cidect 2- structural stability of hollow sections

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Construction with Hollow Steel Sections - Structural stability of hollow sections

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p]ITHCONSTRUCTIONHOLLOW STEEL SECTIONS

Edited by: Comite International pour le Developpement et IEtudeAuthors: Jacques Rondal, Universityof Liegede la Construction TubulaireKarl-GerdWurker, Consulting engineerDipak Dutta, Chairman of the Technical Commission CIDECTJaap Wardenier, Delft University f TechnologyNoel Yeomans, Chairmanof the Cidect Working GroupJoints behaviour and Fatigue-resistance


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J. Rondal, K.-G. Wurker, D. Dutta, J. Wardenier,N. Yeomans

Verlag TUV Rheinland


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I Die Deutsche Bibliothek - CIP EinheitsaufnahmeStructural stability of hollow sections / [ComiteInternational pour le Developpement et IEtude de laConstruction Tubulaire]. J. Rondal . . . - Koln: Verl.TUV Rheinland, 1992

(Construction with hollow steel sections)Dt. Ausg. u.d.T.: Knick- und Beulverhalten vonHohlprofilen (rund und rechteckig).- Franz. Ausg.u.d.T.: Stabilite des structures en profils creuxISBN 3-8249-0075-0, Reprinted editionNE: Rondal, Jasques; Comite International pour leDeveloppement et IEtude de la ConstructionTubulaire

ISBN 3-8249-0075-00 by Verlag TUV Rheinland GmbH, CologneEntirely made by: Verlag TUV Rheinland GmbH, ColognePrinted in GermanyFirst edition 1992Reprinted with corrections 1996


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The objectiveof this design manual s to present theuide lines for the design and calculationof steel structuresconsisting of circular and rectangular hollow sections dealingn particularwith the stability of these structural elements. This book describes in acondensed form theglobal, local and lateral-torsional uckling behaviour of hollow sections as wells the methodsto determine effective buckling lengths of chords and bracings in lattice girders built withthem. Nearlyall design rules and procedures recommended here are basedn the results ofthe analytical investigations and practical tests, which were initiated and sponsored byCIDECT. These research works were carried outn the universities and institutes in variousparts of the world.The technical dataevolving from these research projects,he results of their evaluation andthe conclusions derived were used o establish the European buckling curves or circularand rectangular hollow sections. This was the outcome of a cooperation between ECCS(European Convention for Constructional Steelwork) and CIDECT. These buckling curveshavenowbeen ncorporated in a number of national standards.Theyhavealsobeenproposed or he buckling design by Eurocode 3,Part 1: General Rules and Rules orBuildings (ENV 1993-1-1).Extensive research works on effective buckling lengths of structural elements of hollowsections in lattice girders inhe late seventies led n 1981to the publication f Monograph No.4 Effective lengthsf lattice girder members by IDECT. A recent statistical evaluation of alldata from this research programme resulted in a ecommendation for thecalculation of thesaid buckling length hich Eurocode3,Annex K Hollow section lattice girder connectionsalso contains.This design guide s the second of a series, which CIDECT has alreadyublishedand alsowillpublish in he coming years:1. Design guide for circular hollow section (CHS) joints under predominantly static loading2. Structural stability of hollow sections (reprinted edition)3. Design guide or rectangular hollow section joints under predominantly static loading4. Design guide for hollow section columns exposed to fire (already published)5 . Design guide for concrete filled hollow section columns under static and seismic loading6. Design guide for structural hollow sections for echanicalapplications (already published)7. Designguide or fabrication, assemblyand erection of hollowsectionstructures (in8. Design guide or circular and rectangular hollow section joints under fatigue loading (inAll these publications are intended to make architects, engineers and constructors familiarwith the simplified design procedures of hollow section structures. Worked-out examplesmake them easyo understand and show howo come to a safe and economic design.Our sincere thanks go to the authors of this book, who belong to the group of wellknownspecialists in the field of structural applications of hollow sections. We express our specialthanks to Prof. Jacques Ronda1 of he University of Liege, Belgium as theain author of hisbook. We thank further Dr. D. Grotmann of the Technical University of Aix-la-Chapelle fornumerous stimulating suggestions. Finally we thank all CIDECT members, whose supportmade this book possible.

(already published)

(already published)

(already published)

preparation)preparation)

Dipak DuttaTechnical CommissionCIDECT

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Quadrangular vierendeel columns

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ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Page. . 9

1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1 Limittates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2imittateesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Steelrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Increase in yield strengthdue o cold working . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Cross section classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Members in axial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1eneral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Designethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3esignids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Membersin bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1Designorateral-torsional buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Members in combined compressionnd bending . . . . . . . . . . . . . . . . . . . . . 285.1eneral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2esignethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.1Design for stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.2Designbasedonstress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2.2.1Stressdesignwithoutconsideringshear oad . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2.2.2 Stress design considering shear load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 Thin-walled sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2Rectangularollowections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2.1Effectivegeometricalproperties of class4crosssections . . . . . . . . . . . . . . . . . 346.2.2esignrocedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.2.3esignids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.3 Circularollowections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Buckling length of members inattice girders . . . . . . . . . . . . . . . . . . . . . . . . 407.1eneral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.2Effective buckling length of chord and bracing memberswith ateralsupport . . 407.3Chords of lattice girders,whose ointsarenotsupported aterally . . . . . . . . . . . 408 Designexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.1 Designof a ectangularhollowsectioncolumn in compression . . . . . . . . . . . . . 438.2 Designof a rectangular hollow section column in combined compression and8.3 Design of a rectangular hollow section column in combined compression anduni-axialbending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43bi-axialbending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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8.4Design of a hin-walled ectangularhollowsectioncolumn in compression . . . 478.5Design of a hin-walled ectangularhollowsectioncolumn in concentriccompressionandbi-axialbending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 References 53CIDECT.nternational Commit tee for the Development and Study of TubularStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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IntroductionIt is very often considered that he problems to be solved while designing a steeltructure areonly related to the calculation nd construction of the members and their connections. Theyconcern mainly thetatic or fatigue strengthnd the stability of the tructural members as wellas the load bearingcapacity of the joints. This point of view is certainly not correct as onecannot ignore the important areas dealing with fabrication, erection and when necessary,protection against fire.It is very mportant o bear in mind that he application of hollow sections, circular andrectangular, necessitates special knowledge n all of the above mentioned areas extendingbeyond that forhe open profiles in onventional structural engineering.This book dealswith theaspect of buckling of circular and rectangular hollow sections, theircalculations and theolutions to he stability problems.The aim ofhis design guide s to provide architects andtructural engineers with design idsbased on the most recent research results in the field of application technology of hollowsections. It is mainly based on the rules given in Eurocode 3 Design of Steel Structures,Part 1: General Rules and Rules for uildings and its annexes [ l 21.Small differences canbe found when compared to some national standards. The reader will find in reference [3]a review of the main differences existing between Eurocode3 and the codes used n othercountries.However,when t is possible,some indications aregivenon the rulesandrecommendations in the codes used in Australia, Canada, Japan and United States ofAmerica as well asn some european countries.

