Cruciforme Sections.pdf

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SCHOOL OF CIVIL

ENGINEERING

RESEARCH REPORT R921

OCTOBER 2011

ISSN 18332781

STRENGTH DESIGN OF

CRUCIFORM STEEL COLUMNS

NICHOLAS S TRAHAIR


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SCHOOL OF CIVIL ENGINEERING

STRENGTH DESIGN OF CRUCIFORM STEEL COLUMNS

RESEARCH REPORT R921

N S TRAHAIR

October 2011

ISSN 1833-2781


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Strength Design of Cruciform Steel Columns

School of Civil Engineering Research Report R921 Page 2The University of Sydney

Copyright Notice

School of Civil Engineering, Research Report R921Strength Design of Cruciform Steel ColumnsN S Trahair BSc BE MEngSc PhD DEngOctober 2011

ISSN 1833-2781

This publication may be redistributed freely in its entirety and in its original form without the consent of thecopyright owner.

Use of material contained in this publication in any other published works must be appropriately referenced,and, if necessary, permission sought from the author.

Published by:School of Civil EngineeringThe University of Sydney

Sydney NSW 2006Australia

This report and other Research Reports published by the School of Civil Engineering are available athttp://sydney.edu.au/civil


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ABSTRACT

Very different strengths are predicted by two different methods of designing steel cruciform columns. Bothmethods require design against local and flexural buckling, and while one method also requires design

against torsional buckling, the other does not.

Investigations of the elastic local and torsional buckling and post-buckling of cruciforms columns show thatthese two modes are virtually identical.

The first yield and inelastic buckling approaches often used to formulate methods of designing columnsagainst flexural buckling are extended to the torsional buckling design of cruciforms. These extensions showthat it is sufficient to use local buckling design to guard against torsional buckling.

It is found that design methods which make separate checks against local and torsional buckling areunnecessarily severe, and are equivalent to making the same strength reduction twice. Instead, it is sufficientto ignore the torsional buckling of cruciforms provided design checks are made against local buckling as wellas flexural buckling.

KEYWORDS

Buckling, Columns, Cruciforms, Design, Flexure, Post-buckling, Steel, Torsion, Yield


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TABLE OF CONTENTS

ABSTRACT .......................................................................................................................................................... 3KEYWORDS ........................................................................................................................................................ 3TABLE OF CONTENTS....................................................................................................................................... 41 INTRODUCTION .......................................................................................................................................... 52 ELASTIC TORSIONAL BUCKLING ............................................................................................................. 53 ELASTIC LOCAL BUCKLING ...................................................................................................................... 6

3.1 Local Buckling ...................................................................................................................................... 63.2 Comparison with Torsional Buckling ................................................................................................... 6

4 POST-BUCKLING BEHAVIOUR .................................................................................................................. 74.1 Torsional Post-Buckling ....................................................................................................................... 74.2 Local Post-Buckling ............................................................................................................................. 7

5 DESIGN AGAINST LOCAL AND FLEXURAL BUCKLING .......................................................................... 75.1 Design Against Local Buckling ............................................................................................................ 75.2 Design Against Flexural Buckling ........................................................................................................ 8

6 DESIGN AGAINST TORSIONAL BUCKLING ............................................................................................. 86.1 Methods of Design ............................................................................................................................... 86.2 Discussion ........................................................................................................................................... 96.3 First Yield Strengths ............................................................................................................................ 96.4 Inelastic Buckling ................................................................................................................................. 9

7 CONCLUSIONS ......................................................................................................................................... 108 REFERENCES ........................................................................................................................................... 119 NOTATION ................................................................................................................................................. 12

9.1 Subscripts .......................................................................................................................................... 129.2 Principal Notation ............................................................................................................................... 12

APPENDIX A - TORSIONAL POST-BUCKLING .............................................................................................. 13APPENDIX B FIRST YIELD OF TWISTED CRUCIFORMS .......................................................................... 15APPENDIX C INELASTIC TORSIONAL BUCKLING ..................................................................................... 16


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1 INTRODUCTION

Very different strengths are predicted by two different methods of designing steel cruciform columns (Fig. 1).Both methods require design against local and flexural buckling, and while one method [1, 2] also requiresdesign against torsional buckling, the other [3] does not. This second method might seem optimistic, becausecruciform columns have very low torsional stiffness and are susceptible to torsional buckling. Instead, it relies

on the local buckling design check to guard against torsional failure.

