Biaxial Bending of Steel Angle Section Beams

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Department of Civil EngineeringSydney NSW 2006

AUSTRALIA

http://www.civil.usyd.edu.au/

Centre for Advanced Structural Engineering

Biaxial Bending of

Steel Angle Section Beams

Research Report No R822

N S Trahair BSc BE MEngSc PhD DEng

September 2002


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Department of Civil Engineering

Centre for Advanced Structural Engineering

http://www.civil.usyd.edu.au/

Biaxial Bending of Steel Angle Section Beams

Research Report No R822

N S Trahair BSc BE MEngSc PhD DEng

September 2002

Abstract:The loads applied to angle beams usually act out of the principal planes so that theycause simultaneous biaxial bending about both principal axes. The general practicefor designing unbraced beams against biaxial bending is to consider the separatefailure modes of either in-plane bending or lateral buckling under bending about themajor principal axis, and in-plane bending about the minor principal axis, and then touse an interaction equation to combine the two principal axis strengths. However,the interaction equations provided by many codes for designing against biaxialbending are largely derived from research on doubly symmetric I-section beams,which may be inappropriate for angle section beams, while the lateral buckling rulesof these codes appear to be in error for non-uniform bending with loads acting awayfrom the shear centre.

This paper investigates the biaxial bending of unbraced steel angle beams. Thebiaxial bending of compact equal angles in uniform bending is considered first, and asimple interaction equation is developed for their design which utilises recentproposals for lateral buckling design. A corresponding interaction equation isdeveloped for the design of semicompact and slender equal angles. Suggestionsare then made for extending these to the biaxial bending of unequal angles undergeneral shear centre loading, and finally, a worked example illustrating the method isprovided.

Keywords:angles, beams, biaxial bending, buckling, design, elasticity, member capacity,moments, steel, torsion.


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Biaxial Bending of Steel Angle Section Beams September, 2002


Research Report No R822 2

Copyright Notice

Department of Civil Engineering, Research Report R822

Biaxial Bending of Steel Angle Section Beams

2002 N. S. Trahair

[email protected]

This publication may be redistributed freely in its entirety and in its original form without the

consent of the copyright 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:

Department of Civil EngineeringThe University of Sydney

Sydney NSW 2006

AUSTRALIA

September 2002

http://www.civil.usyd.edu.au


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INTRODUCTION

This paper is concerned with the member design of unbraced steel angle sectionbeams against biaxial bending. Previous related papers were concerned with thesection design of braced angle beams against biaxial bending (Trahair, 2002a) andagainst bearing, shear, and torsion (Trahair, 2002b), and the member design ofunbraced beams against lateral buckling (Trahair, 2002c).

The loads applied to steel angle beams usually act out of the principal planes andeccentrically from the shear centre as shown in Fig. 1, so that they causesimultaneous biaxial bending about both principal axes and torsion. However, mostdesign codes do not provide general guidance on design against torsion, nor on thespecific design of angle beams against biaxial bending and torsion, and so there is aneed to develop a rational set of rules for the general design of angle section beams.Part of such a set must be rules for the biaxial bending of angle section beams whichare loaded through the shear centre so that there are no primary torsion actions.The purpose of this paper is to provide suggestions for the design of such beams.

The general practice for designing unbraced beams against biaxial bending is to

consider the separate failure modes of either in-plane bending or lateral bucklingunder bending about the major principal axis, and in-plane bending about the minorprincipal axis, and then to use an interaction equation to combine the two principalaxis strengths. However, the interaction equations provided by many design codessuch as those of AISC (2000a,b), BSI (2000), and SA (1998) for designing againstbiaxial bending are largely derived from research on doubly symmetric I-sectionbeams, which may be inappropriate for angle section beams, which are either mono-symmetric or asymmetric. In addition, while the lateral buckling rules of these codesallow for mono-symmetry effects in beams under uniform bending, they appear to bein error for non-uniform bending with loads acting away from the shear centre. Arecent study (Trahair, 2002c) has developed proposals for the lateral buckling designof steel angle section beams which are loaded in the major axis principal plane, at oraway from the shear centre.

This paper investigates the biaxial bending of unbraced steel angle beams. Thebiaxial bending of compact equal angles in uniform bending is considered first, and asimple interaction equation is developed for their design which utilises the recentlateral buckling design proposals (Trahair, 2002c). A corresponding interactionequation is developed for the design of semi-compact and slender equal angles.Suggestions are then made for extending these to the biaxial bending of unequalangles under general shear centre loading. Finally, a worked example illustrating themethod is provided.


