 ks is t ,

E = 30.000 ksi

and '1-' = 0.3

oJ:" y:: 0.5

The results are summarized in Table II. The most im­

portant factor for the buckling strength of outstanding flanges

is. Gt • 'It is seen that Bleich's semi-rational theory comesclcsest

to the above computed values. However, it may be added that

:Bleich did not int~nd'to apply his theory to the strain-hardening

range of structural steel.

VI. Tests on Wide-Flange Sect~

In order to, investigate the actual behavior of WF shapes

with regard to local buckling 6 shapes were'testedunder two

extreme loading conditions:

(1) Axial 'compression (Test DI, D2, D3, D4, D5, D6)

(2) Fure bending (Test Bl, B2, B3 ~ BI+ ~ 135,B6)

The test set-ups for both kinds of tests are shown in

Figure ,. The length of each specimen was divided into three

gage lengths over which the change in length was measured directly

with 0.0001" Ames dials. Along the edges of the flanges and the

center of the~eb,def1ectionmeasurements were taken as shown in

205"E.5' -12

tpe same figure. For the pending tests the lateral rotation was

measured at the loading points (which were supported against

-lateral rotation) and near the center line of the beam-. The

dimensions of all specimens are given in T~ble III.

Figure 6 shOvTS the results of tests Bl and DI. First

of all PIA vs cav for the compression test and M/z vs €av for

the bending test are plotted

P: compressive load

M= bending moment

A = area of cross-section

Z; plastic section modulus (twice the static moment of

half the section abmlt the x-axis)

cav : average strain at center of compressed flange.

Furthermore maximum flange and web deflections are plotted as a

fUnction of the average strain and for the bending test the

lateral rotation. From these curves the critical strains were ob­

tained and are indicated by arrows D The critical strain is

defined as the average strain at which the deflection of flange

or web starts to increase more rapidly than it did -initially.

The results of the tests on the other sections are

presented in the same way in Figures 7 - 11.

Figures 12 - 17 show the strain distribution along the

length of the specimens. together with flange deflections, 'lrreb

deflections and lateral rotation.

~nterial properties were obtained from OQupon tests,

the results of which are given in Table IV.

205E.5 ..13

"

However, webs of sections are only uniformly compressed

in the case of axially loaded columns. For these columns loads

rather than deformations are important. Thus critical stresses

vs d/w ratios for the webs are p19tted in Figure 23.

205E.5 -14

VIII. Tentative Recommendations for the Geometry of Wide-•

Flange Shapes

a. Flange BUck:l:ing .

The requirements for the rotation of a plastic hinge

depend on the type of structure and the location of the plastic

hinge in the structure.

In Figure 21 all test results are summarized. It is

seen that the critical strains increase rapidly near a value

bit =17. For this value the critical strain of a hinged flange

just reaches the strain-hardening range G

From a study, which is now being made at Fritz Laboratory

on the required rotation capacity of plastic hinges ,. it is known

that in general it is sUfficient for the strain of the flanges ...

just to reach the strain-hardening :range. Furthermore, it is

seen from the test results that in this case no rapid drop of the

moment occurs.

Therefore, it can be recommended tentatively to specify

bit ~. 17. This will giva satisfactory performance of a plastic

hinge with respect to local buckling except ";in 'specia1 cases of

large required rotations producing strains in the' flanges far

beyond strain-hardening.

b. Web Buckling

For the wide flange shapes subjected to ~~bending'

buckling of the web did not occur for the range of 'shapes tested~

dl!w= 27 to 40

whered 1= d ... 2t

205E.5 -15

•

Hm'1ever, under T.lY.!'_~_.•cQl!1nte..§..§J..Qn 1!Teb buckling did occur.

Taking as the condition that the section can be compresseo up to

strain-hardening gives

d' /' 30- ......., .....w .

besides the above specified value of bit < 17. However, this"

loading condition only occurs in case of axially loaded columns

for which the load rather than the deformation is imnortant.

Thus it 'lt10uld be sufficient for the column to reach the yield

stress. This happened for test n6 1'Tith d'/w =39.6 and

bit =18.2.

Tentatively it can therefore be recommended:

to be determj~ed in the future*

d/w =34

d/w = 43fo r € - €cr - yIi!__._..__._,__._¥ . .._._..__ ~. _ _ .. ._ __ .._._~__._ _..__.__.__ _ .._L

.~.~1'----- --- -_ -.- -_ -~-- 4,_ ----.-•••••• -••••-- _-- ••_ •••• --_•••••• -- _ .•- ••_.~- •• " •.• - •••• ~"'-"'--'-. --," ,-._ •••,'--'-' .---- .----~-•••-.-.--.----••••-.--,.-~.,

i Tentative SnecificationsIi---co;~-re ~;l-~;-f-~:~9e-;_.- ---"-'- ----. --.."--.---" ----f ~-~- ~-~-·r--·;;-~-:~-""- "-b/t'--;-', 7

II Web subjected to pure bendingiiII Web subjected to pure compressionII

I

.'

