BEAM-COLUMNS SUMMARY: OBJECTIVES: REFERENCES: CONTENTS:

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1 BEA-COLUS SUARY: Structural members subjected to axial compression and bending are nown as beam columns. The interaction o normal orce and bending ma be treated elasticall or plasticall using equilibrium or the classiication o cross-section. The behaviour and design o beam-columns are presented within the context o members subjected to uniaxial bending, i.e. deormation taes place onl in the plane o the applied moments. In the case o beam-columns which are susceptible to lateral-torsional bucling, the out-o-plane lexural bucling o the column has to be combined with the lateral-torsional bucling o the beam using the relevant interaction ormulae. For beam-columns with biaxial bending, the interaction ormula is expanded b an additional term. OBJECTIVES: Evaluate the in-plane bending and axial compression orce or beam-columns. Calculate the lateral-torsional bucling o beam-columns. Calculate the biaxial bending and axial compression orce or beam-columns. REFERECES: : Design o steel structures Part 1.1 General rules and rules or buildings. Chen W F and Atsuta T: Theor o Beam-Columns Vols. 1 &, cgraw-hill, Trahair. & Bradord : Behaviour and Design o Steel Structures, nd edt., Chapman & Hall, Dowling P J, Owens G W & Knowles P: Structural Steel Design, Butterworths, ethercot D A: Limit State Design o Structural Steelwor, nd edition, Chapman and Hall, COTETS: 1. Introduction.. In-plane behaviour o beam-columns..1 Cross-sectional behaviour.. Overall stabilit..3 Treatment in..4 The role o. 3. Lateral-torsional behaviour o beam-columns. 3.1 Lateral-torsional bucling. 3. The design process in. 3.3 The role o. 4. Biaxial bending o beam-columns. 4.1 Design or biaxial bending and compression. 4. Cross-section checs. 5. Veriication methods or isolated members and whole rames. 6. Concluding summar.

2 1. ITRODUCTIO. Beam-columns are deined as members subject to combined bending and compression. In principle, all members in rame structures are actuall beam-columns, with the particular cases o beams ( = 0) and columns ( = 0) simpl being the two extremes. Depending upon the exact wa in which the applied loading is transerred into the member, the orm o support provided and the member's cross-sectional shape, dierent orms o response will be possible. The simplest o these involves bending applied about one principal axis onl, with the member responding b bending solel in the plane o the applied moment.. I-PLAE BEHAVIOUR OF BEA-COLUS. When a beam-column is subjected to in plane bending (igure 1a), its behaviour shows an interaction between beam bending and compression member bucling, as indicated in igure 1b. Curve 1 shows the beam elastic linear behaviour. Figure 1 In-plane behaviour o beam-columns. Curve 6 shows the limiting behaviour o a rigid-plastic beam at the ull plastic moment pl. Curve shows the transition o real elastic-plastic beams rom curve 1 to curve 6. The elastic bucling load o a concentricall loaded compression member, is shown in curve 4. cr Curve 3 shows the interaction between bending and bucling in elastic members, and allows or the traditional moment v exerted b the axial load. Curve 7 shows the interaction between bending moment and axial orce causing the member to become ull plastic. This curve allows or the reduction rom the ull plastic moment pl to pr caused b the axial load, and or the additional moment v. The actual behaviour o a beam-column is shown b curve 5 which provides a transition rom curve 3 or elastic members to curve 7 or ull plasticit.

3 .1 CROSS-SECTIOAL BEHAVIOUR..1.1 Bending and axial orce or class 1 and cross-sections. I ull plasticit is allowed to occur, then the ailure condition will be as shown in igure and the combination o axial load and moment giving this condition will be: Figure Full plasticit under axial load and moment. a. For h t ) / n ( neutral axis in web = bt = t w n h t ( h t ) n tw (1) b. For > h t ) / neutral axis in lange n ( h = tw ( h t ) b t n h = b n ( h n ) t () Figure 3 compares Eqs. (1) and () with the approximation used in o: = pl. (1 n)/(1 0,5 ) but. Rd pl. Rd (3) (5.5) or eq Rd a in which = / pl Rd is the ratio o axial load to squash load ( A), and a = ( A bt )/ A 0, 5 n. For cross-sections without bolt holes, the ollowing approximations ma be used or axis moments: or n > a :. Rd = pl.. Rd n a or n > a :. Rd = pl.. Rd 1 1 a where = / pl Rd and a = ( A bt ) / A but a 0,5. n (5.6) or eq and 6.38