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FULLY PLASTIC BIAXIAL BENDING STRENGTHS OF EQUAL ANGLE BEAMS

General

The biaxial bending of compact simply supported equal angle beams in uniformbending is considered in this section. An approximate elastic non-linear analysis ofthe twist rotations of beams with initial twists is used to predict the maximum

principal plane bending moments. When these maximum moments reach the fullyplastic moment combinations, the beams are considered to have failed.

This simplistic method is an extension of a first yield method of strength prediction,which takes approximate account of the additional strength beyond first yield ofcompact beams which can reach full plasticity. It apparently ignores the effects ofresidual stresses and initial crookedness which cause early yielding and reducestrength. This is compensated for by using initial twists which are increasedsufficiently so that the analysis will predict the lateral buckling design strengthsproposed in Trahair (2002c).

The failure moments predicted by this method are used to develop a simple

interaction equation for the biaxial bending of compact equal angle beams whichcombines the lateral buckling design strengths of beams bent in the major axisprincipal plane with the full plastic moments of beams bent about the minor principalaxis.

The capacities of an equal angle section beam to resist bearing and shear may bechecked separately by comparing the appropriate design actions (which may bedetermined by a simple first-order analysis of the beam) with the correspondingdesign capacities recommended in Trahair (2001b).

Elastic Non-Linear Analysis For Small Rotations.

An elastic simply supported equal angle section beam of lengthL and initial twist

0 = 0 sin z/L (1)

in which 0 is the mid-span value of 0 and z is the distance along the beam, isshown in Fig. 2. The beam has equal and opposite end moments Mx, My whichcause uniform bending in the yz, xz principal planes. The small rotation non-lineardifferential equations of equilibrium for biaxial bending and torsion are

in whichE is the Youngs modulus of elasticity, G is the shear modulus,Ix andIy arethe second moments of area about the x, y principal axes, J is the torsion sectionconstant, u and v are the shear centre displacements parallel to the x,y principal axis

directions, is the angle of twist rotation, and indicates differentiation with respectto the distance z. In these equations, the left hand sides represent the internalresistances to bending and torsion, while the right hand sides represent the first- and

)2(

)2()(

)2()(

0

0

cvMuMGJ

bMMuEI

aMMvEI

yx

xyy

yxx

+=

+=

++=


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second-order actions resulting from the applied actions Mx and My and the smalldeflections and twist rotations.

The rotations u, v may be eliminated from Equation 2c by using Equations 2a and2b, whence

{ } { } )3()()( 00 +++= yxxy

xy

y

x

MMEI

M

MMEI

M

GJ

Approximate solutions 2 of this equation may be obtained by assuming that

2 / 2 = sin z/L (4a)

Mx My =Mx My sin z/L (4b)

whence

)5(

1

11

2

2

2

2

222

2

2

2

0

2

+

=

xz

y

yz

x

xzyz

yx

xz

y

yz

x

M

M

M

M

MMMM

MM

MM

in which

Myz= (2EIy GJ/L

2) (6a)

Mxz= (2EIx GJ/L

2) (6b)

It should be noted that the greatest absolute values of 2 will correspond to negative

values of 0. The second-order bending moments are greatest at mid-span, and canbe obtained using Equations 2a and 2b as

)7()(

)7()(

022

022

bMMM

aMMM

xyy

yxx

+=

++=


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Fully Plastic Moment Combinations.

The combinations of principal axis moments Mpx, Mpy which cause full plasticity of acompact equal angle are given by the fully plastic interaction equations (Trahair,2002a)

Mpy/Mpym = 1 (Mpx /Mpxm)2 (8)

in which the principal axis full plastic momentsMpxm,Mpym are given by

Mpxm = 2Mpym =fy b2t/ 2 (9)

in whichfy is the yield stress and b and tare the leg length and thickness of the equalangle section. These combinations are shown by the solid curve in Fig. 3.

Elastic Lateral Buckling and Lateral Buckling Design Proposals.

Elastic lateral buckling

The value of the major axis uniform bending moment at elastic buckling Myz is givenby Equation 6a (Trahair, 2002c).