The methods of [1, 2] use a unified approach to column buckling to allow for torsional buckling. In this unifiedapproach, the common method of designing against flexural buckling is extended to torsional (and flexural-torsional) buckling by replacing the elastic flexural buckling load in the design formulations by the elastictorsional (and flexural-torsional) buckling load. When this method is applied to low stiffness cruciforms, itproduces significant reductions below the section capacity (as governed by yielding and local bucklingeffects). These reductions do not occur with the second method [3].

The purposes of this paper are to compare these two different methods and to find reasons for preferring onemethod over the other.

Firstly, the torsional and local buckling and post-buckling behaviour of cruciform columns are reviewed and

investigated. Secondly, the bases for the design of columns against local and flexural buckling are reviewed.Thirdly, the two torsional design methods are compared, and the justifications that are needed for these arediscussed. These are investigated by extending the first yield and inelastic buckling design bases for flexuralbuckling to torsional buckling.

2 ELASTIC TORSIONAL BUCKLING

The elastic torsional buckling resistance Noz of a simply supported doubly symmetric column of length L isgiven by [4-6]

20

22 /)/( rLEIGJN woz (1)

in which GJis the uniform torsional rigidity, EIwis the warping rigidity andAIIr yx /)(

20 (2)

in whichIx, Iyare the principal axis second moments of area andA is the area of the cross-section. For thin-walled open sections the torsion section constant

3/3btJ (3)is small, while for concurrent sections (such as angles, tees, and cruciforms) the warping section [6]

9/33tbIw (4)is very small and often neglected.

The variations of the dimensionless torsional buckling loads Noz/Ny of cruciforms with b/t = 10, 20, 30 andfy= 235 N/mm

2with the modified minor axis flexural slenderness

)/( oyyoy NN (5)

are shown in Fig. 2, in which the squash load is

yy AfN (6)

and the minor axis elastic flexural buckling load is22 /LEIN yoy (7)

Also shown in Fig. 2 is the variation of the dimensionless minor axis flexural buckling loadNoy/Ny.


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3 ELASTIC LOCAL BUCKLING

3.1 LOCAL BUCKLING

The elastic local buckling load of a cruciform column may be expressed as

olol AfN (8)in which the local buckling stress [5] is given by

22

2

)/()1(12 tb

kEfol

(9)

in which is Poissonsratio (commonly taken as 0.3 for metals) and the buckling coefficient is22

2 4255.0

)1(6

L

b

L

bk

(10)

When kis approximated by 0.4255, then the dimensionless local buckling load may be expressed as

2/1/ olyol NN (11)in which the local buckling modified slenderness is

23509.18

/ y

ol

y

ol

ftb

N

N (12)

The variation of the dimensionless local buckling loadNol/Nywith the local buckling slenderness olis shown inFig. 3.

3.2 COMPARISON WITH TORSIONAL BUCKLING

The local buckling load given by Equations 8-10 may be transformed to

202

22

/)1(

/r

LEIGJN wol

(13)

by using

)1(2

EG (14)

and = 0.3. This is almost identical to Equation 1 for the torsional buckling loadNoz, the difference being the

(1- 2) term in (13). This term is a Poissons ratio effect which is important in two dimensional plates but

negligible in one dimensional beams. The variations of the dimensionless local buckling loads Nol/Ny of

cruciforms with b/t = 10, 20, 30 andfy= 235 N/mm2with the modified minor axis slenderness oyare shown in

Fig. 2.


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4 POST-BUCKLING BEHAVIOUR

4.1 TORSIONAL POST-BUCKLING

There are torsional post-buckling reserves of strength which result from redistributions of axial stress or othersecondary effects, but these are commonly ignored. An analysis of the post-buckling strengths Npz ofcruciforms (with negligible Iw) is made in Appendix A where it is shown that the dimensionless post-bucklingstrength is given by

9

4

9

5

y

oz

y

pz

N

N

N

N (15)

The variation of the dimensionless torsional post-buckling strength Npz/Ny with the modified width-thicknessratio

ololy

yNN

ftb )/(

23509.18

/ (16)

is shown in Fig. 3.