•

*All tested shapes "Cd '/w = 27 - 40) showed satisfactory p~r...

formance 1.'Tith respect to '''eb buckling and the critical value of

d'/w is expected to be considerably higher than 40.

IX. ;Future Ngrk

A proposal for continuation of the project, 205E,

"Inelastic Instability of iATF Shapes", was submitted to the Lehigh

Project Subcommittee and was approved at its meeting on August l3~

1954. A resume of the program is as follows:

a. BuckliniL.9.-f..J:L~}JF.=§~~Under Pure Jj'ome.!lt

A few tests will be necessary to determine the limiting

value of the d/w ratio such that web buckling occurs '"Then the

strain of the flanges reaches the strain•.hardening range.

205E.5 -16

-,

c. Buckling of Flanges an~ Webs Under Moment Gradient

Up till now all bending tests were on WF-sections sub­

jected to pure bending. For the flanges this is probably the

most severe loading condition and tests may reveal that under

·a moment gradient a higher width over thickness ratio could be

allowed. In the case of web buckling this loading condition may

be more severe than pure bending due to the influence of shear.

d •. Stiffening of \"rF -Sections

·If the tests reveal that a large number of available

shapes would perform unsatisfactorily, the effect of stiffening

devices should be investigated.

e. Lateral Buckling of WF-Beams

The bending tests which were carried out revealed that

failure mostly occurred due to a combination of local and lateral

buckling. In order to be able to specify the spacing of lat~ral

support more information is needed.

x. Acknowledgements

The authors wish to express their appreciation to Dr.

Lynn S. Beedle for his helpful suggestions. The work of Mr.

Alfredo T. Gozum on the design of test set-ups and the perfor­

mance of the tests together with Messrs. Satoru Niimoto and

Yuzuru Fujita is gratefully acknowledged. Mr. FUjita also

checked the mathematical derivations presented in this report.

The helpful criticisms of members of the Welding Researoh

Council, Lehigh Project Subcommittee (Mr. T. R. Higgins, Chair­

man) and the Column Research Council Research Committee C (Dr. G.

Winter, Chairman) are sincerely appre~cated.

205E.5 -17

! Mr •. Kenneth R. Harpel, foreman, with his Fritz Laboratory

mechanics and technicians prepared the test set-ups and specimens

and assisted in every step of the testing program.

This work has been carried out as part of the project')

"\'!elded Continuous Frames and Their Components", being conducted

under the general direction of Dr. Lynn S. Beedle.

205E. , -18

XI.

5.

8.

10.

11.

~i~i.pf References

ThU~limlJ1nP, B., Haaijer, G., "Buckling of Steel Angles in'the Pl~p-t:Lc Range", Progress Report T,. Fritz Laboratory,Lehigh, Un.1.versity, August, 19530 . .

Haaijer, G., "Compression Tests on Short Steel Columns ofRectangular Cross-Section", Progress Report S ,FritzLaboratory, Lehigh University, June, 1953.

Girkmann, K., "FltlchentragvTerke", 2nd Edition, Springer­Verlag, Vienna, 1948, pp. 475-~78~

Bleich, F 0' HBuckling strength of Metal structures ll , McGra"Ti-.Hill, New York, 1952, PP. 308-310g

Lundquist, E.E o, Stowell, E.Z,q "Critical Compressiv.e stressfor outstanding Flanges", NoA.C.A., Report 734,194-2.

Lundquist, E ..E., Stowell, E.. Z o ~ "Critical Compressive Stressfor Flat Rectangular Plates Supported Along all Edges andElastically Restrained Against Rotation Along the UnloadedEdges", N.A.C.A. Report No. 733, 1942.

Kaufmann, vi" "Bemerkungen zur StabiliUit DUnnwandigerKreiszylindrischer Schalen Oberhalb del' ProportionalitlitsGrenzell , Ingenief-Archiv, 1935, Vol. VI No.6.

Bij1aard, P.P., "Some Contributions to the Theory of Elasticand Plastic Stability", PUbs I) Intern. Assoc. 'for Bridgeand Structural Engineering, Vol. VIII, 1947e

Ilyushin, A.A., "Stability of Plates and Shells Beyond theProportional Limit"7 NACA TM-1116, October, 1947.