Lateral buckling design strength

It has been proposed (Trahair, 2002c) that the nominal design lateral buckling

moment capacityMb of an angle section beam should be obtained from

in which

as shown in Fig. 4, in which Msxm and Msym are the major and minor axis maximum

section moment capacities, m is a moment modification factor which allows for thevariation of the bending moment distribution (Trahair, 2002c), and Mquy is themaximum moment in the beam at elastic buckling. For a simply supported compact

equal angle in uniform bending,Msxm =Mpxm,Msym =Mpym,Mquy =Myzand m = 1.

)10()(

)10()()(

)()(

)10()(

cMM

bMMMM

aMM

eeysymb

eyeex

exey

exesymsxmsxmb

exesxmb

=

=

=

)13(/

)12(/

)11()7.0(

22.099.0

quysxme

symsxmey

m

ex

MM

MM

==

=


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The modified slenderness limit ex in Equation 11 is an approximation for the value of

(Msxm / Myz) at which Mb = Msxm according to the Australian design code AS 4100

(SA, 1998). Equation 10a uses the major axis section capacity Msxm for low

slenderness beams ( e ex), while Equation 10c uses the minor axis section

capacity Msym for high slenderness beams ( ey e), which is based on the finding of

Trahair (2002c) that the moment capacity is never less than Msym. Equation 10b

provides a simple linear interpolation between Msxm and Msym for beams ofintermediate slenderness ( ex e ey), which provides a close but conservative

approximation to the predictions of Trahair (2002c).

Equivalent Initial Twists.

It is desirable that the initial twist 0 of Equation 1 should be sufficiently large that itwill represent the effects of residual stresses and initial crookednesses and twists onthe strengths of real beams when it is used with the elastic second-order predictionsof Equations 7 to determine the biaxial bending strengths of equal angle sectionbeams. Such initial twists will also predict the lateral buckling design strengths of

unbraced beams bent in their major axis principal plane. Thus the magnitudes 0 ofthe initial twist of a compact beam can be determined by using My = 0 in Equation 5and substituting the second-order moments of Equations 7 into the fully plasticinteraction equations of Equations 8 and 9, whence

)14(/1

22

002

yzb MM=+

and

Mb(2 + 0) /Mpym = 1 (Mb /Mpxm)2 (15)

so that

)16(/2

})/(1}{)/(1{242

0

pxmb

pxmbepxmb

MM

MMMM =

The variation of0 with the modified slenderness e is shown by the circled points inFig. 5, together with the close approximation

0 = 0.1116 + 0.3612 e + 0.3551 e2 0.3935 e

3 (17)


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Maximum Moments At Full Plasticity.

An example of the variations of the maximum moments in an equal angle beam

having a modified slenderness of e = 0.835 so that Mb / Mpxm = 0.75 and

0 = 0.212 rad. is shown in Fig. 6 for first-order biaxial bending moments defined byMx/ My = 2.414. The variations of the first-order momentsMx,My are shown by thedotted straight line, and those of the elastic second-order moments Mx2, My2 by the

curved dashed line. The curved line reaches the solid line representing the fullyplastic moment combinations of Equations 8 and 9 when Mx2 = 0.475 Mpxm,

My2 = 0.388 Mpxm, (the circled Point A) corresponding to first-order moments ofMx = 0.54Mpxm, My = 0.225Mpxm (the squared Point B).

The results for similar determinations of the values of the first-order momentcombinations for which the elastic second-order moment combinations cause fullplasticity are shown in Figs 3 and 6 for first-order bending moments defined by

Mx/My = , 2.414, 1.0, and 0.

Proposed Design Interaction Equation.

Close approximations for the first-order moment combinations Mx,My shown in Fig. 6whose second-order moment combinations Mx2, My2 cause full plasticity can beobtained by reducing the full plastic momentsMpx,Mpy toMx,My given by

)18(2/

11

==

pxm

b

py

y

px

x

M

M

M

M

M

M

in which= tan 1 (Mpx /Mpy) (19)

andMb is the appropriate lateral buckling moment strength obtained from Equations10-13.

This approximation is shown by the dash-dot lines in Figs 3 and 6, and is comparedwith the first-order moment combinations for which the approximate second-ordermoment combinations cause full plasticity (the squared points). It is suggested thatthis approximation can also be used for compact equal angle beams with general

loading through the shear centre by using the moment modification factor m andmaximum moment Mquy at elastic buckling given by Trahair (2002c) in the

determination of the lateral buckling strengthMb.


