buckling and torsion of steel angle section beams

7/30/2019 Buckling and Torsion 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

Buckling and Torsion of Steel Angle

Section Beams

Research Report No R833

N S Trahair BSc BE MEngSc PhD DEng

October 2003


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Department of Civil Engineering

Centre for Advanced Structural Engineering

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

Buckling and Torsion of Steel Angle Section Beams

Research Report No R833

N S Trahair BSc BE MEngSc PhD DEng

October 2003

Abstract:

Although steel single angle sections are commonly used as beams to supportdistributed loads which cause biaxial bending and torsion, their behaviour may beextremely complicated, and the accurate prediction of their strengths very difficult.Further, many design codes do not have any design rules for torsion, while somerecommendations are unnecessarily conservative, or are of limited application, or failto consider some effects which are thought to be important.

This paper is one of a series on the behaviour and design of single angle sectionsteel beams. Two previous papers have studied the biaxial bending behaviour ofrestrained beams, a third has studied the lateral buckling of unrestrained beams, anda fourth the biaxial bending of unrestrained beams. In each paper, simple designmethods have been developed.

In this present paper, an approximate method of predicting the second-orderdeflections and twist rotations of steel angle section beams under major axis bendingand torsion is developed. This method is then used to determine the approximatemaximum biaxial bending moments in such beams, which are then used with the

section moment capacity proposals of the first paper of the series and the lateralbuckling proposals of the third paper to approximate the member capacities.

Keywords:

Angles, beams, bending, buckling, design, elasticity, member capacity, moments,section capacity, steel, torsion.


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Buckling and Torsion of Steel Angle Section Beams October 2003

Department of Civil EngineeringResearch Report No R833

2

Copyright Notice

Department of Civil Engineering, Research Report R833

Buckling and Torsion of Steel Angle Section Beams

2003 N. S. [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 Engineering

The University of Sydney

Sydney NSW 2006

AUSTRALIA

October 2003

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


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3

1 INTRODUCTION

Although the geometry and loading of single angle section steel beams are usuallycomparatively simple as shown in Fig. 1, their behaviour may be extremelycomplicated, and the accurate prediction of their strengths very difficult.

A single angle section beam is commonly loaded eccentrically in a plane inclined tothe principal planes (Fig. 1), so that the beam undergoes primary bending and shearabout both principal axes, torsion, and bearing at the supports. Even in the veryunusual cases when only one of these actions occurs, the primary fully plastic or firstyield capacities may be reduced by local or lateral buckling effects, or increased bystiffening resistances which become important at large rotations (Farwell andGalambos, 1969; Pi and Trahair, 1995). When all of these actions occur, as theyusually do, there are first- and second-order interactions between them, some ofwhich are very difficult to predict.

This paper is one of a series on the behaviour and design of single angle section

steel beams. The first of these (Trahair, 2002a) considered the first-order (smalldeformation) elastic analysis of the biaxial bending (without torsion) of angle sectionbeams including the effects of elastic restraints, and developed proposals for thesection moment capacities which approximate the effects of full plasticity in compactsections, first yield in semi-compact sections, and local buckling in slender sections.A companion paper (Trahair, 2002b) developed proposals for the bearing, shear,and uniform torsion section capacities. The proposals of these two papers may beused to design steel single angle section beams which are laterally restrained asshown in Fig. 2a, so that lateral buckling or second-order effects are unimportant.

A third paper (Trahair, 2003a) considered the lateral buckling strengths ofunrestrained steel single angle section beams which are loaded in the majorprincipal plane as shown in Fig. 2b, so that there are no primary minor axis bendingor torsion effects. That paper developed an approximate method of predicting theelastic second-order twist rotations of an equal angle beam in uniform bending,whose accuracy was investigated subsequently in a study (Trahair, 2003c) of theelastic non-linear large rotation behaviour of beams under non-uniform torsion. Theapproximate predicted rotations were used with the earlier formulations (Trahair,2002a) of the fully plastic biaxial bending moment capacities to investigate the lateralbuckling strengths of angle section beams with initial twist rotations whichapproximated the effects of geometrical imperfections (initial crookedness and twist)

and residual stresses. The investigation included simple design proposals whichapproximated the predicted lateral buckling strengths.