Stowell, E.Z g , "A Unified Theory of Plastic Buckling ofColumns and Plates", NACA Report 898,1948.

Handleman, GoH a , Prager, W., ilPlastic Buckling of a .Rectangular Plate \-lith Edge Thrust", NACA. TN-1530, 1948.

205E.5NOMENCLATURE-------~----

-180.

,A, 8,

A =

E =

E =sec

E t =

Etx=

E ty =Est =

I =

Dx =

D =y

...H =

G =t

('; ts =

M =

P =

Z =€ =x

€ =Y

€ =av

€ =Y

€ =st

y xy -

CT =x

CT =Y

CT =0

al

, a2

, a3

, (1' (2' C3 , C~, (5' C6 , C7 are constants

area of cross-section

modulus of elasticitysecant modulus

tangent modulus

tangent modulus in x direction

tangent modulus in y direction

strain hardening modulus

moment of inertia per unit width of plate = t 3/12

E Itx---T-lIl1_. x y

~~_ Dx_~__ ~~_?1~._~_~_!~1~_2

modulus of rigidity

modulus of rigidity vTith material compressed to strainhardeningbending moment

compressive load

plastic section modulus

strain in x direction

strain in y direction

average strain at center of compressed flange

strain at yield pointstrain at strain hardening point

angular strain in xy plane

stress in x direction

stress in y direction

average compressive stress

-18b

d , ==w

w =

edge moment per unit length to produce unit rotation ofedge

bQ= width of plate with one edge free

b = ,,,idth of flange of 1:lF shape

t = plate thickness

t= thickness of flange of ~& shape

d = width of plate supporte0 on both edges

d = depth of 1& shape

d-2tthickness o~ web of vW 'shape

deflection of plate

V y =p =

\j.; =

205E..5

u y = yield stress

u cr = critical buckling stress

'Txy = shear stress on xy plane

Vx = Poisson's ratio for strain in x direction

Poisson's ratio for strain in y direction

BfA

Y

x --1.) coordinate axes

7, = half wave length of buckled shape

L = length, of plate

.

205E.5 -19

A review of the stress-strain relationships, used in

different theories of plate buckling in the plastic range.

1. Bleich's Theory

Bleich(4) obtains his solution directly from the

differential equation (4) taking quite arbitrarilyE J

D. = .-.-!-• l-v 2

EID =

y I-v 2 (22)

The stress-strain relationships corresponding to the anisotropic

behaviour expressed by this differential equation can be derived

from equation (22) putting:

Et x = Et \iE~·-v~ = v 1--

Ety E. E

(2J);>~=r-E-

I'-~--'---"-'

\!E~_ v = v\I-·-G = y \I Ett 2( I +v)

Bleich states that this theory must be regarded as a

semi-rational theory which can find its justification only by com­

parison of the theoretical pred'ictions 1'Tith the results of tests 0

2. Kaufmann's Theorv

Kaufmann(?) ap~roaches the problem by taking

. (24)

Using the compatibility and equilibrium conditions for the state

of plane stress he derives

__EE t -­

( I + 11) ( E + Et ). (25)

-20

3. ~~format1on Theories

The theories of Bljlaard(8)~. Ilyushin(9) and Stowell (10)

are usually referred to as deformation theories.

Their basic assumption is that the relationship betvleen

the intensity of stress

and the intensity of strain

•,=A, \l~:-:-~::··, ~±y:~

· (26)

· (27)

is a uniquely defined, single-valued function for any given

material if a, increases in magnitude (loading condit+on)

For a simple coupon test

and €I = €x for Poisson's ratio = 0.5

Therefore,a, = Esee E, (28)

;

Esee

- secant modulus (see Figure 1)

The stress-strain relations compatible with equati9n? (26) a~d

(27) areI.--

Esee

10" - :!!-·a )x y

l-i a + 0- ) · (29)f y = Eseex· y I . ';

3I'xy = --

E ..,.see x y

205E.5 ..21

The relationships between the increments of stress and

strain can be obtained by differentiating equation (29).

The coefficients of these incremental stress-strain

relationships will be a function ofdo- 1

o-x ' o-y , 7" , E and E t =For x y sec d l; I

0- :: 0- and 0- ='7" :: 0x 0 y x y

one obtains

:: -_._._- ...._-_..__..'.-

I 3 E_. + '-.~

4 4 Esec

I 3 Et.- + _.,-_..