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FIRST YIELD BIAXIAL BENDING STRENGTHS OF EQUAL ANGLE BEAMS

General

The first yield of simply supported equal angle beams in uniform biaxial bending isconsidered in this section. An approximate elastic non-linear analysis of the twistrotations of beams with initial twists is used to predict the maximum principal plane

bending moments. The small elastic second-order rotations 2 are given byEquation 5, and the maximum second-order moments by Equations 7.

When these maximum moments reach the moment combinations corresponding tothe first yield moment combinations, the beams are considered to have failed. Thefailure moments predicted by this method are used to develop a simple interactionequation for the first yield biaxial bending strengths of equal angle beams whichcombines the lateral buckling design strengths of beams bent in the major axisprincipal plane with the first yield moments of beams bent about the minor principalaxis.

First Yield Moment Combinations

The combinations of principal axis moments Myx, Myy which cause first yield of anequal angle are given by the interaction equations (Trahair, 2002a)

Myy/Myym = 1 Myx /Myxm (20)

in which the principal axis first yield momentsMyxm,Myym are given by

Myxm = 2Myym =fy b2t( 2 / 3) (21)

These combinations are shown by the solid curve in Fig. 7.

Equivalent Initial Twists

It is desirable that the initial twist 0 of Equation 1 should be sufficiently large that itwill represent the effects of residual stresses and initial crookednesses and twists onthe strengths of real beams when it is used with the elastic second-order predictionsof Equations 7 to determine the biaxial bending strengths of equal angle sectionbeams. Such initial twists will also predict the lateral buckling design strengths of

unbraced beams bent in their major axis principal plane. Thus the magnitude 0 of

the initial twist of a compact beam can be determined by using My = 0 in Equation 5and substituting the second-order moments of Equations 7 into the first yieldinteraction equations of Equations 20 and 21, which leads to Equation 14 and

Mb(2 + 0) /Myym = 1 Mb /Myxm (22)

so that

)23(/2

}/1}{)/(1{42

0

yxmb

yxmbeyxmb

MM

MMMM =


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The variation of0 with the modified slenderness e is shown by the triangled pointsin Fig. 5, together with the close approximation

0 = 0.0358 + 0.0499 e + 0.4262 e2 0.3142 e

3 (24)

Maximum Moments At First Yield

The values of the first-order moment combinations for which the elastic second-ordermoment combinations cause first yield are shown by the squared points in Fig. 7 for

first-order bending moments defined byMx/My = , 2.414, 1.0, and 0.

Proposed Design Interaction Equation.

Close approximations for the first-order moment combinations Mx,My shown in Fig. 7whose second-order moment combinationsMx2,My2 cause first yield can be obtainedby reducing the first yield momentsM

yx,M

yytoM

x,M

ygiven by

)25(15.05.2

=

yym

y

yxm

y

yxm

b

yxm

b

yxm

x

M

M

M

M

M

M

M

M

M

M

This approximation is shown by the dash-dot lines in Fig. 7, and is compared withthe first-order moment combinations for which the approximate second-ordermoment combinations cause first yield (the squared points). It is suggested that thisapproximation can also be used for equal angle beams with general loading through

the shear centre by using the moment modification factorm and maximum moment

Mquy at elastic buckling given by Trahair (2002c) in the determination of the lateralbuckling strengthMb.


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LOCAL BUCKLING EFFECTS

Section Classification and Capacities

The effects of local buckling on the section moment capacities of angle sectionbeams have been discussed in Trahair (2002a). In that paper, sections areclassified as being plastic, compact, semi-compact or slender by comparing their

long leg plate slendernesses

with the limiting slenderness values given in Table 1.

Table 1 Local Buckling Slenderness Limits

Bending moment Plastic limit p Compact limit c Yield limit yMx 12 16 26M

y 10 14 23

A plastic section must have sufficient rotation capacity to maintain a plastic hingeuntil a plastic collapse mechanism develops. A plastic section satisfies

p (27)

in which p is the plasticity limit. A compact section must be able to form a plastic

hinge. A compact section satisfies

p < c (28)

in which c is the compact limit. The nominal section moment capacityMs of a plasticor compact section is equal to its fully plastic capacityMp, so that

Ms =Mp (29)

A slender section has its moment capacity reduced below the first yield moment bylocal buckling effects. A slender section satisfies

y < (30)

in which y is the yield limit. The nominal section moment capacity of a slender

sectionMs is approximated by

Ms =My ( y / )2 (31)

in whichMy is the first yield capacity.