4.2 LOCAL POST-BUCKLING

There are local post-buckling reserves of strength which also result from redistributions of axial stress. Ananalysis similar to that in Appendix 3 for torsional post-buckling may be made for local post-buckling. Asimple approximation [7] for this is given by

olyy

pl

ftbN

N

1235

/

09.18 (17)

The variation of this dimensionless local post-buckling strength Npl/Ny with the modified local buckling

slenderness olis shown in Fig. 3. It can be seen that this approximate local post-buckling strength is close tothe torsional post-buckling strength of Equation 15.

5 DESIGN AGAINST LOCAL AND FLEXURAL BUCKLING

5.1 DESIGN AGAINST LOCAL BUCKLING

A column is first designed against local buckling [7] by comparing the width-thickness ratio (b/t)of its plate

elements against design code yield slenderness limits ey. These limits are based on elastic local bucklingstresses folwhich have been modified to account for the effects of material and geometric imperfections. For

a lightly welded cruciform with a yield stress of fy = 235 N/mm2, the limiting width-thickness ratio of [1]corresponds approximately to b/t= 15, compared with the value of 18.09 at which the elastic buckling stress isequal to the yield stress. If the width-thickness ratio is less than 15, then the cross-section is fully effectiveand the section capacity is

ys NN (18)

If not, then the section capacity is reduced to

y

ysftb

NN 235

/

15

(19)

which is based on the local post-buckling strength. The variation of Ns/Ny with the modified local buckling

slenderness olfor a cruciform is shown in Fig. 3.


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5.2 DESIGN AGAINST FLEXURAL BUCKLING

A column is designed against flexural buckling [7] by reducing the section capacity Nsby a reduction factor which depends principally on the relative magnitude of the section capacity Ns and the elastic minor axisflexural buckling loadNoy, as expressed by the modified slenderness

oysey NN / (20)

Thus

)(/ eysd fnNN (21)

in whichfn(ey) allows for the effects of geometrical imperfections and residual stresses.

The effects of geometrical imperfections are usually allowed for by using the load which causes first yield in acolumn with initial crookedness as the nominal strength. Thus, for example [7],

2

22222

1

2

/)4/1(1

2

/)4/1(1

ey

eyeyeyey

s

fy

N

N

(22)

as shown in Fig. 4. This method ignores the negative effects of residual stresses, and the positive effects ofpost-yielding and strain-hardening.

The effects of residual stresses are usually allowed for by using the inelastic tangent modulus buckling loadas the nominal strength. Thus, for example [7],

24/1 2 eyeys

i

N

N while (23)

as shown in Fig. 4. This method ignores the negative effects of geometrical imperfections, and the positiveeffects of the tangent modulus theory and strain-hardening.

Design codes usually modify one or other of these methods in the light of experimental evidence. The

variation ofNd/Nswith eyaccording to [1] for lightly welded cruciforms is also shown in Fig. 4.

6 DESIGN AGAINST TORSIONAL BUCKLING

6.1 METHODS OF DESIGN

The method of [1, 2] for designing against torsional (and flexural-torsional) buckling is to use the same form

for the slenderness reduction factor as for flexural buckling, but with the modified flexural slenderness eyreplaced by the modified flexural-torsional slenderness

)/( oftseft NN (24)in which Noft is the lowest elastic flexural-torsional buckling load. The effect of this on the variations of the

dimensionless cruciform design strengths Nd/Nsaccording to [1] with the modified flexural slenderness ey isshown in Fig. 5 by the solid lines.

The method of [3] is to not to make any specific reductions in strength to allow for torsional buckling. Theeffect of doing this on the design of cruciforms according to [1] is shown by the dashed line in Fig. 5. It can beseen that using the method of [1] to allow for torsional buckling leads to significant reductions in the designstrength, even for cruciforms with low b/tratios and correspondingly high torsional resistances.


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6.2 DISCUSSION

A rational explanation needs to be found for or against the significant reductions shown in Fig. 5. Possibleexplanations might be derived from the rationales used for the methods of designing against flexural buckling.

These include the effects of first yield on columns with initial crookedness, and the effects of residual stresseson inelastic buckling.