2 2 Esec

(3°)

EG - .._~-S

t - 3

4. :Flo~r\ Theor:y;Randleman and prager(ll) applied the so-called flow

theory to the plate buckling problem~ The strain increments are

thought to consist of re'versible (elastic) and permanent (plastic)

componentsI II

:: dl;' + dEx x

dl;y

:: dl; I. + dE' I

Y y (31)

dy :: dy' + dy'lxy xy xy

For the elastic components Rook's law holds

E dE' do- - vdax x y

E dE' :: - vad + do-y x y

E dl;;:: - vdo- x - vdo- y

E._--- -_. dy :: d'T2( I+v) xy xy

(32 )

-22

and for the piastic components

EdEll = aida + bldax . • y

E dE I I = ?I)·'d a + b I I doy ....'. .y

E d E ~ I = a II I do x + b 11 I day

E dy II = 2C 1 d-r• y . • y

The coefficients a', a", a l II, b',bll , b'll and c l depend on

the existing stress 0o

It is possible to determine aI, a l I and a l I.' considering

that:

1. Plastic deformations do not cause any change in

volume

d € I 'd€ I I + d € 11• + y 2 =0 (34)

2. The y~ and z-axis are symmetric with respect to

the direction of compression x

a".~ a"' ~

3. For a simple coupon' test

(35)

EThe result is a' =- 2a ll =- 2a l II = -1Et

Next the criterion for neutral changes of stresses

is considered. It is assumed that

4._ The criterion for loading or unloa.ding is fu,rnished

by the sign of the work ~v which the existing stresses, ',. ,

do on the change of shape produced by the ~ncremel1tSl

of stress.

5. The total strain increments will be cont1nuou~ ~n

the region which marks the transition from un~oading

through the neutral state to loading. (For

neutral changes of stress no plastic deformations

occur).

205£.5

This furnishes the conditions

a I + 2b ' = 0

a II + 2b" = 0

alII + 2b lll = 0

c' =0

Combining equations 32 and 33 finally gives

-23

<'38)

4EE -

Et y = E+iE-;

_ Etl2v - I) + E

v - ----------------x - 2E

2 ~ t I 2v - I) + E]v = ------------------------------y E + 3E t

i

I\)

o':"'1.t:r.j•

TABLE I

--.----1-------·--····;·--···-----·-----·;·--·----·-------·--·-r--·~-----_·_-_·_-----~·-l~ I ; I t J

Theory !Etx I Ety : Gt i· ZlX ; 1Iy I:_." --- --.----..-~-. -_ -_. _._~--- .---- --·-·----+--·---1-··--··-·-------r--!:=..:.:.:=·.-=:.:--·~ --- ~--. __-._-.:.w,_-_..,._ -..-.P.,- ···1·--·-----'--~ ~··_·-···--·-----1

! i: i \j E Et I' IEt--- ,IE IBleich (4) ! E' E' I 11\~E 11\1_ Ii t 2(1+v)!· \fE JEt i

! I

11

2 [Et (2l1-1)+E]

E+3Et

1-2

Et (211-1 ).J..E

2EI

1!_.

E

; E12(1+11)I

. .-L _

iI,ii

! Et:_---! 1 3 Eti -+--! 4 4 EseciIi. 4EEtI! E+3EtJ

IlEtI!

!,

Kaufmann (7 )

Handelmann (11)Prager (11)

Bij1aard (8)Ilyushin (9)Stowell (10)

-L _

-25

TABLE II .

0 0 5I 0~3I

I O~981

I

0.,1,110900 2,000

Bleioh

Kaufmann

B1jlaardIlyushinStowell

- I .Etxl Ety Gt 1----··_-1 . T_heo.ry ---l._._k_Si I--k_S_1_._-+-_k_Si__-r-_1I~.__+._1I_~__

11: 0 0 5 I 900 30 ,000 1,730 I 0.09 j 2 0 39

11 = 0 0 3 900 30,000 2,000 0 .. 05 I 1~73

11 = OQ5 I 900 30,000 580 0 0 5

11 = 0.3 I 900 30 ,000 670 0.3,J = 35 ks1 1 900 1,760 860f: =13 x10-3 Ja y =45 ksi .. JE, =,13 X 10-3 J

1(,82HandelmanPrager

------

900 I 3,300 11,500 I 0.49.-'--__ J .. ~ - --..

TABLE III

-j' . __n_ --r -.. - .. "'-'T---~' ... T--- -'i===.-·--:--------·-;--· :---- : ~-·1--1----;----!--..·--_j

