ICS ; Supersedes ENV :1992. English version

Transcription

1 EUROPEA STADARD ORE EUROPÉEE EUROPÄISCHE OR E a 005 ICS ; Supersedes EV :199 English version Eurocode 3: Design of seel srucures - Par 1-1: General rules and rules for buildings Eurocode 3: Calcul des srucures en acier - Parie 1-1: Règles générales e règles pour les bâimens This European Sandard was approved b CE on 16 April 004. Eurocode 3: Bemessung und Konsrukion von Sahlbauen - Teil 1-1: Allgemeine Bemessungsregeln und Regeln für den Hochbau CE members are bound o compl wih he CE/CEELEC Inernal Regulaions which sipulae he condiions for giving his European Sandard he saus of a naional sandard wihou an aleraion. Up-o-dae liss and bibliographical references concerning such naional sandards ma be obained on applicaion o he Cenral Secrearia or o an CE member. This European Sandard exiss in hree official versions (English, French, German). A version in an oher language made b ranslaion under he responsibili of a CE member ino is own language and noified o he Cenral Secrearia has he same saus as he official versions. CE members are he naional sandards bodies of Ausria, Belgium, Cprus, Cech Republic, Denmark, Esonia, Finland, France, German, Greece, Hungar, Iceland, Ireland, Ial, Lavia, Lihuania, Luxembourg, ala, eherlands, orwa, Poland, Porugal, Slovakia, Slovenia, Spain, Sweden, Swierland and Unied Kingdom. EUROPEA COITTEE FOR STADARDIZATIO COITÉ EUROPÉE DE ORALISATIO EUROPÄISCHES KOITEE FÜR ORUG anagemen Cenre: rue de Sassar, CE All righs of exploiaion in an form and b an means reserved worldwide for CE naional embers. B-1050 Brussels Ref. o. E :005: E

2 E : 005 (E) Conens Page 1 General Scope ormaive references Assumpions Disincion beween principles and applicaion rules Terms and definiions Smbols Convenions for member axes... 0 Basis of design....1 Requiremens Basic requiremens Reliabili managemen Design working life, durabili and robusness.... Principles of limi sae design Basic variables Acions and environmenal influences aerial and produc properies Verificaion b he parial facor mehod Design values of maerial properies Design values of geomerical daa Design resisances Verificaion of saic equilibrium (EQU) Design assised b esing aerials General Srucural seel aerial properies Ducili requiremens Fracure oughness Through-hickness properies Tolerances Design values of maerial coefficiens Connecing devices Faseners Welding consumables Oher prefabricaed producs in buildings Durabili Srucural analsis Srucural modelling for analsis Srucural modelling and basic assumpions... 9

3 E : 005 (E) 5.1. Join modelling Ground-srucure ineracion Global analsis Effecs of deformed geomer of he srucure Srucural sabili of frames Imperfecions Basis Imperfecions for global analsis of frames Imperfecion for analsis of bracing ssems ember imperfecions ehods of analsis considering maerial non-lineariies General Elasic global analsis Plasic global analsis Classificaion of cross secions Basis Classificaion Cross-secion requiremens for plasic global analsis Ulimae limi saes General Resisance of cross-secions General Secion properies Tension Compression Bending momen Shear Torsion Bending and shear Bending and axial force Bending, shear and axial force Buckling resisance of members Uniform members in compression Uniform members in bending Uniform members in bending and axial compression General mehod for laeral and laeral orsional buckling of srucural componens Laeral orsional buckling of members wih plasic hinges Uniform buil-up compression members General Laced compression members Baened compression members Closel spaced buil-up members Serviceabili limi saes General Serviceabili limi saes for buildings Verical deflecions Horional deflecions Dnamic effecs Annex A [informaive] ehod 1: Ineracion facors k ij for ineracion formula in 6.3.3(4)

4 E : 005 (E) Annex B [informaive] ehod : Ineracion facors k ij for ineracion formula in 6.3.3(4) Annex AB [informaive] Addiional design provisions...81 Annex BB [informaive] Buckling of componens of building srucures

5 Foreword E : 005 (E) This European Sandard E 1993, Eurocode 3: Design of seel srucures, has been prepared b Technical Commiee CE/TC50 «Srucural Eurocodes», he Secrearia of which is held b BSI. CE/TC50 is responsible for all Srucural Eurocodes. This European Sandard shall be given he saus of a aional Sandard, eiher b publicaion of an idenical ex or b endorsemen, a he laes b ovember 005, and conflicing aional Sandards shall be wihdrawn a laes b arch 010. This Eurocode supersedes EV According o he CE-CEELEC Inernal Regulaions, he aional Sandard Organiaions of he following counries are bound o implemen hese European Sandard: Ausria, Belgium, Cprus, Cech Republic, Denmark, Esonia, Finland, France, German, Greece, Hungar, Iceland, Ireland, Ial, Lavia, Lihuania, Luxembourg, ala, eherlands, orwa, Poland, Porugal, Slovakia, Slovenia, Spain, Sweden, Swierland and Unied Kingdom. Background of he Eurocode programme In 1975, he Commission of he European Communi decided on an acion programme in he field of consrucion, based on aricle 95 of he Trea. The objecive of he programme was he eliminaion of echnical obsacles o rade and he harmoniaion of echnical specificaions. Wihin his acion programme, he Commission ook he iniiaive o esablish a se of harmonied echnical rules for he design of consrucion works which, in a firs sage, would serve as an alernaive o he naional rules in force in he ember Saes and, ulimael, would replace hem. For fifeen ears, he Commission, wih he help of a Seering Commiee wih Represenaives of ember Saes, conduced he developmen of he Eurocodes programme, which led o he firs generaion of European codes in he 1980s. In 1989, he Commission and he ember Saes of he EU and EFTA decided, on he basis of an agreemen 1 beween he Commission and CE, o ransfer he preparaion and he publicaion of he Eurocodes o he CE hrough a series of andaes, in order o provide hem wih a fuure saus of European Sandard (E). This links de faco he Eurocodes wih he provisions of all he Council s Direcives and/or Commission s Decisions dealing wih European sandards (e.g. he Council Direcive 89/106/EEC on consrucion producs CPD and Council Direcives 93/37/EEC, 9/50/EEC and 89/440/EEC on public works and services and equivalen EFTA Direcives iniiaed in pursui of seing up he inernal marke). The Srucural Eurocode programme comprises he following sandards generall consising of a number of Pars: E 1990 Eurocode: Basis of srucural design E 1991 Eurocode 1: Acions on srucures E 199 Eurocode : Design of concree srucures E 1993 Eurocode 3: Design of seel srucures E 1994 Eurocode 4: Design of composie seel and concree srucures E 1995 Eurocode 5: Design of imber srucures E 1996 Eurocode 6: Design of masonr srucures E 1997 Eurocode 7: Geoechnical design E 1998 Eurocode 8: Design of srucures for earhquake resisance 1 Agreemen beween he Commission of he European Communiies and he European Commiee for Sandardisaion (CE) concerning he work on EUROCODES for he design of building and civil engineering works (BC/CE/03/89). 5

6 E : 005 (E) E 1999 Eurocode 9: Design of aluminium srucures Eurocode sandards recognie he responsibili of regulaor auhoriies in each ember Sae and have safeguarded heir righ o deermine values relaed o regulaor safe maers a naional level where hese coninue o var from Sae o Sae. Saus and field of applicaion of Eurocodes The ember Saes of he EU and EFTA recognie ha Eurocodes serve as reference documens for he following purposes : as a means o prove compliance of building and civil engineering works wih he essenial requiremens of Council Direcive 89/106/EEC, paricularl Essenial Requiremen 1 - echanical resisance and sabili - and Essenial Requiremen - Safe in case of fire; as a basis for specifing conracs for consrucion works and relaed engineering services; as a framework for drawing up harmonied echnical specificaions for consrucion producs (Es and ETAs) The Eurocodes, as far as he concern he consrucion works hemselves, have a direc relaionship wih he Inerpreaive Documens referred o in Aricle 1 of he CPD, alhough he are of a differen naure from harmonied produc sandard 3. Therefore, echnical aspecs arising from he Eurocodes work need o be adequael considered b CE Technical Commiees and/or EOTA Working Groups working on produc sandards wih a view o achieving a full compaibili of hese echnical specificaions wih he Eurocodes. The Eurocode sandards provide common srucural design rules for everda use for he design of whole srucures and componen producs of boh a radiional and an innovaive naure. Unusual forms of consrucion or design condiions are no specificall covered and addiional exper consideraion will be required b he designer in such cases. aional Sandards implemening Eurocodes The aional Sandards implemening Eurocodes will comprise he full ex of he Eurocode (including an annexes), as published b CE, which ma be preceded b a aional ile page and aional foreword, and ma be followed b a aional annex (informaive). The aional Annex (informaive) ma onl conain informaion on hose parameers which are lef open in he Eurocode for naional choice, known as aionall Deermined Parameers, o be used for he design of buildings and civil engineering works o be consruced in he counr concerned, i.e. : values for parial facors and/or classes where alernaives are given in he Eurocode, values o be used where a smbol onl is given in he Eurocode, geographical and climaic daa specific o he ember Sae, e.g. snow map, he procedure o be used where alernaive procedures are given in he Eurocode, references o non-conradicor complemenar informaion o assis he user o appl he Eurocode. Links beween Eurocodes and produc harmonied echnical specificaions (Es According o Ar. 3.3 of he CPD, he essenial requiremens (ERs) shall be given concree form in inerpreaive documens for he creaion of he necessar links beween he essenial requiremens and he mandaes for hes and ETAGs/ETAs. 3 According o Ar. 1 of he CPD he inerpreaive documens shall : a) give concree form o he essenial requiremens b harmoniing he erminolog and he echnical bases and indicaing classes or levels for each requiremen where necessar ; b) indicae mehods of correlaing hese classes or levels of requiremen wih he echnical specificaions, e.g. mehods of calculaion and of proof, echnical rules for projec design, ec. ; c) serve as a reference for he esablishmen of harmonied sandards and guidelines for European echnical approvals. The Eurocodes, de faco, pla a similar role in he field of he ER 1 and a par of ER. 6

7 and ETAs) E : 005 (E) There is a need for consisenc beween he harmonied echnical specificaions for consrucion producs and he echnical rules for works 4. Furhermore, all he informaion accompaning he CE arking of he consrucion producs which refer o Eurocodes should clearl menion which aionall Deermined Parameers have been aken ino accoun. Addiional informaion specific o E E 1993 is inended o be used wih Eurocodes E 1990 Basis of Srucural Design, E 1991 Acions on srucures and E 199 o E 1999, when seel srucures or seel componens are referred o. E is he firs of six pars of E 1993 Design of Seel Srucures. I gives generic design rules inended o be used wih he oher pars E o E I also gives supplemenar rules applicable onl o buildings. E comprises welve subpars E o E each addressing specific seel componens, limi saes or maerials. I ma also be used for design cases no covered b he Eurocodes (oher srucures, oher acions, oher maerials) serving as a reference documen for oher CE TC s concerning srucural maers. E is inended for use b commiees drafing design relaed produc, esing and execuion sandards, cliens (e.g. for he formulaion of heir specific requiremens) designers and consrucors relevan auhoriies umerical values for parial facors and oher reliabili parameers are recommended as basic values ha provide an accepable level of reliabili. The have been seleced assuming ha an appropriae level of workmanship and quali managemen applies. 4 See Ar.3.3 and Ar.1 of he CPD, as well as clauses 4., 4.3.1, 4.3. and 5. of ID 1. 7

8 E : 005 (E) aional annex for E This sandard gives values wih noes indicaing where naional choices ma have o be made. Therefore he aional Sandard implemening E should have a aional Annex conaining all aionall Deermined Parameers o be used for he design of seel srucures o be consruced in he relevan counr. aional choice is allowed in E hrough he following clauses:.3.1(1) 3.1() 3..1(1) 3..(1) 3..3(1) 3..3(3)B 3..4(1)B 5..1(3) 5..(8) 5.3.(3) 5.3.(11) 5.3.4(3) 6.1(1) 6.1(1)B 6.3..() (1) () (1)B ()B 6.3.3(5) 6.3.4(1) 7..1(1)B 7..(1)B 7..3(1)B BB.1.3(3)B 8

9 E : 005 (E) 1 General 1.1 Scope Scope of Eurocode 3 (1) Eurocode 3 applies o he design of buildings and civil engineering works in seel. I complies wih he principles and requiremens for he safe and serviceabili of srucures, he basis of heir design and verificaion ha are given in E 1990 Basis of srucural design. () Eurocode 3 is concerned onl wih requiremens for resisance, serviceabili, durabili and fire resisance of seel srucures. Oher requiremens, e.g. concerning hermal or sound insulaion, are no covered. (3) Eurocode 3 is inended o be used in conjuncion wih: E 1990 Basis of srucural design E 1991 Acions on srucures Es, ETAGs and ETAs for consrucion producs relevan for seel srucures E 1090 Execuion of Seel Srucures Technical requiremens E 199 o E 1999 when seel srucures or seel componens are referred o (4) Eurocode 3 is subdivided in various pars: E Design of Seel Srucures : General rules and rules for buildings. E Design of Seel Srucures : Seel bridges. E Design of Seel Srucures : Towers, mass and chimnes. E Design of Seel Srucures : Silos, anks and pipelines. E Design of Seel Srucures : Piling. E Design of Seel Srucures : Crane supporing srucures. (5) E o E refer o he generic rules in E The rules in pars E o E supplemen he generic rules in E (6) E General rules and rules for buildings comprises: E E E E E E E E E Design of Seel Srucures : General rules and rules for buildings. Design of Seel Srucures : Srucural fire design. Design of Seel Srucures : Cold-formed hin gauge members and sheeing. Design of Seel Srucures : Sainless seels. Design of Seel Srucures : Plaed srucural elemens. Design of Seel Srucures : Srengh and sabili of shell srucures. Design of Seel Srucures : Srengh and sabili of planar plaed srucures ransversel loaded. Design of Seel Srucures : Design of joins. Design of Seel Srucures : Faigue srengh of seel srucures. E Design of Seel Srucures : Selecion of seel for fracure oughness and hrough-hickness properies. E Design of Seel Srucures : Design of srucures wih ension componens made of seel. E Design of Seel Srucures : Supplemenar rules for high srengh seel. 9

10 E : 005 (E) 1.1. Scope of Par 1.1 of Eurocode 3 (1) E gives basic design rules for seel srucures wih maerial hicknesses 3 mm. I also gives supplemenar provisions for he srucural design of seel buildings. These supplemenar provisions are indicaed b he leer B afer he paragraph number, hus ( )B. OTE For cold formed hin gauge members and plae hicknesses < 3 mm see E () The following subjecs are deal wih in E : Secion 1: General Secion : Basis of design Secion 3: aerials Secion 4: Durabili Secion 5: Srucural analsis Secion 6: Ulimae limi saes Secion 7: Serviceabili limi saes (3) Secions 1 o provide addiional clauses o hose given in E 1990 Basis of srucural design. (4) Secion 3 deals wih maerial properies of producs made of low allo srucural seels. (5) Secion 4 gives general rules for durabili. (6) Secion 5 refers o he srucural analsis of srucures, in which he members can be modelled wih sufficien accurac as line elemens for global analsis. (7) Secion 6 gives deailed rules for he design of cross secions and members. (8) Secion 7 gives rules for serviceabili. 1. ormaive references This European Sandard incorporaes b daed or undaed reference, provisions from oher publicaions. These normaive references are cied a he appropriae places in he ex and he publicaions are lised hereafer. For daed references, subsequen amendmens o or revisions of an of hese publicaions appl o his European Sandard onl when incorporaed in i b amendmen or revision. For undaed references he laes ediion of he publicaion referred o applies (including amendmens) General reference sandards E 1090 E ISO 1944 E 1461 Execuion of seel srucures Technical requiremens Pains and varnishes Corrosion proecion of seel srucures b proecive pain ssems Ho dip galvanied coaings on fabricaed iron and seel aricles specificaions and es mehods 1.. Weldable srucural seel reference sandards E :004 E 1005-:004 Ho-rolled producs of srucural seels - Par 1: General deliver condiions. Ho-rolled producs of srucural seels - Par : Technical deliver condiions for nonallo srucural seels. E :004 Ho-rolled producs of srucural seels - Par 3: Technical deliver condiions for normalied / normalied rolled weldable fine grain srucural seels. 10