6.3 FIRST YIELD STRENGTHS

The use of first yield for design against flexural buckling [7] is based on the assumption of an initialcrookedness of the same form as the buckling mode. This crookedness is magnified by incipient flexuralbuckling effects, and additional normal stresses are generated which add to the normal stresses due to theaxial load, leading to early first yield. The corresponding geometrical imperfection for cruciforms that fail bytorsional buckling is initial twist, which will be increased by incipient torsional buckling effects, so that torsionalshear stresses will be developed. The combination of these with the normal axial stresses will lead to earlyyield.

An analysis of the effects of initial twist on the first yield of cruciform columns is given in Appendix B. Thedimensionless first yield loads Nfy/Nsare shown in Fig. 5. For the cruciform with b/t= 10 (which according to[1] is fully effective against local buckling), the torsional first yield load is virtually equal to the section capacityNs, and first yield is governed by initial crookedness and flexural buckling effects.

For the cruciform with b/t= 20, the torsional first yield load is noticeably lower than the torsional buckling load,but substantially greater than the design strength of [1] predicted by using No= Noz. For the cruciform withb/t= 30, the first yield load is slightly lower than the torsional buckling load, but substantially greater than thedesign strength of [1] predicted by usingNo=Noz.

It can be seen that there is no first yield justification for the substantial reductions of the method of [1] shownin Fig. 5.

It may be noted that first yield loads do not allow for the significant post-buckling reserves of strength thatoccur at low slendernesses, as shown in Fig. 3, and so it can be expected that the first yield loads will provideconservative estimates of the torsional buckling strength.

6.4 INELASTIC BUCKLING

The use of inelastic buckling for design against flexural buckling [7] is based on the assumption of residualnormal stresses, which when combined with normal stresses due to the axial load lead to early yield andreductions in the effective modulus below the Youngs modulus E, and corresponding reductions in thebuckling resistance. This early yield will also cause reductions in the shear modulus below the elastic valueG, and corresponding reductions in the torsional buckling resistance.

An analysis of the inelastic torsional buckling of cruciform columns is given in Appendix C. The reducedinelastic buckling loads are compared in Fig. 3 with the first yield loads. It can be seen that they are generallya little lower for low slendernesses, but not markedly lower than the local buckling strengths.

These inelastic buckling loads do not allow for the significant post-buckling reserves of strength that occur atlow slendernesses, as shown in Fig. 3, and so it can be expected that the inelastic buckling loads will provideconservative estimates of the torsional buckling strength. It can be concluded that there is no inelasticbuckling justification for the substantial reductions of the method of [1] shown in Fig. 5.


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7 CONCLUSIONS

The principal justification for the method of designing columns against torsional, flexural, or flexural-torsionalbuckling by adapting the methods for flexural buckling is that this corrects for the inability of the flexuralbuckling method to allow for the low flexural-torsional buckling resistances of some types of section, such as

lipped channels and lipped angles. It provides a seemingly unified common method of design for a completerange of cross section types.

However, the application of this proposal to cruciform columns leads to significant reductions in their lowslenderness design strengths, which cannot be justified by modifying for torsional buckling either the first yieldor the inelastic buckling approach often used for the design of columns against flexural buckling. This paperhas shown that torsional and local buckling and post-buckling analyses of cruciform columns lead to virtuallyidentical results, so that design against local buckling can be regarded as simultaneously designing againsttorsional buckling.

This virtual identity between torsional and local buckling in cruciform columns leads to the conclusion that themethod of [1] for designing against torsional buckling allows for this twice, once in designing against torsionalbuckling, and a second time in designing against local buckling. This leads to the significant strength

reductions predicted by the method.

On the other hand, the investigations in this paper of the effects of post-buckling, first yield and inelasticbuckling show that it is appropriate to ignore the effects of torsional buckling, since these are accounted for bythe allowances made for local buckling, as in the design method of [3].


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8 REFERENCES

[1] BSI. Eurocode 3: Design of Steel Structures: Part 1-1: General Rules and Rules for Buildings, BS EN 1993-1-1. British Standards Institution, London, 2005.

[2] AISC. Specification for Structural Steel Buildings.American Institute of Steel Construction, Chicago, 2010.

[3] SA. AS 4100-1998 Steel Structures. Standards Australia, Sydney, 1998.