Lift shaft with tubular frames9


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1 General1 l Limit statesMost design codes for see1 structures are, at the present time, based on limit state design.Limit states are those beyond which the structure no longer satisfies the design performancerequirements.Limit state conditions are classified nto- ultimate limit state- serviceability imit stateUltimate limit states are those associated with collapse of a structure or with other failuremodes, which endanger the safety of human life. For the sake of simplicity, states prior tostructural collapse are classified and treated as ultimate limit states in place of the collapseitself.Ultimate limit states, which may equire consideration, include:- Loss of equilibrium of a structureor a partof it, considered as a rigid body- Loss of load bearingcapacity, as for example, rupture, nstability, fatigue or other agreedServiceability limit states correspond to states beyond whichpecified service criteria are nolonger met. They include:- Deformations or deflections which affect the appearance or effective use of the structure(including the malfunction of machines or services) or cause damage to finishes or non-structural elements- Vibration which causes discomfort to people, damage to the building or its contents or

which limits its functional effectivenessRecent national and international design standards recommend procedures proving limitstateresistance.This mplies, in particular for stability analysis, hat the imperfections,mechanical and eometrical, which nfluence he behaviourof a structure ignificantly, mustbe aken into account.Mechanical imperfections are, orexample, residual stresses instructural membersndonnections.Geometricalmperfectionsreossible pre-deformations in members and cross sections as wells tolerances.

limiting states, such as excessive deformations and stresses

1.2 Limit state designIn the Eucrocode3 ormat, when considering a imit state, it shall be verified that:

whereyF = Partial safety factor for the action FyU = Partial safety factor for he resistance RF = Value of an actionR = Value of a resistance for a relevant limit stateyF F = Fdis called the designoad whileR l r M = R, is designated as the design resistance.It is not within the scopeof this book to discuss in detail these general provisions. They can betaken from Eurocode nd othernationalcodes, whichcan sometimes show small deviationsfrom one another. As for example, the calculations in the recent US-codes are made with6 = ? h M .10


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1.3 Steel gradesTable 1 gives the grades of the generally used structural steels with the nominal minimumvalues of the yield strength f range of the ultimate tensile strength and elongations. Thesteel grades correspond to the hot-rolled hollow sections as well as to the basic materials fcold-formed hollow sections. The designationsf the steel gradesn Table 1 are in accordancewith EN 10 0251311. They can be different in other standards. For hot-rolled hollow sections(circular and rectangular), the european code EN 10 210, Part1201, 1994 is available.

Table 1 - Steel grades or structural steelsmin. yield strength

f, (N/mm2), (Nlmrn')tensile strength min. percentage elongationsteel grade L = 5.654

longitudinal transverseS 235

15750. . .72060460 20290..63055355

20 2210.. ,56075275 24 2640. . ,47035

Table 2 contains the recommended physical properties valid forll structural steels.

Table 2 - Physical propertiesof structural steelsmodulusflasticity: E = 210 000 N/mm2shearodulus: G = 81 000 N/mm2po iso n co-efficient: U = 0.3co-efficient of inear expansion: Q = 12 . 10 6 / o Cdensity: e = 7850 kg/m3

E2(1 + U)

1.4 Increase in yield strength due to cold work ingCold rolling of profiles provides an increasen the yield strength due to strain hardening, whichmay be usedn the design by means f the rules givenn Table 3. However, this increase canbe used only forHS in tension or compression elements and cannot be taken into accountfthe members are subjected to bending (see Annex Af Eurocode 3121).For cold rolled square and rectangular hollow sections, eq.1.2) can be simplifiedk = 7 for allcold-forming of hollow sections and n 4) resulting in:

14tfya = fyb + b h (fu-fyb),

5 1.2 . ybFig. 1 allows a quick estimationf the average yield strength after cold-forming, for square andrectangular hollow sections for the four basic structural steels.

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Table 3 - ncrease of yield strength due to cold-formingf RHS profilesAverage yield strength:The average yield strength may be determined from full size section testsor as follows [19, 321:f = f + (k . n . tz/A) . (f, - (14where f = specified ensile yield strength and ultimate ensile strength of the basic material(N/mm2)

t = materialhicknessmm)A = gross ross-sectionalareamm2)k = co-efficient depending on the type of forming (k = 7 for cold rolling)n = number of90 bends in the sectionwithan nternal adius


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2 Cross sectionclassificationDifferent models can be used for the analysis of steel structures and for the calculationf thestress resultants (normal force, shear force, bending moment and torsional moment in themembers of a structure).For an ultimateimit state design, the designers faced mainly with three design methods (seeFig. 2). The cross section classesand 4 with the procedure elastic-elastic iffer from eachother only by the requirement for local buckling for class.Procedure plastic-plasticCross section class 1This procedure deals with the plastic design and the formationf plastic hinges and momentredistribution in the structure. Full plasticity is developed n the cross section (bi-rectangularstress blocks). .The cross section can form a plastic hinge with the rotation capacity requiredfor plastic analysis. The ultimate imit state is reached when the numberof plastic hinges issufficient to produce a mechanism. The system must remainn static equilibrium.Procedure elastic-plasticCross section class 2In this procedure the stress resultants are determined following an elastic analysis and theyare compared to the plastic resistance capacities of the member cross sections. Crosssections can develop their plastic resistance, but have limited rotation capacity. Ultimateimitstate is achieved by the formationf the first plastic hinge.Procedure elastic-elasticCross section class 3This procedure consistsf pure elastic calculation f the stress resultants and the resistancecapacities of the member cross sections. Ultimate limit state is achieved by yielding of theextreme fibresof a cross section. The calculated stressn the extreme compression fibre fthe member cross section can reachts yield strength, but local bucklings liableto prevent thedevelopment of the plastic moment resistance.Procedure elastic-elasticCross section class 4The cross section is composedf thinner walls than those of class. It is necessary to makeexplicit allowances for the effectsf local buckling while determining the ultimate moment orcompression resistance capacity f the cross section.The application of the irst three above mentioned procedures is based on the presumptionthat the cross sections or their parts do not buckle locally before achieving their ultimateimitloads; that means, the cross sections must not be thin-walled.n order to fulfil this condition,the blt-ratio or rectangular hollow sections or thelt-ratio or circular hollow sections must notexceed certain maximum values. They are different for the cross section classesthrough 3as given in Tables 4, 5 and 6.A cross section must be classified according to the least favourable (highest) class of theelements under compression andlor bending.Tables 4 through 6 give the slenderness limits blt or d/t for different cross section classesbased on Eurocode 3 [ l 21.Other design codes show slightly different values (compareTables 8 and 9).

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cross section class 1classes

load resistancecapacity

stress distributionand rotationcapacity

full plasticity inthe cross sectionfull rotationcapacity

procedure for thedetermination ofthe stressresultants

plastic

procedure for thedeterminationof the ultimateresistancecapacity of asection

plastic

class 2

full plasticity inthe cross sectionrestricted rotationcapacity

- Vf Y

plastic

Fig. 2 - Cross section classification and design methods

class 3

elastic crosssectionyield stress in theextreme fibre

elastic

elastic

class 4

elastic crosssectionlocal bucklingobe taken intoaccount

- f v1+ f y

elastic

elastic

Table 4 - Limiting d/t ratios for cir cular hollow sections8 2

cross section class

2dlt S 50t2

compression andlor bending

dlt S 90e2dlt 5 70e2

f (Nlmm2) 46055 275 235e

0.51.66.8520.72 0.81.92

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Table 5 - Limit ing h,/t-ratios for webs of rectangularhollow sectionswebs: (internal element perpendicular to the xis of bending)h, = h - 3t

classcompressionending

web subject oeb subject to

stress distributionin element(compression positive)h3 / h 1

f V - 2+1 h,/t 5 72c h,lt 33 e

2 h,lt 5 83c h,lt 5 386

I Istress distributionin elementpositive)(compression DTh1 2yv - hai if v

3 h,/t 2 ~,lt 5 1 2 4 ~

h, = h - 31web subject o bending and compression

f V -when CY > 0.5h,/t 5 396e/(13~~-)when CY < 0.5

h,/t 36clawhen CY > 0.5h, l t 456~1(13~~-)when CY < 0.5

h,lt S 41.5cla

PT* t v -when 11 > - 1h,/t 5 42d (0.67 + 0.3311.)

when 11.< - 1h,/t 5 62 t ( l - 1.m

fY 235 460 355 275 0.72.81.92

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Table 6 - Limiti ng b,/t-ratios or flanges of rectangular hollow sectionsflanges: (internal elements parallel to the axisof bending)b, = b 3t

class I I sectionnending sectionnompressionstress distributionnelement and cross section(compression positive)

n~

I I b,/t 5 33e I b,/t 2e2 1 I b,/t 8c I b,/t 5 42 c

stress distribution nelement and cross section(compression positive)