4 Figure 3 Full plasticit interaction major axis bending o HEA 450 section. Further simpliications or a range o common cross-sectional shapes are provided in Table 1. Table 1 Expressions or reduced plastic moment resistance (otation: n = / pl.rd). Cross-section Shape Expression or Rolled I or H, = 1,11 pl. (1 n) (5.7)

5 , 1,56 pl. (1 n)(0,6 n) = (5.8) Square hollow section, 1,6 pl (1 n) = (5.31) Rectangular hollow section Circular hollow section, 1,33 pl. (1 n), = (5.3) = pl. 1 n ht 0,5 A 1,7, 1,04 pl (1 n ) (5.33) = (5.34) In all cases the value o should, o course, not exceed that o pl..1. Bending and axial orce or Class 3 cross-sections. Figure 4 shows a point somewhere along the length o an H-shape column where the applied compression and moment about the axis produce the uniorm and varing stress distribution shown in igures 4a and 4b. Figure 4 Elastic behaviour o cross-section in compression and bending. For elastic behaviour the principle o superposition ma be used to simpl add the two stress distributions as shown in igure 4c. First ield will thereore develop at the edge where the maximum compressive bending stress occurs and will correspond to the condition: = σ σ where: is the material ield stress, h is the overall depth o section and I is the second moment o area about the axis. c b

6 σ = A is the stress due to the compressive load c / h / σ b = is the maximum compressive stress due to the moment. I Class 3 cross-sections will be satisactor i the maximum longitudinal stress σ x.ed satisies the criterion: ; d = / γ (5.31) or eq. 6.4 σ x. Ed d Bending and axial orce or class 4 cross-sections. Class 4 cross-sections will be satisactor i the maximum longitudinal stress σ x.ed calculated using the eective widths o the compression elements (5.3..() o EC3) satisies the criterion: ; d = / γ (5.39) or eq σ x. Ed d 0. OVERALL STABILITY. The treatment o cross-sectional behaviour in the previous section too no account o the exact wa in which the moment at the particular cross-section under consideration was generated. Figure 5 shows a beam-column undergoing lateral delection as a result o the combination o compression and equal and opposite moments applied at the ends. Figure 5 Primar and secondar moments. The moment at an point within the length ma convenientl be regarded as being composed o: primar moment secondar moment v. Using elastic strut theor gives the maximum delection at the centre (Trahair & Bradord, 1988) as:

7 where P E v max = π sec P E 1 π EI = is the Euler critical load or major axis bucling, and the maximum moment is: L (4) max π = sec (5) P E In both equations the secant term ma be replaced b noting that the irst order delection (due onl to the end moments) and the irst order moment (ordinar beam theor) are approximatel ampliied b: as shown in igure / P E (6) Figure 6 aximum delection and moment in beam-columns with equal moments. Thus: v max L 1 = (7) 8EI 1 / max P E 1 = (8) 1 / P E Since the maximum elastic stress will be: Eq. (9) ma be rewritten as: max σ max = σ c σ b (9) σ c σ b (1 / P E = 1,0 ) (10)

8 Eq. (10) ma be solved or values o σ c and σ b that just cause ield, taing dierent values o P (which is dependent on the slenderness L/r ). This gives a series o curves as shown in igure 7, which indicate that as σ b 0, σ c tends to the value o material strength. E (a) (b) (c) Figure 7 Form o Eq. (10) eect o: (a) slenderness (b) cross sectional shape (c) moment gradient. Eq. (10) does not recognise the possibilit o bucling under pure axial load at a stress σ E given b: PE π EI π E σ E = = = (11) A AL λ Use o both Eq. (10) and Eq. (11) ensures that both conditions are covered as shown in igure 8..3 TREATET I EUROCODE 3. Figure 8 Combination o Eqs. (10) and (11) Eqs. (10) and (11) are written in terms o stresses and originate rom the concept o ailure being deined as either the attainment o irst ield or elastic bucling o the perect member. Limit state design codes, as, normall tae ultimate load as the design criterion when considering resistance under static loading.