11 E : 005 (E) E :004 Ho-rolled producs of srucural seels - Par 4: Technical deliver condiions for hermomechanical rolled weldable fine grain srucural seels. E :004 Ho-rolled producs of srucural seels - Par 5: Technical deliver condiions for srucural seels wih improved amospheric corrosion resisance. E :004 E 10164:1993 Ho-rolled producs of srucural seels - Par 6: Technical deliver condiions for fla producs of high ield srengh srucural seels in he quenched and empered condiion. Seel producs wih improved deformaion properies perpendicular o he surface of he produc - Technical deliver condiions. E :1994 Ho finished srucural hollow secions of non-allo and fine grain srucural seels Par 1: Technical deliver requiremens. E :1997 Cold formed hollow secions of srucural seel - Par 1: Technical deliver requiremens. 1.3 Assumpions (1) In addiion o he general assumpions of E 1990 he following assumpions appl: fabricaion and erecion complies wih E Disincion beween principles and applicaion rules (1) The rules in E 1990 clause 1.4 appl. 1.5 Terms and definiions (1) The rules in E 1990 clause 1.5 appl. () The following erms and definiions are used in E wih he following meanings: frame he whole or a porion of a srucure, comprising an assembl of direcl conneced srucural elemens, designed o ac ogeher o resis load; his erm refers o boh momen-resising frames and riangulaed frames; i covers boh plane frames and hree-dimensional frames 1.5. sub-frame a frame ha forms par of a larger frame, bu is be reaed as an isolaed frame in a srucural analsis pe of framing erms used o disinguish beween frames ha are eiher: semi-coninuous, in which he srucural properies of he members and joins need explici consideraion in he global analsis coninuous, in which onl he srucural properies of he members need be considered in he global analsis simple, in which he joins are no required o resis momens global analsis he deerminaion of a consisen se of inernal forces and momens in a srucure, which are in equilibrium wih a paricular se of acions on he srucure 11

12 E : 005 (E) ssem lengh disance in a given plane beween wo adjacen poins a which a member is braced agains laeral displacemen in his plane, or beween one such poin and he end of he member buckling lengh ssem lengh of an oherwise similar member wih pinned ends, which has he same buckling resisance as a given member or segmen of member shear lag effec non-uniform sress disribuion in wide flanges due o shear deformaion; i is aken ino accoun b using a reduced effecive flange widh in safe assessmens capaci design design mehod for achieving he plasic deformaion capaci of a member b providing addiional srengh in is connecions and in oher pars conneced o i uniform member member wih a consan cross-secion along is whole lengh 1.6 Smbols (1) For he purpose of his sandard he following smbols appl. () Addiional smbols are defined where he firs occur. Secion 1 x-x - - u-u v-v b h d w f r r 1 r Secion P k G k 1 OTE Smbols are ordered b appearance in E Smbols ma have various meanings. axis along a member axis of a cross-secion axis of a cross-secion major principal axis (where his does no coincide wih he - axis) minor principal axis (where his does no coincide wih he - axis) widh of a cross secion deph of a cross secion deph of sraigh porion of a web web hickness flange hickness radius of roo fille radius of roo fille oe radius hickness nominal value of he effec of presressing imposed during erecion nominal value of he effec of permanen acions

13 X K X n R d R k γ γ i γ f η a d Secion 3 f f u R eh R m A 0 ε ε u Z Z Rd E G ν α Secion 5 α cr F F cr H V characerisic values of maerial proper nominal values of maerial proper design value of resisance characerisic value of resisance general parial facor paricular parial facor parial facor for faigue conversion facor design value of geomerical daa ield srengh ulimae srengh ield srengh o produc sandards ulimae srengh o produc sandards original cross-secion area ield srain ulimae srain E : 005 (E) required design Z-value resuling from he magniude of srains from resrained meal shrinkage under he weld beads. available design Z-value modulus of elasici shear modulus Poisson s raio in elasic sage coefficien of linear hermal expansion facor b which he design loads would have o be increased o cause elasic insabili in a global mode design loading on he srucure elasic criical buckling load for global insabili mode based on iniial elasic siffnesses design value of he horional reacion a he boom of he sore o he horional loads and ficiious horional loads oal design verical load on he srucure on he boom of he sore δ H, horional displacemen a he op of he sore, relaive o he boom of he sore h λ φ φ 0 α h h sore heigh non dimensional slenderness design value of he axial force global iniial swa imperfecion basic value for global iniial swa imperfecion reducion facor for heigh h applicable o columns heigh of he srucure 13

14 E : 005 (E) α m m e 0 L η ini η cr e 0,d reducion facor for he number of columns in a row number of columns in a row maximum ampliude of a member imperfecion member lengh ampliude of elasic criical buckling mode shape of elasic criical buckling mode design value of maximum ampliude of an imperfecion Rk characerisic momen resisance of he criical cross secion Rk characerisic resisance o normal force of he criical cross secion α imperfecion facor EI η bending momen due o η cr a he criical cross secion " cr χ reducion facor for he relevan buckling curve α ul,k minimum force amplifier o reach he characerisic resisance wihou aking buckling ino accoun α cr minimum force amplifier o reach he elasic criical buckling q equivalen force per uni lengh δ q q d in-plane deflecion of a bracing ssem equivalen design force per uni lengh design bending momen k facor for e 0,d ε srain σ sress σ com, maximum design compressive sress in an elemen l lengh ε coefficien depending on f c widh or deph of a par of a cross secion α porion of a par of a cross secion in compression ψ sress or srain raio k σ plae buckling coefficien d ouer diameer of circular ubular secions Secion 6 γ 0 γ 1 γ parial facor for resisance of cross-secions whaever he class is parial facor for resisance of members o insabili assessed b member checks parial facor for resisance of cross-secions in ension o fracure σ x, design value of he local longiudinal sress σ, design value of he local ransverse sress τ design value of he local shear sress design normal force, design bending momen, - axis, design bending momen, - axis Rd design values of he resisance o normal forces 14

15 ,Rd design values of he resisance o bending momens, - axis,rd design values of he resisance o bending momens, - axis s p n d 0 e E : 005 (E) saggered pich, he spacing of he cenres of wo consecuive holes in he chain measured parallel o he member axis spacing of he cenres of he same wo holes measured perpendicular o he member axis number of holes exending in an diagonal or ig-ag line progressivel across he member or par of he member diameer of hole shif of he cenroid of he effecive area A eff relaive o he cenre of gravi of he gross cross secion addiional momen from shif of he cenroid of he effecive area A eff relaive o he cenre of gravi of he gross cross secion A eff effecive area of a cross secion,rd design values of he resisance o ension forces pl,rd design plasic resisance o normal forces of he gross cross-secion u,rd design ulimae resisance o normal forces of he ne cross-secion a holes for faseners A ne ne area of a cross secion ne,rd design plasic resisance o normal forces of he ne cross-secion c,rd design resisance o normal forces of he cross-secion for uniform compression c,rd design resisance for bending abou one principal axis of a cross-secion W pl plasic secion modulus W el,min minimum elasic secion modulus W eff,min minimum effecive secion modulus A f area of he ension flange A f,ne ne area of he ension flange V design shear force V c,rd design shear resisance V pl,rd plasic design shear resisance A v η S I A w A f T T Rd shear area facor for shear area firs momen of area second momen of area area of a web area of one flange design value of oal orsional momens design resisance o orsional momens T, design value of inernal S. Venan orsion T w, design value of inernal warping orsion τ, design shear sresses due o S. Venan orsion τ w, design shear sresses due o warping orsion σ w, design direc sresses due o he bimomen B B bimomen V pl,t,rd reduced design plasic shear resisance making allowance for he presence of a orsional momen 15

16 E : 005 (E) ρ reducion facor o deermine reduced design values of he resisance o bending momens making allowance for he presence of shear forces V,,Rd reduced design values of he resisance o bending momens making allowance for he presence of shear forces,,rd reduced design values of he resisance o bending momens making allowance for he presence of normal forces n a α β e, e, raio of design normal force o design plasic resisance o normal forces of he gross cross-secion raio of web area o gross area parameer inroducing he effec of biaxial bending parameer inroducing he effec of biaxial bending shif of he cenroid of he effecive area A eff relaive o he cenre of gravi of he gross cross secion (- axis) shif of he cenroid of he effecive area A eff relaive o he cenre of gravi of he gross cross secion (- axis) W eff,min minimum effecive secion modulus b,rd design buckling resisance of a compression member χ Φ reducion facor for relevan buckling mode value o deermine he reducion facor χ a 0, a, b, c, d class indexes for buckling curves cr i λ 1 λ T elasic criical force for he relevan buckling mode based on he gross cross secional properies radius of graion abou he relevan axis, deermined using he properies of he gross cross-secion slenderness value o deermine he relaive slenderness relaive slenderness for orsional or orsional-flexural buckling cr,tf elasic orsional-flexural buckling force cr,t elasic orsional buckling force b,rd design buckling resisance momen χ LT Φ LT α LT reducion facor for laeral-orsional buckling value o deermine he reducion facor χ LT imperfecion facor λ LT non dimensional slenderness for laeral orsional buckling cr elasic criical momen for laeral-orsional buckling λ LT,0 plaeau lengh of he laeral orsional buckling curves for rolled secions β correcion facor for he laeral orsional buckling curves for rolled secions χ LT,mod modified reducion facor for laeral-orsional buckling f k c ψ L c λ f i f I eff,f modificaion facor for χ LT correcion facor for momen disribuion raio of momens in segmen lengh beween laeral resrains equivalen compression flange slenderness radius of graion of compression flange abou he minor axis of he secion effecive second momen of area of compression flange abou he minor axis of he secion 16

17 A eff,f effecive area of compression flange A eff,w,c effecive area of compressed par of web λ c0 slenderness parameer k fl modificaion facor momens due o he shif of he cenroidal - axis momens due o he shif of he cenroidal - axis χ χ k k k k reducion facor due o flexural buckling (- axis) reducion facor due o flexural buckling (- axis) ineracion facor ineracion facor ineracion facor ineracion facor λ op global non dimensional slenderness of a srucural componen for ou-of-plane buckling χ op reducion facor for he non-dimensional slenderness λ op E : 005 (E) α ul,k minimum load amplifier of he design loads o reach he characerisic resisance of he mos criical cross secion α cr,op minimum amplifier for he in plane design loads o reach he elasic criical resisance wih regard o laeral or laeral orsional buckling Rk characerisic value of resisance o compression,rk characerisic value of resisance o bending momens abou - axis,rk characerisic value of resisance o bending momens abou - axis Q m local force applied a each sabilied member a he plasic hinge locaions L sable sable lengh of segmen L ch h 0 a α i min A ch buckling lengh of chord disance of cenrelines of chords of a buil-up column disance beween resrains of chords angle beween axes of chord and lacings minimum radius of graion of single angles area of one chord of a buil-up column ch, design chord force in he middle of a buil-up member I design value of he maximum momen in he middle of he buil-up member I eff S v n A d d A V I ch I b effecive second momen of area of he buil-up member shear siffness of buil-up member from he lacings or baened panel number of planes of lacings area of one diagonal of a buil-up column lengh of a diagonal of a buil-up column area of one pos (or ransverse elemen) of a buil-up column in plane second momen of area of a chord in plane second momen of area of a baen µ efficienc facor 17

18 E : 005 (E) i Annex A C m C m radius of graion (- axis) equivalen uniform momen facor equivalen uniform momen facor C mlt equivalen uniform momen facor µ facor µ facor cr, elasic flexural buckling force abou he - axis cr, elasic flexural buckling force abou he - axis C C C C w w n pl facor facor facor facor facor facor facor λ max maximum of λ and b LT c LT d LT e LT ψ facor facor facor facor C m,0 facor C m,0 facor a LT I T I raio of end momens (- axis) facor λ S. Venan orsional consan second momen of area abou - axis i, (x) maximum firs order momen δ x maximum member displacemen along he member Annex B α s α h C m facor facor Annex AB γ G G k γ Q Q k equivalen uniform momen facor parial facor for permanen loads characerisic value of permanen loads parial facor for variable loads characerisic value of variable loads 18

19 Annex BB λ eff,v effecive slenderness raio for buckling abou v-v axis λ eff, effecive slenderness raio for buckling abou - axis λ eff, effecive slenderness raio for buckling abou - axis L L cr S I w C ϑ,k K υ K ϑ ssem lengh buckling lengh shear siffness provided b sheeing warping consan roaional siffness provided b sabiliing coninuum and connecions facor for considering he pe of analsis facor for considering he momen disribuion and he pe of resrain E : 005 (E) C ϑr,k roaional siffness provided b he sabiliing coninuum o he beam assuming a siff connecion o he member C ϑc,k roaional siffness of he connecion beween he beam and he sabiliing coninuum C ϑd,k roaional siffness deduced from an analsis of he disorsional deformaions of he beam cross secions L m L k L s C 1 C m C n a B 0 B 1 B η i s β R 1 R R 3 R 4 R 5 sable lengh beween adjacen laeral resrains sable lengh beween adjacen orsional resrains sable lengh beween a plasic hinge locaion and an adjacen orsional resrain modificaion facor for momen disribuion modificaion facor for linear momen gradien modificaion facor for non-linear momen gradien disance beween he cenroid of he member wih he plasic hinge and he cenroid of he resrain members facor facor facor raio of criical values of axial forces radius of graion relaed o cenroid of resraining member raio of he algebraicall smaller end momen o he larger end momen momen a a specific locaion of a member momen a a specific locaion of a member momen a a specific locaion of a member momen a a specific locaion of a member momen a a specific locaion of a member R E maximum of R 1 or R 5 R s c h h maximum value of bending momen anwhere in he lengh L aper facor addiional deph of he haunch or aper h max maximum deph of cross-secion wihin he lengh L h min minimum deph of cross-secion wihin he lengh L 19

20 E : 005 (E) h s L h L verical deph of he un-haunched secion lengh of haunch wihin he lengh L lengh beween resrains 1.7 Convenions for member axes (1) The convenion for member axes is: x-x along he member - axis of he cross-secion - axis of he cross-secion () For seel members, he convenions used for cross-secion axes are: generall: cross-secion axis parallel o he flanges - cross-secion axis perpendicular o he flanges for angle secions: axis parallel o he smaller leg - axis perpendicular o he smaller leg where necessar: u-u v-v - major principal axis (where his does no coincide wih he axis) - minor principal axis (where his does no coincide wih he axis) (3) The smbols used for dimensions and axes of rolled seel secions are indicaed in Figure 1.1. (4) The convenion used for subscrips ha indicae axes for momens is: "Use he axis abou which he momen acs." OTE All rules in his Eurocode relae o principal axis properies, which are generall defined b he axes - and - bu for secions such as angles are defined b he axes u-u and v-v. 0

21 E : 005 (E) Figure 1.1: Dimensions and axes of secions 1

22 E : 005 (E) Basis of design.1 Requiremens.1.1 Basic requiremens (1)P The design of seel srucures shall be in accordance wih he general rules given in E () The supplemenar provisions for seel srucures given in his secion should also be applied. (3) The basic requiremens of E 1990 secion should be deemed be saisfied where limi sae design is used in conjuncion wih he parial facor mehod and he load combinaions given in E 1990 ogeher wih he acions given in E (4) The rules for resisances, serviceabili and durabili given in he various pars of E 1993 should be applied..1. Reliabili managemen (1) Where differen levels of reliabili are required, hese levels should preferabl be achieved b an appropriae choice of quali managemen in design and execuion, according o E 1990 Annex C and E Design working life, durabili and robusness General (1) Depending upon he pe of acion affecing durabili and he design working life (see E 1990) seel srucures should be designed agains corrosion b means of suiable surface proecion (see E ISO 1944) he use of weahering seel he use of sainless seel (see E ) deailed for sufficien faigue life (see E ) designed for wearing designed for accidenal acions (see E ) inspeced and mainained Design working life for buildings (1)B The design working life should be aken as he period for which a building srucure is expeced o be used for is inended purpose. ()B For he specificaion of he inended design working life of a permanen building see Table.1 of E (3)B For srucural elemens ha canno be designed for he oal design life of he building, see.1.3.3(3)b Durabili for buildings (1)B To ensure durabili, buildings and heir componens should eiher be designed for environmenal acions and faigue if relevan or else proeced from hem.