I : 1 A ! Z ! bit i d !' w I L f L1 i c i. i I rI 's' . 2 !. 3 I. 'in ' i I • I in I 1 '. Ibit' I d' / I d/ 1 '! Spec.! hape \ J.n ~ J.n ! J.n \ \ n i m j n I J.n i i W. ~ i. '1- t---·---i----------·l------t·-·.--.t----- .-. --·1..··-._;~- .. -r-- .--- ··--t- ---;- --- -r-C'---r--- '.- --1'- ------l--~--.:-tI Bl Dl i 10 1rJF 33 I' 9,,66 138.56 ! 7,,95 10.429 9.80 100294 i 32 132 j 38 11805 ! 300)4 r 33.31I : ' : i l i I j i ! I ! IIB2 D2 I 8 ltJF 24 I 6.83 t 22.56 ! 6.. 55 10•383 8 0 0li 00 236 i 26 126 !38 !17~l i30.7 !33 .. 9\I I I I l 56! 8 i 88! 8 I I I 48 I 6! t ,'I B3 D3 i 10 vJF 39 \11034 'I :- .. 3!1 .02 10.512 9. !0.32 I' 32 r 32 ! !15~! 27~0 !300l!

I I . f ' ! I J i , !IB4 n4 1 12 vJF 50 !14.25 170028 \ 8 .. 18 \0 ..620. 12.19 \0 0351 1 32 i32 i 52 i13 u2 !31<02 i34071

; B5 D5! 8WF 35 11110.00 \ 33 .. 68 !. 8.08 r.476 8.13 II; 0.308 I32 !32 144 117.0 123~3 j 26.4!I I I,. : . I , I I : I

IB6 D6 11,10 WF 21 i 5.84 \22 ..45 \ 5.. 77 10.318 9.82 10,,232 '23 !26 !30 ,i 18.2 139.. 6 !42 0 311

I I I ' Iii' ,I----...1----.-.-__- ,~,._......._.......__.~.,, ;"~ . ._~__._.._.__.... ..!___ ~.._._.__._ .-----.-- ";-_.- .-~,-'*-_..-- _*_. ~__ .. ' .__..~ ._._. . _.J

•d =d - 2t

(

TABLE IV

Results of Coupon Tests

1\)o\J\ttj..\J\

.5. to

3'4

Locl1t1.on ofCOUILQl1S

All couponstested in Baldwin60,000# Hydr-aulicMachine.Valve openingcorresponding totesting speed of1 micro";'ino/in"per sec. in theelastic range.

*See Figure

I *ICoupon Section Buckling Loca- Yield Stress Strain at Strain- Type ofTests tion o-y ksi Strain- Hard·· Loading,

~ardenin~ ening;IjI I

I Cst x 10 .. ModulusI II IEst ksiIi~..

T 6 1 1 35~5 16~5 675 tensionT 7

i5 35G O 141t 7 7'50 tens'ionI

. tIO WF 33 BI-Dl I

C 14 l 2 40 0 0 140 5 855 compressicnC 15 \ 6 370 0 13 0 8 805 compression I<oJ

I1 35 0 4 18 0 4 530 tension.T 21 Ja WF 241

: T 22 B2-D2 2 35.6 ·181)0 600 tensionT 23 3 36 0 3 19.3 470 tension

T 31 1 1 35~6 14 0 3. 525 tension. T 32 jl0 WF 39 B3-D3 2 36 .. 8 18 a 9 580 tensionIT 33 3 37.8 16.3 580 tensionI

IT 41 112 vJF1 37.1 18 0 0 500 tension

! T 42 5~· B4-n4 2 36.9 18.1 530 tensionI T lt3 J 3 39.4 1509 580 tensionI

j T 51) 8 WF 35

1 37 0 6 16.9 560 tensionI, 52 B4-D5 2 37,,3 .16.6 465 tensioni I' . -iT 53 3 39.9 19.6 600 tension

IT 61J10 ,IF 21

1 38 0 0 20.8 520 tensionT 62 B6-D6 2 34~2 23.4 570 tensionT 63 3 44.2 23.6 490 tension

!

-28

TABLE V

Test Results

l·u I~ cr ks i -----J

~ Flange! Web I.5 I 34 0 2 34.2

07 34~o 34.0

~o 39 0 0 39 0 0

0 0 36 0 8 35~4

QO 38 a O 38~0

~6 33~8· 37.2

IL-_~ ...L-__...L.--__...L--__--'-- -.J

Ccr .103I

Test u y ksi Flange We

D 1 34.4 8.5 8

D 2 34.0 13.5 12

D 3 I 35 0 2 19.0 19D 4 35.0 18.5 5D 5 36.6 171)0 ~7

D 6 .