)26(250

yf

t

b=


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A semi-compact section must be able to reach the first yield moment, but localbuckling effects may prevent it from forming a plastic hinge. A semi-compact sectionsatisfies

c < y (32)

The nominal section moment capacity Ms of a semi-compact section is given by the

linear interpolation between the full plastic and first yield capacities of

Biaxial Bending Strengths

The approximation of the biaxial bending strengths of compact equal angle beams

( c) by the modified full plasticity interaction equation of Equation 18 is

demonstrated in Fig. 3.

It is suggested that the biaxial bending strengths of semi-compact and slender equal

angle beams ( y ) may be approximated by further modifying the first yield

interaction equation of Equation 25 to

)34(15.05.2

=

sym

y

sxm

y

sxm

b

sxm

b

sxm

x

M

M

M

M

M

M

M

M

M

M

in which Msxm is obtained from Equation 33 or 31 and Mb includes an allowance forthe effect of local buckling through the use of reduced capacities Msxm and Msym inEquation 10.

BIAXIAL BENDING OF UNEQUAL ANGLE BEAMS

Some dimensionless full plastic and first yield moment combinations (Trahair, 2002a)for an extreme unequal angle section b x 0.5b x t are shown in Fig. 8. Thesecombinations are point symmetric about the origin rather than symmetric about theaxes as a result of the asymmetry of the unequal angle section. Also shown areindications of the correspondences between the regions of the figure and thedirections of the resultant loads which cause the moment combinations. Thedirections of the corresponding moment vectors are perpendicular to the loaddirections. It can be seen that the full plastic combinations are substantially largerthan the first yield combinations.

)33()(

)()(

py

p

ypps MMMM

=


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It is suggested that the first-order moment combinations Mx,My whose second-ordermoment combinations Mx2, My2 cause failure may be approximated by using thegreater of the sets of values obtained from

)35(2

4.02

4.111

2

==

sxm

b

sy

y

sx

x

M

M

M

M

M

M

and

)36(1

=

sym

y

sxm

b

sxm

x

M

M

M

M

M

M

These approximations are shown in Fig. 8 for the extreme cases where Mb =Msym for

uniform bending of unequal angle section beams with = 0.5. They will beconservative if applied to equal angle section beams instead of Equation 18 (forcompact beams) or 34 (for semi-compact or slender beams).

EXAMPLE

Problem

A 150 x 100 x 12 unequal angle beam is shown in Fig. 9. The section propertiescalculated using THIN-WALL (Papangelis and Hancock, 1997) for the thin-wall

assumption ofb = 144 mm, b = 94 mm, and t= 12 mm are shown in Fig. 9b. The

unbraced beam is simply supported over a span ofL = 6 m, and has a designuniformly distributed vertical load ofq* = 6 kN/m acting parallel to the long leg andthrough the shear centre at the leg junction, as shown in Fig. 9b.

The first-order analysis of the beam, the determination of the elastic buckling

moment of the beam and the lateral buckling design strength, and the check of thebiaxial bending capacity are summarised below. The checking of the bearing andshear capacities are summarised in Trahair (2002 b).

Elastic Analysis

The design major axis bending moments are

Mx* = (q* L2 / 8) cos = 24.7 kNm,My* = (q* L

2 / 8) sin = 10.9 kNm.

Moment at Elastic Buckling

The elastic buckling moment calculated in Trahair (2002c) for a load acting at 19.0mm below the shear centre isMquy = 30.6 kNm. However, in the present example,the load acts at the shear centre, and so the elastic buckling moment decreases to

Mquy = 29.9 kNm.


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Lateral Buckling Design Strength

The angle section has been shown to be compact (Trahair, 2002a).The lateral buckling moment strength calculated in Trahair (2002c) forMquy = 30.6kNm isMb = 25.0 kNm. However, in the present example with a reduced value of

Mquy = 29.9 kNm, the lateral buckling moment strength decreases toMb = 24.7 kNm.