A fourth paper (Trahair, 2003b) considered the biaxial bending of unrestrained steelangle section beams which are loaded through the shear centre as shown in Fig. 2c,so that there are no primary torsion actions. This paper also developed anapproximate method of analyzing the elastic second-order twist rotations of an equalangle section beam in uniform bending, and used the predicted rotations to developsimple approximations for the biaxial bending design strengths of equal or unequalangle section beams.


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4

The present paper is an extension of the third paper (Trahair, 2003a) on the lateralbuckling strengths of single angle section beams (Fig. 2b). In this extension, thebeam is considered to be loaded in a plane parallel to the major principal plane buteccentric from the shear centre as shown in Fig. 2d, so that there are primary majoraxis bending and torsion actions, but no primary minor axis bending actions. Again,an approximate analysis is developed for predicting the elastic second-order twist

rotations of an equal angle section beam under idealized loading, and used topredict the effects of torsion on beam lateral buckling strengths and to developsimple proposals for the design of beams under major axis bending and torsion.This paper may serve as a starting point for research into the general biaxial bendingand torsion of unrestrained single angle steel beams, such as that shown in Fig. 1.

2 MEMBER DESIGN METHODOLOGY

The uniform bending and torsion of unrestrained simply supported equal angle steel

beams is considered in the following sections. An approximate elastic non-linearanalysis of the small twist rotations of beams with initial twists is used to predict themaximum principal plane bending moments. Compact beams (Trahair, 2002a) areconsidered to have failed when these maximum moments reach the fully plasticmoment combinations, and semi-compact beams (Trahair, 2002a) when thesemoments reach the first yield combinations.

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. Similar methods have been used topredict the lateral buckling and biaxial bending strengths of single angle beams

(Trahair, 2003a,b). The method apparently ignores the effects of residual stressesand initial crookedness which cause early yielding and reduce strength. It alsomakes small rotation approximations which generally overestimate the principalplane moments. These are compensated for by using initial twists which areincreased sufficiently so that the small rotation analysis will predict the lateralbuckling design strengths proposed in Trahair (2003a).

This method also ignores the reductions in the torque resultants of eccentricallyapplied loads which occur at finite rotations. For example, the load eccentricityshown in Fig. 3a decreases towards zero as the twist rotation of the equal anglesection increases towards 45o. A different example is shown in Fig. 3b, where the

point of application of the eccentric load (which does not rotate) moves suddenlytowards the shear centre at the leg junction when twist rotation commences. It isconservative to ignore these reductions.

The failure moments predicted by this method are used to develop a simple methodof designing unrestrained equal angle beams against major axis bending and torsionwhich combines the lateral buckling design strengths Mb of beams bent in the majoraxis principal plane with the first-order maximum twist rotations 1.

The capacities of equal angle section beams to resist bearing, shear, and uniformtorsion may be checked separately by comparing the appropriate design actions(which may be determined by a simple first-order analysis of the beam) with thecorresponding design capacities recommended in Trahair (2002b).


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5

3 BUCKLING AND TORSION OF EQUAL ANGLE BEAMS

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

0 = 0 sin(z/L) (1)

is shown in Fig. 4. The beam has equal and opposite major end moments Mcausing uniform bending Mx =M in the yzprincipal plane and a uniformly distributedtorque per unit length m. The small deformation differential equations of equilibriumfor biaxial bending and torsion are

in whichMz= m L / 2 m z (3)

is the variation of the axial torque caused by the distributed torque m,Eand G arethe Youngs and shear moduli of elasticity, Ix andIy are the second moments of areaabout the principal x, y axes, J is the torsion section constant, u and v are the shear

centre deflections in thex,y directions, is the angle of twist rotation, and ' indicatesdifferentiation with respect to the distancezalong the beam.

In Equations 2, the left hand sides represent the internal resistances to bending andtorsion, while the right hand sides represents the first- and second-order actions

resulting from the applied actions Mx and Mz and the small deflections u, v and twistrotations . The third of Equations 2 omits a large twist rotation resistanceEIn(')

3/2(Trahair, 2003c) because the non-linear Wagner section constant In is quite smallfor equal angle sections (In = b

5t / 90, rather than the erroneous 8 b5t / 45 used inTrahair (2003a), in which b is the leg length and tthe thickness of the section).