23 E : 005 (E) ()B The effecs of deerioraion of maerial, corrosion or faigue where relevan should be aken ino accoun b appropriae choice of maerial, see E and E , and deails, see E , or b srucural redundanc and b he choice of an appropriae corrosion proecion ssem. (3)B If a building includes componens ha need o be replaceable (e.g. bearings in ones of soil selemen), he possibili of heir safe replacemen should be verified as a ransien design siuaion.. Principles of limi sae design (1) The resisance of cross-secions and members specified in his Eurocode 3 for he ulimae limi saes as defined in E 1990, 3.3 are based on ess in which he maerial exhibied sufficien ducili o appl simplified design models. () The resisances specified in his Eurocode Par ma herefore be used where he condiions for maerials in secion 3 are me..3 Basic variables.3.1 Acions and environmenal influences (1) Acions for he design of seel srucures should be aken from E For he combinaion of acions and parial facors of acions see Annex A o E OTE 1 The aional Annex ma define acions for paricular regional or climaic or accidenal siuaions. OTE B For proporional loading for incremenal approach, see Annex AB.1. OTE 3B For simplified load arrangemen, see Annex AB.. () The acions o be considered in he erecion sage should be obained from E (3) Where he effecs of prediced absolue and differenial selemens need o be considered, bes esimaes of imposed deformaions should be used. (4) The effecs of uneven selemens or imposed deformaions or oher forms of presressing imposed during erecion should be aken ino accoun b heir nominal value P k as permanen acions and grouped wih oher permanen acions G k from a single acion (G k + P k ). (5) Faigue acions no defined in E 1991 should be deermined according o Annex A of E aerial and produc properies (1) aerial properies for seels and oher consrucion producs and he geomerical daa o be used for design should be hose specified in he relevan Es, ETAGs or ETAs unless oherwise indicaed in his sandard..4 Verificaion b he parial facor mehod.4.1 Design values of maerial properies (1) For he design of seel srucures characerisic values X K or nominal values X n of maerial properies should be used as indicaed in his Eurocode..4. Design values of geomerical daa (1) Geomerical daa for cross-secions and ssems ma be aken from produc sandards he or drawings for he execuion o E 1090 and reaed as nominal values. 3

24 E : 005 (E) () Design values of geomerical imperfecions specified in his sandard are equivalen geomeric imperfecions ha ake ino accoun he effecs of: geomerical imperfecions of members as governed b geomerical olerances in produc sandards or he execuion sandard; srucural imperfecions due o fabricaion and erecion; residual sresses; variaion of he ield srengh..4.3 Design resisances (1) For seel srucures equaion (6.6c) or equaion (6.6d) of E 1990 applies: R R 1 k d = = R k ( η1x k1; ηix ki; a d ) (.1) γ γ where R k is he characerisic value of he paricular resisance deermined wih characerisic or nominal values for he maerial properies and dimensions γ is he global parial facor for he paricular resisance OTE For he definiions of η 1, η i, X k1, X ki and a d see E Verificaion of saic equilibrium (EQU) (1) The reliabili forma for he verificaion of saic equilibrium in Table 1. (A) in Annex A of E 1990 also applies o design siuaions equivalen o (EQU), e.g. for he design of holding down anchors or he verificaion of uplif of bearings of coninuous beams..5 Design assised b esing (1) The resisances R k in his sandard have been deermined using Annex D of E () In recommending classes of consan parial facors γ i he characerisic values R k were obained from R k = R d γ i (.) where R d are design values according o Annex D of E 1990 γ i are recommended parial facors. OTE 1 The numerical values of he recommended parial facors γ i have been deermined such ha R k represens approximael he 5 %-fracile for an infinie number of ess. OTE For characerisic values of faigue srengh and parial facors γ f for faigue see E OTE 3 For characerisic values of oughness resisance and safe elemens for he oughness verificaion see E (3) Where resisances R k for prefabricaed producs should be deermined from ess, he procedure in () should be followed. 4

25 E : 005 (E) 3 aerials 3.1 General (1) The nominal values of maerial properies given in his secion should be adoped as characerisic values in design calculaions. () This Par of E 1993 covers he design of seel srucures fabricaed from seel maerial conforming o he seel grades lised in Table 3.1. OTE For oher seel maerial and producs see aional Annex. 3. Srucural seel 3..1 aerial properies (1) The nominal values of he ield srengh f and he ulimae srengh f u for srucural seel should be obained a) eiher b adoping he values f = R eh and f u = R m direc from he produc sandard b) or b using he simplificaion given in Table 3.1 OTE The aional Annex ma give he choice. 3.. Ducili requiremens (1) For seels a minimum ducili is required ha should be expressed in erms of limis for: he raio f u / f of he specified minimum ulimae ensile srengh f u o he specified minimum ield srengh f ; he elongaion a failure on a gauge lengh of 5,65 A o (where A 0 is he original cross-secional area); he ulimae srain ε u, where ε u corresponds o he ulimae srengh f u. OTE The limiing values of he raio f u / f, he elongaion a failure and he ulimae srain ε u ma be defined in he aional Annex. The following values are recommended: f u / f 1,10; elongaion a failure no less han 15%; ε u 15ε, where ε is he ield srain (ε = f / E). () Seel conforming wih one of he seel grades lised in Table 3.1 should be acceped as saisfing hese requiremens Fracure oughness (1) The maerial should have sufficien fracure oughness o avoid brile fracure of ension elemens a he lowes service emperaure expeced o occur wihin he inended design life of he srucure. OTE The lowes service emperaure o be adoped in design ma be given in he aional Annex. () o furher check agains brile fracure need o be made if he condiions given in E are saisfied for he lowes emperaure. 5

26 E : 005 (E) (3)B For building componens under compression a minimum oughness proper should be seleced. OTE B The aional Annex ma give informaion on he selecion of oughness properies for members in compression. The use of Table.1 of E for σ = 0,5 f () is recommended. (4) For selecing seels for members wih ho dip galvanied coaings see E Table 3.1: ominal values of ield srengh f and ulimae ensile srengh f u for ho rolled srucural seel Sandard and seel grade E mm ominal hickness of he elemen [mm] 40 mm < 80 mm f [/mm ] f u [/mm ] f [/mm ] f u [/mm ] S S S S E S 75 /L S 355 /L S 40 /L S 460 /L E S 75 /L S 355 /L S 40 /L S 460 /L E S 35 W S 355 W E S 460 Q/QL/QL

27 E : 005 (E) Table 3.1 (coninued): ominal values of ield srengh f and ulimae ensile srengh f u for srucural hollow secions Sandard and seel grade E mm ominal hickness of he elemen [mm] 40 mm < 80 mm f [/mm ] f u [/mm ] f [/mm ] f u [/mm ] S 35 H S 75 H S 355 H S 75 H/LH S 355 H/LH S 40 H/HL S 460 H/LH E S 35 H S 75 H S 355 H S 75 H/LH S 355 H/LH S 460 H/LH S 75 H/LH S 355 H/LH S 40 H/LH S 460 H/LH Through-hickness properies (1) Where seel wih improved hrough-hickness properies is necessar according o E , seel according o he required quali class in E should be used. OTE 1 Guidance on he choice of hrough-hickness properies is given in E OTE B Paricular care should be given o welded beam o column connecions and welded end plaes wih ension in he hrough-hickness direcion. OTE 3B The aional Annex ma give he relevan allocaion of arge values Z according o 3.() of E o he quali class in E The allocaion in Table 3. is recommended for buildings: Table 3.: Choice of quali class according o E Targe value of Z according o E Required value of Z Rd expressed in erms of design Z-values according o E Z < Z 0 Z 15 0 < Z 30 Z 5 Z > 30 Z 35 7

28 E : 005 (E) 3..5 Tolerances (1) The dimensional and mass olerances of rolled seel secions, srucural hollow secions and plaes should conform wih he relevan produc sandard, ETAG or ETA unless more severe olerances are specified. () For welded componens he olerances given in E 1090 should be applied. (3) For srucural analsis and design he nominal values of dimensions should be used Design values of maerial coefficiens (1) The maerial coefficiens o be adoped in calculaions for he srucural seels covered b his Eurocode Par should be aken as follows: modulus of elasici E = / mm E shear modulus G = / mm² (1 + ν) Poisson s raio in elasic sage ν = 0, 3 6 coefficien of linear hermal expansion α = 1 10 perk (for T 100 C) OTE For calculaing he srucural effecs of unequal emperaures in composie concree-seel 6 srucures o E 1994 he coefficien of linear hermal expansion is aken as α = per K. 3.3 Connecing devices Faseners (1) Requiremens for faseners are given in E Welding consumables (1) Requiremens for welding consumables are given in E Oher prefabricaed producs in buildings (1)B An semi-finished or finished srucural produc used in he srucural design of buildings should compl wih he relevan E Produc Sandard or ETAG or ETA. 4 Durabili (1) The basic requiremens for durabili are se ou in E () The means of execuing he proecive reamen underaken off-sie and on-sie should be in accordance wih E OTE E 1090 liss he facors affecing execuion ha need o be specified during design. (3) Pars suscepible o corrosion, mechanical wear or faigue should be designed such ha inspecion, mainenance and reconsrucion can be carried ou saisfacoril and access is available for in-service inspecion and mainenance. 8

29 (4)B For building srucures no faigue assessmen is normall required excep as follows: a) embers supporing lifing appliances or rolling loads b) embers subjec o repeaed sress ccles from vibraing machiner c) embers subjec o wind-induced vibraions d) embers subjec o crowd-induced oscillaions E : 005 (E) (5) For elemens ha canno be inspeced an appropriae corrosion allowance should be included. (6)B Corrosion proecion does no need o be applied o inernal building srucures, if he inernal relaive humidi does no exceed 80%. 5 Srucural analsis 5.1 Srucural modelling for analsis Srucural modelling and basic assumpions (1) Analsis should be based upon calculaion models of he srucure ha are appropriae for he limi sae under consideraion. () The calculaion model and basic assumpions for he calculaions should reflec he srucural behaviour a he relevan limi sae wih appropriae accurac and reflec he anicipaed pe of behaviour of he cross secions, members, joins and bearings. (3) The mehod used for he analsis should be consisen wih he design assumpions. (4)B For he srucural modelling and basic assumpions for componens of buildings see also E and E Join modelling (1) The effecs of he behaviour of he joins on he disribuion of inernal forces and momens wihin a srucure, and on he overall deformaions of he srucure, ma generall be negleced, bu where such effecs are significan (such as in he case of semi-coninuous joins) he should be aken ino accoun, see E () To idenif wheher he effecs of join behaviour on he analsis need be aken ino accoun, a disincion ma be made beween hree join models as follows, see E , 5.1.1: simple, in which he join ma be assumed no o ransmi bending momens; coninuous, in which he behaviour of he join ma be assumed o have no effec on he analsis; semi-coninuous, in which he behaviour of he join needs o be aken ino accoun in he analsis (3) The requiremens of he various pes of joins are given in E Ground-srucure ineracion (1) Accoun should be aken of he deformaion characerisics of he suppors where significan. OTE E 1997 gives guidance for calculaion of soil-srucure ineracion. 9

30 E : 005 (E) 5. Global analsis 5..1 Effecs of deformed geomer of he srucure (1) The inernal forces and momens ma generall be deermined using eiher: firs-order analsis, using he iniial geomer of he srucure or second-order analsis, aking ino accoun he influence of he deformaion of he srucure. () The effecs of he deformed geomer (second-order effecs) should be considered if he increase he acion effecs significanl or modif significanl he srucural behaviour. (3) Firs order analsis ma be used for he srucure, if he increase of he relevan inernal forces or momens or an oher change of srucural behaviour caused b deformaions can be negleced. This condiion ma be assumed o be fulfilled, if he following crierion is saisfied: α α cr cr where α cr F F cr F = F F = F cr cr for elasic analsis for plasic analsis (5.1) is he facor b which he design loading would have o be increased o cause elasic insabili in a global mode is he design loading on he srucure is he elasic criical buckling load for global insabili mode based on iniial elasic siffnesses OTE A greaer limi for α cr for plasic analsis is given in equaion (5.1) because srucural behaviour ma be significanl influenced b non linear maerial properies in he ulimae limi sae (e.g. where a frame forms plasic hinges wih momen redisribuions or where significan non linear deformaions from semi-rigid joins occur). Where subsaniaed b more accurae approaches he aional Annex ma give a lower limi for α cr for cerain pes of frames. (4)B Poral frames wih shallow roof slopes and beam-and-column pe plane frames in buildings ma be checked for swa mode failure wih firs order analsis if he crierion (5.1) is saisfied for each sore. In hese srucures α cr ma be calculaed using he following approximaive formula, provided ha he axial compression in he beams or rafers is no significan: α cr where H V H = V h δ H, (5.) is he design value of he horional reacion a he boom of he sore o he horional loads and ficiious horional loads, see 5.3.(7) is he oal design verical load on he srucure on he boom of he sore δ H, is he horional displacemen a he op of he sore, relaive o he boom of he sore, when he frame is loaded wih horional loads (e.g. wind) and ficiious horional loads which are applied a each floor level h is he sore heigh 30

31 E : 005 (E) Figure 5.1: oaions for 5..1() OTE 1B For he applicaion of (4)B in he absence of more deailed informaion a roof slope ma be aken o be shallow if i is no seeper ha 1: (6 ). OTE B For he applicaion of (4)B in he absence of more deailed informaion he axial compression in he beams or rafers ma be assumed o be significan if A f λ 0,3 (5.3) where is he design value of he compression force, λ is he inplane non dimensional slenderness calculaed for he beam or rafers considered as hinged a is ends of he ssem lengh measured along he beams of rafers. (5) The effecs of shear lag and of local buckling on he siffness should be aken ino accoun if his significanl influences he global analsis, see E OTE For rolled secions and welded secions wih similar dimensions shear lag effecs ma be negleced. (6) The effecs on he global analsis of he slip in bol holes and similar deformaions of connecion devices like suds and anchor bols on acion effecs should be aken ino accoun, where relevan and significan. 5.. Srucural sabili of frames (1) If according o 5..1 he influence of he deformaion of he srucure has o be aken ino accoun () o (6) should be applied o consider hese effecs and o verif he srucural sabili. () The verificaion of he sabili of frames or heir pars should be carried ou considering imperfecions and second order effecs. (3) According o he pe of frame and he global analsis, second order effecs and imperfecions ma be accouned for b one of he following mehods: a) boh oall b he global analsis, b) pariall b he global analsis and pariall hrough individual sabili checks of members according o 6.3, c) for basic cases b individual sabili checks of equivalen members according o 6.3 using appropriae buckling lenghs according o he global buckling mode of he srucure. 31

32 E : 005 (E) (4) Second order effecs ma be calculaed b using an analsis appropriae o he srucure (including sep-b-sep or oher ieraive procedures). For frames where he firs swa buckling mode is predominan firs order elasic analsis should be carried ou wih subsequen amplificaion of relevan acion effecs (e.g. bending momens) b appropriae facors. (5)B For single sore frames designed on he basis of elasic global analsis second order swa effecs due o verical loads ma be calculaed b increasing he horional loads H (e.g. wind) and equivalen loads V φ due o imperfecions (see 5.3.(7)) and oher possible swa effecs according o firs order heor b he facor: α cr provided ha α cr 3,0, where α cr ma be calculaed according o (5.) in 5..1(4)B, provided ha he roof slope is shallow and ha he axial compression in he beams or rafers is no significan as defined in 5..1(4)B. OTE B For α cr < 3,0 a more accurae second order analsis applies. (6)B For muli-sore frames second order swa effecs ma be calculaed b means of he mehod given in (5)B provided ha all sores have a similar disribuion of verical loads and disribuion of horional loads and disribuion of frame siffness wih respec o he applied sore shear forces. OTE B For he limiaion of he mehod see also 5..1(4)B. (7) In accordance wih (3) he sabili of individual members should be checked according o he following: a) If second order effecs in individual members and relevan member imperfecions (see 5.3.4) are oall accouned for in he global analsis of he srucure, no individual sabili check for he members according o 6.3 is necessar. b) If second order effecs in individual members or cerain individual member imperfecions (e.g. member imperfecions for flexural and/or laeral orsional buckling, see 5.3.4) are no oall accouned for in he global analsis, he individual sabili of members should be checked according o he relevan crieria in 6.3 for he effecs no included in he global analsis. This verificaion should ake accoun of end momens and forces from he global analsis of he srucure, including global second order effecs and global imperfecions (see 5.3.) when relevan and ma be based on a buckling lengh equal o he ssem lengh (8) Where he sabili of a frame is assessed b a check wih he equivalen column mehod according o 6.3 he buckling lengh values should be based on a global buckling mode of he frame accouning for he siffness behaviour of members and joins, he presence of plasic hinges and he disribuion of compressive forces under he design loads. In his case inernal forces o be used in resisance checks are calculaed according o firs order heor wihou considering imperfecions. OTE The aional Annex ma give informaion on he scope of applicaion. 5.3 Imperfecions Basis (1) Appropriae allowances should be incorporaed in he srucural analsis o cover he effecs of imperfecions, including residual sresses and geomerical imperfecions such as lack of vericali, lack of (5.4) 3

33 E : 005 (E) sraighness, lack of flaness, lack of fi and an minor eccenriciies presen in joins of he unloaded srucure. () Equivalen geomeric imperfecions, see 5.3. and 5.3.3, should be used, wih values which reflec he possible effecs of all pe of imperfecions unless hese effecs are included in he resisance formulae for member design, see secion (3) The following imperfecions should be aken ino accoun: a) global imperfecions for frames and bracing ssems b) local imperfecions for individual members 5.3. Imperfecions for global analsis of frames (1) The assumed shape of global imperfecions and local imperfecions ma be derived from he elasic buckling mode of a srucure in he plane of buckling considered. () Boh in and ou of plane buckling including orsional buckling wih smmeric and asmmeric buckling shapes should be aken ino accoun in he mos unfavourable direcion and form. (3) For frames sensiive o buckling in a swa mode he effec of imperfecions should be allowed for in frame analsis b means of an equivalen imperfecion in he form of an iniial swa imperfecion and individual bow imperfecions of members. The imperfecions ma be deermined from: a) global iniial swa imperfecions, see Figure 5.: φ = φ 0 α h α m (5.5) where φ 0 is he basic value: φ 0 = 1/00 α h is he reducion facor for heigh h applicable o columns: h α h = bu α h 1, 0 h 3 is he heigh of he srucure in meers α m is he reducion facor for he number of columns in a row: α m m = 1 0,5 1 + m is he number of columns in a row including onl hose columns which carr a verical load no less han 50% of he average value of he column in he verical plane considered Figure 5.: Equivalen swa imperfecions b) relaive iniial local bow imperfecions of members for flexural buckling e 0 / L (5.6) where L is he member lengh de rabajo OTE The values e 0 / L ma be chosen in he aional Annex. Recommended values are given in Table 5.1. Documeno 33