[4] Wagner, H. Verdrehung und knickung von offenen profilen (Torsion and buckling of open sections). 25th

Anniversary Publication, Technische Hochschule, Danzig, 1936; Translated as Technical Memorandum No.87, National Committee for Aeronautics.

[5] Timoshenko, SP, and Gere, JM. Theory of Elastic Stability. 2nd ed., McGraw-Hill, New York, 1961.

[6] Trahair, NS. Flexural-Torsional Buckling of Structures. E & FN Spon, London, 1993.

[7] Trahair, NS, Bradford, MA, Nethercot, DA, and Gardner, L. The Behaviour and Design of Steel Structures to

EC3. Taylor and Francis, London, 2008.

[8] Trahair, NS. Inelastic lateral buckling of beams. Beams and Beam-Columns: Stability and Strength, AppliedScience Publishers, 1983; 35-69.

[9] Trahair, NS. Inelastic buckling of monosymmetric I-beams. Research Report R921, School of CivilEngineering, University of Sydney, September 2011.


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9 NOTATION

9.1 SUBSCRIPTS

o, i Elastic or inelastic bucklingl, ft, x, y Buckling modem Maximum value

9.2 PRINCIPAL NOTATION

A Area of cross sectionb Leg widthE Youngs modulus of elasticityf Stressfb Notional stress at the end of a legf

e Equivalent von Mises stress

fr Residual stressfy Yield stressG Shear modulus of elasticityIx, Iy Second moments of area aboutx, yaxesIw Warping section constantJ Uniform torsion constantk Local buckling coefficientL Column lengthN Axial compressionNd Design strengthNfy First yield loadNpl Local post- buckling load

Npz Torsional post- buckling loadNs Design section capacityNsz Torsional post- buckling strengthNy Squash loadr0 Polar radius of gyrationt Leg thicknessv, w Displacements iny, zdirectionsW A stress resultant of axial stresseswf Displacement due to axial strainingws Axial shorteningx, y Principal axis coordinatesxy Value ofxfor whichf = fyz Distance along column

Design slenderness reduction factor Initial crookedness

Twist rotation

0 Initial twist rotation

o Modified slenderness

eft Design modified slenderness for flexural-torsional buckling

ey Design modified slenderness for flexural buckling

Poissons ratio

Maximum twist rotation

0 Maximum initial twist rotation

Shear stress


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APPENDIX A - TORSIONAL POST-BUCKLING

After torsional buckling, a simply supported cruciform undergoes twist rotations

Lzm /sin (A1)

as shown in Fig. A1. It is assumed that the axial end loadNacts through rigid end platens so that the enddisplacements ware constant. These displacements are combinations of those due to elastic axial strainingand to axial shortening caused by the twist rotations.

The shortening displacements are

dzdz

dvw

L

s

2

02

1

(A2)

in which

xv (A3)whence

22

22

2

2 xL

Lw ms

(A4)

The displacements due to axial straining are

sf www (A5)

so that the elastic compression stresses are

22//

22

2

2 xL

LL

ELEwLEwf mf

(A6)

The axial compression force is

6

4

2/

32

2

2 tbL

LL

ELEAwfdAN m

A

(A7)

so that

262

222

2

2 xbL

LL

E

A

Nf m

(A8)

and the maximum compression stress is

62

22

2

2bL

LL

E

A

Nf mm

(A9)

First yield atN = Nfyoccurs whenfm= fyso that

62

22

2

2 AbL

LL

ENN myfy

(A10)

The stressesfcause torsional post-buckling atN=Npzwhen

A

dAyxfGJ )( 22 (A11)

whence

tbbL

LL

ErNGJ mpz

1018

42

552

2

220

(A12)

so that

20

52

2

2

4452

2 rtbL

LLENN mozpz (A13)


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If the post-buckling strength is taken as the value ofNpzwhich causes first yield so that

fypzsz NNN (A14)

then

))(5/4( szyozsz NNNN (A15)

after using

3/220 br (A16)for a cruciform section. Thus

9

4

9

5

y

oz

y

sz

N

N

N

N (A17)


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APPENDIX B FIRST YIELD OF TWISTED CRUCIFORMS

The first yield loadNfyof a simply supported cruciform column with initial twists