I

38 0 0 4.3 1B 1 7a O

B 2 23 a O

B 3 22 0 5B 4 29.0

B 5 22 0 0

B 6 14.0

modulus)

I~

~... '

105 E.5

upII)II)(J)

M.pCI)

(J)

bOasM(J)

~

tanOC' = E (modulus of elasticity)

tan(3 = Et (tangent modulus)

tan?= EseJ secant modulus)

Average Strain

Fig. 1

------------------r~III

E (strain-hazldeningst I

IIIIII

Average Strain

Fig. 2

205 E.!:>

//

""~IO--_----,b.....I,-,2",,--_~., //

/.1"6-----10.1....... ---'l"i'~/

Fig. 3

Fig. 4

t = thickness of plate

Z. = half' wave length of'buckled shape 0

L = total length of'plate

w = thickness of' plate

205 f.5

Lateral'Supp0r"ts

~Web and - JRange..l?uckh

'77--;~-r-7-r7-r-r-~'T7"7-r-;~~"'7"""7"';,.....y/ Oeflec.+lon5

BENDING TEST- B SPECIME.NS

COMPRE.SS ION TEST - D SPE.CIMENS .

Figure 5. Test Set-upso

'vNeb~

~}Web and. Flange

Bucklin3 .

105 E S.

~_: I.---1

I,

Ii -,,- I I

I, • ,

-- t- -LI

- ~--- - .- . f·--- . --- 0-

10 I Iks! 33 I. .It I I

I I I".. 1+0-n Q k., , I t ! 1i

I t

40 '-- dtw=~3.3 I I .I -1__ J

~- - .. -. , -- h- .. - r-- - -I

Flange 1 uck11ng-Bl. p I I --j . .

IA ' .Flange 1 uokl1ng-DlI i , ---i I, ,

Dll1 , i "tn, -=M I

I I- , ,~

!' - i . I! II

--, 5$ _~ .1' -.j- .- 1------I- ~- --I ~o-~ (1) pend1p.g te !it Bl.. - . .I ~~.\.~ I

I I ... ~~I - . . --..I ~~

~.t:J "T

__ ... .1,.....,-~ .........,a~ -.LT.:L

I I j-,.- .- ~ -- r

I l· I-~ ~--t- -- .. . .- --- - .., I :r-T EJ

I f- -t 30 - \. I"I 1

( i I ,, I T 10 2f 130 I 4 ) X 10-3

I I IH', .. l .

~1lY 1~+ < I---4- r1 1 -_.,.- :.. -f-- -r l'~-

- -- -._~_ .'0' '[

,~. - ~- .- --- - - - - .. - - ·1- -

i .-1_,

: I tI

_! o.~.1_ '-'.~

~l I I I--l .~max~ web!-de.f" bA ... 4 .... ~

-I-- -r-j-;- - .1- t: I -r 'I • :

. ,i - -o,~~H "'"

~ 1\:)' I i r I II~- -

~.- -

r- - - _.. --+-_. _. - -""--rlI I t II I I

! -'l\/flX I , I 1t • flatlge d ~f'l;eo l.i1 n ! _.. +. I - I-- , -- - -~ .. - ~. 1.I) •