34 E : 005 (E) Table 5.1: Design values of iniial local bow imperfecion e 0 / L Buckling curve elasic analsis plasic analsis acc. o Table 6.1 e 0 / L e 0 / L a 0 1 / / 300 a 1 / / 50 b 1 / 50 1 / 00 c 1 / 00 1 / 150 d 1 / / 100 (4)B For building frames swa imperfecions ma be disregarded where H 0,15 V (5.7) (5)B For he deerminaion of horional forces o floor diaphragms he configuraion of imperfecions as given in Figure 5.3 should be applied, where φ is a swa imperfecion obained from (5.5) assuming a single sore wih heigh h, see (3) a). Figure 5.3: Configuraion of swa imperfecions φ for horional forces on floor diaphragms (6) When performing he global analsis for deermining end forces and end momens o be used in member checks according o 6.3 local bow imperfecions ma be negleced. However for frames sensiive o second order effecs local bow imperfecions of members addiionall o global swa imperfecions (see 5..1(3)) should be inroduced in he srucural analsis of he frame for each compressed member where he following condiions are me: a leas one momen resisan join a one member end A f λ > 0,5 (5.8) where is he design value of he compression force and λ is he in-plane non-dimensional slenderness calculaed for he member considered as hinged a is ends OTE Local bow imperfecions are aken ino accoun in member checks, see 5.. (3) and

35 E : 005 (E) (7) The effecs of iniial swa imperfecion and local bow imperfecions ma be replaced b ssems of equivalen horional forces, inroduced for each column, see Figure 5.3 and Figure 5.4. iniial swa imperfecions iniial bow imperfecions Figure 5.4: Replacemen of iniial imperfecions b equivalen horional forces (8) These iniial swa imperfecions should appl in all relevan horional direcions, bu need onl be considered in one direcion a a ime. (9)B Where, in muli-sore beam-and-column building frames, equivalen forces are used he should be applied a each floor and roof level. (10) The possible orsional effecs on a srucure caused b ani-smmeric swas a he wo opposie faces, should also be considered, see Figure 5.5. A A 1 (a) Faces A-A and B-B swa in same direcion B B 1 ranslaional swa roaional swa A A (b) Faces A-A and B-B swa in opposie direcion Figure 5.5: Translaional and orsional effecs (plan view) B B 35

36 E : 005 (E) (11) As an alernaive o (3) and (6) he shape of he elasic criical buckling mode η cr of he srucure ma be applied as a unique global and local imperfecion. The ampliude of his imperfecion ma be deermined from: where: and η = cr e0 Rk ini e0 " cr " cr EI η η = η cr,max λ EI η (5.9) cr,max e Rk 1 0 Rk χλ 1 γ = α ( λ 0,) for λ > 0, (5.10) 1 χλ α ul,k λ = is he relaive slenderness of he srucure (5.11) α cr α is he imperfecion facor for he relevan buckling curve, see Table 6.1 and Table 6.; χ is he reducion facor for he relevan buckling curve depending on he relevan cross-secion, see 6.3.1; α ul,k is he minimum force amplifier for he axial force configuraion in members o reach he characerisic resisance Rk of he mos axiall sressed cross secion wihou aking buckling ino accoun α cr is he minimum force amplifier for he axial force configuraion in members o reach he elasic criical buckling Rk is he characerisic momens resisance of he criical cross secion, e.g. el,rk or pl,rk as relevan Rk is he characerisic resisance o normal force of he criical cross secion, i.e. pl,rk EI η " cr,max is he bending momen due o ηcr a he criical cross secion η cr is he shape of elasic criical buckling mode OTE 1 For calculaing he amplifiers α ul,k and α cr he members of he srucure ma be considered o be loaded b axial forces onl ha resul from he firs order elasic analsis of he srucure for he design loads. OTE The aional Annex ma give informaion for he scope of applicaion of (11) Imperfecion for analsis of bracing ssems (1) In he analsis of bracing ssems which are required o provide laeral sabili wihin he lengh of beams or compression members he effecs of imperfecions should be included b means of an equivalen geomeric imperfecion of he members o be resrained, in he form of an iniial bow imperfecion: e 0 = α m L / 500 (5.1) where L is he span of he bracing ssem and α m = 1 0,5 1 + m in which m is he number of members o be resrained. () For convenience, he effecs of he iniial bow imperfecions of he members o be resrained b a bracing ssem, ma be replaced b he equivalen sabiliing force as shown in Figure 5.6: e0 + δ q qd = 8 (5.13) L 36

37 E : 005 (E) where δ q is he inplane deflecion of he bracing ssem due o q plus an exernal loads calculaed from firs order analsis OTE δ q ma be aken as 0 if second order heor is used. (3) Where he bracing ssem is required o sabilie he compression flange of a beam of consan heigh, he force in Figure 5.6 ma be obained from: = / h (5.14) where is he maximum momen in he beam and h is he overall deph of he beam. OTE Where a beam is subjeced o exernal compression should include a par of he compression force. (4) A poins where beams or compression members are spliced, i should also be verified ha he bracing ssem is able o resis a local force equal o α m / 100 applied o i b each beam or compression member which is spliced a ha poin, and o ransmi his force o he adjacen poins a which ha beam or compression member is resrained, see Figure 5.7. (5) For checking for he local force according o clause (4), an exernal loads acing on bracing ssems should also be included, bu he forces arising from he imperfecion given in (1) ma be omied. e 0 imperfecion q d equivalen force per uni lengh 1 bracing ssem The force is assumed uniform wihin he span L of he bracing ssem. For non-uniform forces his is slighl conservaive. Figure 5.6: Equivalen sabiliing force 37

38 E : 005 (E) Φ Φ 1 Φ Φ Φ Φ = α m Φ 0 : Φ 0 = 1 / 00 Φ = α m / splice bracing ssem Figure 5.7: Bracing forces a splices in compression elemens ember imperfecions (1) The effecs of local bow imperfecions of members are incorporaed wihin he formulas given for buckling resisance for members, see secion 6.3. () Where he sabili of members is accouned for b second order analsis according o 5..(7)a) for compression members imperfecions e 0 according o 5.3.(3)b), 5.3.(5) or 5.3.(6) should be considered. (3) For a second order analsis aking accoun of laeral orsional buckling of a member in bending he imperfecions ma be adoped as ke 0,d, where e 0,d is he equivalen iniial bow imperfecion of he weak axis of he profile considered. In general an addiional orsional imperfecion need no o be allowed for. OTE The aional Annex ma choose he value of k. The value k = 0,5 is recommended. 5.4 ehods of analsis considering maerial non-lineariies General (1) The inernal forces and momens ma be deermined using eiher a) elasic global analsis b) plasic global analsis. OTE For finie elemen model (FE) analsis see E () Elasic global analsis ma be used in all cases. 38

39 E : 005 (E) (3) Plasic global analsis ma be used onl where he srucure has sufficien roaion capaci a he acual locaions of he plasic hinges, wheher his is in he members or in he joins. Where a plasic hinge occurs in a member, he member cross secions should be double smmeric or single smmeric wih a plane of smmer in he same plane as he roaion of he plasic hinge and i should saisf he requiremens specified in 5.6. Where a plasic hinge occurs in a join he join should eiher have sufficien srengh o ensure he hinge remains in he member or should be able o susain he plasic resisance for a sufficien roaion, see E (4)B As a simplified mehod for a limied plasic redisribuion of momens in coninuous beams where following an elasic analsis some peak momens exceed he plasic bending resisance of 15 % maximum, he pars in excess of hese peak momens ma be redisribued in an member, provided, ha: a) he inernal forces and momens in he frame remain in equilibrium wih he applied loads, and b) all he members in which he momens are reduced have Class 1 or Class cross-secions (see 5.5), and c) laeral orsional buckling of he members is prevened Elasic global analsis (1) Elasic global analsis should be based on he assumpion ha he sress-srain behaviour of he maerial is linear, whaever he sress level is. OTE For he choice of a semi-coninuous join model see 5.1.() o (4). () Inernal forces and momens ma be calculaed according o elasic global analsis even if he resisance of a cross secion is based on is plasic resisance, see 6.. (3) Elasic global analsis ma also be used for cross secions he resisances of which are limied b local buckling, see Plasic global analsis (1) Plasic global analsis allows for he effecs of maerial non-lineari in calculaing he acion effecs of a srucural ssem. The behaviour should be modelled b one of he following mehods: b elasic-plasic analsis wih plasified secions and/or joins as plasic hinges, b non-linear plasic analsis considering he parial plasificaion of members in plasic ones, b rigid plasic analsis neglecing he elasic behaviour beween hinges. () Plasic global analsis ma be used where he members are capable of sufficien roaion capaci o enable he required redisribuions of bending momens o develop, see 5.5 and 5.6. (3) Plasic global analsis should onl be used where he sabili of members a plasic hinges can be assured, see (4) The bi-linear sress-srain relaionship indicaed in Figure 5.8 ma be used for he grades of srucural seel specified in secion 3. Alernaivel, a more precise relaionship ma be adoped, see E Figure 5.8: Bi-linear sress-srain relaionship 39

40 E : 005 (E) (5) Rigid plasic analsis ma be applied if no effecs of he deformed geomer (e.g. second-order effecs) have o be considered. In his case joins are classified onl b srengh, see E (6) The effecs of deformed geomer of he srucure and he srucural sabili of he frame should be verified according o he principles in 5.. OTE The maximum resisance of a frame wih significanl deformed geomer ma occur before all hinges of he firs order collapse mechanism have formed. 5.5 Classificaion of cross secions Basis (1) The role of cross secion classificaion is o idenif he exen o which he resisance and roaion capaci of cross secions is limied b is local buckling resisance Classificaion (1) Four classes of cross-secions are defined, as follows: Class 1 cross-secions are hose which can form a plasic hinge wih he roaion capaci required from plasic analsis wihou reducion of he resisance. Class cross-secions are hose which can develop heir plasic momen resisance, bu have limied roaion capaci because of local buckling. Class 3 cross-secions are hose in which he sress in he exreme compression fibre of he seel member assuming an elasic disribuion of sresses can reach he ield srengh, bu local buckling is liable o preven developmen of he plasic momen resisance. Class 4 cross-secions are hose in which local buckling will occur before he aainmen of ield sress in one or more pars of he cross-secion. () In Class 4 cross secions effecive widhs ma be used o make he necessar allowances for reducions in resisance due o he effecs of local buckling, see E , 5... (3) The classificaion of a cross-secion depends on he widh o hickness raio of he pars subjec o compression. (4) Compression pars include ever par of a cross-secion which is eiher oall or pariall in compression under he load combinaion considered. (5) The various compression pars in a cross-secion (such as a web or flange) can, in general, be in differen classes. (6) A cross-secion is classified according o he highes (leas favourable) class of is compression pars. Excepions are specified in 6..1(10) and 6...4(1). (7) Alernaivel he classificaion of a cross-secion ma be defined b quoing boh he flange classificaion and he web classificaion. (8) The limiing proporions for Class 1,, and 3 compression pars should be obained from Table 5.. A par which fails o saisf he limis for Class 3 should be aken as Class 4. (9) Excep as given in (10) Class 4 secions ma be reaed as Class 3 secions if he widh o hickness raios are less han he limiing proporions for Class 3 obained from Table 5. when ε is increased b / f σ γ 0 com, σ com,, where is he maximum design compressive sress in he par aken from firs order or where necessar second order analsis. 40

41 E : 005 (E) (10) However, when verifing he design buckling resisance of a member using secion 6.3, he limiing proporions for Class 3 should alwas be obained from Table 5.. (11) Cross-secions wih a Class 3 web and Class 1 or flanges ma be classified as class cross secions wih an effecive web in accordance wih (1) Where he web is considered o resis shear forces onl and is assumed no o conribue o he bending and normal force resisance of he cross secion, he cross secion ma be designed as Class, 3 or 4 secions, depending onl on he flange class. OTE For flange induced web buckling see E Cross-secion requiremens for plasic global analsis (1) A plasic hinge locaions, he cross-secion of he member which conains he plasic hinge should have a roaion capaci of no less han he required a he plasic hinge locaion. () In a uniform member sufficien roaion capaci ma be assumed a a plasic hinge if boh he following requiremens are saisfied: a) he member has Class 1 cross-secions a he plasic hinge locaion; b) where a ransverse force ha exceeds 10 % of he shear resisance of he cross secion, see 6..6, is applied o he web a he plasic hinge locaion, web siffeners should be provided wihin a disance along he member of h/ from he plasic hinge locaion, where h is he heigh of he cross secion a his locaion. (3) Where he cross-secion of he member var along heir lengh, he following addiional crieria should be saisfied: a) Adjacen o plasic hinge locaions, he hickness of he web should no be reduced for a disance each wa along he member from he plasic hinge locaion of a leas d, where d is he clear deph of he web a he plasic hinge locaion. b) Adjacen o plasic hinge locaions, he compression flange should be Class 1 for a disance each wa along he member from he plasic hinge locaion of no less han he greaer of: d, where d is as defined in (3)a) he disance o he adjacen poin a which he momen in he member has fallen o 0,8 imes he plasic momen resisance a he poin concerned. c) Elsewhere in he member he compression flange should be class 1 or class and he web should be class 1, class or class 3. (4) Adjacen o plasic hinge locaions, an fasener holes in ension should saisf 6..5(4) for a disance such as defined in (3)b) each wa along he member from he plasic hinge locaion. (5) For plasic design of a frame, regarding cross secion requiremens, he capaci of plasic redisribuion of momens ma be assumed sufficien if he requiremens in () o (4) are saisfied for all members where plasic hinges exis, ma occur or have occurred under design loads. (6) In cases where mehods of plasic global analsis are used which consider he real sress and srain behaviour along he member including he combined effec of local, member and global buckling he requiremens () o (5) need no be applied. 41

42 E : 005 (E) Table 5. (shee 1 of 3): aximum widh-o-hickness raios for compression pars Class Sress disribuion in pars (compression posiive) c c Par subjec o bending f - + f c Inernal compression pars c c Par subjec o compression 1 c / 7ε c / 33ε c / 83ε c / 38ε Sress disribuion in pars (compression posiive) f - + f c/ 3 c / 14ε c / 4ε c f f f c c c c c c Axis of bending Axis of bending Par subjec o bending and compression f - + when α > 0,5 : when α 0,5 : when α > 0,5 : when α 0,5 : when ψ 1 + f αc c 396ε c / 13α 1 36ε c / α 456ε c / 13α 1 41,5ε c / α - ψ f 4ε when ψ > 1: c / 0,67 + 0,33ψ *) f c : c / 6ε(1 ψ) ε = 35/ f f ε 1,00 0,9 0,81 0,75 0,71 *) ψ -1 applies where eiher he compression sress σ f or he ensile srain ε > f /E ( ψ) 4

43 E : 005 (E) Table 5. (shee of 3): aximum widh-o-hickness raios for compression pars Class Sress disribuion in pars (compression posiive) c Rolled secions c Par subjec o compression + c Ousand flanges c Welded secions Par subjec o bending and compression Tip in compression Tip in ension αc αc 1 c / 9ε c / 9ε c / 9ε α α α c / 10ε c / 10ε c / 10ε α α α Sress disribuion in pars c (compression c c posiive) 3 c / 14ε c / 1ε k σ For k σ see E ε = 35/ f f ε 1,00 0,9 0,81 0,75 0,71 - c + + c - c 43

44 E : 005 (E) Table 5. (shee 3 of 3): aximum widh-o-hickness raios for compression pars Refer also o Ousand flanges (see shee of 3) Angles Class Secion in compression Sress disribuion across + f secion (compression posiive) + 3 h / 15ε : b + h 11, 5ε Tubular secions ε = Class h d b Does no appl o angles in coninuous conac wih oher componens Secion in bending and/or compression d / 50ε d / 70ε d / 90ε OTE For d / > 90ε see E f ε 1,00 0,9 0,81 0,75 0,71 ε 1,00 0,85 0,66 0,56 0, / f 44