)/sin(00 Lzm (B1)

may be determined by solving the torsional equilibrium equation)( '0

'20

'''' NrEIGJ w (B2)in which indicates differentiation with respect to the distance z along the column. If Equation B1 and thesolution

)/sin( Lzm (B3)

are substituted into Equation B2, then m can be obtained from

oz

oz

m

m

NN

NN

/1

/

0

(B4)

in which

20

22 /

r

LEIGJN woz

(B5)

is the torsional buckling load. The maximum shear stress is given by

tL

GtG mmm

(B6)

The normal stress due to the axial load

ANf / (B7)and the torsional shear stress may be combined as an equivalent von Mises stress

23 ffe (B8)First yield occurs when

ye ff (B9)

so that

)3( 22 yfy fAN (B10)

The maximum initial twist may be taken as

b

Lm

2

1000/0 (B11)

which is consistent with the maximum initial crookedness of L/1000 often assumed for first yield in columnsthat fail by flexural buckling.


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APPENDIX C INELASTIC TORSIONAL BUCKLING

The inelastic torsional buckling of a cruciform column with the normal residual stresses

)/21(3.0 bxff yr (C1)

shown in Fig. 1c for one leg may be analysed by using a reduced shear modulus in the yielded regionsshown.

If the applied compressive load is defined by a notional stress fb at the end of the leg, then the stressdistribution is given by

bxxff

xxbxfff

yy

yyb

while

while 0)/21(3.0 (C2)

in whichxyis given by

6.0

)/3.1( yby ff

b

x (C3)

and the axial compression by

2

2)(3.0

b

xb

f

fbtfN

y

y

byi (C4)

The stressesfandfrhave a stress resultant W[4] which is given by

A

r dAyxffW ))(( 22

(C5)

whence

3

1

20

33.1

3

143

3

b

x

b

x

f

ftbfW

yy

y

by (C6)

When the column twists, this stress resultant exerts a disturbing torque [4-6] which is resisted by the inelastictorsional stiffness

3

))(()(

3txbGGxGJ

yiy

i

(C7)

in which the elastic and inelastic [6, 8, 9] shear moduli for steel may be taken as

MPa20761

MPa76923

iG

G (C8)

For inelastic torsional buckling

iGJW )( (C9)This equation can be solved iteratively for the inelastic buckling load Niwhich corresponds to a given set ofvalues of b, t,andfy.


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Fig. 1 Cruciform Section and Properties

t= 10 mm

b/t= 10, 20, 30

E= 2E5 N/mm2

Ei= 6E3 N/mm2

G= 76923 N/mm2

Gi= 20761 N/mm2

fy= 235 N/mm2

0.3fy

0.3fy

b

b

b

b

(a) Section (b) Properties

(c) Residual stresses

C C

T


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0 0.2 0.4 0.6 0.8 1.0 1.2

Modified slenderness oy= (Ny/Noy)

Fig. 2 Torsional and Local Buckling Loads

4.0

3.5

3.0

2.5

2.0

2.5

1.0

0.5

0

DimensionlessbucklingloadN

o/N

y

b/t= 10

b/t= 20

b/t= 30

TorsionalNoz/NyLocalNol/Ny

FlexuralNoy/Ny


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1.0

0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Modified slenderness ol= (Ny/Nol) = (Ny/Noz) = oz

Fig. 3 Local Buckling and Torsional Strengths

N

/Ny

Ns/Ny

Nol/Ny

Nfy/Ny

Ni/Ny

Npl/Ny

Equation 17

Equation 15


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Nd/Ns

Noy/Ns

Nfy/Ns

Ni/Ns

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Modified flexural slenderness ey= (Ns/Noy)

Fig. 4 Design Against Flexural Buckling

N/N

s

1.0

0.8

0.6

0.4

0.2

0


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0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Modified flexural slenderness ey=(Ns/Noy)

Fig. 5 Torsional Design [1] and First Yield

1.0

0.8

0.6

0.4

0.2

0

b/t= 10b/t= 20b/t= 30

Design (No= Noz)Design (No= Noy)

Noz/ Ns

Nfy/ NsNoy/ Ns

N/N

s


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w

ws

wf= w - ws

N

N

x

z

x

y

v = x b

b

b b

(a) Elevation

(b) Section

Fig. A1 Torsional Post-Buckling

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