9 Thus these equations must be re-written in terms o orces and moments. In doing this it is also necessar to mae some allowance or those eects present in real structures that have not so ar been explicitl allowed or, i.e. initial lac o straightness, residual stresses, etc. For consistenc in design it is essential that the interaction equation or combined loading reduces down to the column and beam design procedures as moment and axial load respectivel reduce to ero..3.1 embers with class 1 and cross-sections. The approach taen in (assuming bending about the axis) is to use: χ A W pl.. (1) 5.5.4(1) (5.51) or eq & 6.6 in which χ is the reduction actor or column bucling, and μ = 1 but 1, 5 χ A where is a modiication actor discussed in section.3.4., and Wpl, μ = λ (β 4) 1 but μ 0, 90 W el, where β is an equivalent uniorm moment actor accounting or the non-uniormit o the moment diagram, see table (moment diagram about axis and restraints in the direction)..3. embers with class 3 cross-sections. embers with class 3 cross-sections subject to bending and axial load shall satis: χ A W el.. (13) 5.5.4(3) (5.53) or eq & 6.6 where and χ is as in Eq. (1) with μ λ ( β 4) but = μ 0,90 Table Equivalent uniorm moment actors β. (C m ) OET DIAGRA EQUIVALET UIFOR OET FACTOR β End moments β ψ = 1,8 0, 7ψ, oments due to in-plane lateral loads For uniorml distributed load: β, Q =1,3

10 For concentrated load: β, Q =1,4 oments due to in-plane lateral loads plus end moments β = β Q, ψ ( β, Q β, ψ Δ where: ) Q = max due to lateral load onl Δ.3.3 embers with class 4 cross-sections. Δ = max = max min and or moment diagram without change o sign embers with class 3 cross-sections subject to bending and axial load shall satis: where sign o moment diagram changes χ A e (. W (14) e. e. ) 5.5.4(3) (5.56) or eq & 6.6 Where: and χ is as in Eq. (1) with μ as in Eq. (13) A e. is the eective cross-sectional area or pure compression W e. is the eective cross-sectional modulus or pure bending e. is the shit in neutral axis comparing the ull cross-section with the eective cross-section (calculated assuming pure compression) used to account or local bucling.3.4 The role o. The value o, as shown b the equations explaining Eq. (1), depends in a rather complex wa on: Level o axial load as measured b the ratio χ A ember slenderness λ argin between the cross-section s plastic & elastic moduli W pl & W el (or class 1 & onl) Pattern o primar moments. When all o this combine in the most severe wa the sae value o is 1,5. The role o is to allow or the secondar bending eect described earlier plus the eects o nonuniorm moment and spread o ield. Figure 5 showed how, or the particular cases o equal and opposite beam moments, the primar moments are ampliied due to the eect o the axial load acting through the lateral displacements v.

11 When the pattern o primar moments is dierent, the two eects will not be so directl additive since maximum primar and secondar moments will not necessaril occur at the same location. Figure 9 illustrates the situation or end moments and ψ, where ψ can adopt values between 1 (uniorm single curvature) and 1 (double curvature). Figure 9 on-uniorm moment case. The particular case shown corresponds to a value ψ 0,5. For the case illustrated the maximum moment still occurs within the member length but the situation is clearl less severe than that o igure 5 assuming all conditions to be identical apart rom the value o ψ. It is customar to recognise this in design b reducing the contribution o the moment term to the interaction relationship. Thus in in Eq. (1) depends upon the ratio ψ. Since the case o uniorm single curvature moment is the most severe, it ollows that a sae simpliication is alwas to use the procedure or ψ = 1,0. Returning to igure 9, it is possible or the point o maximum moment to be at the end at which the larger primar moment is applied. This would usuall occur i the axial load was small and/or slenderness was low so that secondar bending eects were relativel slight. In such cases design will be controlled b ensuring adequate cross-sectional resistance at this end. The ormula, table, or the particular shape o cross-section being used, should thereore be emploed. In cases where onl the uniorm moment (ψ = 1.0) arrangement is being considered, the overall bucling chec o Eq. (1) will alwas be more severe than (or in the limit equal to) the cross-sectional chec which, and thereore this latter chec need not be perormed separatel.