i ~_ I I II .1

~~~ -j --- -- -- fr i ,-r II I : r :\.I No .,. I.atert~ -ro ~at-1on -n&t -meas-uredr - ~- -

n te~t Blo II

OA"S '..j:J" .... I~ "t:.K:::

~I I I TT. . I -- -

w~b-hepe_f--L.. - -

-4rDU:\.X. ~nt I

~--

,.~I I_ ,O..ll" I 0....... ! .!... It- - - - - ~ - I- I t

I . I . : I II

I.- I I I f,

I

~.- 'j I- 4 .. - - - ~ t - -+ .-f 1 : ~. na~e de tLeet10n - -l-~-I T I K

I I t I ·,1- .. . • e- . - I -. - - t ,. ,I · ~ - ,

,j

I I j

I r 1l:J ; , I

II· II ff- ~ - I- - -j- . . r .. - 1 -1-- •I '1 --fr-t

,f-------!-. I I ,

~ .-iI t II I I

,I I

l?!FrI

~l - - -,- ~t~ I r- - • 6 0 're .1; Results .. D10t I

I !,I 1 I I

c:

\.

•

c

cz

z... aCl ..JN IIf-w

o

,-----------------

I·

f--~_l___-"---I--- -

205E 5I

I I

1I

- ,I I

-·~r-I- -J

I i

.'f

- 1

1.

!

II

• . . - ,

I I

-i-·-1- 1 ~ -

I , I

i

I[

.-. , I l -.,

S- WF -2-4- 1_ iI

Ib/t= '7.1 j , I···

I

I. ,-

II

~-J-40-~1> I

~I:~-+-~' - - ---T -

v'!'

Z

L1'J."'

oz

--t

,

zW II:

N ll.

~ 0o NI'

o ~N 'I

c:('7'"

II:

'"C.<la.1:c.( T

II: U

Z

/

.. .

.

- 40 I--

, kg!

I

pI

'20S E ~S__ .--_-.,.-_'---r__~_- --~---r-

~-~ , . r ,i I '1-,_-1__ - - ~o ~~ 39- I l - -+. - I-

I I . 1 1 • ~ Ib t= 5.6 I i

d/w=~O.l : I I II i~N-~ Flange bllokllng-B3.-. :?~~ ._. --Flange -b~kling-D3

I . . . ..~.- Web buok~lng - D3

I I Ei· BI

~ :t-I

-':~"'b'"- ...!:~~~= ~W' -.- - :-<:> cllend~ng 1;<.st Be- 1-. ~':' ':'- < • -~-

! I 1 [~I~' ~ I I I 8 oomp:re s,sl ~n teat D3

--

I •

I•I

i

0.1 If J1 I :t¥"""jG--s..,Z1 ~_ .'\10 ..---L ... j ~~ I ~ t I-' U ... ..:.. .. , I I I ':

0.1" t ~x..- nmlgb- -- ~~~ - -< .- --t-:- - I

I I I detleotloo. ~l ~ I I I 1 I

,

. L• .!

1-- ,

I~. : i ! : I ~ !: .. , - : i-I-----r--- -----j- 1--- - I -t---i-,-----i- -- --r

I I .) . ,-1...._: -:50 ~-~-+_+-_+_--:-_+--+-__i---+__:_---_+_~--___t

I I I I

.J..

oL

11« ~ll' •l:l U

7.

zwc: w

11...'"o '

1<

ll'..110(:l.

Fianga- uokl1ng-B.flAnge uokling-DWeb buo ling - D4

40

105 E. S,-__----,r--~-._r-__-,-_

~--~-~ D40

~I .

I:t

z.oJ

Q:wa.<l

Ia.~ r~ ~

Z l!.W W~ a.~ 0

o :o ­!Ii

:ro

c,

,------------------------------------- ----

D5

B5

I

I- uCkling -B$

uck11ng-D$·....... _1~~ ~_D5

--~--

I~'+-_.:--+--~---+--+--+-------t--.......---_i

40

Iksl

z

'205 f. S,- ------

f

lk

0.~

0.

:rQ.

~ I• oJ

Z

Z... lk

.JN ll.f-...w

"I

:t')

205.E .5. ' It . 1I •-r -~-

j ,I

:I

- -; .. - f __ Fl~~e 1uokl ~-B€----1__...-1-+--;....; -r:ti!iiSer: uoKI ng-D-e

! I; _cc ,- we~ b~C} ling 1- DE

I I I I

!I..-_t--'-'....,I cc ' : · cl :I - r ~- --t------+----t- Tf--r - il

-- J. V tv 1O~ -- I

- t·, -' --1- ,-_. ,;--1cai' : b/t-1I-8 41 2 I

f-· -+ - +

,_ .• 1-.1

- ~, rl\:- +L_;- _.--, - ..- L 'f~ - -- t- - -, ~-.' ", .--L; I .L-- _ -- --- r----t-- I' Ii ~ c· i • -- i

J I [I, !: ,,- • c, ~ ' I I ,I! I -3

I· ! I I I. '3/,,! I j 30 _ .l._ '_j ~i) ~ 10 rL I~ - 10 'i, 1- ; -- ~~ -17 Co - ±tt' I •., .-~ 1~"4 I I I Co. ' ,-+- __

,; '<::::! . , + I ~ I I I

I . ;,. - ,.---.- -1----1 ! " + : ' .. -4--- .. !__ _ .,..... T~:- '- ~~ -...,. -- c_ c_ i --j _c -T _c_ I I I !