3 1 I b,/t 5 42 e I b,lt S 42 tf (N/mm2) 460557535

t 0.72.81.92In Table 7 the blt, hlt and dlt limiting values for the different cross section classes, crosssection types and stress distributions are given for a quick determinationf the cross sectionclass of a hollow section. The values for width b and height hf a rectangular hollow sectionare calculated by using the relationshipblt = b,lt + 3 and hlt = h,lt + 3.For the application f the procedures plastic-plastic (class )and elastic-plastic (class2),the ratio of the specified minimum tensile strength, to yield strength, must be not less than1.2.Further, according to Eurocode [ l ,21, the minimum elongation at failure on a gauge lengthl, = 5.65nowhere A, is the original cross section area) is,noto be less than15%.For he application of the procedure plastic-plastic (full rotation), the strain E corrres-ponding to the ultimate ensile strength f, must be at east 20 times he yield strain Ecorresponding to the yield strengthThe steel gradesn Table 1 for hot formedRHS and hot or cold formedHS may be acceptedas satisfying these requirements.Tables 8 and ive, for circular hollow sections and for square or rectangular hollow sectionsrespectively, the imiting bl t and hlt ratios, which are recommendedn various national codesaround the world [3].Table 8 shows that there are significant differencesn dlt limits recommended by the nationalcodes, when a circular hollow sections under bending.In particular, this is clear in the case of the ecentamericancode AlSC 86.For heconcentrically loaded circular hollow sections, the deviations are significantly smaller (lessthan about 10%).Table 9 shows that the differences in blt limits for rectangular hollow sections between thenational codes are, n general, not as large s those for circular hollow sections.16


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Table 7 - blt- ,hlt- and dlt limitsor the cross section lasses 1,2 and 3 with blt= b,lt + 3 nd hlt = h,lt + 3r I 3r class 1 23545

6

75 55-7535

45

1

6.0

0.0

355 460 35 275 3556.6

6.6

03.3

9.6

460__32.2

2.2

0.8

6.0

cross section I element41.6 36.6 32.2 41.6 36.6 32.2 45 41.6HP compression' compression 08 33.3 29.3 25.7 37.9

__79.5

33.4 29.3 45 41.6HP mending compression

75 69.3 61 l 53.6 70.0 61.5 127 1 17.3

6.9

RHP bending bending m

50 42.7 33.1 25.5 59.8

-46.3 35.8 90.0CHS

-compressionandlor bending

There is no difference between blt and hlt limits or the classes 1 , 2 and 3, hen the whole cross section is only under compression.


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Table 8 - Max. d/t limit s or circular hollow sections by country and codet =e , in N/mm*)

axial compression

Tabelle 9 - Max. b,/t limi ts or rectangular hollow sections by country and code-E =dy , in N/mm2)

Y

country codeI

Australia ASDR 87 164

Belgium I NBN51-002 (08.88)Canada I CANICSA-S16.1M89

Germany I DIN 18800, Part 1 (1.90)Japan AIJ 80

Netherlands I NEN 6770, publ. draft 08.89)United BS950art 1 (1985)

U.S.A.lSClLRFD1986)Europeanurocode [ l ]Community

bendingaxial compression plastic limit yield limit

(class 2) (class 3)4 0 . 2 ~ ' 4 0 . 2 ~ '9.9 t45.4t" 45.4"

42 t

42.2 t4.6t2.2 t42 t4 t2 t

47.8 t7.8 t37.8 t7 t7.8 t43.6 t4 . 2 ~7.6642 c4 t

4 0 . 8 ~42 c8 t2 t

4 0 . 8 ~

for cold formed non-stress relieved hollow sections* * for hot-formed and cold-formed stress relieved hollow sections18


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3 Members in axial com pression3.1 GeneralThis chapter of the book is devoted to the bucklingof compressed hollow section membersbelonging to the cross section classes , 2 and 3. Thinwalled cross sections (class) will bedealt with n chapter 6.The buckling of a concentrically compressed column is, historically speaking, the oldestproblem of stability and was already investigated by Euler andater by many other researchers151. At thepresent ime, hebucklingdesign of asteelelementundercompression sperformed by using the so called European buckling curves in most european countries.They are based on many extensive experimental and theoretical nvestigations, which, inparticular, take mechanical (as for example residual stress, yield stress distribution) andgeometrical (as for example, linear deviation) imperfectionsn the members into account.

00 0 0 5 1.0 1 5

Fig. 3 - European buckling curves l ]0

A detailed discussion on the differences between buckling curves usedn codes around theworld is given in reference [3]. oth design methods, allowable stress design andimit statedesign, have been covered. For ultimate limit state design, multiple buckling curves aremostly used (as for example, Eurocode 3 with a a, b, c curves, similarly in Australia andCanada). Other standards adopt a single buckling curve, presumably due to the fact thatemphasis is placed on simplicity. Differences up to5% can be observed between the variousbuckling curves n the region of medium slenderness X).

3.2 Design methodAt present, a large numberf design codes exist and the recommended procedures are oftenvery similar. Eurocode3 [ l ,21 is referred to in the following.For hollow sections, the only buckling mode to be considered is flexural buckling. It is notrequired to take account of lateral-torsional buckling, since very large torsional rigidity of ahollow section prevents any torsional buckling.

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The design buckling load of a compression member s given by the condition;Nd Nb,Rdwhere Nd = Design oad of the compressed member 7 imes working oad)= Design buckling resistance capacity of the member

fY MNb,Rd= X .A . (3.1)

A is the area of the cross section;X is the reduction actor of the relevantbucklingcurve (Fig. 3,Tables 1 1 through 14)fy is the yield strength of the material used;-yM is the partial safety factor on the resistance side (in U.S.A.: l/y, = 6)The eduction actor X is the ratio of thebuckling esistance Nb,Rd to theaxialplasticresistance N

dependent on he non-dimensional slenderness ; of a column;

x = - - -Nb.Rd fb,RdNpl.Rd fy,d-fb,,d = design buckling stress = b,RdAfY,d = design yield strength = MThe non-dimensional slenderness ; is determined by

fY

with X = (Ib = effective buckling length; i = radius of gyration)bIX = P .fi Eulerian slenderness)E = 210 000 N/mm2

Table 10a - Eulerian slenderness or various structural steelssteelgrade1, (N/rnmz)

S 6035527523546055 275 235

IX 67.16.4 86.8 93.9

The selectionof the buckling urve (a through cn Fig. 3) depends on the cross section type.This is mainly based on the various evels of residual stresses occurring due to differentmanufacturing processes. Table10b shows the curves for hollow sections.20


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Table 10b - Buckling curves accordingo manufacturing processf,, = Yield strength of the basic not cold-formed) materialf,, = Yield strengthof the material after cold-formingI cross section 1 manufacturing process buckling curves I

Table 1 1 - Reduction factor - buckling curvea,h

0.00.l0.20.30.40.50.60.70.80.90

1 .oo1.101.201.301.401.501.60l .701.801.902.002.102.202.302.402.502.602.702.802.903.003.103.203.303.403.503.60

01 .oooo1 .oooo1 .oooo0.98590.97010.95130.92760.89610.85330.79610.72530.64820.57320.50530.44610.39530.35200.31500.28330.25590.23230.21 170.1 9370.1 7790.16390.15150.14040.13050.12160.11360.10630.09970.09370.08820.08320.07860.0744

11 .oooo1 .oooo0.99860.98450.96840.94920.92480.89240.84830.78950.71780.64050.56600.49900.44070.39070.34800.31 160.28040.25340.23010.20980.19200.17640.16260.15030.13940.12960.12070.11280.10560.09910.09310.08770.08280.07820.0740

21 .oooo 1 .oooo1 .oooo 1 .oooo0.9973 0.99590.9829 0.98140.9667 0.96490.9470 0.94480.9220 0.91910.8886 0.88470.8431 0.83770.7828 0.77600.7101 0.70250.6329 0.62520.5590 0.55200.4927 0.48660.4353 0.43000.3861 0.38160.3441 0.34030.3083 0.30500.2775 0.27460.2509 0.24850.2280 0.22580.2079 0.20610.1904 0.18870.1749 0.17350.1613 0.16000.1491 0.14800.1383 0.13730.1286 0.12770.1199 0.11910.1120 0.11130.1049 0.10430.0985 0.09790.0926 0.09200.0872 0.08670.0823 0.08180.0778 0.07730.0736 0.0732