45 E : 005 (E) 6 Ulimae limi saes 6.1 General (1) The parial facors γ as defined in.4.3 should be applied o he various characerisic values of resisance in his secion as follows: resisance of cross-secions whaever he class is: γ 0 resisance of members o insabili assessed b member checks: γ 1 resisance of cross-secions in ension o fracure: γ resisance of joins: see E OTE 1 For oher recommended numerical values see E 1993 Par o Par 6. For srucures no covered b E 1993 Par o Par 6 he aional Annex ma define he parial facors γ i ; i is recommended o ake he parial facors γ i from E OTE B Parial facors γ i for buildings ma be defined in he aional Annex. The following numerical values are recommended for buildings: γ 0 = 1,00 γ 1 = 1,00 γ = 1,5 6. Resisance of cross-secions 6..1 General (1) The design value of an acion effec in each cross secion should no exceed he corresponding design resisance and if several acion effecs ac simulaneousl he combined effec should no exceed he resisance for ha combinaion. () Shear lag effecs and local buckling effecs should be included b an effecive widh according o E Shear buckling effecs should also be considered according o E (3) The design values of resisance should depend on he classificaion of he cross-secion. (4) Elasic verificaion according o he elasic resisance ma be carried ou for all cross secional classes provided he effecive cross secional properies are used for he verificaion of class 4 cross secions. (5) For he elasic verificaion he following ield crierion for a criical poin of he cross secion ma be used unless oher ineracion formulae appl, see 6..8 o σ f γ x, 0 σ + f γ, 0 σ f γ x, 0 σ f γ, 0 3 τ + f γ where σ is he design value of he local longiudinal sress a he poin of consideraion x,, 0 1 σ is he design value of he local ransverse sress a he poin of consideraion τ is he design value of he local shear sress a he poin of consideraion OTE The verificaion according o (5) can be conservaive as i excludes parial plasic sress disribuion, which is permied in elasic design. Therefore i should onl be performed where he ineracion of on he basis of resisances Rd, Rd, V Rd canno be performed. (6.1) 45

46 E : 005 (E) (6) The plasic resisance of cross secions should be verified b finding a sress disribuion which is in equilibrium wih he inernal forces and momens wihou exceeding he ield srengh. This sress disribuion should be compaible wih he associaed plasic deformaions. (7) As a conservaive approximaion for all cross secion classes a linear summaion of he uiliaion raios for each sress resulan ma be used. For class 1, class or class 3 cross secions subjeced o he combinaion of,, and, his mehod ma be applied b using he following crieria: Rd,, (6.),Rd,Rd where Rd,,Rd and,rd are he design values of he resisance depending on he cross secional classificaion and including an reducion ha ma be caused b shear effecs, see OTE For class 4 cross secions see (). (8) Where all he compression pars of a cross-secion are a leas Class, he cross-secion ma be aken as capable of developing is full plasic resisance in bending. (9) Where all he compression pars of a cross-secion are Class 3, is resisance should be based on an elasic disribuion of srains across he cross-secion. Compressive sresses should be limied o he ield srengh a he exreme fibres. OTE The exreme fibres ma be assumed a he midplane of he flanges for ULS checks. For faigue see E (10) Where ielding firs occurs on he ension side of he cross secion, he plasic reserves of he ension one ma be uilied b accouning for parial plasificaion when deermining he resisance of a Class 3 cross-secion. 6.. Secion properies Gross cross-secion (1) The properies of he gross cross-secion should be deermined using he nominal dimensions. Holes for faseners need no be deduced, bu allowance should be made for larger openings. Splice maerials should no be included e area (1) The ne area of a cross-secion should be aken as is gross area less appropriae deducions for all holes and oher openings. () For calculaing ne secion properies, he deducion for a single fasener hole should be he gross cross-secional area of he hole in he plane of is axis. For counersunk holes, appropriae allowance should be made for he counersunk porion. (3) Provided ha he fasener holes are no saggered, he oal area o be deduced for fasener holes should be he maximum sum of he secional areas of he holes in an cross-secion perpendicular o he member axis (see failure plane ➁ in Figure 6.1). OTE The maximum sum denoes he posiion of he criical fracure line. 46

47 E : 005 (E) (4) Where he fasener holes are saggered, he oal area o be deduced for faseners should be he greaer of: a) he deducion for non-saggered holes given in (3) s b) nd 0 4p where s is he saggered pich, he spacing of he cenres of wo consecuive holes in he chain measured parallel o he member axis; p is he spacing of he cenres of he same wo holes measured perpendicular o he member axis; is he hickness; n is he number of holes exending in an diagonal or ig-ag line progressivel across he member or par of he member, see Figure 6.1. d 0 is he diameer of hole (5) In an angle or oher member wih holes in more hen one plane, he spacing p should be measured along he cenre of hickness of he maerial (see Figure 6.) Shear lag effecs Figure 6.1: Saggered holes and criical fracure lines 1 and Figure 6.: Angles wih holes in boh legs (1) The calculaion of he effecive widhs is covered in E () In class 4 secions he ineracion beween shear lag and local buckling should be considered according o E OTE For cold formed hin gauge members see E (6.3) 47

48 E : 005 (E) Effecive properies of cross secions wih class 3 webs and class 1 or flanges (1) Where cross-secions wih a class 3 web and class 1 or flanges are classified as effecive Class cross-secions, see 5.5.(11), he proporion of he web in compression should be replaced b a par of 0ε w adjacen o he compression flange, wih anoher par of 0ε w adjacen o he plasic neural axis of he effecive cross-secion in accordance wih Figure ε 4 0 ε f 1 compression ension 3 plasic neural axis 4 neglec w w Figure 6.3: Effecive class web Effecive cross-secion properies of Class 4 cross-secions (1) The effecive cross-secion properies of Class 4 cross-secions should be based on he effecive widhs of he compression pars. () For cold formed hin walled secions see 1.1.(1) and E (3) The effecive widhs of planar compression pars should be obained from E (4) Where a class 4 cross secion is subjeced o an axial compression force, he mehod given in E should be used o deermine he possible shif e of he cenroid of he effecive area A eff relaive o he cenre of gravi of he gross cross secion and he resuling addiional momen: = e OTE The sign of he addiional momen depends on he effec in he combinaion of inernal forces and momens, see (). (5) For circular hollow secions wih class 4 cross secions see E f (6.4) 48

49 E : 005 (E) 6..3 Tension (1) The design value of he ension force a each cross secion should saisf:,rd 1,0 (6.5) () For secions wih holes he design ension resisance,rd should be aken as he smaller of: a) he design plasic resisance of he gross cross-secion pl,rd A f = (6.6) γ 0 b) he design ulimae resisance of he ne cross-secion a holes for faseners 0,9A f ne u u,rd = (6.7) γ (3) Where capaci design is requesed, see E 1998, he design plasic resisance pl,rd (as given in 6..3() a)) should be less han he design ulimae resisance of he ne secion a faseners holes u,rd (as given in 6..3() b)). (4) In caegor C connecions (see E , 3.4.(1), he design ension resisance,rd in 6..3(1) of he ne secion a holes for faseners should be aken as ne,rd, where: ne,rd A ne f = (6.8) γ 0 (5) For angles conneced hrough one leg, see also E , Similar consideraion should also be given o oher pe of secions conneced hrough ousands Compression (1) The design value of he compression force a each cross-secion should saisf: c,rd 1,0 (6.9) () The design resisance of he cross-secion for uniform compression c,rd should be deermined as follows: c,rd c,rd A f = for class 1, or 3 cross-secions (6.10) γ 0 A eff f = for class 4 cross-secions (6.11) γ 0 (3) Fasener holes excep for oversie and sloed holes as defined in E 1090 need no be allowed for in compression members, provided ha he are filled b faseners. (4) In he case of unsmmerical Class 4 secions, he mehod given in should be used o allow for he addiional momen due o he eccenrici of he cenroidal axis of he effecive secion, see 6...5(4). 49

50 E : 005 (E) 6..5 Bending momen (1) The design value of he bending momen a each cross-secion should saisf: c,rd 1,0 (6.1) where c,rd is deermined considering fasener holes, see (4) o (6). () The design resisance for bending abou one principal axis of a cross-secion is deermined as follows: c,rd c,rd c,rd Wpl f = pl,rd = for class 1 or cross secions (6.13) γ 0 Wel,min f = el,rd = for class 3 cross secions (6.14) γ 0 0 Weff,min f = for class 4 cross secions (6.15) γ where W el,min and W eff,min corresponds o he fibre wih he maximum elasic sress. (3) For bending abou boh axes, he mehods given in 6..9 should be used. (4) Fasener holes in he ension flange ma be ignored provided ha for he ension flange: A f,ne γ 0,9 f u A γ f f 0 where A f is he area of he ension flange. OTE The crierion in (4) provides capaci design (see 1.5.8) in he region of plasic hinges. (6.16) (5) Fasener holes in ension one of he web need no be allowed for, provided ha he limi given in (4) is saisfied for he complee ension one comprising he ension flange plus he ension one of he web. (6) Fasener holes excep for oversie and sloed holes in compression one of he cross-secion need no be allowed for, provided ha he are filled b faseners Shear (1) The design value of he shear force V a each cross secion should saisf: V V c,rd 1,0 (6.17) where V c,rd is he design shear resisance. For plasic design V c,rd is he design plasic shear resisance V pl,rd as given in (). For elasic design V c,rd is he design elasic shear resisance calculaed using (4) and (5). () In he absence of orsion he design plasic shear resisance is given b: V pl,rd ( f / 3) A v = (6.18) γ 0 where A v is he shear area. 50

51 (3) The shear area A v ma be aken as follows: a) rolled I and H secions, load parallel o web f ( w ) f b) rolled channel secions, load parallel o web f ( w ) f c) rolled T-secion, load parallel o web 0,9 ( A ) E : 005 (E) A b + + r bu no less han η h w w A b + + r d) welded I, H and box secions, load parallel o web η ( h ) b f w w e) welded I, H, channel and box secions, load parallel o flanges A- ( h ) f) rolled recangular hollow secions of uniform hickness: load parallel o deph load parallel o widh Ah/(b+h) Ab/(b+h) g) circular hollow secions and ubes of uniform hickness A/π where A is he crosssecional area; b is he overall breadh; h is he overall deph; h w is he deph of he web; r is he roo radius; f is he flange hickness; w is he web hickness (If he web hickness in no consan, w should be aken as he minimum hickness.). η see E OTE η ma be conservaivel aken equal 1,0. (4) For verifing he design elasic shear resisance V c,rd he following crierion for a criical poin of he cross secion ma be used unless he buckling verificaion in secion 5 of E applies: τ ( 3 γ ) f 0 1,0 where τ ma be obained from: where V is he design value of he shear force S I w w (6.19) τ = V S I (6.0) is he firs momen of area abou he cenroidal axis of ha porion of he cross-secion beween he poin a which he shear is required and he boundar of he cross-secion is second momen of area of he whole cross secion is he hickness a he examined poin OTE The verificaion according o (4) is conservaive as i excludes parial plasic shear disribuion, which is permied in elasic design, see (5). Therefore i should onl be carried ou where he verificaion on he basis of V c,rd according o equaion (6.17) canno be performed. 51

52 E : 005 (E) (5) For I- or H-secions he shear sress in he web ma be aken as: V τ = if A f / A w 0,6 (6.1) A w where A f is he area of one flange; A w is he area of he web: A w = h w w. (6) In addiion he shear buckling resisance for webs wihou inermediae siffeners should be according o secion 5 of E , if h w w ε > 7 η For η see secion 5 of E OTE η ma be conservaivel aken equal o 1,0. (6.) (7) Fasener holes need no be allowed for in he shear verificaion excep in verifing he design shear resisance a connecion ones as given in E (8) Where he shear force is combined wih a orsional momen, he plasic shear resisance V pl,rd should be reduced as specified in 6..7(9) Torsion (1) For members subjec o orsion for which disorional deformaions ma be disregarded he design value of he orsional momen T a each cross-secion should saisf: T 1,0 (6.3) T Rd where T Rd is he design orsional resisance of he cross secion. () The oal orsional momen T a an cross- secion should be considered as he sum of wo inernal effecs: T = T, + T w, (6.4) where T, is he inernal S. Venan orsion; T w, is he inernal warping orsion. (3) The values of T, and T w, a an cross-secion ma be deermined from T b elasic analsis, aking accoun of he secion properies of he member, he condiions of resrain a he suppors and he disribuion of he acions along he member. (4) The following sresses due o orsion should be aken ino accoun: he shear sresses τ, due o S. Venan orsion T, he direc sresses σ w, due o he bimomen B and shear sresses τ w, due o warping orsion T w, (5) For he elasic verificaion he ield crierion in 6..1(5) ma be applied. (6) For deermining he plasic momen resisance of a cross secion due o bending and orsion onl orsion effecs B should be derived from elasic analsis, see (3). (7) As a simplificaion, in he case of a member wih a closed hollow cross-secion, such as a srucural hollow secion, i ma be assumed ha he effecs of orsional warping can be negleced. Also as a simplificaion, in he case of a member wih open cross secion, such as I or H, i ma be assumed ha he effecs of S. Venan orsion can be negleced. 5

53 E : 005 (E) (8) For he calculaion of he resisance T Rd of closed hollow secions he design shear srengh of he individual pars of he cross secion according o E should be aken ino accoun. (9) For combined shear force and orsional momen he plasic shear resisance accouning for orsional effecs should be reduced from V pl,rd o V pl,t,rd and he design shear force should saisf: V V pl,t,rd 1,0 (6.5) in which V pl,t,rd ma be derived as follows: for an I or H secion: V pl,t,rd for a channel secion: V pl,t,rd τ = 1 (6.6) 1,5, Vpl, Rd ( f / 3) /γ 0 τ, τ w, 1 Vpl, Rd 1,5 ( f / 3) /γ 0 ( f / 3) /γ 0 = (6.7) for a srucural hollow secion: V pl,t,rd τ, 1 Vpl, Rd ( f / 3) /γ 0 = (6.8) where V pl,rd is given in Bending and shear (1) Where he shear force is presen allowance should be made for is effec on he momen resisance. () Where he shear force is less han half he plasic shear resisance is effec on he momen resisance ma be negleced excep where shear buckling reduces he secion resisance, see E (3) Oherwise he reduced momen resisance should be aken as he design resisance of he cross-secion, calculaed using a reduced ield srengh (1 ρ) f (6.9) for he shear area, V where ρ = 1 and V V pl,rd is obained from 6..6(). pl,rd OTE See also 6..10(3). V (4) When orsion is presen ρ should be obained from ρ = V as 0 for V 0,5V pl,t,rd. pl,t,rd 1, see 6..7, bu should be aken 53

54 E : 005 (E) (5) The reduced design plasic resisance momen allowing for he shear force ma alernaivel be obained for I-cross-secions wih equal flanges and bending abou he major axis as follows:,v,rd ρa w Wpl, f 4 w = bu,v,rd,c, Rd (6.30) γ 0 where,c,rd is obained from 6..5() and A w = h w w (6) For he ineracion of bending, shear and ransverse loads see secion 7 of E Bending and axial force Class 1 and cross-secions (1) Where an axial force is presen, allowance should be made for is effec on he plasic momen resisance. () For class 1 and cross secions, he following crierion should be saisfied:,rd (6.31) where,rd is he design plasic momen resisance reduced due o he axial force. (3) For a recangular solid secion wihou fasener holes,rd should be aken as:,rd pl,rd [ 1 ( / ) ] = (6.3) pl,rd (4) For doubl smmerical I- and H-secions or oher flanges secions, allowance need no be made for he effec of he axial force on he plasic resisance momen abou he - axis when boh he following crieria are saisfied: 0,5 pl,rd 0 and (6.33) 0,5h w wf (6.34) γ For doubl smmerical I- and H-secions, allowance need no be made for he effec of he axial force on he plasic resisance momen abou he - axis when: h w w f (6.35) γ 0 (5) For cross-secions where fasener holes are no o be accouned for, he following approximaions ma be used for sandard rolled I or H secions and for welded I or H secions wih equal flanges:,,rd = pl,,rd (1-n)/(1-0,5a) bu,,rd pl,,rd (6.36) for n a:,,rd = pl,,rd (6.37) n a for n > a:,,rd = pl,,rd 1 (6.38) 1 a where n = / pl.rd a = (A-b f )/A bu a 0,5 54