1~

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~

CJ

wZw.'J

I

l

!

: -.

I I I. I I

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! ' , i,! I . i l,__~ -I -T' "J . I I • I I . j ..... -_. -' t- I7'1 -I-::-:+-, -- ~r.-11----·1 "I --'--4--~I-~T -I- I • I I,' I -I,---r--I' . I I I I + ~ I j + I

I "1 . 'Il~+, . r '-: ~ ....._.-':hc 1'091; R"n4t. c jle I.. De.' :'1 .I-- 'i:~' .... :EJ.&-~~~l; !-"'e II,·· t I I, 1

," -. ,I ~M ]. ...L. --,.. ~I-.' --Ji -'- -+-1--'i-I .-..l I' --tI - f • , :" I ' -:-4!. - -- ..- ," - -' Ie', --,! ,- I • I I'. 'c 'c ii'" I I

-- It lL l~N' I, I I • I t'o T 1, -f- .O. ~ l' . 1---- ~ --, -- t - - T. - l' , --++'i !.: - . -- T h a'M,",dl 41,,*4"1: dO"." I'\h 'I • r 1I _-;_

I .. max, ,I -I! :• ~ rt ll' ~ + l\~ I , J _ _+ t-=-;"'-'f-a- I __O. I .~. .LS - --1--1 ....,.-....,1-;- 1:- --l I , I

h~ -1-.' i I- -... ,. EI . I 'I' I , I I

~ .1l:l WN Q

~ 0Ci I

oN (\j

IIo"1"l

DZ

, I

II

- -==tIl!E:il!ll:;t::==it-s· 10

nOY"3:scite

T

,

!

.- ~

TtII

Ul- i--'1-0 -

" I

o -VeN.Scale .

- 1-

,- -!

__ L _

I

_I

La era \

I

-

')fyQln 5.f!a1~,I I

Ii I L-,---~-.....-I----!--+---;--r1

-. ;We$t~ I , f )£1\0 KG 11;5T b1Ldadlna _. 1.. - - ~.3- foint

Ji -j G ~- af eav• = "05 x 10 _

I I I _~.1- : 9 '0-I ..... - - e - -- I G - - -€> alf' e~" _ 13· 3~_

~~ - ... ~ 'T-----.... "'~ ,," liJ'

-!.-. I _ _ ~'l...,- ~-(1' c__' ~I - --;-1l+~\ - '5' I -,..-j' Dc: flec+i¢n of N. Flang~ I 1 I I I i

I I I I <~ + .

---!P' - '1:l

befled on'S - =!).Fl!!:9~- I- --;-- -+..__-+--+-]_ ' , 1- l. . ~ll f--i-

,it I --- ---- --1:_ ---~ ----10 I tI I

_ .J

!as LoadingPoinf

....... - ..,. ,T ; :.1>' r I ,_e ~{~c~ o~- tFlsnc~! +~I -- I.' -- •• <1-<-1: f--... -1. !,

I I, I _~ l.--.l- --- ~ ~I::/ $ J----4 ,", .- • _I- I": I I' I;, ""'-.;~ ,. j

, t ~ ,'-" ~+t ......+ It • I, I _~ I, , '" ' T

--- .-

.-

r

7

I

I I I~gUlt&,L lBo"-~li'l'fng~ ~ Web ~u t

I

....------

I ~ I-- r ~

~ --

I ,,;'4

Det(edion~- N.Flange -1- - ~:t-~~ t' I- - I l I ~ t

r-~- ------ I

I I3o,.IO~

_.-1 I~-1- -

-~-_ .... _...-to

I

~ ~tra'ln- 5.F nge. • II

II 19\1r 13. Fl,nge +ndL. .!.--

~p

~---I---.:-+-~~ottlen--4fkb.----1--f----t---r--+-l'----!. --:--4---f--:--f--+-_._--l

l.z...

I I

,J

I133 -

1 .t

111. ,/'-g.'/

r-l--------I • -j-.........~-~-;.---.

I "------t--....,t

- ---......;----

--~--.,..@-'

Deflec.tions - N.Flange II I ;-------- .. --lt ~ I :

. ! I :I

'205 J: . S~~. I _~-~-- -~-l..oad)n~ _-Point l--- ..., -...,..........

"."... ....

stra'IYl +! We5!

t J

..

e.. v.· 11- 2.'f )C K5'~-J

~J2A."'lt1O

l-I

tz.

, Top r

+w-N~S

p I/"1

.•,.. J __~_"'I• J

1R-.L-_-t!J.=1 at e.a - l06 10::A _,...J. . lA"'~

,.. ... -a ,~~ ....

I -,.

-I

-1

---

...... -...G- ....... .....r - I -- .....

~

I

!I

I I III

i -I- I!I I i

I IStrain - op Fl9nge. If- - I --

~fledion - N.FlanqeIf. . II I J L _, _. __

+ -:- ------r---=-r---- .... --, II I L __ --r-----,.

I--- , t I I Ij stya n - N. Filange i r"-. __._ _

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