41 .oooo1 .oooo0.99450.97990.96310.94250.91610.88060.83220.76910.69480.61760.54500.48060.42480.37720.33650.30170.27190.24610.22370.20420.18710.17210.15870.14690.13630.12680.11830.11060.10360.09720.09150.08620.08140.07690.0728

51 .oooo1 .oooo0.99310.97830.96120.94020.91300.87640.82660.76200.68700.61010.53820.47460.41970.37280.33280.29850.26910.24370.221 70.20240.18550.1 7070.15750.14580.13530.12590.1 1750.10980.10290.09660.09090.08570.08090.07650.0724

6l .oooo1 .oooo0.9910.97670.95930.93780.90990.87210.82080.75490.67930.60260.53140.46870.41470.36850.32910.29540.26640.24140.21960.20060.18400.16930.15630.14470.13430.12500.11670.10910.10230.09600.09040.08520.08040.07610.0720

-- 71 .oooo1 .oooo0.99030.97510.95740.93540.90660.86760.81480.74760.67150.59510.52480.46290.40970.36430.32550.29230.26370.23900.21760.19890.18240.1 6790.1 5500.14360.13330.12420.11590.10840.10160.09550.08980.08470.08000.07560.071 7

81 .oooo1 .oooo0.98890.97350.95540.93280.90320.86300.80870.74030.66370.58770.51820.45720.40490.36010.32190.28920.2610.23680.21 560.19710.18090.16650.15380.14250.13240.12330.11510.10770.10100.09490.08930.08420.07950.07520.0713

91.00001 .oooo0.98740.97180.95340.93020.89970.85820.80250.73290.65600.58040.51 170.45160.40010.35600.31840.28620.25850.23450.21360.19540.17940.16520.15260.14140.13140.12240.11430.10700.10030.09430.08880.08370.07910.07480.0709

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Table 12 - Reduction factor - buckling curve a--1.00.l0.20.30.40.50.60.70.80.90

1 .oo1.101.201.301.401.501.601.70l B O1.902.002.102.202.302.402.502.602.702.802.903.003.103.203.303.403.503.60

-

-

01 .oooo1 .oooo1 .oooo0.97750.95280.92430.89000.84770.79570.73390.66560.59600.53000.47030.41790.37240.33320.29940.27020.24490.22290.20360.18670.17170.1 5850.14670.13620.12670.1 1820.1 1050.10360.09720.09150.08620.08140.07690.0728

11 .oooo1 .oooo0.99780.97510.95010.9210.88620.84300.78990.72730.65860.58920.52370.46480.41300.36820.32960.29630.26750.24260.22090.20180.18510.1 7040.1 5730.14560.13520.12580.1 1740.10980.10290.09660.09090.08570.08090.07650.0724

21 .oooo1 .oooo0.99560.97280.94740.91790.88230.83820.78410.72060.65160.58240.51750.45930.40830.36410.32610.29330.26490.24030.21880.20010.1 8360.16900.15600.14450.13420.1 2500.11660.10910.10220.09600.09040.08520.08040.07610.0721

31 .oooo1 .oooo0.99340.97040.94470.91470.87830.83320.77810.71390.64460.57570.51 140.45380.40360.36010.32260.29020.26230.23800.21680.19830.18200.16760.15480.14340.13320.12410.11580.10840.10160.09540.08980.08470.08000.07570.071

41 .oooo1 .oooo0.99120.96800.94190.91 140.87420.82820.77210.70710.63760.56900.50530.44850.39890.35610.31910.28720.25970.23580.21490.19660.1 8050.16630.15360.14240.13230.12320.11500.10770.10100.09490.08930.08420.07950.07520.0713

51 .oooo1 oooo0.98890.96550.93910.90800.87000.82300.76590.70030.63060.56230.49930.4432,039430.35210.31570.28430.25710.23350.21290.19490.17900.16490.15240.14130.13130.12240.11430.10700.10030.09430.08880.08370.07910.07480.0709

61 .oooo1 .oooo0.98670.96300.93630.90450.86570.81780.75970.69340.62360.55570.49340.43800.38980.34820.31240.28140.25460.23140.21 100.19320.17750.16360.15130.14030.13040.12150.11350.10630.09970.09370.08820.08320.07860.07440.0705

71 .oooo1 .oooo0.98440.96050.93330.90100.86140.81240.75340.68650.61670.54920.48750.43290.38540.34440.30910.27860.25220.22920.20910.19150.1 7600.16230.15010.13920.12950.12070.1 1280.10560.0990.09310.08770.08280.07820.07400.0702

81 .oooo1 .oooo0.98210.95800.93040.89740.85690.80690.74700.67960.60980.54270.48170.42780.38100.34060.30580.27570.24970.22710.20730.18990.17460.16100.14900.1 3820.12850.11980.11200.10490.09850.09260.08720.08230.07780.07360.0698

91 .oooo1 .oooo0.97980.95540.92730.89370.85240.80140.74050.67260.60290.53630.47600.42280.37670.33690.30260.27300.24730.22500.20540.1 8830.17320.15980.14780.13720.12760.11900.11130.10420.09780.09200.08670.0810.07730.07320.0694

The buckling curves can be described analyticallyfor computer calculations) by the equation:1

x = b + P but x 1with 4 = 0,5 [l + a x 0,2) x*]The mperfection actor a (in equation 3.4) for he corresponding buckling curve can beobtained from he following table:

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Table 13 - Reduction factorX - buckling curve b--x0.00.l0.20.30.40.50.60.70.80.90

1 .oo1.101.201.301.401.501.601.701.801.902.002.102.202.302.402.502.602.702.802.903.003.103.203.303.403.503.60

-

-

01 .oooo1 .oooo1 .oooo0.96410.92610.88420.83710.78370.72450.66120.59700.53520.47810.42690.38170.34220.30790.27810.25210.22940.20950.19200.1 7650.1 6280.15060.13970.12990.12110.11320.10603.09940.09350.08800.08310.07850.0743D.0704

11 .oooo1 .oooo0.99650.96040.92210.87980.83200.77800.71830.65470.59070.52930.47270.42210.37750.33860.30470.27530.24960.22720.20760.19030.17510.16150.14940.13870.12900.12030.11240.10530.09880.09290.08750.08260.07810.07390.0700

21 .oooo1 .00000.99290.95670.91 810.87520.82690.77230.71200.64830.58440.52340.46740.41 740.37340.33500.3010.27260.24730.22520.20580.18870.1 7360.1 6020.14830.1 3760.12810.11950.11170.10460.09820.09240.08700.08210.07760.07350.0697

31 .oooo1 .oooo0.98940.95300.91400.87070.82170.76650.70580.64190.57810.51 750.46210.41270.36930.33140.29850.26990.24490.22310.20400.18710.17220.15900.14720.13660.12720.11860.11090.10390.09760.09180.08650.08160.07720.07310.0693

4 51 .oooo 1 oooo1 .oooo 1 .00000.9858 0.98220.9492 0.94550.9099 0.90570.8661 0.86140.8165 0.81120.7606 0.75470.6995 0.69310.6354 0.62900.571 0.56570.51 17 .50600.4569 0.45170.3653 0.36130.3279 0.32450.2955 0.29250.2672 0.26460.2426 0.24030.221 0.21910.2022 0.20040.1855 0.18400.1708 0.16940.1577 0.15650.1461 0.14500.1356 0.13470.1263 0.12540.1178 0.11700.1102 0.10950.1033 0.10260.0970 0.09640.0912 0.09070.0860 0.08550.0812 0.08070.0768 0.07630.0727 0.07230.0689 0.0686

0.4081 -0.4035

61 .oooo1 .oooo0.97860.94170.9010.85660.80580.74880.68680.62260.55950.50030.44660.39910.35740.3210.28950.26200.23810.21 710.19870.18250.16810.15530.14390.1 3370.1 2450.1 1620.1 0880.10200.09580.09020.08500.08030.07590.07190.0682

71 oooo1 .oooo0.97500.93780.89730.85180.80040.74280.68040.61620.55340.49470.44160.39460.35350.31770.28660.25950.23590.21520.19700.18090.16670.15410.14280.13270.12370.11550.10810.10130.09520.08960.08450.07980.07550.07150.0679

81 .oooo1 .00000.97140.93390.89300.84700.79490.73670.67400.60980.54730.48910.43660.39030.34970.31440.28370.25700.23370.21320.19530.17940.16540.1 5290.14180.13180.1 2280.1 1470.10740.10070.09460.08910.08400.07940.07510.07120.0675

91 .oooo1 oooo0.96780.93000.88860.84200.78930.73060.66760.60340.54120.48360.4310.38600.34590.31 10.28090.25450.23150.21 130.19360.1 7800.16410.15170.14070.13080.12190.11390.10670.10010.09400.08860.08350.07890.07470.07080.0672

Eurocode 3, Annex D allows the use of the higher buckling curve ao instead of a forcompressed members of I-sections of certain dimensions and steel gradeS 460 [6]. This isbased on the fact that, in case of high strength steel, the mperfections (geometrical andstructural) play a less detrimental role on the buckling behaviour, as shown by numericalcalculations and experimental tests on I-section columns of S460. As a consequence hotformed hollow sections using S 460 steel grade may be designed with respect to bucklingcurve a, instead of a.