Biaxial Bending Capacity

Using Trahair (2002c),Mpxm = 38.4 kNm,Mpym = 15.5 kNm.Using Trahair (2002a),MpX = 27.4 kNm andMpY= 0 kNm.

ThusMpx = 25.0 kNm,Mpy = 11.1 kNm, = 1.154 rad., and 2 / = 0.734.

Using Equation 35,Mx = 25.0 x {1 (1 24.7 / 38.4) (1.4 x 0.734 0.4 x 0.7342)}

= 17.8 kNmUsing Equation 36,Mx = 24.7 {1-Mx x (11.1 / 25.0) / 15.5} = 14.5 kNm < 17.8 kNm.

ThusMx

= 17.8 kNm and Mx

= 16.0 kNm (using = 0.9).This is less than the design momentMx* = 24.7 kNm, and the beam is inadequate.

CONCLUSIONS

This paper develops rational, consistent, and economical design methods fordetermining the biaxial bending strengths of unbraced steel angle section beamsloaded through the shear centre, and illustrates their use in a design example.

An approximate small rotation non-linear elastic analysis is used to predict themaximum moments in equal angle beams in uniform bending. The beams have initial

twists. The maximum strengths are assumed to be reached when the maximumpredicted moments cause either full plasticity or first yield of the cross-section(Trahair, 2002a). The magnitudes of the initial twists are chosen so that thepredicted strengths of beams bent in the major principal plane are equal to recentrecommendations for the lateral buckling strengths (Trahair, 2002c).

The biaxial bending strengths of compact section beams are predicted assuming fullplasticity at the maximum moment section, and a simple design approximation isdeveloped. The biaxial bending strengths of semi-compact and slender sectionbeams are based on predictions assuming first yield at the maximum momentsection, and a simple design approximation is developed. These designapproximations are formulated so that they can be used for equal angle beams

under general loading.

The first yield and fully plastic behaviour of unequal angle section beams is thenconsidered, and conservative design approximations are developed from those forequal angle section beams.

Proposals have been made elsewhere (Trahair, 2002b) for checking the bearing andshear capacities of angle section beams.


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APPENDIX 1 REFERENCES

AISC (2000a), Specification for Load and Resistance Factor Design of Single-Angle Members,American Institute of Steel Construction, Chicago.

AISC (2000b), Commentary on theSpecification forLoad and Resistance FactorDesign of Single-Angle Members, American Institute of Steel Construction,

Chicago.

BSI (2000), BS5950 Structural Use of Steelwork in Building. Part 1:2000. Code ofPractice for Design in Simple and Continuous Construction: Hot Rolled Sections,British Standards Institution, London.

Papangelis, JP and Hancock, GJ (1997), THIN-WALL Cross-section Analysisand Finite Strip Buckling Analysis of Thin-Walled Structures, Centre for AdvancedStructural Engineering, University of Sydney.

SA (1998),AS 4100-1998 Steel Structures, Standards Australia, Sydney.

Trahair, NS (2002a), Moment Capacities of Steel Angle Sections, Journal ofStructural Engineering, ASCE, 128 (11), in press.

Trahair, NS (2002b), Bearing, Shear, and Torsion Capacities of Steel AngleSections, Journal of Structural Engineering, ASCE, 128 (11), in press.

Trahair, NS (2002c), Lateral Buckling Strengths of Steel Angle Section Beams,Journal of Structural Engineering, ASCE, accepted for publication.

Trahair, NS, Bradford, MA, and Nethercot, DA (2001), The Behaviour and Designof Steel Structures to BS5950, 3rd British edition, E & FN Spon , London.


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APPENDIX 2 NOTATION

b long leg lengthE Youngs modulus of elasticitye eccentricity of load from the shear centre

fy yield stress

G shear modulus of elasticityIx,Iy second moments of area about thex,y principal axesJ torsion section constantL span lengthMb lateral buckling moment strengthMpx,Mpy values ofMx,My at full plasticityMpxm,Mpym maximum values ofMpx,MpyMquy maximum moment at elastic lateral bucklingMs nominal section moment capacityMsx,Msy values ofMx,My at section capacityMsxm,Msym maximum values ofMsx,MsyMx,My moments about thex,y principal axesMx*,My* design moments about thex,y principal axesMxz,Myz uniform bending elastic buckling momentsMx2,My2 second-order moments about thex,y principal axesMyx,Myy values ofMx,Myat first yieldMyxm,Myym maximum values ofMyx,Myyq intensity of uniformly distributed loadq* design intensity of uniformly distributed loadt leg thicknessu, v shear centre deflections parallel to thex,y principal axes