55 E : 005 (E) For cross-secions where fasener holes are no o be accouned for, he following approximaions ma be used for recangular srucural hollow secions of uniform hickness and for welded box secions wih equal flanges and equal webs:,,rd = pl,,rd (1 - n)/(1-0,5a w ) bu,,rd pl,.rd (6.39),,Rd = pl,,rd (1 - n)/(1-0,5a f ) bu,,rd pl,,rd (6.40) where a w = (A - b)/a bu a w 0,5 for hollow secions a w = (A-b f )/A bu a w 0,5 for welded box secions a f = (A - h)/a bu a f 0,5 for hollow secions a f = (A-h w )/A bu a f 0,5 for welded box secions (6) For bi-axial bending he following crierion ma be used:,,,rd α +,,,Rd β 1 in which α and β are consans, which ma conservaivel be aken as uni, oherwise as follows: I and H secions: α = ; β = 5n bu β 1 circular hollow secions: α = ; β = recangular hollow secions: 1,66 α = β = bu α = β 6 1 1,13n where n = / pl,rd Class 3 cross-secions (6.41) (1) In he absence of shear force, for Class 3 cross-secions he maximum longiudinal sress should saisf he crierion: where f σ x, (6.4) γ σ x, 0 is he design value of he local longiudinal sress due o momen and axial force aking accoun of fasener holes where relevan, see 6..3, 6..4 and Class 4 cross-secions (1) In he absence of shear force, for Class 4 cross-secions he maximum longiudinal sress σ x, calculaed using he effecive cross secions (see 5.5.()) should saisf he crierion: where f σ x, (6.43) γ σ x, 0 is he design value of he local longiudinal sress due o momen and axial force aking accoun of fasener holes where relevan, see 6..3, 6..4 and

56 E : 005 (E) () The following crierion should be me: A eff where A eff f / γ 0 + W, + eff,,min f e / γ 0 + W, + eff,,min f e / γ 0 1 is he effecive area of he cross-secion when subjeced o uniform compression (6.44) W eff,min is he effecive secion modulus (corresponding o he fibre wih he maximum elasic sress) of he cross-secion when subjeced onl o momen abou he relevan axis e is he shif of he relevan cenroidal axis when he cross-secion is subjeced o compression onl, see 6...5(4) OTE The signs of,,,, and i = e i depend on he combinaion of he respecive direc sresses Bending, shear and axial force (1) Where shear and axial force are presen, allowance should be made for he effec of boh shear force and axial force on he resisance momen. () Provided ha he design value of he shear force V does no exceed 50% of he design plasic shear resisance V pl.rd no reducion of he resisances defined for bending and axial force in 6..9 need be made, excep where shear buckling reduces he secion resisance, see E (3) Where V exceeds 50% of V pl.rd he design resisance of he cross-secion o combinaions of momen and axial force should be calculaed using a reduced ield srengh (1-ρ)f (6.45) for he shear area where ρ= (V / V pl.rd -1) and V pl,rd is obained from 6..6(). OTE Insead of reducing he ield srengh also he plae hickness of he relevan par of he cross secion ma be reduced. 6.3 Buckling resisance of members Uniform members in compression Buckling resisance (1) A compression member should be verified agains buckling as follows: b,rd where 1,0 (6.46) is he design value of he compression force; b,rd is he design buckling resisance of he compression member. () For members wih non-smmeric Class 4 secions allowance should be made for he addiional momen due o he eccenrici of he cenroidal axis of he effecive secion, see also 6...5(4), and he ineracion should be carried ou o or

57 (3) The design buckling resisance of a compression member should be aken as: b,rd b,rd 1 E : 005 (E) χ A f = for Class 1, and 3 cross-secions (6.47) γ χ A eff f = for Class 4 cross-secions (6.48) γ 1 where χ is he reducion facor for he relevan buckling mode. OTE For deermining he buckling resisance of members wih apered secions along he member or for non-uniform disribuion of he compression force second order analsis according o 5.3.4() ma be performed. For ou-of-plane buckling see also (4) In deermining A and A eff holes for faseners a he column ends need no o be aken ino accoun Buckling curves (1) For axial compression in members he value of χ for he appropriae non-dimensional slenderness λ should be deermined from he relevan buckling curve according o: 1 χ = bu χ 1,0 (6.49) Φ + Φ λ [ + ] where Φ =,51+ α( λ 0,) 0 λ Af λ = for Class 1, and 3 cross-secions cr A eff f λ = for Class 4 cross-secions α cr cr is an imperfecion facor is he elasic criical force for he relevan buckling mode based on he gross cross secional properies. () The imperfecion facor α corresponding o he appropriae buckling curve should be obained from Table 6.1 and Table 6.. Table 6.1: Imperfecion facors for buckling curves Buckling curve a 0 a b c d Imperfecion facor α 0,13 0,1 0,34 0,49 0,76 (3) Values of he reducion facor χ for he appropriae non-dimensional slenderness λ ma be obained from Figure 6.4. (4) For slenderness λ 0, or for 0, 04 he buckling effecs ma be ignored and onl cross cr secional checks appl. 57

58 E : 005 (E) Rolled secions Welded I-secions Hollow secions Welded box secions U-, T- and solid secions L-secions h Table 6.: Selecion of buckling curve for a cross-secion Cross secion h f f b b f w f h/b > 1, h/b 1, Limis f 40 mm 40 mm < f 100 f 100 mm f > 100 mm f 40 mm f > 40 mm Buckling abou axis Buckling curve S 35 S 75 S 460 S 355 S 40 a b b c b c d d b c c d a 0 a 0 ho finished an a a 0 cold formed an c c generall (excep as below) hick welds: a > 0,5 f b/ f < 30 h/ w <30 a a a a c c b c c d an b b an c c an c c an b b 58

59 E : 005 (E) Reducion facor χ 1,1 1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0, 0,1 a 0 a b c d 0,0 0,0 0, 0,4 0,6 0,8 1,0 1, 1,4 1,6 1,8,0,,4,6,8 3, Slenderness for flexural buckling on-dimensional slenderness λ Figure 6.4: Buckling curves (1) The non-dimensional slenderness λ is given b: Af L cr 1 λ = = for Class 1, and 3 cross-secions (6.50) i λ A cr 1 1 eff eff cr λ = = for Class 4 cross-secions (6.51) f cr L i A A λ where L cr is he buckling lengh in he buckling plane considered i is he radius of graion abou he relevan axis, deermined using he properies of he gross cross-secion E λ1 = π = 93, 9ε f f 35 ε = (f in /mm ) OTE B For elasic buckling of componens of building srucures see Annex BB. () For flexural buckling he appropriae buckling curve should be deermined from Table

60 E : 005 (E) Slenderness for orsional and orsional-flexural buckling (1) For members wih open cross-secions accoun should be aken of he possibili ha he resisance of he member o eiher orsional or orsional-flexural buckling could be less han is resisance o flexural buckling. () The non-dimensional slenderness λ T for orsional or orsional-flexural buckling should be aken as: where Af λ T = for Class 1, and 3 cross-secions (6.5) A cr eff λ T = for Class 4 cross-secions (6.53) cr f cr = bu < cr,tf cr cr, T cr,tf is he elasic orsional-flexural buckling force; cr,t is he elasic orsional buckling force. (3) For orsional or orsional-flexural buckling he appropriae buckling curve ma be deermined from Table 6. considering he one relaed o he -axis Uniform members in bending Buckling resisance (1) A laerall unresrained member subjec o major axis bending should be verified agains laeralorsional buckling as follows: b,rd 1,0 (6.54) where is he design value of he momen b,rd is he design buckling resisance momen. () Beams wih sufficien resrain o he compression flange are no suscepible o laeral-orsional buckling. In addiion, beams wih cerain pes of cross-secions, such as square or circular hollow secions, fabricaed circular ubes or square box secions are no suscepible o laeral-orsional buckling. (3) The design buckling resisance momen of a laerall unresrained beam should be aken as: b,rd f = χ LTW (6.55) γ 1 where W is he appropriae secion modulus as follows: W = W pl, for Class 1 or cross-secions W = W el, for Class 3 cross-secions W = W eff, for Class 4 cross-secions χ LT is he reducion facor for laeral-orsional buckling. OTE 1 For deermining he buckling resisance of beams wih apered secions second order analsis according o 5.3.4(3) ma be performed. For ou-of-plane buckling see also OTE B For buckling of componens of building srucures see also Annex BB. 60

61 (4) In deermining W holes for faseners a he beam end need no o be aken ino accoun Laeral orsional buckling curves General case E : 005 (E) (1) Unless oherwise specified, see , for bending members of consan cross-secion, he value of χ LT for he appropriae non-dimensional slenderness λ LT, should be deermined from: 1 χ LT = bu χ LT 1,0 (6.56) Φ + Φ λ LT LT where Φ =,51 + α ( λ LT 0, ) LT [ LT + ] LT 0 λ LT α LT is an imperfecion facor λ LT = W f cr cr is he elasic criical momen for laeral-orsional buckling () cr is based on gross cross secional properies and akes ino accoun he loading condiions, he real momen disribuion and he laeral resrains. OTE The imperfecion facor α LT corresponding o he appropriae buckling curve ma be obained from he aional Annex. The recommended values α LT are given in Table 6.3. Table 6.3: Recommended values for imperfecion facors for laeral orsional buckling curves Buckling curve a b c d Imperfecion facor α LT 0,1 0,34 0,49 0,76 The recommendaions for buckling curves are given in Table 6.4. Table 6.4: Recommended values for laeral orsional buckling curves for crosssecions using equaion (6.56) Cross-secion Limis Buckling curve Rolled I-secions h/b a h/b > b Welded I-secions h/b c h/b > d Oher cross-secions - d (3) Values of he reducion facor χ LT for he appropriae non-dimensional slenderness λ LT ma be obained from Figure 6.4. (4) For slendernesses λ LT λlt,0 (see ) or for λlt,0 (see ) laeral orsional buckling cr effecs ma be ignored and onl cross secional checks appl. 61

62 E : 005 (E) Laeral orsional buckling curves for rolled secions or equivalen welded secions (1) For rolled or equivalen welded secions in bending he values of χ LT for he appropriae nondimensional slenderness ma be deermined from χ LT 1,0 1 χ bu 1 LT = (6.57) χ LT Φ LT + Φ LT βλ LT λ LT Φ LT =,51 [ + α LT ( λ LT λ LT,0 ) + β ] LT 0 λ OTE The parameers λ LT, 0 and β and an limiaion of validi concerning he beam deph or h/b raio ma be given in he aional Annex. The following values are recommended for rolled secions or equivalen welded secions: λ LT,0 = 0,4 (maximum value) β = 0,75 (minimum value) The recommendaions for buckling curves are given in Table 6.5. Table 6.5: Recommendaion for he selecion of laeral orsional buckling curve for cross secions using equaion (6.57) Cross-secion Limis Buckling curve Rolled I-secions h/b b h/b > c Welded I-secions h/b c h/b > d () For aking ino accoun he momen disribuion beween he laeral resrains of members he reducion facor χ LT ma be modified as follows: χ LT χ LT,mod = bu χ LT,mod 1 (6.58) f OTE The values f ma be defined in he aional Annex. The following minimum values are recommended: f = 1 0,5(1 k )[1,0( λ c LT 0,8) k c is a correcion facor according o Table 6.6 ] bu f 1,0 6

63 E : 005 (E) Table 6.6: Correcion facors k c omen disribuion ψ = 1-1 ψ 1 k c 1,0 1 1,33 0, 33ψ 0,94 0,90 0,91 0,86 0,77 0, Simplified assessmen mehods for beams wih resrains in buildings (1)B embers wih discree laeral resrain o he compression flange are no suscepible o laeralorsional buckling if he lengh L c beween resrains or he resuling slenderness λ f of he equivalen compression flange saisfies: k c c c,rd λ f = λ c0 (6.59) i f, L λ 1, where, is he maximum design value of he bending momen wihin he resrain spacing c,rd = W γ f 1 W is he appropriae secion modulus corresponding o he compression flange k c is a slenderness correcion facor for momen disribuion beween resrains, see Table 6.6 i f, is he radius of graion of he equivalen compression flange composed of he compression flange plus 1/3 of he compressed par of he web area, abou he minor axis of he secion λ c,0 is a slenderness limi of he equivalen compression flange defined above E λ1 = π = 93, 9ε f f 35 ε = (f in /mm ) 63

64 E : 005 (E) OTE 1B For Class 4 cross-secions i f, ma be aken as Ieff,f i f, = 1 A eff,f + A eff,w,c 3 where I eff,f A eff,f is he effecive second momen of area of he compression flange abou he minor axis of he secion is he effecive area of he compression flange A eff,w,c is he effecive areas of he compressed par of he web OTE B The slenderness limi λ c0 ma be given in he aional Annex. A limi value λ c 0 = λ LT,0 + 0,1 is recommended, see ()B If he slenderness of he compression flange resisance momen ma be aken as: where b,rd = k f l χ c,rd b.rd c. Rd χ k fl λ f exceeds he limi given in (1)B, he design buckling bu (6.60) is he reducion facor of he equivalen compression flange deermined wih λ is he modificaion facor accouning for he conservaism of he equivalen compression flange mehod OTE B The modificaion facor ma be given in he aional Annex. A value k f l = 1, 10 is recommended. (3)B The buckling curves o be used in ()B should be aken as follows: curve d for welded secions provided ha: curve c for all oher secions h f 44ε where h is he overall deph of he cross-secion f is he hickness of he compression flange OTE B For laeral orsional buckling of componens of building srucures wih resrains see also Annex BB Uniform members in bending and axial compression (1) Unless second order analsis is carried ou using he imperfecions as given in 5.3., he sabili of uniform members wih double smmeric cross secions for secions no suscepible o disorional deformaions should be checked as given in he following clauses, where a disincion is made for: members ha are no suscepible o orsional deformaions, e.g. circular hollow secions or secions resrain from orsion members ha are suscepible o orsional deformaions, e.g. members wih open cross-secions and no resrain from orsion. () In addiion, he resisance of he cross-secions a each end of he member should saisf he requiremens given in 6.. OTE 1 The ineracion formulae are based on he modelling of simpl suppored single span members wih end fork condiions and wih or wihou coninuous laeral resrains, which are subjeced o compression forces, end momens and/or ransverse loads. f 64

65 E : 005 (E) OTE In case he condiions of applicaion expressed in (1) and () are no fulfilled, see (3) For members of srucural ssems he resisance check ma be carried ou on he basis of he individual single span members regarded as cu ou of he ssem. Second order effecs of he swa ssem (P- -effecs) have o be aken ino accoun, eiher b he end momens of he member or b means of appropriae buckling lenghs respecivel, see 5..(3)c) and 5..(8). (4) embers which are subjeced o combined bending and axial compression should saisf: χ γ 1 χ γ 1 Rk Rk + k + k, χ LT, χ LT + γ,rk 1 + γ,rk 1,, + k + k,, + γ,rk 1 + γ,rk 1,, 1 1 (6.61) (6.6) where,, and, are he design values of he compression force and he maximum momens abou he - and - axis along he member, respecivel,,, are he momens due o he shif of he cenroidal axis according o for class 4 secions, see Table 6.7, χ and χ are he reducion facors due o flexural buckling from χ LT is he reducion facor due o laeral orsional buckling from 6.3. k, k, k, k are he ineracion facors Table 6.7: Values for Rk = f A i, i,rk = f W i and i, Class A i A A A A eff W W pl, W pl, W el, W eff, W W pl, W pl, W el, W eff,, e,, e, OTE For members no suscepible o orsional deformaion χ LT would be χ LT = 1,0. (5) The ineracion facors k, k, k, k depend on he mehod which is chosen. OTE 1 The ineracion facors k, k, k and k have been derived from wo alernaive approaches. Values of hese facors ma be obained from Annex A (alernaive mehod 1) or from Annex B (alernaive mehod ). OTE The aional Annex ma give a choice from alernaive mehod 1 or alernaive mehod. OTE 3 For simplici verificaions ma be performed in he elasic range onl General mehod for laeral and laeral orsional buckling of srucural componens (1) The following mehod ma be used where he mehods given in 6.3.1, 6.3. and do no appl. I allows he verificaion of he resisance o laeral and laeral orsional buckling for srucural componens such as single members, buil-up or no, uniform or no, wih complex suppor condiions or no, or plane frames or subframes composed of such members, 65

66 E : 005 (E) which are subjec o compression and/or mono-axial bending in he plane, bu which do no conain roaive plasic hinges. OTE The aional Annex ma specif he field and limis of applicaion of his mehod. () Overall resisance o ou-of-plane buckling for an srucural componen conforming o he scope in (1) can be verified b ensuring ha: χ op γ α 1 ul,k 1,0 (6.63) where α ul,k is he minimum load amplifier of he design loads o reach he characerisic resisance of he mos criical cross secion of he srucural componen considering is in plane behaviour wihou aking laeral or laeral orsional buckling ino accoun however accouning for all effecs due o in plane geomerical deformaion and imperfecions, global and local, where relevan; χ op is he reducion facor for he non-dimensional slenderness laeral and laeral orsional buckling. λ op, see (3), o ake accoun of (3) The global non dimensional slenderness λ op for he srucural componen should be deermined from α ul,k λ op = (6.64) α cr,op where α ul,k is defined in () α cr,op is he minimum amplifier for he in plane design loads o reach he elasic criical resisance of he srucural componen wih regards o laeral or laeral orsional buckling wihou accouning for in plane flexural buckling OTE In deermining α cr,op and α ul,k Finie Elemen analsis ma be used. (4) The reducion facor χ op ma be deermined from eiher of he following mehods: a) he minimum value of χ for laeral buckling according o χ LT for laeral orsional buckling according o 6.3. each calculaed for he global non dimensional slenderness λ op. OTE For example where α ul,k is deermined b he cross secion check mehod leads o: Rk γ 1 +,Rk, γ 1 χ op 1 α ul,k = Rk +,,Rk his (6.65) b) a value inerpolaed beween he values χ and χ LT as deermined in a) b using he formula for α ul,k corresponding o he criical cross secion OTE For example where α ul,k is deermined b he cross secion check mehod leads o: 1 α ul,k = Rk +,,Rk his 66