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Table 14- Reduction factorX - buckling curve ch

0.00.l0.20.30.40.50.60.70.80.90

1 .oo1.101.201.301.401.501.601.701.801.902.002.102.202.302.402.502.602.702.802.903.003.103.203.303.403.503.60

01 .oooo1 .oooo1 .oooo0.94910.89730.84300.78540.72470.66220.59980.53990.48420.43380.38880.34920.31450.28420.25770.23450.21410.19620.18030.16620.15370.14250.13250.12340.11530.10790.10120.09510.08950.08440.07970.07540.07150.0678

11 .oooo1 .oooo0.99490.94400.89200.83740.77940.71850.65590.59370.53420.47900.42900.38460.34550.31 130.28140.25530.23240.21220.19450.1 7880.16490.15250.14150.13150.12260.11450.10720.10060.09450.08900.08390.07930.07500.0710.0675

21 .oooo1 .oooo0.98980.93890.88670.8310.77350.71230.64960.58760.52840.47370.42430.38050.34190.30810.27860.25280.23020.21040.19290.17740.16360.15140.14040.13060.1 210.1 1370.10650.09990.09390.08850.08350.07890.07460.07070.0671

31 .oooo1 .oooo0.98470.93380.8810.82610.76750.70600.64330.58150.52270.46850.41 970.37640.33830.30500.27590.25040.22810.20850.19120.1 7590.16230.15020.13940.12970.12090.1 1300.10580.09930.09340.08790.08300.07840.07420.07030.0668

41 .oooo1 .oooo0.97970.92860.87600.82040.76140.69980.63710.57550.51710.46340.41510.37240.33480.30190.27320.24810.22600.20670.18960.17450.16110.14910.13840.12870.12010.1 1220.10510.09870.09280.08740.08250.07800.07380.07000.0664

51 .oooo1 .oooo0.97460.92350.87050.81460.75540.69350.63080.56950.51 150.45830.41 060.36840.33130.29890.27050.24570.22400.20490.18800.17310.15980.14800.13740.12780.1 1930.1 1150.10450.09810.09220.08690.08200.07750.07340.06960.0661

61 .oooo1 .oooo0.96950.91830.86510.80880.74930.68730.62460.56350.50590.45330.40610.36440.32790.29590.26790.24340.22200.20310.18640.17170.15850.14680.13640.12690.11840.11080.10380.09750.09170.08640.08160.07710.07300.06920.0657

71 .oooo1 .oooo0.96440.91310.85960.80300.74320.68100.61840.55750.50040.44830.40170.36060.32450.29290.26530.2410.22000.2010.18490.1 7030.15730.14570.13540.12600.11760.1 1000.10310.09690.0910.08590.0810.07670.07260.06890.0654

81 .oooo1 .oooo0.95930.90780.85410.79720.73700.67470.61220551 60.49500.44340.39740.35670.3210.29000.26270.23890.21800.19960.18330.16890.15610.14460.13440.12520.11680.10930.10250.09630.09060.08540.08060.07630.07220.06850.0651

91 .oooo1 .oooo0.95420.90260.84860.79130.73090.66840.60600.54580.48960.43860.39310.35290.31 780.28710.26020.23670.21 610.19790.18180.16760.15490.14360.13340.12430.11610.10860.10180.09570.09010.08490.08020.07590.07190.06820.0647

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Tubular triangular arched trussor the roof structure of a stadium

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4 Members nbendingIn general, ateral-torsional buckling resistance need not be checked or circular hollowsections and rectangular hollow sections normally usedn practice (b/h 0.5). his is due tothe fact that their polar moment of intertia I, is very large in comparison with that of openprofiles.

4.1 Design for laterial-torsional bucklingThe critical lateral-torsionalmoment decreaseswith increasing length of a beam.Table 15 shows he length of a beam (of various steel grades) exceeding which lateral-torsional failure occurs.The values are based onhe relation:I 113 400-


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5 Members in combined compression and bending5.1 GeneralBesides concentrically compressed columns, structural elementsaremost often loadedsimultaneously by axial compression and bending moments. This chapter s devoted oclasses 1,2and 3 beam-columns. Thin-walled members (class ) are consideredn chapter 6.

5 .2 Designmethod5.2.1 Design forstabilityLateral-torsional buckling is not a potential failure mode for hollow sections (see chapter).According to Eurocode 3 [ l ] he relation is based on he following linear interaction formulae:N,, My.Sd+ K- Mz,Sd+ K,- Mz,R d 5 1Nb.Rdwhere:NSd = Design value of axial compression (yFtimes oad)

X = min X , , X , ) = Reduction factor (smaller of x, and xz ,see chapter 3.2A = Crossectional reaf, = Yield strengthyM = Partial safety actor or esistanceMy,Sd, Mz,SdMaximum absolute design valueof the bending moment about y-yor z-z axisaccording to the first order theory)

M = W,,,, .f by elastic utilization of a cross section (class3)YMor M = W,,,, .f by plastic utilization of a cross section (class 1 and 2)Y M

Mz,Rd W,,,, .f by elastic utilization of a cross section (class3 .YMor = W,,,, f by plastic utilization ofacrosssection class 1 and 2)YM

) increment of bending moments according to the second order theory is considered by determiningand X by buckling lengths f whole structural system28


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K, = 1 ~x , .Np, p z , however K, .5 5 .6 )

(2: ;), howeverp, .9 (5.7)For elastic sections (class 3) the value n the equations (5.5) and (5.7) is taken to be~ Z = 2 @ M , z - 4 ) + - WPIZ .Wll,Zequal to 1.PM,,and PM,,are equivalentuniform moment factors ccording to Table 16,column2, in orderto determine the form of the bending moment distribution M and M,.Remark 1For uni-axial bending with axial force, the reduction factor X is related to the loaded bendingaxis, as for example,X , for the applied M with M, = 0.Then the following additional requirement has to be fulfilled:

Table 16 - Equivalent uniformmoment factors pp and p1

moment diagram

end moments

1 < * < l

moment from lateral loadMMa

moment due to combinedlateral loadplus endmoments

2equivalent uniform momentfactor pu

M = I max M due tolateral load onlyAM = I max M I for momentdiagram withoutchange of sign

I maxM + I minMIwhere sign of momentchanges

3equivalent uniform momentfactor p ,Dm,+= 0.66+ 0.44 4however p,, 1 -and p,, 0.44

NNKi

P,,, = 1.0

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Remark 2:A further design method for the loading case of bending moment and axial compression isavailable in the literature [21,22, 231, which is called substituting member method [24,51.It is based on he ormula or uni-axial bending moment and axial force), which is usedfrequently:

N S d P m 1+-- . 5 1y Np l , Rdd 1 ~N Ywhere, besides the definitions already described,

(5.9)

r 2 .ElN =-- N p l (Eulerianbuckling oad)x2P, = Equivalent uniform moment factor from Table 16, column 3.