The first-order solutions v1, u1, 1 of Equations 2 which satisfy the boundary

conditions for simple supports are obtained by ignoring the terms u', v', and (+ 0)on the right hand sides, whence

in which

)2(

1'

')(

'1

'

"

"

0

+

=

z

x

y

x

M

M

u

v

u

GJ

uEI

vEI

)5(8/

)5(0

)5(8/

2

1

1

2

1

cGJmL

b

aEIML

u

xv

=

==

)4()/4/4(

)4()/4/4()4()/4/4(

22

11

22

11

22

11

cLzLz

bLzLzuaLzLzv

u

v

=

==


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6

Approximate second-order solutions may be obtained by assuming that

Mzu' =Mzv' = 0 (6a)

v2 /v2 = u2 /u2 =2 / 2 = sin (z/L) (6b)

and by making the replacements

Equations 6a are based on the finding of Trahair and Teh (2001) that the effects ofthese second-order moment components of applied torques are small and can beignored. The sine wave shape approximations of Equations 6b are close to the first

order shapes of Equations 4. The replacements of Equations 7 incorporate sine orcosine shapes for terms which are either constant or else are approximately of sineor cosine shape.

Thus

in which

Myz = (2EIyGJ/L

2) (9)

The approximate Equations 8 disagree slightly with the first-order solutions ofEquations 5 when the second-order quantities are ignored. Agreement can beobtained by adjusting Equations 8 to

The bending moments are greatest at mid-span, and can be obtained usingEquations 2 and 6a as

)11()(

)11(

022

2

bMM

aMM

y

x

+==

)7(

')/cos()/(

)/sin(

)/cos()2/(

)/sin(

2

2

Mu

M

M

M

for

LzLM

LzM

LzmL

LzM

z

u

)8(/1

)/(2/

)8()//(

)8()//(

22

0

222

2

22

22

22

2

cMM

MMGJmL

bLEIM

aLEIM

yz

yz

yu

xv

+=

=

=

)10(/1

)/(

)10()//(

)10(

22

0

22

1

2

22

22

12

cMM

MM

bLEIM

a

yz

yz

yu

vv

+==

=


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7

4 FULLY PLASTIC BUCKLING AND TORSION STRENGTHS OF EQUALANGLE BEAMS

4.1 Fully Plastic Moment Combinations.

The combinations of principal axis moments Mpx,Mpy which cause full plasticity of an

equal angle are given by the fully plastic interaction equation (Trahair, 2002a)

Mpy/Mpym = 1 (Mpx /Mpxm)2 (12)

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

Mpxm = 2Mpym =fy b2t/ 2 (13)

in whichfy is the yield stress.

4.2 Elastic Lateral Buckling and Lateral Buckling Design Proposals.

4.2.1 Elastic lateral buckling

The value of the major axis uniform bending moment at elastic buckling Myz is givenby Equation 9 (Trahair, 1993, 2003a).

4.2.2 Lateral buckling design strength

It has been proposed (Trahair, 2003a) that the nominal design lateral bucklingmoment capacityMb of an angle section beam should be obtained from

in which

as shown in Fig. 5, 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, 1993, 2003a), and Mquy is themaximum moment in the beam at elastic buckling (Trahair, 2003a). For a simplysupported compact equal angle in uniform bending, M

sxm= M

pxm, M

sym= M

pym,

Mquy =Myz and m = 1.

)14()(

)14()()(

)()(

)14()(

cMM

bMMMM

aMM

eeysymb

eyeex

exey

exesymsxmsxmb

exesxmb

=

=

=

)17(/

)16(/

)15()7.0(

22.099.0

quysxme

symsxmey

m

ex

MM

MM

=

=

=


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8

The modified slenderness limitex in Equation 15 is an approximation for the value of(Msxm / Myz) at whichMb =Msxm according to the Australian design code AS 4100(SA, 1998). Equation 14a uses the major axis section capacity Msxm for low

slenderness beams (e ex), while Equation 14c uses the minor axis sectioncapacity Msym for high slenderness beams (ey e), which is based on the finding ofTrahair (2003a) that the moment capacity is never less than Msym. Equation 14b

provides a simple linear interpolation between Msxm and Msym for beams ofintermediate slenderness (ex e ey), which provides a close but conservativeapproximation to the predictions of Trahair (2003a).