67 χ Rk γ 1 + χ LT,,Rk γ Laeral orsional buckling of members wih plasic hinges General E : 005 (E) (6.66) (1)B Srucures ma be designed wih plasic analsis provided laeral orsional buckling in he frame is prevened b he following means: a) resrains a locaions of roaed plasic hinges, see , and b) verificaion of sable lengh of segmen beween such resrains and oher laeral resrains, see ()B Where under all ulimae limi sae load combinaions, he plasic hinge is no-roaed no resrains are necessar for such a plasic hinge Resrains a roaed plasic hinges (1)B A each roaed plasic hinge locaion he cross secion should have an effecive laeral and orsional resrain wih appropriae resisance o laeral forces and orsion induced b local plasic deformaions of he member a his locaion. ()B Effecive resrain should be provided for members carring eiher momen or momen and axial force b laeral resrain o boh flanges. This ma be provided b laeral resrain o one flange and a siff orsional resrain o he cross-secion prevening he laeral displacemen of he compression flange relaive o he ension flange, see Figure 6.5. for members carring eiher momen alone or momen and axial ension in which he compression flange is in conac wih a floor slab, b laeral and orsional resrain o he compression flange (e.g. b connecing i o a slab, see Figure 6.6). For cross-secions ha are more slender han rolled I and H secions he disorsion of he cross secion should be prevened a he plasic hinge locaion (e.g. b means of a web siffener also conneced o he compression flange wih a siff join from he compression flange ino he slab). Figure 6.5: Tpical siff orsional resrain 1 compression flange Figure 6.6: Tpical laeral and orsional resrain b a slab o he compression flange 1 67

68 E : 005 (E) (3)B A each plasic hinge locaion, he connecion (e.g. bols) of he compression flange o he resising elemen a ha poin (e.g. purlin), and an inermediae elemen (e.g. diagonal brace) should be designed o resis o a local force of a leas,5% of f, (defined in (5)B) ransmied b he flange in is plane and perpendicular o he web plane, wihou an combinaion wih oher loads. (4)B Where i is no pracicable providing such a resrain direcl a he hinge locaion, i should be provided wihin a disance of h/ along he lengh of he member, where h is is overall deph a he plasic hinge locaion. (5)B For he design of bracing ssems, see 5.3.3, i should be verified b a check in addiion o he check for imperfecion according o ha he bracing ssem is able o resis he effecs of local forces Q m applied a each sabilied member a he plasic hinge locaions, where; Q f, = 1,5 (6.67) 100 m α m where f, is he axial force in he compressed flange of he sabilied member a he plasic hinge locaion; α m is according o 5.3.3(1). OTE For combinaion wih exernal loads see also 5.3.3(5) Verificaion of sable lengh of segmen (1)B The laeral orsional buckling verificaion of segmens beween resrains ma be performed b checking ha he lengh beween resrains is no greaer han he sable lengh. h For uniform beam segmens wih I or H cross secions wih f 40ε under linear momen and wihou significan axial compression he sable lengh ma be aken from L L sable sable where ε = = 35 ε i f = for 0,65 ψ 1 ( 60 40ψ) ε i for 1 ψ 0, [ / mm ] pl,rd,min ψ = = raio of end momens in he segmen OTE B For he sable lengh of a segmen see also Annex BB.3. (6.68) ()B Where a roaed plasic hinge locaion occurs immediael adjacen o one end of a haunch, he apered segmen need no be reaed as a segmen adjacen o a plasic hinge locaion if he following crieria are saisfied: a) he resrain a he plasic hinge locaion should be wihin a disance h/ along he lengh of he apered segmen, no he uniform segmen; b) he compression flange of he haunch remains elasic hroughou is lengh. OTE B For more informaion see Annex BB.3. 68

69 E : 005 (E) 6.4 Uniform buil-up compression members General (1) Uniform buil-up compression members wih hinged ends ha are laerall suppored should be designed wih he following model, see Figure 6.7. L 1. The member ma be considered as a column wih a bow imperfecion e 0 = 500. The elasic deformaions of lacings or baenings, see Figure 6.7, ma be considered b a coninuous (smeared) shear siffness S V of he column. OTE For oher end condiions appropriae modificaions ma be performed. () The model of a uniform buil-up compression member applies when 1. he lacings or baenings consis of equal modules wih parallel chords. he minimum numbers of modules in a member is hree. OTE This assumpion allows he srucure o be regular and smearing he discree srucure o a coninuum. (3) The design procedure is applicable o buil-up members wih lacings in wo planes, see Figure 6.8. (4) The chords ma be solid members or ma hemselves be laced or baened in he perpendicular plane. e 0 = L/500 Figure 6.7: Uniform buil-up columns wih lacings and baenings 69

70 E : 005 (E) L ch = 1,5a L ch = 1,8a L ch = a Figure 6.8: Lacings on four sides and buckling lengh L ch of chords (5) Checks should be performed for chords using he design chord forces ch, from compression forces and momens a mid span of he buil-up member. (6) For a member wih wo idenical chords he design force ch, should be deermined from: where + h A 0 ch ch, = 0,5 (6.69) Ieff cr e = 1 0 cr + S I v π EIeff = is he effecive criical force of he buil-up member L is he design value of he compression force o he buil-up member is he design value of he maximum momen in he middle of he buil-up member considering second order effecs I h 0 A ch is he design value of he maximum momen in he middle of he buil-up member wihou second order effecs is he disance beween he cenroids of chords is he cross-secional area of one chord I eff is he effecive second momen of area of he buil-up member, see 6.4. and S v is he shear siffness of he lacings or baened panel, see 6.4. and

71 E : 005 (E) (7) The checks for he lacings of laced buil-up members or for he frame momens and shear forces of he baened panels of baened buil-up members should be performed for he end panel aking accoun of he shear force in he buil-up member: V = π (6.70) L 6.4. Laced compression members Resisance of componens of laced compression members (1) The chords and diagonal lacings subjec o compression should be designed for buckling. OTE Secondar momens ma be negleced. () For chords he buckling verificaion should be performed as follows: ch, b,rd 1,0 (6.71) where ch, is he design compression force in he chord a mid-lengh of he buil-up member according o 6.4.1(6) and b,rd is he design value of he buckling resisance of he chord aking he buckling lengh L ch from Figure 6.8. (3) The shear siffness S V of he lacings should be aken from Figure 6.9. (4) The effecive second order momen of area of laced buil-up members ma be aken as: I = 0,5h eff 0 A ch Ssem S V nea d d 3 ah 0 nea n is he number of planes of lacings A d and A V refer o he cross secional area of he bracings d d 3 ah 0 nea d ah A + dh 0 d 1 3 A Vd Figure 6.9: Shear siffness of lacings of buil-up members Consrucional deails (6.7) (1) Single lacing ssems in opposie faces of he buil-up member wih wo parallel laced planes should be corresponding ssems as shown in Figure 6.10(a), arranged so ha one is he shadow of he oher. 71

72 E : 005 (E) () When he single lacing ssems on opposie faces of a buil-up member wih wo parallel laced planes are muuall opposed in direcion as shown in Figure 6.10(b), he resuling orsional effecs in he member should be aken ino accoun. (3) Tie panels should be provided a he ends of lacing ssems, a poins where he lacing is inerruped and a joins wih oher members. chord chord Lacing on face A Lacing on face B Lacing on face A Lacing on face B a) Corresponding lacing ssem (Recommended ssem) b) uuall opposed lacing ssem (o recommended) Figure 6.10: Single lacing ssem on opposie faces of a buil-up member wih wo parallel laced planes Baened compression members Resisance of componens of baened compression members (1) The chords and he baens and heir joins o he chords should be checked for he acual momens and forces in an end panel and a mid-span as indicaed in Figure OTE For simplici he maximum chord forces ch, ma be combined wih he maximum shear force V. 7

73 E : 005 (E) Figure 6.11: omens and forces in an end panel of a baened buil-up member () The shear siffness S V should be aken as follows: S ch ch v = (6.73) I a ch h 0 a 1 + 4EI ni b a π EI (3) The effecive second momens of area of baened buil-up members ma be aken as: I = 0,5h A + µ I eff 0 ch ch where I ch = in plane second momen of area of one chord I b = in plane second momen of area of one baen µ = efficienc facor from Table 6.8 n = number of planes of lacings Table 6.8: Efficienc facor µ Crierion Efficienc facor µ λ λ 75 < λ < 150 µ = 75 λ 75 1,0 L I1 where λ = ; i 0 = ; I 1 = 0,5h 0A ch + Ich A i 0 ch (6.74) 73

74 E : 005 (E) Design deails (1) Baens should be provided a each end of a member. () Where parallel planes of baens are provided, he baens in each plane should be arranged opposie each oher. (3) Baens should also be provided a inermediae poins where loads are applied or laeral resrain is supplied Closel spaced buil-up members (1) Buil-up compression members wih chords in conac or closel spaced and conneced hrough packing plaes, see Figure 6.1, or sar baened angle members conneced b pairs of baens in wo perpendicular planes, see Figure 6.13 should be checked for buckling as a single inegral member ignoring he effec of shear siffness (S V = ), when he condiions in Table 6.9 are me. Figure 6.1: Closel spaced buil-up members Table 6.9: aximum spacings for inerconnecions in closel spaced buil-up or sar baened angle members Tpe of buil-up member embers according o Figure 6.1 conneced b bols or welds embers according o Figure 6.13 conneced b pair of baens *) cenre-o-cenre disance of inerconnecions i min is he minimum radius of graion of one chord or one angle aximum spacing beween inerconnecions *) 15 i min 70 i min () The shear forces o be ransmied b he baens should be deermined from (1). (3) In he case of unequal-leg angles, see Figure 6.13, buckling abou he - axis ma be verified wih: i0 i = (6.75) 1,15 where i 0 is he minimum radius of graion of he buil-up member. 74

75 7 Serviceabili limi saes 7.1 General Figure 6.13: Sar-baened angle members E : 005 (E) (1) A seel srucure should be designed and consruced such ha all relevan serviceabili crieria are saisfied. () The basic requiremens for serviceabili limi saes are given in 3.4 of E (3) An serviceabili limi sae and he associaed loading and analsis model should be specified for a projec. (4) Where plasic global analsis is used for he ulimae limi sae, plasic redisribuion of forces and momens a he serviceabili limi sae ma occur. If so, he effecs should be considered. 7. Serviceabili limi saes for buildings 7..1 Verical deflecions (1)B Wih reference o E 1990 Annex A1.4 limis for verical deflecions according o Figure A1.1 should be specified for each projec and agreed wih he clien. OTE B The aional Annex ma specif he limis. 7.. Horional deflecions (1)B Wih reference o E 1990 Annex A1.4 limis for horional deflecions according o Figure A1. should be specified for each projec and agreed wih he clien. OTE B The aional Annex ma specif he limis Dnamic effecs (1)B Wih reference o E 1990 Annex A1.4.4 he vibraions of srucures on which he public can walk should be limied o avoid significan discomfor o users, and limis should be specified for each projec and agreed wih he clien. OTE B The aional Annex ma specif limis for vibraion of floors. v v v v 75

76 E : 005 (E) Annex A [informaive] ehod 1: Ineracion facors k ij for ineracion formula in 6.3.3(4) Ineracion facors k k k k Auxiliar erms: 1 cr, µ = 1 χ µ w w 1 cr, = 1 χ W = W W = W pl, el, pl, el, cr, cr, 1,5 1,5 n pl = Rk / γ 1 C m see Table A. IT a LT = 1 0 I C wih C Table A.1: Ineracion facors k ij (6.3.3(4)) Design assumpions elasic cross-secional properies plasic cross-secional properies class 3, class 4 class 1, class µ µ 1 C mc mlt C mc mlt C 1 1 = 1 + b wih c C wih d C wih e LT = 1 + LT = 1 + LT = 1 + LT C C m C m C m 1 mlt ( w 1) = 0,5 a µ cr, µ 1 µ 1 LT ( w 1) = 10 a LT ( w 1) = a LT ( w 1) = 1,7 a LT cr, cr, cr, 1,6 C w, λ 0 χ LT C 14 λ λ 4 C m C 14 λ 0 4 0,1 + λ C m pl,,rd m w χ m m 1,6 C w λ 0 4 0,1 + λ C λ 5 w LT λ 5 χ m m λ max max, max LT λ χ, 1,6 w, pl,,rd n pl pl,,rd n max LT, pl pl,,rd 1,6 w C c C d pl,,rd C C C m m C m LT LT m m 1 mlt λ C max µ cr, µ 1 m 0,6 0,6, λ 1 C cr, µ 1 n pl,,rd max pl w w w w n pl cr, 1 C cr, b W W W W e LT el, pl, el, pl, LT 0,6 1 C 0,6 w w W W W W w w el, pl, el, pl, 76

77 λ = max max λ λ Table A.1 (coninued) E : 005 (E) λ 0 = non-dimensional slenderness for laeral-orsional buckling due o uniform bending momen, i.e. ψ =1,0 in Table A. λ LT = non-dimensional slenderness for laeral-orsional buckling If λ 0 0, C cr, : cr,tf Cm = C m,0 C m = C m,0 C mlt = 1,0 If 0 > 0, C1 4 1 λ 1 cr, : C m = C m,0 + ( 1 C m,0 ) cr,tf el, C = m C m,0 a LT C mlt = Cm cr, cr,t, A ε = W for class 1, and 3 cross-secions A, eff ε = for class 4 cross-secions Weff, cr, = elasic flexural buckling force abou he - axis cr, = elasic flexural buckling force abou he - axis cr,t = elasic orsional buckling force I T I = S. Venan orsional consan = second momen of area abou - axis 1+ ε ε a LT a LT 77

78 E : 005 (E) Table A.: Equivalen uniform momen facors C mi,0 omen diagram C mi, 0 1 ψ 1 1 ψ 1 (x) (x) C C mi,0 mi,0 = 0,79 + 0,1ψ π EI = 1+ L i i, i + 0,36( ψ δ x 1 (x) i 0,33) i, (x) is he maximum momen, or, δ x is he maximum member displacemen along he member C mi,0 = 1 0,18 C mi,0 = 1+ 0,03 cr.i cr.i cr.i cr.i 78

79 E : 005 (E) Annex B [informaive] ehod : Ineracion facors k ij for ineracion formula in 6.3.3(4) Table B.1: Ineracion facors k ij for members no suscepible o orsional deformaions Ineracion facors k k k k Tpe of secions I-secions RHS-secions I-secions RHS-secions I-secions RHS-secions I-secions RHS-secions Design assumpions elasic cross-secional properies plasic cross-secional properies class 3, class 4 class 1, class m 1+ 0,6λ χ C m 1+ 0,6 χ Rk C C m 1+ ( λ 0,) Rk / γ 1 k / γ 1 χ C m 1+ 0,8 χ Rk / γ 0,6 k 0,8 k 0,6 k C m 1+ 0,6λ χ Rk / γ C m 1+ 0,6 χ Rk / γ 1 1 ( λ 0,6) Rk 1 C m 1+ χ C m 1+ 1,4 χ Rk / γ 1 ( λ 0,) C m 1+ χ C m 1+ 0,8 χ Rk / γ For I- and H-secions and recangular hollow secions under axial compression and uniaxial bending, he coefficien k ma be k = 0. Table B.: Ineracion facors k ij for members suscepible o orsional deformaions Design assumpions Ineracion facors elasic cross-secional properies plasic cross-secional properies class 3, class 4 class 1, class k k from Table B.1 k from Table B.1 k k from Table B.1 k from Table B.1 0,05λ 0,1λ 1 1 ( C mlt 0,5) χ Rk / γ 1 ( C mlt 0,5) χ Rk / γ 1 0,05 0,1 1 1 ( C mlt 0,5) χ Rk / γ 1 ( C mlt 0,5) χ Rk / γ 1 k for λ < 0,4 : k = 0,6 + λ 1 0,1λ Rk Rk 1 / γ / γ / γ ( C mlt 0,5) χ Rk / γ 1 79