P, < 1, allowed only for fixed endsf a member and constant compression withoutlateral loadM according to equation 5.3) (elastic or plastic)Equation (5.9)can be written conservatively n a simplified manner:

NSd P m M y. S dy Np l , Rd

+ 5 0.9 (5.9a)

5.2.2 Designbased on stressA compressed member has to be designed on the basisf the most stressed cross sectionnaddition to stability. Axial force, bending moments M, and M, and shear force have to beconsidered simultaneously. According to Eurocode3 [ l ] an applied shear forceV,, can beneglected, when the following conditions fulfilled:Sd 0.5 pI,Rd (5.10)where v,,,,, = Design plastic shear resistanceof a cross section

= 2d-f i . hfi Y M

f, for CHS (5.11)f= 2 t . h 2 (5.12)

for RHS (b, instead of h when shear force s parallel to b)A = 21 dm or 2 t . h

) Corresponding formulaefor uni-or bi-axial bending and axial force are given in21, 23130


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Equation(5.10)s satisfied in nearly all practical cases.In some other codes[21]he limiting values for , up to which the shear force can bedisregarded, is significantly lower than.5.

'SdVpl,Rd

5.2.2.1 Stress design without Considering shear load [l]The following relationship s valid forplastic design (cross section classes and 2):

where cy = p = 2 or CHSc y = @ = - , however 5 61 1.13n2

(5.13)

(5.14)

(5.15)Y M

M and MNz,Rd are he educedplastic esistancemoments akingaxial orces ntoaccount. These reduced moments are describedy the relations given below.For rectangular hollow sections:

= Mpl,y,Rd (l n), however Mpl,y,Rd (5.16)MNz.Rd = Mpl.z.Rd 0.5(+ h 1 however Mpl,z,Rdl - n

For square hollow sections:M = 1,26Mpl,Rd(1 n), however 5 MP,,,,For circular hollow sections:M = 1,04. P,(1- n'.'), however 5 MP,

(5.1 )

(5.18)

(5.19)For circular hollow sections, the following exact and simple equation23]s also valid insteadof the equation(5.19):MSd

Mpl,Rd(5.20)

where MSd = i (5.21)But the shear force must be limitedo ~ 5 0.25Forelastic design the following simple linear equation can be applied insteadf the equation(5.13):

'Sd'pl.Rd

Sd My Sd SdL C 1 1A . yd wel.y . yd fyd (5.22)where fyd = fJyMThis equation can also be used, as a lower bound, but more simple to use, for plastic designof cross section classes and 2 nstead of the equation(5.13).

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5.2.2.2 Stress design considering shear load [ l ]If the shear load Vs, exceeds50% of the plastic design resistancef the cross sectionthe design resistanceof the cross section to combinationsf moment and axial force shall becalculated using a reduced yield strength for the shear area, where:red. f, = (1 - e ) ,

e = 6 -l> (5.23)(5.24)Vpl,Rdis according to equation(5.1 1)r (5.12).For circular hollow section: A, = AFor rectangular hollow section:- shear load parallel o depth: A, =

hb + h

- shear load parallel to width: A, =-bb + hFor circular hollow section, the following xact but simple equation can be givenaking alsothe shear force nto account (231:a

MSdMpl,Rd

where 7 =

(5.25)

(5.26)

Sd = i-i (5.27)V is according to the equation (5.1 1).M is according to the equation (5.21).No reduction for f, as shown in the equation (5.23) as to be made.

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Uni-planar tubular broken-off truss

Tubular supports for a canvas roof construction33


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6 Thin-walled sections6.1 GeneralThe optimisation of the buckling behaviour of hollow sections leads, for a constant value fcross sectional area, to profiles f large dimensions and small thicknesses (large momentfinertia).Small thicknesses (relative to outer dimensions) can cause failure, before reaching yieldstrength in the outer fibres, by local buckling. The unavoidable imperfections f the profilesinvolve an interaction between local bucklingn the cross section and flexural bucklingn thecolumn. This decreases the resistance to both typesf buckling.By keeping within thedlt or blt limits for the respective cross section classes givenn Tables4, 5 and 6, it s not required to check local buckling.Only when exceeding the dl t or blt limits for class 3 sections, does the influence of localbuckling on the load bearingapacityof the structural members have to be taken into account.The cross section thus involved shall be classifieds class 4 (see Fig.2).It should be noted thathe phenomenon of local buckling can become moreritical by applyingand utilizing higher yield strength, o that smaller b/t ratios have to be selected (see Tablesand 5, last line).Eurocode 3 [ l ] akes accountof local buckling by the determininghe load bearing capacityusing effective cross section dimensions, which are smaller than the real ones.In the structures, which are dealt with in this book, circular hollow sections with a dlt ratiohigher than theimiting values given n Table 4 are seldom used;n general, dlt values are50at hehighest. In consequence, hischapter is mainlydevoted oclass 4 squareandrectangular hollow sections.

6.2 Rectangular hollow sections6.2.1 Effective geometrical properties of class 4 cross sectionsThe effective cross section properties of class 4 cross sections are based on the effectivewidths of the compression elements.The effective widthsof flat compression elements shall be obtained using Table 17.The plate buckling reduction facor 4 shall be calculated by means of the relations given inTable 18. or the sake f simple calculation, the equation6.2)nd (6.1) are describedn Fig. 8( e = f(X)) and Fig. 9 (k, = f(4)).In order to determine the effective widthf a flange element, the stressatio 4 used in Table17 shall be basedn he properties of the gross (not reduced) cross section.o calculate theeffective depth (h,) of web elements, the effective areaf the compressed flangebe,, . ) butthe gross areaof the webs h . ) has to be used. This simplification allows a direct calculationof effective widths.Strictly speaking, an exact calculation of the effective width of a web element requires aniterative procedure.Under bending moment loadingt is possible that the effective (reduced) width becomes validonly for one flange. This results in a mono-symmetrical cross section with a correspondingshift of the neutral axis. s a consquence, the effective section modulas has to be calculatedwith reference to the new neutral axis.Note: Eurocode 3 [ l , 21 is not consistent regarding the definition of a so-called thin-walledprofile.34


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Table 17- Effective widths and buckling factorsor thin-walled rectangular ho llow sectionsstress distribution compression positive)b , = h - 3 t o r b - 3 t effective widthben

ben = e .btbel = 0.5 be,be, = 0.5 ben

$ = a2/a, 01 > > 0lbuckling factor

Alternatively: for 1 2 $ z - 13.24.0km 1.05 - $ 7.81

k, = 16h1+ )2+0.112(1 -$ )2+ (1 +$)

ben = e .bcbel = 0.4benbe, = 0.6 be,

0 > > - 1 1 - 1 - 1 > > - 27.81 - 6.29 + 9.78$? 5.98 1 - 4)'3.9

Plate buckl lng reduc ton actor p B u ck l i n gfactor K,1 . 0 , , r , 1 60

Nond om ension al slenderness x pFig. 8 - Plate buckling reductionactor p

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Table 18 - Plate buckling reduction factor

I where xp , the non-dimensional slenderness of the flat compression element, is given by:where f is the critical plate bucklingtressand k, is the plate buckling factor (see Table 17 and Fig.)with t =e nd f, = yield strength in N/mm2Reference [2] considers that the influenceof the internal corner radius need noto be taken intoaccount provided that:r s 5 t

These conditions are fulfilled by practically allctually produced square and rectangular hollowsections.

The b,/t limit above which local bucklingneeds to be takennto account according to Tables8 and9 is blt > 42 E for a uniformly compressed flange. However equation (6.2)n Table 18 foran identically oaded flange givesX, > 0.673; this results in b,/t > 38.2 E some what smallerthan the 42E above.It is well known, that the equation (6.3) for plate buckling gives conservative results. Onaccount of his, possible local bucklingf thin-walled sections haso be considered first, whenthe b,/t limits given in Tables 5 hrough 7 are exceeded.