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16 3. LATERAL-TORSIOAL BEHAVIOUR OF BEA-COLUS. When an unrestrained beam-column is bent about its major axis (igure 10a), it ma bucle b delecting laterall & twisting at a load signiicantl less than the maximum load predicted b an inplane analsis. The most general situation is illustrated in Figure 10b. When bending is applied about both principal axes the member's response will be 3-dimensional in nature, involving biaxial bending and twisting. (a) (b) Figure 10 Lateral-torsional behaviour.

17 This lateral-torsional bucling ma occur while the member is still elastic (curve 1 o igure 11), or ater some ielding (curve ) due to in-plane bending and compression has occurred. Figure 11 Lateral-torsional bucling o beam-columns. 3.1 LATERAL-TORSIOAL BUCKLIG. Considering the lateral-torsional behaviour o an unrestrained I section beam-column bent about its major axis it can be assumed the elastic behaviour and the arrangement o applied loading and support conditions given in igure 1. Figure 1 Basic case or lateral-torsional bucling. The critical combinations o and ma be obtained rom the solution o (Chen & Atsuta, 1976): i P P 0 E E0 in which 1 P 1 PE = E 0 I I i0 = is the polar radius o gration A π EI PE = is the minor axis critical load L GI = t π EI w P E0 1 is the torsional bucling load. i0 GIt L (15)

18 Eq. (15) reduces to the bucling o a beam when 0 and to the bucling o a column in either lexure (P E) or torsion (P E0) as 0. In the irst case the critical value o will be given b: π EI 1 π w cr = EI GIt (16) L L GIt in which: EI is the minor axis lexural rigidit GI t is the torsional rigidit EI w is the warping rigidit. In deriving Eq. (15) no allowance was made or the ampliication o the in-plane moments b the axial load acting through the in-plane delections. This ma be approximated as 1 / P E. Eq. (15) can, thereore, be modiied to: i P P 0 E E0 1 P 1 P 1 PE = E E 0 (17) oting the relative magnitudes o P, P and P, and re-arranging gives the ollowing approximation: E E E0 or P E 1 1 / P P E E i 1 1 / P 0 E P E cr P E0 = 1 = 1 (18) (19) 3. THE DESIG PROCESS I EUROCODE 3. For design purposes it is necessar to mae suitable allowances or eects such as initial lac o straightness, partial ielding, residual stresses, etc., as has been ull discussed in earlier lectures in the context o columns and beams. Thus some modiication to Eq. (19) is necessar to mae it suitable or design. In particular, the end points (corresponding to the cases o = 0 and = 0) must conorm to the established procedures or columns and beams 3..1 embers with class 1 and cross-sections. uses the interaction equation: χ A χ W. pl. (0) 5.5.4() (5.5) or eq & 6.6 in which χ is the reduction actor or column bucling around the minor axis, χ is the reduction actor or lateral-torsional beam bucling, and