4.3 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 predictions

of Equations 11 to determine the buckling and torsion strengths of equal anglesection beams. Such initial twists will also predict the lateral buckling designstrengths of unbraced beams bent in their major axis principal plane. The

magnitudes 0 of these initial twists of compact beams have been determined inTrahair (2003b), and are closely approximated by

0 = 0.1116 + 0.3612 e + 0.3551 e2 0.3935 e3 0 (18)

4.4 Member Actions for Full Plasticity

The values of the dimensionless applied moment M / Mpxm and the maximum first-order elastic twist rotation 1 at which the second-order principal axis moments Mx2,

My2 cause full plasticity can be determined by combining Equations 10c, 11, 12, and13 as

pxm

pxmepxm

MM

MMMM

/2

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

242

01

=+ (19)

in which 0 is given by Equation 18, and by solving iteratively. The variations ofM/Mpxm withe for given values of1 are shown in Fig. 5. For this figure, the values

ofM / Mpxm which are less than 0.5 have been replaced by 0.5. This is consistentwith the use of dimensionless lateral buckling strengths Mb / Mpxm which are neverless than 0.5 (Trahair, 2003a). This use is appropriate because in the worst casewhen the twist rotation reaches 90o, the applied moment will act about the minorprincipal axis, in which case the resistance will be equal to the minor axis full plasticmomentMpym = 0.5Mpxm.


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9

The values ofM/Mpxm shown in Fig. 5 for1 = 0 are very close to the dimensionlesslateral buckling strengths of Equations 14-17, demonstrating the ability of theequivalent initial twists of Equation 18 to represent the effects of residual stressesand initial crookednesses and twists. For the special case ofe = 0 and 0 = 0, thedimensionless values ofM / Mpxm vary with the maximum first order elastic twistrotation 1 according to

(M/Mpxm)0 = (12 + 1) - 1 (20)

Simple but conservative approximations of the values ofM / Mpxm and 1 at fullplasticity can be obtained by using

)21(

0

=

pxm

pym

pxm

b

pxmpxm M

M

M

M

M

M

M

M

as shown in Fig. 5.

5 FIRST YIELD BUCKLING AND TORSION STRENGTHS OF EQUAL ANGLESECTION BEAMS

5.1 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 (22)

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

Myxm = 2Myym =fy b2t(2 / 3) (23)

5.2 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 predictions

of Equations 11 to determine the buckling and torsion strengths of equal anglesection beams. The magnitudes 0 of the initial twists of semi-compact unbracedbeams bent in their major axis principal plane which will predict their lateral bucklingdesign strengths have been determined in Trahair (2003b), and are closelyapproximated by

0 = 0.0358 + 0.0499 e + 0.4262 e2 0.3142 e3 0 (24)


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10

5.3 Member Actions for First Yield

The values of the dimensionless applied moment M / Myxm and the maximum first-order elastic twist rotation 1 at which the second-order principal axis moments Mx2,

My2 cause first yield can be determined by combining Equations 10c, 11, 22, and 23as

)25(/2

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

24

01

yxm

yxmeyxm

MM

MMMM

=+

in which 0 is given by Equation 24, and by solving iteratively. The variations ofM/Myxm with e for given values of1 are shown in Fig. 6. For this figure, the valuesofM / Myxm which are less than 0.5 have again been replaced by 0.5, as were thecorresponding values for full plasticity in Fig. 5.

The values ofM/Myxm shown in Fig. 6 for1 = 0 are very close to the dimensionlesslateral buckling strengths of Equations 14-17, demonstrating the ability of the

equivalent initial twists of Equation 24 to represent the effects of residual stressesand initial crookednesses and twists. For the special case ofe = 0 and 0 = 0, thedimensionless values ofM / Myxm vary with the maximum first order elastic twistrotation 1 according to

(M/Myxm)0 = 1 / (1 + 2 1) (26)

Simple but conservative approximations of the values ofM/Myxm and 1 at first yieldcan be obtained by using

)27(

0

=

yxm

yym

yxm

b

yxmyxm M

M

MM

MM

MM

as shown in Fig. 6.

6 LOCAL BUCKLING EFFECTS

6.1 Section Classification and Moment Capacities

The effects of local buckling on the section moment capacities of angle sectionbeams has been discussed in Trahair (2002a). In that paper, sections wereclassified as being plastic, compact, semi-compact or slender (BSI, 2000, Trahair etal, 2001) by comparing their long leg plate slendernesses

with limiting slenderness values.