6.2.2 Design procedureWhen the effective geometricalroperties of a class 4 cross section, e. . effective areaAeffective radius f gyration eff, ffective section modulusWeff ,have been calculated,t is easyto check the stability and the resistance. Indeed, it is just necessary to use these effectiveproperties n place of the geometrical propertiesf the gross section n class 3 calculations.For dimensioning thin-walled ross section, equation (5.22) s replaced by the relation:

NSd + + Mz.SdAeff Iyd fyd weff,z fyc 5 14with fyd = Y M

Hollow sectionshave two axes of symmetrynd therefore there s no shift of the neutral axiswhen the crosssection is subject ouniformcompression.This eads o an importantsimplificationof class 4 eam-column equations, becausedditional bending momentsdue tothis shift do not exist n the case of structural hollow section.The use of effective geometrical propertiesof thin-walled sections is recommended in thecodes of the most countries around the world. Only in the japanese code, the load bearingcapacity of a thin-walled ectangular hollow section s given by the smaller of the maximumplate buckling load nd global buckling oad.At last, as shown n reference 101, he lateral-torsional bucklingan alsobe disregarded forthin-walled hollow sections of class 4.36


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6.2.3 Design aidsFor practical application, the transition from the cross section classo class 4 is of specialimporance showing the lt limits, below which ocal buckling can be disregarded. With = 1 ,the equation 6.2) leads to he limit x,, S 0.673.Fig. 10gives -on base of the depth or width-to-thickness ratio and of the coefficient(Table17) as well as of the yield strength , - the possibility of a quick checkof the zone where noallowance for ocal buckling s necessary. The area o the leftof the curves belongs o crosssection class, hile thato the right covers class , all of them lying in the elastic range. Whenblt limits given by the curves are exceeded (local buckling), the plateuckling reduction factore according to the equation (6.2) has to be determined.

K O fv (N/mm21 = 460 355 275 235

50

40

3023.9

20

1040

25 50 75 1002550l o r .L

Fig. 10 - b,/t or h,/t limits, below which localbuckling can be disregarded

P1.000.900.800.700.600 50

0.400.300.200 . 1 0

0 0 10 20 3 0 40 5 0 60 70 Ib l i t

Fig. 1 1 - Plate buckling curves

Fig. 12- Effective RHS cross section under axial forceN and bending momentsM,, M,37


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Plate buckling reduction factor e vs. or various structural steel grades is drawn inFig. 11 (see equation6.3).Effective geometrical alues for the cross sectionsof class 4 can be calculated by meansfthe formulae given in Table 19. The notations in Table 19 are explained n Fig. 12.

b,/tf i o

Table 19 - Effective geometrical propertiesaxial force:

Aen 2t (ben+ h" + 41)

bending moments:

W,,, = t I (be,, + 2t)(> 6 - 2 (+ - 6> (h,,, + ben+ 2t)

6.3 Circular hollow sectionsFor hin-walled circular hollowsections, it is more difficult to judge the ocal bucklingbehaviour, especially the interaction etween global and local buckling, than in the case ofplates.This is due to the localnstability behaviour of cylindrical shells, theirhigh susceptibilityto imperfectionsand sudden reduction of load bearing apacity without reserve[23].Local buckling has alsoo be considered forCHS,when the d/timits for the cross sectionareexceeded (see Tables and 7 ) .38


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Circular hollow sections, which are applied n practice, do not or seldom, possess dlt ratiosexceeding those given n Tables4 and 7; in general dlt S 50.In cases, wherehin-walled circular ollow sections arepplied, he procedureof substitutingthe yield strength f in the already mentioned formulae by the real buckling stresses. for ashort cylinder, can be used.These buckling stresses can be calculated by he procedure shown in [26] or [27]. Theprocedures in bothcases are simple; however, there s no equation describing the bucklingstress explicitly.

* U in [26]; uXS,RKn [ 2 7 ]39


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7 Buc k l ing leng th of m em bers in lat t ice g irders7.1 GeneralChord and bracingmembers of welded lattice girder are artially fixed at the nodes,lthoughthe static calculation of the forces in the members is carried out assuming the joints to behinged.As a consequence of this partial restraint, a reduction f the system length I is made to obtainthe effective buckling lengthb .

7.2 Effective buckling lengtho f chord and bracing members with lateral supportThe buckling of hollow sections in lattice girders has been treated in [14, 15, 281. Based onthis, Eurocode3 l , 2 - Annex K ] recommends the buckling lengths for hollow sections inlattice girders as follows:Chords:- in-plane: I, = 0.9 x system length between joints- out-of-plane: = 0.9 x system length between the lateral supportsBracings:- in- and out-of-plane:b = 0.75 x system length between joints.When the ratio of the outer diameter or widthf a bracing tohat of a chords smaller than 0.6,the buckling length f the bracingmember canbe determinedn accord with Table20.The equations given are only valid for bracing members, which are welded on the chordsalong theull perimeter length without cropping or flattening of the ends of the members. Dueto the fact that no test results are, at present time, available on fully overlapped joints, theequation given in able 20 cannot be applied o this type of oint.

Fully overlappedpmls

In bothof the last cases,a buckling length qual to the system ength of the bracingmemberhas to be used.

7.3 Chords of lattice g irders, whose joints are not supported laterallyThe calculation is difficult and lengthy.Therefore, it is convenient to usea computer.For laterally unsupported truss chords the effective buckling ength can be considerablysmaller than the actual unsupportedength.References(12, 151 give two calculation methods for the casef compression chordsn latticegirders without lateral support. Both methods are based on anterative method and equire heuse of a computer.However, in order to facilitate the application or commonly encounteredcases (laterally restrained in direction), 64 design charts have been drawn and appear asappendices in CIDECT Monograph no.4 [15].The effective buckling lengthof a bottom chord loaded in compression (as for example, byuplift loading) depends on the loading in the chord, the torsional rigidity of the truss, the40


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bending rigidity of the purlins and the purlin to truss connections. For detailed information,reference is given to 12, 151.For the example given in the following figure, the buckling length of the unsupported bottomchord can be reducedo 0.32 times the chord lengthL.

buckl lng lengthbottom c ho rd l t ~ - 0.32 L

IPE 140Q 1 3 9 . 7 ~ 4Q 6 0 x 3Q 1 3 9 7 x 4

Lateral buckllngof laterally UnSUpPOrIedchords

Table 20 - Buckling length of a bracing member in a latticegirderdo: outer diameter of a circular chord memberdl: outer diameter of a circular bracing memberbo: externalwidth of a square chord member dlllb,: external width of square bracing member

for all P : Ib/1 0.75when p < 0.6,n general 0.55 0.75calculate with:

IbI

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Lattice girderof square hollow sections supported by a cable construction

General view of a RHS roof structure42


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8 Design examples8.1 Design o f a rectangular hollow section column in compression

Nqd= 1150 k N 1150 Nrl/2= 4m_I

Y - V

v@m

l I t

Fig. 13 - Column under concentric compressionA column is to be designed using a rectangular hollow section 300 x 200 x 7.1 mm, hot-formed with a yield trength of 235 N/mm2 (steel grade 35).The length of the column is 8 m. It has hinged support at both ends. An intermediate supportat the middle of the column length exists against buckling about the weak axis2-2.Given: Concentric compression (design load) NSd 1150 kNbucklingength: = 8 mlb ,z= 4 msteelrade: S235; f = 235/mm2geometric properties: A = 67.7 cm2; iy = 11.3 cm; iz = 8.24cm

b,max.- 300 71 = 39.25 < 42 (compare with Tab. 5 and 6)t 7.1X = - 8oo = 70.8; X = 48.6 < X y00Y 11.3.24 - = - - L -70 - 0.754seeTab. loa)y 93.9

xY = 0.821 (Tab. 12, buckling curve a)Acc. to equation (3.1):N = 0.821 .6770. = 1187 kN > 1150kN.Thereforecolumnokay.351.18.2 Design of a rectangular hollow section column in combined compression and uni-axial bending

Fig. 14 -

NSd= 800k N

2 - 2 ,rdColumn under combined compression and uni-axial bending

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given: hot-formed rectangular hollow section column 300 x 200 x 8 mmcompressionsd = 800 kNbending moment My,Sd 60 kNm or 18 kNm at both endsbuckling length l b . , = = 8.0msteelgrade S275; = 275 Nlmm'W, = 634cm3; W, = 510 cm3WPl,, = 765 cm3; Wpl,z= 580 cm3

geometric properties: A = 75.8 m'; i, = 11.2 m; iz = 8.20 m

- -, 200 3.8 = 22t 8- -, 300 3.8 = 34.5 (Tables 5 and 6)