19 μ χ A = 1 but, 0 with μ 0,15( λ β, 1) but 0, 90 = μ where β, is a actor accounting or the non-uniormit o the moment diagram, see Table (moment diagram about axis and restrains in the direction). 3.. embers with class 3 cross-sections. embers with class 3 cross-section should satis the ollowing criterion: χ A χ W. el. (1) 5.5.4(4) (5.54) or eq & embers with class 4 cross-sections. embers with class 3 cross-section should satis the ollowing criterion: χ A χ. W e. e, () 5.5.4(5) (5.57) or eq & The role o. The value o, as shown b the equations explaining Eq. (0), depends on: the level o axial load as measured b the ratio the member slenderness λ the pattern o primar moments. χ A For the most severe combination adopts the value o unit, corresponding to a linear combination o the compressive and bending terms. This relects the reduced scope or ampliication eects in this case, since the value o cannot exceed χ A, which will, in turn, be signiicantl less than the elastic critical load or in-plane bucling P E. It is, o course, also necessar to ensure against the possibilit o in-plane ailure b excessive delection in the plane o the web at a lower load than that given b Eq. (0). This might occur, or example, in situations where dierent bracing and/or support conditions are provided in the x and x planes as illustrated in igure 13.

20 Figure 13 Column with dierent support conditions in x and x planes. Such cases should be treated b checing, in addition to Eq. (0), an in-plane equation o the orm:. (3) χ A W min pl. in which χ min depends on the in-plane conditions. Usuall, however, Eq. (0) will govern. 4. BIAXIAL BEDIG OF BEA-COLUS. Analsis or the ull three-dimensional case, even or the simple elastic version, is extremel complex and closed-orm solutions are not available. Rather than starting analticall it is more convenient to approach the question o a suitable design approach rom considerations o behaviour and the use o the methods alread derived or the simpler cases o igure 14. Figure 14 Biaxial bending. Figure 15 presents a diagrammatic version o the design requirement.

21 Figure 15 Interaction diagram or biaxial bending. The - and - axes correspond to the two uniaxial cases alread examined. Interaction between the two moments and corresponds to the horiontal plane. When all the three load components, and are present the resulting interaction plots somewhere in the three-dimensional space represented b the diagram. An point alling within the boundar corresponds to a sae combination o loads. Assuming proportional loading, an combination ma be regarded as a straight line starting at the origin, the orientation o which depends upon the relative sies o the three load components. Increasing the loads extends this line rom the origin until it just reaches and exceeds the boundar. In each case the axes have been taen as the ratio o the applied component to the member s resistance under the load component alone, e.g. /χ min A in the case o the compressive loading. Thus igure 15 actuall represents the situation or one particular example with particular values o cross-sectional properties, slenderness and load arrangement. 4.1 DESIG FOR BIAXIAL BEDIG AD COPRESSIO. embers with class 1 and cross-sections subject to combined biaxial bending and axial compression should satis the ollowing criterion: χ A min W pl,, W pl,, (4) 5.5.4(1) (5.51) or eq & 6.6 where is a actor similar to, see Eq. (1). embers with class 1 and cross-sections subject to combined biaxial bending and axial compression where lateral-torsional bucling is relevant, should also satis the ollowing criterion:

22 χ A χ W, pl, W pl,, (5) 5.5.4() (5.5) or eq & 6.6 embers with class 3 cross-sections subject to combined biaxial bending and axial compression: χ A min W el,, W el,, (6) 5.5.4(3) (5.53) or eq & 6.6 embers with class 3 cross-sections subject to combined biaxial bending and axial compression where lateral-torsional bucling is relevant, should also satis the ollowing criterion: χ A χ W, el, W el,, (7) 5.5.4(4) (5.54) or eq & 6.6 embers with class 4 cross-sections subject to combined biaxial bending and axial compression: χ A min e (, W e, e ) (, W e, e ) (8) 5.5.4(5) (5.56) or eq.6.44, 6.61 & 6.6 embers with class 4 cross-sections subject to combined biaxial bending and axial compression where lateral-torsional bucling is relevant, should also satis the ollowing criterion: χ A e χ, W e, e (, W e, e ) (9) 5.5.4(6) (5.57) or eq.6.44, 6.61 & 6.6 An important point to note rom the deinition o A e and W e above is that the calculation o crosssectional properties, and thus also cross-sectional classiication, should be undertaen on a separate basis or each o the three load components, and. This does, o course, mean that the same member ma be classiied as (sa) class 1 or major axis bending, class or minor axis bending and class 3 or compression. The sae design approach is to chec all beam-columns using the least avourable class procedures. 4. CROSS-SECTIO CHECKS. Class 4 I allowance has been made when determining the actors (through the use o β ) or the less severe eect o patterns o moment other than the uniorm single curvature bending, it is necessar urther to