)28(250

yf

t

b=


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11

A plastic section must have sufficient rotation capacity to maintain a plastic hingeuntil a plastic collapse mechanism develops. A plastic section satisfies (Trahair,2002a)

12 (29)

A compact section must be able to form a plastic hinge. A compact section satisfies

12 < 16 (30)

The nominal section moment capacity Msx of a plastic or compact section is equal toits fully plastic capacityMpxm, so that

Msx =Mpxm (31)

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

26< (32)

The nominal section moment capacity of a slender sectionMsx is approximated by

Msx =Myxm (y / )2 (33)

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

16< 26 (34)

The nominal section moment capacity Msx of a semi-compact section isapproximated by the linear interpolation between the full plastic and first yieldcapacities given by

6.2 Buckling and Torsion Strengths

The effects of local buckling on the buckling and torsion strengths of equal anglesection beams can be approximated as shown in Fig. 7, by using

)36(

0

=

sx

sy

sx

b

sxsx M

M

M

M

M

M

M

M

in which (M/Msx)0 for plastic and compact sections is given by the value of (M/Mpxm)0

obtained from Equation 20, and for semi-compact and slender sections by the value of

(M/Myxm)0 from Equation 26. The general Equation 36 reduces to Equation 21 for plastic andcompact sections ( 16), and to Equation 27 for sections with =26.

)35(10

)16()(

=

yxmpxmpxmsx MMMM


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12

7 UNEQUAL ANGLE BEAMS

7.1 Plastic and Compact Beams

It is proposed that the buckling and torsion strengths of plastic and compact unequalangle beams should be approximated by using Equation 36 developed for equal

angle beams with two modifications.

The first modification is for the maximum moment Mquy at elastic lateral bucklingwhich is used in Equations 14-17 for the nominal design lateral buckling capacity Mb.Approximations for the values ofMquy for unequal angle section beams are given inTrahair (2003a).

The second modification is associated with the need to replace Equations 12 and 13for the combinations of the full plastic principal axis moment combinations Mpx and

Mpy for equal angles. The combinations ofMpx / fyb2t and Mpy / fyb

2t for unequal

angles with 0.5 1.0 (in which is the leg length ratio) shown in Fig. 8 wereobtained from the formulations in Trahair (2002a). In this figure, the dashed linescorrespond to the approximations proposed for the cases where the angle sectioncapacity cannot reach full plasticity. It can be seen that the fully plastic momentcombinations for moments of opposite sign are less than those for the same sign.The principal axis values Mpxm and Mpym ofMpx and Mpy may be approximated using(Trahair, 2003a)

)38(117.0546.0075.0/

)37(371.0001.0337.0/

22

22

+=

+=

tbfM

tbfM

ypym

ypxm

The dimensionless applied moments (M/Mpxm)0 for which first-order rotations 1cause full plasticity in zero slenderness beams (e = 0) can be obtained by using Fig.8 and the first-order relationships

)39()(

)39()(

101

01

bMM

aMM

y

x

==

The variations of (M/Mpxm)0 with 1 for unequal angle beams are shown by the solidand dashed lines in Fig. 9. They may be approximated by using

(M/Mpxm)0 =1 + a11 + a212 + a31

3 (40)

in which either

)41(

1

57.1257.3038.24300.6

88.1248.3135.25245.6

085.2103.4424.25900.0

3

2

3

2

1

a

a

a

a

=

whenMx andMy are of the same sign (1 negative), as shown in Fig. 10b, or


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13

)41(

1

428.779.1986.19566.7

96.1504.4349.4291.15

615.616.1935.20810.8

3

2

3

2

1

b

a

a

a

=

whenMx andMy are of opposite sign (1 positive), as shown in Fig. 10a.

The signs ofMx andMy depend on the attitude of the unequal angle, the sense of thetwist rotation 1, and the principal axis directions x and y, as shown in Fig. 10. Withthe short leg down, they axis initially vertically down, an eccentric load q through thelegs causes a positive (clockwise) rotation 1 as shown in Fig. 10a. In this case, theapplied moment Mhas components Mx and My of opposite sign, so that the smallervalues of (M/Mpxm)0 obtained using Equation 41b with Equation 40 are appropriate.When the short leg is up and the y axis is initially vertically up as shown in Fig. 10b,then an eccentric load through the legs causes a negative (anticlockwise) rotation,and the applied moment M has components of the same sign, in which case thelarger values of (M/Mpxm)0 obtained using Equation 41a with Equation 40 areappropriate.

The values of (M/Mpxm)0 given by Equations 40 and 41 should be used for (M/Msx)0 inEquation 36 to approximate the buckling and torsion strengths of plastic or compactunequal angle beams.