- < 38.0.92= 35for class 2 cross section of S275-t 8

a) Calculation for flexural buckling: /

A = ~ = 71.4;00

y 11.2 A = ~ = 97.6008.2-A = -71'4 0.823 see Tab. loa); - 97.686.886.8 X = 1.124

i c y = 0.782 see Tab. 12, uckling curve "a"); X , = 0.580Acc. toTab le16:~ , , ,=1 .8-0 .7~0.3=1.59Acc. to equation (5.5):, = 0.823 (21.59- 4)+ 765 634 = 0.468 0.9634Acc. to equation (5.4):, = 1 ( - 0'468) ' O3 = 1.23 1.50.782.7580.275Calculation for the stability about y-y axis acc. to equation (5.1):800.103.1.1 23 6o l o 6 ' = 0.540+ 0.386 0.926< 1 .O0.782 7580.275 765. o3. 75

Calculation for buckling about z-z axis:NSd Nb.z.Rd800 < 0.580 7580 275 = 1099.1kN.Thereforecolumnokay.

b) Calculation for the load bearing capacityShear load V: 'y.Sd = ~ - - 5.25 N8Acc. toequation (5.1 : = 2 8 (300 8) 275 = 674 kN4 3 . 1 . 1- - - - 5 25 - 0.008< 0.5

Pl.y,Rd - 674The shear load can be disregarded.Acc. to equation (5.13): My,Sd I

(MNy,Rd)M = 60 kNm max)ff= 1.66 NSd 800. .1 = o,422n = -1 - 1.13 n2pl Rd 75.8 27.5

- 1.661 - 1.13. .422'= 2.07

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ACC.o equation (5.16): M,,,,, = 1.33 765o3 1 175 ( - 800-103-1.1)580 275= 147. lo6Nmm= 147kNm

( = ( ~ ) ' '0.156c 1.0. Therefore column okay.MNy,Rd8.3 Design of a rectangular hollow section column in combined compressionandbi-axial bending

Nld = 1000 k N 1000 k N

2 - 2 My,rd v - v Mz. rdFig. 15- Column under combined compression and

bi-axial bending

Given: Hot formed rectangular hollow section column300 x 200 x 8.8mmThe length of the column s 8 m.Both ends of the columns have hinged support about the strong axis y-y and fixedsupport at the foot end about he weak axisz-z.

Compression N = 1000 kNBending momentM,,, = 60 kNmMz,Sd 50 kNmSteel grade: S355; f, = 355 Nlrnm'Buckling ength: lb., = 8 m

lb., = 0.7.8.0 5.6 mGeometric properties:A = 82.9cm3

W, = 689 cm3; W, = 553 cm3Wp,,y834 cm3; W,,,, = 632cm3iy = 11.2 m; i, = 8.16 m

The cross sectionust satisfies the requirements for the classof S355 Tables5 and 6).45


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a) Calculation or the global bucklingacc. to equation 5.1)X = 71.4 X = 68.66011.2 8.16- 71.4X = 0.935 X = 0.89868 676.4 76.4X , = 0.711 ( = X,,,) x Z = 0.735 (buckling curve a)Acc. oequation 5.2): N = 0.711 .8290. = 1902 kN ( = min Nb,R,,)551.1

Nb,d = 0.735 8290.F i o - 3 = 1966 kNACC. o equation 5.3): MP,,,,,, = 834. o3. = 269 kNm

= 632. lo3. = 204 kNm3551 l

Acc.oab. 16: OM,, = 1.8Acc. o equation 5.5): p y = 0.935 2. 1.8 - 4) + g l) = 0.164 < 0.9Acc.oquation 5.4): K, = 1 ( - 0.164) 1000. lo30.711 .a290 355 = 1.078< 1.5Acc.oab. 16: OM,, = 1.8 0. 7 - 0.5)= 2.15Acc. to equation 5.7): = 0.898 2 . 2.15 - 4) + g l) = 0.412 < 0.9Acc. oequation 5.6): K, = 1 - 0.4121000 . lo3 = o.809 < ,50.735.8290 * 355Finally, acc. to equation 5.1):-902 269Oo0 + 6o + * 50 = 0.526 + 0.240 + 0.198204 = 0.964< 1.0

b) Calculation or load bearing capacityIn order toobtain sufficient load bearingapacity of the cross sectionhe elasticequation5.22) s applied conservatively (all values in kN and mm):

1000~1:1 60.103.1.18290 .0.355 689 . O3 .0.355 553 . l O3 .0.35550 * lo3. = 0.374 + 0.270 + 0.280= 0.924 < 1.OIf this calculation would not have led to a satisfactory result (that means 1.O), hen thecalculation must be carried out using equation 5.13).The assumption to neglect shearoad in quations 5.13)and 5.22) s V,, 0.5V,,,,,, seeequation 5.10)[ l 21.The shear resistance acc. to equation 5.12) s decisive in this case:Vpl,z,Rd = 2 8.8 200 8.8) 355d 3 - 1 . 1= 627 kN

Sdvpl,Rd- - - 0.015 < 0.5. Therefore shear is not critical.

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8.4 Design of a thin-walled rectangular hollow section column in compression

2 2 V - V

Fig. 16 - Thin-walledcolumn under concentriccompressionGiven: Cold-formed rectangular hollow section column 400 200 x 4 mm (acc. o IS0 4019

11 71)The length of the column is 10m.Both ends of the column have hinged support about the strong axis y-y and fixedsupports at both ends about the weak axis-z.

Steelgrade:S275, f = 275N/mm2 basichot olled strip)Buckling ength: lb., = 10m

= 5 m0

N = 500 kNCross sectional area A= 46.8 cm2

1. Calculation of average increased yield strength after cold-formingAcc. to equation (1.2): = 275 + 400 + 2004 4 (410 275)

= 287.6 N/mm2.:1.2.275 = 330 Nlmrn

2. Cross section classificationLongside: -Shortside: r

h, 400 - 3 . 4t - 4

=g7142c 38.8 (Tables 5 and 6)b, 200 3 . 4 = 47- 4The cross sections thin-walled (class4) and the calculation shall be made using effectivewidth.According to Fig. 8, the limit for plate buckling: Xp, ,lml, = 0.673 xp acc. to equation (6.2)with e = 1.0).Non-dimensionalslenderness akingyieldstrength of thebasicmaterial fb acc. oequation (6.3):- 97

= 28.4 . dTi- - 1.85 > 0.673-Ap,z = 28.4 47 - 0.90 > 0.673

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Non-dimensional slenderness taking average increased yield strength f (287.6 N/mm2)after cold-forming:- 97hp. , =

xp,, =28.4. izi- = 1.89 > 0.673

- 4728.4. dTd235/287.6 = 0.92 > 0.673In all cases, the cross section belongs to class 4.

3. Effective geometric valuesa) With yield strength f the basic material (275 N/mm2) and , = 4 (simplecompression):

e , = 0.476e , = 0.840 acc. to equation (6.2)heft = 0.476 (400 - 3.4) = 184.7 mmbe, = 0.840 (200 3.4) = 157.7 mmA,, = 28.69 cm2ieff,y17.50cmacc. oTab.19i,,,, = 8.32cm

ace. to Tab. 17

b) With average increased yield strengthafter cold forming (fya = 287.6 N/mm2)e = 0.468e , = o.827 ] acc. to equation (6.2)h = 0.468 (400 3 .4) = 181.6 mmbe, = 0.827 (200 3 . 4) = 155.5 mmAetf = 28.25 cm2ie+,y = 17.60 cmief,., = 8.33cm

acc. to Tab. 17

4. Design for global bucklinga) With yield strengthof the basic material f y b = 275 N/mm2):

Strong axisx - 17.5Oo0 - 57.1-X - -- 86.8 =71 0.66 (seeTab. loa)x , = 0.806 (acc. toTab.13,curve b)Nb = 0.806.2869 .11= 578 kN (see equation (3.1))275Weak axisx, = - -8.3200 - 60.1- 60.1X = 86 8 = 0.69x , = 0.7893 acc. oTab.13,curve b)Nb,Rd = 0.7893.2869.- 566 kN.2751.1

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