80 E : 005 (E) k k from Table B.1 k from Table B.1 Table B.3: Equivalen uniform momen facors C m in Tables B.1 and B. omen diagram range C m and C m and C mlt uniform loading concenraed load -1 ψ 1 0,6 + 0,4ψ 0,4 0 α s 1-1 ψ 1 0, + 0,8α s 0,4 0, + 0,8α s 0,4-1 α s < 0 0 ψ 1 0,1-0,8α s 0,4-0,8α s 0,4-1 ψ < 0 0,1(1-ψ) - 0,8α s 0,4 0,(-ψ) - 0,8α s 0,4 0 α h 1-1 ψ 1 0,95 + 0,05α h 0,90 + 0,10α h -1 α h < 0 0 ψ 1 0,95 + 0,05α h 0,90 + 0,10α h -1 ψ < 0 0,95 + 0,05α h (1+ψ) 0,90-0,10α h (1+ψ) For members wih swa buckling mode he equivalen uniform momen facor should be aken C m = 0,9 or C = 0,9 respecivel. C m, C m and C mlt should be obained according o he bending momen diagram beween he relevan braced poins as follows: momen facor bending axis poins braced in direcion C m - - C m - - C mlt

81 E : 005 (E) Annex AB [informaive] Addiional design provisions AB.1 Srucural analsis aking accoun of maerial non-lineariies (1)B In case of maerial non-lineariies he acion effecs in a srucure ma be deermined b incremenal approach o he design loads o be considered for he relevan design siuaion. ()B In his incremenal approach each permanen or variable acion should be increased proporionall. AB. Simplified provisions for he design of coninuous floor beams (1)B For coninuous beams wih slabs in buildings wihou canilevers on which uniforml disribued loads are dominan, i is sufficien o consider onl he following load arrangemens: a) alernaive spans carring he design permanen and variable load (γ G G k + γ Q Q k ), oher spans carring onl he design permanen load γ G G k b) an wo adjacen spans carring he design permanen and variable loads (γ G G k + γ Q Q k ), all oher spans carring onl he design permanen load γ G G k OTE 1 a) applies o sagging momens, b) o hogging momens. OTE This annex is inended o be ransferred o E 1990 in a laer sage. 81

82 E : 005 (E) Annex BB [informaive] Buckling of componens of building srucures BB.1 Flexural buckling of members in riangulaed and laice srucures BB.1.1 General (1)B For chord members generall and for ou-of-plane buckling of web members, he buckling lengh L cr ma be aken as equal o he ssem lengh L, see BB.1.3(1)B, unless a smaller value can be jusified b analsis. ()B The buckling lengh L cr of an I or H secion chord member ma be aken as 0,9L for in-plane buckling and 1,0L for ou-of-plane buckling, unless a smaller value is jusified b analsis. (3)B Web members ma be designed for in-plane buckling using a buckling lengh smaller han he ssem lengh, provided he chords suppl appropriae end resrain and he end connecions suppl appropriae fixi (a leas bols if boled). (4)B Under hese condiions, in normal riangulaed srucures he buckling lengh L cr of web members for in-plane buckling ma be aken as 0,9L, excep for angle secions, see BB.1.. BB.1. Angles as web members (1)B Provided ha he chords suppl appropriae end resrain o web members made of angles and he end connecions of such web members suppl appropriae fixi (a leas wo bols if boled), he eccenriciies ma be negleced and end fixiies allowed for in he design of angles as web members in compression. The effecive slenderness raio λ eff ma be obained as follows: λ eff,v = 0,35 + 0, 7λ v for buckling abou v-v axis λ eff, = 0,50 + 0, 7λ for buckling abou - axis (BB.1) λ eff, = 0,50 + 0, 7λ for buckling abou - axis where λ is as defined in ()B When onl one bol is used for end connecions of angle web members he eccenrici should be aken ino accoun using 6..9 and he buckling lengh L cr should be aken as equal o he ssem lengh L. BB.1.3 Hollow secions as members (1)B The buckling lengh L cr of a hollow secion chord member ma be aken as 0,9L for boh in-plane and ou-of-plane buckling, where L is he ssem lengh for he relevan plane. The in-plane ssem lengh is he disance beween he joins. The ou-of-plane ssem lengh is he disance beween he laeral suppors, unless a smaller value is jusified b analsis. ()B The buckling lengh L cr of a hollow secion brace member (web member) wih boled connecions ma be aken as 1,0L for boh in-plane and ou-of-plane buckling. (3)B For laiced girders wih parallel chords and braces, for which he brace o chord diameer or widh raio β is less han 0,6 he buckling lengh L cr of a hollow secion brace member wihou cropping or flaening, welded around is perimeer o hollow secion chords, ma generall be aken as 0,75L for boh in-plane and ou-of-plane buckling, unless smaller values ma be jusified b ess or b calculaions. OTE The aional Annex ma give more informaion on buckling lenghs. 8

83 E : 005 (E) BB. Coninuous resrains BB..1 Coninuous laeral resrains (1)B If rapeoidal sheeing according o E is conneced o a beam and he condiion expressed b equaion (BB.) is me, he beam a he connecion ma be regarded as being laerall resrained in he plane of he sheeing. S where S π L π L 70 h EI w + GI + EI 0,5h (BB.) I w I I L h is he shear siffness (per uni of beam lengh) provided b he sheeing o he beam regarding is deformaion in he plane of he sheeing o be conneced o he beam a each rib. is he warping consan is he orsion consan is he second momen of area of he cross secion abou he minor axis of he cross secion is he beam lengh is he deph of he beam If he sheeing is conneced o a beam a ever second rib onl, S should be subsiued b 0,0S. OTE Eqaion (BB.) ma also be used o deermine he laeral sabili of beam flanges used in combinaion wih oher pes of cladding han rapeoidal sheeing, provided ha he connecions are of suiable design. BB.. Coninuous orsional resrains (1)B A beam ma be considered as sufficienl resrain from orsional deformaions if Cϑ > K, k pl,k K ϑ EI υ (BB.3) where C ϑ,k = roaional siffness (per uni of beam lengh) provided o he beam b he sabiliing coninuum (e.g. roof srucure) and he connecions K υ = 0,35 for elasic analsis K υ = 1,00 for plasic analsis K ϑ = facor for considering he momen disribuion see Table BB.1 and he pe of resrain pl,k = characerisic value of he plasic momen of he beam 83

84 E : 005 (E) Table BB.1: Facor K ϑ for considering he momen disribuion and he pe of resrain Case omen disribuion wihou ranslaional resrain wih ranslaional resrain 1 4,0 0 a b 3,8 0 3,5 0,1 0,3 4 1,6 1,0 5 1,0 0,7 ()B The roaional siffness provided b he sabiliing coninuum o he beam ma be calculaed from 1 C = + + (BB.4) ϑ, k Cϑ R,k Cϑ C,k Cϑ D,k where C ϑr,k = roaional siffness (per uni of he beam lengh) provided b he sabiliing coninuum o he beam assuming a siff connecion o he member C ϑc,k = roaional siffness (per uni of he beam lengh) of he connecion beween he beam and he sabiliing coninuum C ϑd,k = roaional siffness (per uni of he beam lengh) deduced from an analsis of he disorsional deformaions of he beam cross secions, where he flange in compression is he free one; where he compression flange is he conneced one or where disorsional deformaions of he cross secions ma be negleced (e.g. for usual rolled profiles) C ϑd,k = OTE For more informaion see E BB.3 Sable lenghs of segmen conaining plasic hinges for ou-of-plane buckling BB.3.1 Uniform members made of rolled secions or equivalen welded I-secions BB Sable lenghs beween adjacen laeral resrains de rabajo (1)B Laeral orsional buckling effecs ma be ignored where he lengh L of he segmen of a member beween he resrained secion a a plasic hinge locaion and he adjacen laeral resrain is no greaer han L m, where: Documeno 84

85 L where 38i E : 005 (E) m = (BB.5) 1 1 W pl, f + 57,4 A 756 C1 AI 35 A is he design value of he compression force [] in he member is he cross secion area [mm²] of he member W pl, is he plasic secion modulus of he member I f C 1 is he orsion consan of he member is he ield srengh in [/mm²] is a facor depending on he loading and end condiions o be aken from lieraure provided ha he member is resrained a he hinge as required b and ha he oher end of he segmen is resrained eiher b a laeral resrain o he compression flange where one flange is in compression hroughou he lengh of he segmen, or b a orsional resrain, or b a laeral resrain a he end of he segmen and a orsional resrain o he member a a disance ha saisfies he requiremens for L s, see Figure BB.1, Figure BB. and Figure BB3. OTE In general L s is greaer han L m. Figure BB.1: Checks in a member wihou a haunch 1 ension flange plasic sable lengh (see BB.3.1.1) 3 elasic secion (see 6.3) 4 plasic hinge 5 resrains 6 bending momen diagram 7 compression flange 8 plasic wih ension flange resrain, sable lengh = L s (see BB.3.1., equaion (BB.7) or (BB.8)) 9 elasic wih ension flange resrain (see 6.3), χ and χ LT from cr and cr including ension flange resrain 85

86 E : 005 (E) 86 1 ension flange elasic secion (see 6.3) 3 plasic sable lengh (see BB.3..1) or elasic (see ()B) 4 plasic sable lengh (see BB.3.1.1) 5 elasic secion (see 6.3) 6 plasic hinge 7 resrains 8 bending momen diagram 9 compression flange 10 plasic sable lengh (see BB.3.) or elasic (see ()B) 11 plasic sable lengh (see BB.3.1.) 1 elasic secion (see 6.3), χ and χ LT from cr and cr including ension flange resrain Figure BB.: Checks in a member wih a hree flange haunch 1 ension flange elasic secion (see 6.3) 3 plasic sable lengh (see BB.3..1) 4 plasic sable lengh (see BB.3.1.1) 5 elasic secion (see 6.3) 6 plasic hinge 7 resrains 8 bending momen diagram 9 compression flange 10 plasic sable lengh (see BB.3.) 11 plasic sable lengh (see BB.3.1.) 1 elasic secion (see 6.3), χ and χ LT from cr and cr including ension flange resrain Figure BB.3: Checks in a member wih a wo flange haunch

87 E : 005 (E) BB.3.1. Sable lengh beween orsional resrains (1)B Laeral orsional buckling effecs ma be ignored where he lengh L of he segmen of a member beween he resrained secion a a plasic hinge locaion and he adjacen orsional resrain subjec o a consan momen is no greaer han L k, provided ha he member is resrained a he hinge as required b and here are one or more inermediae laeral resrains beween he orsional resrains a a spacing ha saisfies he requiremens for L m, see BB.3.1.1, where L k 600f h 5,4 + i E f = (BB.6) f h 5,4 1 E f ()B Laeral orsional buckling effecs ma be ignored where he lengh L of he segmen of a member beween he resrained secion a a plasic hinge locaion and he adjacen orsional resrain subjec o a linear momen gradien and axial compression is no greaer han L s, provided ha he member is resrained a he hinge as required b and here are one or more inermediae laeral resrains beween he orsional resrains a a spacing ha saisfies he requiremens for L m, see BB.3.1.1, where pl,,rk L s = Cm L k (BB.7),,Rk + a C m a is he modificaion facor for linear momen gradien, see BB.3.3.1; is he disance beween he cenroid of he member wih he plasic hinge and he cenroid of he resrain members; pl,,rk is he characerisic plasic momen resisance of he cross secion abou he - axis,,rk is he characerisic plasic momen resisance of he cross secion abou he - axis wih reducion due o he axial force (3)B Laeral orsional buckling effecs ma be ignored where he lengh L of a segmen of a member beween he resrained secion a a plasic hinge locaion and he adjacen orsional resrain subjec o a non linear momen gradien and axial compression is no greaer han L s, provided ha he member is resrained a he hinge as required b and here are one or more inermediae laeral resrains beween he orsional resrains a a spacing ha saisfies he requiremens for L m, see BB3.1.1 where L = C L (BB.8) s n k C n is he modificaion facor for non-linear momen gradien, see BB.3.3., see Figure BB.1, Figure BB. and Figure BB.3. 87

88 E : 005 (E) BB.3. Haunched or apered members made of rolled secions or equivalen welded I- secions BB.3..1 Sable lengh beween adjacen laeral resrains (1)B Laeral orsional buckling effecs ma be ignored where he lengh L of he segmen of a member beween he resrained secion a a plasic hinge locaion and he adjacen laeral resrain is no greaer han L m, where for hree flange haunches (see Figure BB.) L 38i m = (BB.9) 1 1 W pl, f + 57,4 A 756 C1 AI 35 for wo flange haunches (see Figure BB.3) L where 0,85 38i m = (BB.10) 1 1 W pl, f + 57,4 A 756 C1 AI 35 W A pl, AI is he design value of he compression force [] in he member is he maximum value in he segmen is he cross secional area [mm²] a he locaion where member W pl, is he plasic secion modulus of he member I f i is he orsional consan of he member is he ield srengh in [/mm²] is he minimum value of he radius of graion in he segmen W pl, AI is a maximum of he apered provided ha he member is resrained a he hinge as required b and ha he oher end of segmen is resrained eiher b a laeral resrain o he compression flange where one flange is in compression hroughou he lengh of he segmen, or b a orsional resrain, or b a laeral resrain a he end of he segmen and a orsional resrain o he member a a disance ha saisfies he requiremens for L s. BB.3.. Sable lengh beween orsional resrains (1)B For non uniform members wih consan flanges under linear or non-linear momen gradien and axial compression, laeral orsional buckling effecs ma be ignored where he lengh L of he segmen of a member beween he resrained secion a a plasic hinge locaion and he adjacen orsional resrain is no greaer han L s, provided ha he member is resrained a he hinge as required b and here are one or more inermediae laeral resrains beween he orsional resrains a a spacing ha saisfies he requiremens for L m, see BB.3..1, 88

89 where for hree flange haunches (see Figure BB.) E : 005 (E) C L L = n k s c (BB.11) for wo flange haunches (see Figure BB.3) C L L = n k s 0,85 c (BB.1) where L k is he lengh derived for a uniform member wih a cross-secion equal o he shallowes secion, see BB.3.1. C n see BB.3.3. c is he aper facor defined in BB BB.3.3 odificaion facors for momen gradiens in members laerall resrained along he ension flange BB Linear momen gradiens (1)B The modificaion facor C m ma be deermined from C in which B 0 B 1 B η = m 1+ 10η = 1+ 0η 1 = (BB.13) B + B β + B β 0 5 η = π + 10 η cre 0,5 0,5 = 1+ π η 1+ 0η cre crt π = EI L 1 L is he disance beween he orsional resrains 1 π EI π a EI w = crt + + GI is he elasic criical orsional buckling force for an I-secion i s L L beween resrains o boh flanges a spacing L wih inermediae laeral resrains o he ension flange. i = i + i + a s where a is he disance beween he cenroid of he member and he cenroid of he resraining members, such as purlins resraining rafers 89

90 E : 005 (E) β is he raio of he algebraicall smaller end momen o he larger end momen. omens ha produce compression in he non-resrained flange should be aken as posiive. If he raio is less han 1,0 he value of β should be aken as 1,0, see Figure BB β 100 = = 0, BB.3.3. on linear momen gradiens bu β Figure BB.4: Value of β (1)B The modificaion facor C n ma be deermined from C n [ R + 3R + 4R + 3R + R + ( R R )] S E β = = ,0 hus β = 1,0 = 1 (BB.14) in which R 1 o R 5 are he values of R according o ()B a he ends, quarer poins and mid-lengh, see Figure BB.5, and onl posiive values of R should be included. In addiion, onl posiive values of (R S R E ) should be included, where R E is he greaer of R 1 or R 5 R s is he maximum value of R anwhere in he lengh L R E R S R E R 1 R R 3 R 4 R 5 R 1 R R 3 R 4 R S = R E R 5 ()B The value of R should be obained from: 90 pl, R E Figure BB.5: omen raios R R 3 R 1 R 5 R 4 R S R E R 1 R R 3 R 4 R5, + a R = (BB.15) f W R S

91 E : 005 (E) where a is he disance beween he cenroid of he member and he cenroid of he resraining members, such as purlins resraining rafers. BB Taper facor (1)B For a non uniform member wih consan flanges, for which h 1,b and h/ f 0 he aper facor c should be obained as follows: for apered members or segmens, see Figure BB.6(a): c 3 h h h 9 f min 1 / 3 max = + (BB.16) 1 for haunched members or segmens, see Figures BB.6(b) and BB.6(c): c where h h 3 h h h 9 f / 3 h h = + (BB.17) 1 s L L is he addiional deph of he haunch or aper, see Figure BB.6; h max is he maximum deph of cross-secion wihin he lengh L, see Figure BB.6; h min h s L h L is he minimum deph of cross-secion wihin he lengh L, see Figure BB.6; is he verical deph of he un-haunched secion, see Figure BB.6; is he lengh of haunch wihin he lengh L, see Figure BB.6; is he lengh beween poins a which he compression flange is laerall resrained. (h/ f ) is o be derived from he shallowes secion. h max L h min h s h h L h L h s L h L (a) Tapered segmen (b) Haunched segmen (c) Haunched segmen x = resrain Figure BB.6: Dimensions defining aper facor h h 91