7.2 Semi-Compact and Slender Beams

7.2.1 First Yield Capacities

Approximations for the buckling and torsion strengths of semi-compact and slenderunequal angle beams may be developed in a similar manner to that used above forplastic and compact beams, provided the full plastic moment combinations arereplaced by first yield combinations which are modified appropriately to account forinelastic and elastic local buckling effects.

The combinations of the first yield principal axis moment combinations Myx /fyb2tand

Myy /fyb2tfor unequal angles with 0.5 1.0 shown in Fig. 11 were obtained from

the formulations in Trahair (2002a). It can be seen that the first yield momentcombinations for moments of opposite sign are generally less than those for thesame sign. The principal axis values Myxm and Myym ofMyx and Myy may beapproximated using

)43(028.0068.0276.0/

)42(354.0460.0577.0/

22

22

+=

+=

tbfM

tbfM

yyym

yyxm


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14

The dimensionless applied moments (M/Myxm)0 for which first-order rotations 1cause first yield in zero slenderness beams (e = 0) can be obtained by using Fig. 11and the first-order relationships of Equations 39. The variations of (M/Myxm)0 with 1for unequal angle beams are shown by the solid lines in Fig. 12. They may beapproximated by using

)44(11

14

3

12

1

0 b

b

b

b

M

M

sx +

=

in which

)45(

1

3.223.650.680.27

830.0880.0930.088.1

62.1510.039.227.2

00000.1

3

2

4

3

2

1

=

b

b

b

b

andMsx = Myxm.

7.2.2 Capacities of Semi-Compact and Slender Beams

The dimensionless applied moments (M/Msx)0 for which first-order rotations 1 causefirst yield in zero slenderness angle beams (e = 0) need to be modified for semi-compact and slender angles to allow for inelastic or elastic local buckling effects.This may be done by replacing the major axis first yield capacity Myxm used forMsx inEquation 44 by the appropriate section capacity Msx determined using Equation 35

for semi-compact angles or Equation 33 for slender angles. The modified values of(M/Msx)0 should be used in Equation 36 to approximate the buckling and torsionstrengths of semi-compact or slender unequal angle beams.

8 EXAMPLE

8.1 Problem

A 150 x 100 x 12 unequal angle beam is shown in Fig. 13. 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. 13b. Theunbraced 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 with aneccentricity ofe = 47 mm from the shear centre at the leg junction, as shown in Fig.13b.

The first-order analysis of the beam, the determination of the lateral buckling designstrength, and the check of the buckling and torsion capacity are summarised below.The determination of the bearing, shear, and torsion capacities is summarised inTrahair (2002 b).


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15

8.2 Elastic Analysis

The design major axis bending moments are

Mx* = (q* L2 / 8) cos = 24.7 kNm, andMy* = (q* L2 / 8) sin = 10.9 kNm.

The design torque is Mz* = (q* e L / 2) = 0.846 kNm. This is much less than the

design capacity ofMu= 2.31 kNm (Trahair, 2002b).

8.3 Lateral Buckling Design Strength

The angle section has been shown to be compact (Trahair, 2002a).The elastic buckling moment and the lateral buckling strength calculated in Trahair(2003a) areMquy = 30.6 kNm andMb = 25.0 kNm.

8.4 Buckling and Torsion Capacity

Using Equation 5c, the maximum first-order twist rotation is1 = 0.6 x 47 x (6E3)

2 / (8 x 8E4 x 0.1371E6) = 0.016 rad., which is small. = 94 / 144 = 0.653Using Fig. 10, the moment components ofMx* are of opposite sign.

Using Equations 41b, a1 = 1.850, a2 = 2.07, a3 = 0.969.Using Equation 40, (M/ Mpxm)0 = 0.972, so that the effect of torsion is small in thiscase.

Using Trahair (2003a) or Equations 37 and 38,Mpxm = 38.4 kNm,Mpym = 15.5 kNm.Using Equation 36 withMsx =Mpxm andMsy =Mpym,

M= 0.972 x 25.0 = 24.3 kNm > 15.5 kNm so that M= 21.9 kNm (using = 0.9).This is less than the major axis design moment Mx* = 24.7 kNm and the beam is

inadequate, even if the minor axis moment My* = 10.9 kNm is ignored.