x,y principal axesX, Y rectangular (geometric) axes

Xc, Yc X, Ydistances from centroid to shear centrez distance along beam

inclination ofx principal axis toXrectangular (geometric) axis

m moment modification factor

leg length ratio

x monosymmetry section constant

long leg local buckling slenderness

c, p, y compact, plasticity, and yield local buckling slenderness limits

e modified slenderness for beam lateral buckling

ex, ey beam lateral buckling slenderness limits

angle of twist rotation, or

capacity factor0 initial angle of twist

2 second-order angle of twist rotation

0 maximum value of0

2 maximum value of2

angle betweenx axis and resultant moment


18/26




z

L

Y

q

(a) Elevation

x

XC

Y

t

y

b

Yc

t

Xc

b

(b) Cross-section

Fig. 1. Eccentrically Loaded Angle Section Beam

(c) Loading

qq

e e

Flange down Flange up


19/26




Mx

20

My

x

yu

v

X

Y

(a) Elevation

L

Mx Mx z

My

My

y

(b) Plan

L

My My z

Mx

Mx

x

(c) Section

Fig. 2. Simply Supported Equal Angle in Uniform Bending


20/26




1.414

1.124

0.835

0.545

1.0

0.8

0.6

0.4

0.2

e 0.257

Approximation (Equation 18)Second-order calculations

Mx

/Mpxm

0 0.1 0.2 0.3 0.4 0.5

My / Mpym

Mx Mx

MyMy

Fig. 3. Biaxial Bending Strengths of Compact Equal Angles


21/26




1.0

0.8

0.6

0.4

0.2

0

0 0.5 1.0 1.5 2.0 2.5 3.0

Modified slendernesse = (Msxm /Mquy)

Fig. 4. Lateral Buckling Design Strengths

Proposeddimensionlessmome

ntcapacityMb

/M

sxm

ex

Msxm /Msxm

Elastic buckling

Mb /Msxm

Mquy /Msxm

Msym /Msxm

e ey


22/26




Compact Slender

Approximate

Accurate

0.25

0.20

0.15

0.10

0.05

0

Modified slenderness e = (Msxm /Myz)0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Fig. 5. Equivalent Initial Twist Rotations

Equivalentinitialtw

istrotation

0

(rad.)


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B

2

0

0 0.1 0.2 0.3 0.4 0.5

My /Mpxm

1.0

2.414

Mx /Mp = 0

Mx /Mp =

Mx

/M

pxm

A

Mb /Mpxm

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.1

e = 0.835Mb /Mpxm = 0.75

Elastic second-orderMy2 /Mpxm, Mx2 /Mpxm

Elastic first-orderMy /Mpxm, Mx /Mpxm

Biaxial bending strength approximation

Fully plasticMpy /Mpxm,Mpx /Mpxm

Fig. 6. Determination of Biaxial Bending Strengthsof Compact Equal Angles


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1.0

0.8

0.6

0.4

0.2

0

1.414

1.124

0.835

0.545

e 0.257

Fig. 7. Biaxial Bending First Yield Strengths of Equal Angles

Mx

/Myxm

My/ Myxm

0 0.1 0.2 0.3 0.4 0.5

Mx Mx

MyMy

Approximation

Second-order calculations


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FullyPlastic

FirstYield

e< 0.26

e> 2.08

0.5

0.4

0.3

0.2

0.1

Mpy / fy b2t, Myy / fy b

2t

0.2 0.1 0 0.1 0.2

Mpx

/fy

b2t,

Myx

/fyb

2

t

= 0.5

Fig. 8. Moment Combinations for an Unequal Angle


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150 x 100 x 12 UA

(b xb x t= 144 x 94 x 12)

E= 200,000 MPaG = 80,000 MPafy = 300 MPa

= 23.91Ix= 7.548 E6 mm

4

Iy = 1.314 E6 mm4

J= 0.1371 E6 mm4

yo = 32.30 mm

x = 78.33 mm

Fig. 9 Example

(b) Section

q*

(a) Elevation

q* = 6.0 kN/m

L = 6000 mm

Documents

Biaxial Bending of Steel Angle Section Beams