23 chec that the cross-section is everwhere capable o locall restraining the combination o compression and primar moment(s) present at an point. Expressions or checing several tpes o cross-section under compression plus uniaxial bending were given in section 1.1. For biaxial bending uses:.. Rd α.. Rd β (30) (5.35) or eq.6.41 in which the values o α and β depend upon the tpe o cross-section as indicated in table 3. Table 3 Values o α and β or use in Eq. (30) (otation: n = / pl.rd). Tpe o cross-section α β I and H sections 5n but 1 Circular tubes Rectangular hollow sections 1,66 1 1,33n 1,66 1 1,33n but 6 but 6 Solid rectangles and plates 1,73 1,8n 3 1,73 1,8n 3 A simpler but conservative alternative is: pl. Rd.. Rd.. Rd (31) (5.36) 5. VERIFICATIO ETHODS FOR ISOLATED EBERS AD WHOLE FRAES. ormall, the design o an individual member in a rame is done b separating it rom the rame and dealing with it as an isolated substructure.

24 The end conditions o the member should then compl with its deormation conditions, in the spatial rame, in a conservative wa, e.g. b assuming a nominall pinned end condition, and the internal action eects, at the ends o the members, should be considered b appling equivalent external end moments and end orces, Figure 16. ethods o veriication or these members are given in Section 5.1. Figure 16 Isolated members A and B rom the plane rame analsis. A more general procedure is given in Section 5., or the case where members cannot be isolated rom the rame structure in the wa described above ethods o veriication or isolated members. For the design o beam-columns, with mono-axial bending onl, two checs must be carried out: the in-plane bucling chec taing into account the in-plane imperections. the out-o-plane bucling chec, including the lateral-torsional bucling veriication that taes account o the out-o-plane imperections (Figure 17). Figure 17 Assumptions or member imperections. It has been ound b test calculations that twist imperections, ρ, o beam-columns that are susceptible to lateral-torsional bucling, can be substituted b lexural imperections, see Figure 18.

25 Figure 18 Twist imperections φ 0 and lexural imperection W0. embers with suicient torsional stiness, i.e. hollow section members, need not be veriied or lateral-torsional bucing. λ When the non-dimensional slenderness 0,4, the reduction coeicient χ need not be taen into account. This rule ma be used or spacing the lateral restraints to resist lateral-torsional bucling. 5.. ethod o veriication o whole rames. Figure 19 gives an example o a portal rame with tapered columns and beams, the external langes o which are laterall supported b the purlins which, due to their lexural stiness, also provide torsional restraint; the beams and columns ma, however, be subject to distortion o the cross-section, due to the lexibilit o the web. Figure 19 Portal rame with tapered beams and columns, with elastic torsional restraints and displacements restraints b the purlins. The cross-section is susceptible to distortion. An accurate veriication o this arrangement should be based on a inite element model which taes the above eects into account. The basic assumptions made regarding the imperections in this model, would be such that the standard veriication given previousl would produce equall avourable results since the standard procedure has been calibrated against test results. A more simpliied procedure is, thereore, given here which is related to the veriication o columns or lexural bucling, and beams or lateral-torsional bucling. The basic principles governing the standard veriication o columns or lexural bucling, and beams or lateral-torsional bucing, are as ollows: 1. The non-dimensional slenderness λ is deined b:

26 λ FB = pl cr ; λ = pl cr Where pl, pl are the characteristic values o the elastic/plastic resistances o the column or beam neglecting an out-o-plane eects; and cr, cr are the critical biurcation values or the column resistance, or the beam resistance, when considering out-o-plane delections and hperelastic behaviour in the equilibrium state.. Using the non-dimensional slenderness, λ, a reduction actor χ can be determined rom the European bucling curves that allows the design value o the resistance o the column or beam to be deined b: bd = χ pl / γ 1 or the column bd = χ pl / γ 1 or the beam. In appling this principle to an loaded structure, see Figure 0, the procedure is as ollows: Figure 0 Stepwise veriication o a structure, assuming in step 1: elastic-plastic in-plane behaviour and no lateral delections, and in step : hperelastic behaviour and lateral delection. 1. As a irst step the structure is analsed or a given load case with an elastic or plastic analsis assuming that an out o plane delections are prevented. B this analsis a multiplier, γ pl, o the given loads is ound that represents the ultimate resistance o the structure.. The structure is then checed assuming hperelastic material behaviour allowing or lateral and torsional delections. This leads to a multiplier γ crit o the given loads that represents the critical elastic resistance o the structure to lateral bucling or lateral-torsional bucing. 3. The overall slenderness, λ, o the structure can then be deined b: λ = γ γ pl crit And b using the reduction coeicient χ rom the relevant European bucling curve, e.g. curve c, the inal saet actor γ can be derived.: γ = χ γ pl This procedure is analogous to the erchant-ranine procedure or the rames non- elastic veriication.

27 In general the procedure described earlier needs a computer program that perorms a planar elasticplastic analsis o the rame and determines the elastic biurcation load o the structure or lateral and torsional delections, including distortion. Such a program, or calculating the elastic biurcation loads, can either be based on inite elements or on a grid model where the langes and stieners are considered as beams and the web is represented b an equivalent lattice sstem that allows or second order eects; such programs are available on PC's. 6. COCLUDIG SUARY. Beam-columns are structural members subjected to axial compression and bending about one or both axes o the cross-section. The behaviour o beam-columns can be understood in three stages: (a) behaviour o the restrained beam-column; (b) uniaxial bending and compression o the unrestrained beam-column; (c) biaxial bending and compression o the unrestrained beam-column. Stage (a) is governed b the behaviour o the cross-section. Stage (b) is governed b an interaction o the cross-section behaviour with in-plane column bucling and/or lateral-torsional bucling. Stage (c) is governed b the same actors as stage (b), but the moment about the other axis must be incorporated into the design equation. For the cross-section, the interaction o normal orce and bending ma be treated elasticall using the principle o superposition or plasticall using equilibrium and the concept o stress blocs. When considering the member as a whole, secondar-bending eects must be allowed or. Strut analsis ma be used as a basis or examining the role o the main controlling parameters. Design is normall based on the use o an interaction equation, an essential eature o which is the resistance o the component as a beam and as a column. The class o cross-section will aect some o the values used in the interaction equations. The biaxial bending case is the most general and includes the two others as simpler and more restricted component cases. A three dimensional rame ma generall be analsed b separating it into plane rames and analsing these on the assumption o no imperections; the individual members o the rame should then be checed with the imperection eects taen into account. The isolated members in general represent beam-columns with either in plane or biaxial bending. In certain cases the standard procedure or the veriication o a beam- column is not applicable and more accurate models must be used. As a non-linear spatial analsis including the eect o imperections is diicult, an alternative procedure is provided b which the overall slenderness o a rame is deined; this allows veriication o the rame using the European bucling curves which tae account o lateral- torsional bucling. ADDITIOAL READIG 1. araanan, R., Editor, "Beams and Beam Columns: Stabilit and Strength", App Sci Pub Chen, W. F. and Atsuta, T. "Theor o Beam Columns Volume 1 &, cgraw Hill Timosheno, S. P. and Gere, J.., "Theor o Elastic Stabilit" nd Edition, cgraw Hill Bleich, F., "Bucling Strength o etal Structures", cgraw Hill Trahair, S Bradord, A, "The Behaviour & Design o Steel Structures", Chapman Hall, Ballio, G. aolani, F.., "Theor and Design o Steel Structures", Chapman Hall, Galambos, T. V., "Guide to Stabilit Deign Criteria or etal Structures", 4th edition, Wile. 8. Dowling, P. J., Owens, G. W. and Knowles, P., "Structural Steel Design", Butterworths, ethercot, D.A., "Limit State Design o Structural Steelwor", nd edition, Chapman Hall, 1991.