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16

9 CONCLUSIONS

This paper develops a rational, consistent, and economical design method fordetermining the buckling and torsion strength of an eccentrically loaded unbracedsteel angle section beam, and illustrates its 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 and torsion. Thebeams have initial twists. 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 buckling and torsion strengths of equal angle beams are predicted by assumingeither full plasticity or first yield at the maximum moment section, and simple designapproximations are developed. The effects of local buckling are considered usingproposed definitions (Trahair, 2002a) of the section capacities of plastic, compact,semi-compact, and slender sections. These definitions are then used with the

simple fully plastic and first yield design approximations to develop a simple generaldesign approximation for equal angle beams. This design approximation isformulated in terms of the maximum first-order moment Mand angle of twist rotation

1, and can be used for equal angle beams under general loading which acts in aplane parallel to the major axis principal plane.

The fully plastic and first yield behaviours of unequal angle section beams are thenconsidered, and design approximations are developed from those for equal anglebeams.

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


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17

APPENDIX 1 REFERENCES

BSI (2000), BS5950 Structural Use of Steelwork in Building. Part1:2000. Code ofPractice for Design in Simple and Continuous Construction: Hot Rolled Sections,British Standards Institution, London.

Farwell, CR and Galambos, TV (1969), Non-uniform torsion of steel beams ininelastic range, Journal of the Structural Division, ASCE, 95 (ST12), 2813-29.

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.

Pi, YL and Trahair, NS (1995a), Inelastic torsion of steel I-beams, Journal ofStructural Engineering, ASCE, 121 (4), 609-620.

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

Trahair, NS (1993), Flexural-Torsional Buckling of Structures, E & FN Spon,London.

Trahair, NS (2002a), Moment capacities of steel angle sections, Journal ofStructural Engineering, ASCE, 128 (11), 1387-93.

Trahair, NS (2002b), Bearing, shear, and torsion capacities of steel anglesections, Journal of Structural Engineering, ASCE, 128 (11), 1394-98.

Trahair, NS (2003a), Lateral buckling strengths of steel angle section beams,Journal of Structural Engineering, ASCE, 129 (6), 784-91.

Trahair, NS (2003b), Biaxial Bending of Steel Angle Section Beams, Journal ofStructural Engineering, ASCE, in press.

Trahair, NS (2003c), Non-linear elastic non-uniform torsion, Research ReportR828, Department of Civil Engineering, University of Sydney.

Trahair, NS, Bradford, MA, and Nethercot, DA (2001), The Behaviour and Design

of Steel Structures to BS5950, 3rd

British edition, E & FN Spon , London.

Trahair, NS and Teh, LH (2001), Second order moments in torsion members,Engineering Structures, 23(6), 631-642.


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18

APPENDIX 2 NOTATIONa1-3 constantsb long leg lengthb1-4 constants

E Youngs modulus of elasticitye eccentricity of load from the shear centre

fy yield stressG shear modulus of elasticity

Ix,Iy second moments of area about the x,y principal axesIn non-linear Wagner section constantJ torsion section constantL span lengthM applied end momentMb lateral buckling moment strengthMpx,Mpy values ofMx,My at full plasticityMpxm,Mpym principal axis values ofMpx,MpyMquy maximum moment at elastic lateral bucklingMsx,Msy principal axis section moment capacitiesMx,My moments about thex,y principal axesMx*,My* design moments about thex,y principal axesMyz uniform bending elastic buckling momentMx2,My2 second-order moments about thex,y principal axesMyx,Myy values ofMx,Myat first yieldMyxm,Myym principal axis values ofMyx,Myym intensity of uniformly distributed torqueq intensity of uniformly distributed loadq* design intensity of uniformly distributed load

t leg thicknessu, v shear centre deflections parallel to thex,y principal axesu1, v1 first-order deflectionsu2, v2 second-order deflections

x,y principal axesX, Y rectangular (geometric) axesXc, 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 constantu1, v1 maximum first-order deflections

u2, v2 maximum second-order deflections

long leg local buckling slenderness

e modified slenderness for beam lateral buckling

ex, ey beam lateral buckling slenderness limits

angle of twist rotation, orcapacity factor

0 initial angle of twist rotation

1 first-order angle of twist rotation

2 second-order angle of twist rotation0 maximum value of0

1 maximum value of12 maximum value of2


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19

S

z

L

Y

q

(a) Elevation

Fig. 1. Eccentrically Loaded Angle Section Beam

(c) Loading

qq

e e

Flange down Flange up

t

x

XC

Y y

b

Yc

t

Xc

b

(b) Cross-section


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20

(b) Lateral buckling

x

Sy

q

Fig. 2. Single Angle Beam Behaviour

q

x

Sy

(c) Unrestrainedbiaxial bending

q

(a) Restrainedbiaxial bending

x

Sy

q

e

(d) Bucklingand torsion

x

Sy


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21

q

(a) Movement of point of application

e sec

e sec cos (+)

= 45o

S

e

(b) Change of point of application

S

q

Fig. 3. Reduction of Load Eccentricity


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22

y

mL/2mL/2m

L

M Mz

Fig. 4. Simply Supported Equal Angle in Uniform Bending and Torsion


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23

M = Mpym

0.15

0.30

0.45

0.60

0.75

Dim

ensionlessmomentcapacityM/M

pxm

Modified slenderness e = (Mpxm /Myz)

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

1.0

0.8

0.6

0.4

0.2

0

1 = 0 rad. M = Mpxm

Equation 19Approximate Equation 21

m = 1.0

Elastic bucklingMyz / Mpxm

Lateral buckling strengthMb / Mpxm

Fig. 5. Fully Plastic Buckling and Torsion Strengths of Equal Angle Beams


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24

M = Myym

0.1

0.2

0.3

0.4

0.5

Dimensionlessm

omentcapacityM/M

yxm

Modified slendernesse = (Myxm /Myz)

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

1.0

0.8

0.6

0.4

0.2

0

1 = 0 rad. M = Myxm

Equation 25Approximate Equation 27

m = 1.0

Elastic bucklingMyz / Myxm

Lateral buckling strengthMb / Myxm

Fig. 6. First Yield Buckling and Torsion Strengths of Equal Angle Beams


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25

M / Msx

0 0.5 1.0 1.5 2.0

Modified slenderness e = (Msx /Mquy)

Fig. 7. Proposed Design Strengths

1.0

0.5

0

DimensionlessmomentcapacityM/M

sx

M = Msx

Elastic bucklingMquy / Msx

Lateral buckling strengthMb / Msx

M = Msy

M = Msy(M / Msx)0

ex ey

M = Msx(M / Msx)0


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26

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1Mpx

/

fyb2t

Mpy/ fyb t

Fig. 8. Fully Plastic Moment Combinations for Unequal Angles

0.3 0.2 0.1 0 0.1 0.2 0.3

= 1.0

0.9

0.8

0.7

0.6

0.5

x

y

Mx

My

y

x

Mx

My

= 1.0

0.9

0.8

0.7

0.6

0.5


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27

Fig. 9. Fully Plastic Strengths for Zero Slenderness

1ve 1 +ve

= 1.0

0.8

0.7

0.6

0.5

= 1.0

0.7

0.6

0.5

1.2 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2

Maximum first-order twist rotation 1 rad.

Dimensi

onlessmoment(M/M

pxm

)0

1.0

0.8

0.6

0.4

0.2

qq

ApproximationsAccurate values

e = 0


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28

(a) Components of opposite sign

1 +ve,Mx +ve,Myve

(b) Components of same sign

1ve,Mxve,Myve

Fig. 10. Signs of Moment Components

11

1

1

1

1

q

q

Mx

Mx

My

My

M

M

x

x

y

y


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29

0.5

0.4

0.3

0.2

0.1Myx

/

fybt

Myy/ fyb t

Fig. 11. First Yield Moment Combinations for Unequal Angles

0.3 0.2 0.1 0 0.1 0.2 0.3

= 0.5

0.9

0.8

0.7

0.6

1.0

x

y

Mx

My

y

x

Mx

My


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30

1.2 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2

Approximations

Accurate

= 1.0

0.7

0.6

0.5

= 0.7

0.6

0.5

0.8

0.9

1.0

Fig. 12. First Yield Strengths for Zero Slenderness

Maximum first-order twist rotation 1 rad.

Dime

nsionlessmoment(M/M

yxm

)0

1.0

0.8

0.6

0.4

0.2

1ve

q

1 +ve

q

e = 0


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Fig. 13. Example

150 x 100 x 12 UA

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

E= 200,000 MPaG = 80,000 MPa

fy= 300 MPa

= 23.91Iy = 1.314 E6 mm

4

J= 0.1371 E6 mm4

yo = 32.30 mm

x= -78.33 mm

(b) Section

q*

e = 47 mm

(a) Elevation

q* = 6.0 kN/m

L= 6000 mm

buckling and torsion of steel angle section beams

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