[{"data":1,"prerenderedAt":208},["ShallowReactive",2],{"article:second-order:en":3},{"title":4,"seoTitle":5,"description":6,"label":7,"lead":8,"blocks":9},"When do steel frames need second-order analysis, and which imperfections apply?","Second-order analysis and imperfections (EC3) · Struct4U","When a steel frame needs second-order analysis (EN 1993-1-1 §5.2), which sway and bow imperfections apply (§5.3), and a worked portal-frame example.","EN 1993-1-1 · §5.2 · §5.3 · §6.3","A frame that sways carries its vertical loads with an extra lever arm, so the moments grow beyond what a first-order analysis finds. EN 1993-1-1 says when that matters (§5.2), how to include the out-of-plumbness and bow that every real frame has (§5.3), and what the member checks of §6.3 still have to cover.",[10,14,17,19,22,24,35,37,45,47,50,52,57,60,78,80,82,84,107,109,112,114,120,122,125,127,134,136,139,142,144,159,161,166,168,177,179,182,184,186,188,193,196,203],{"type":11,"id":12,"text":13},"h2","effects","What second-order effects are",{"type":15,"text":16},"p","A first-order analysis calculates the internal forces on the undeformed structure. In reality the structure deflects, and the vertical loads then act on a displaced geometry: a column that leans over by *δ* carries its axial force *N* with an extra moment *N* · *δ*. That moment increases the deflection, which increases the moment again. In a stiff frame this converges quickly and the effect is small. In a flexible frame it does not, and at the elastic critical load the frame buckles.",{"type":15,"text":18},"There are two levels. Global effects (P-Δ) come from the sway of the frame as a whole; member effects (P-δ) come from the bending of each member between its ends. Both come from axial compression acting on a deformed shape, and both start from an imperfect one.",{"type":11,"id":20,"text":21},"alpha-cr","When a first-order analysis is enough: *α*_{cr}",{"type":15,"text":23},"Clause 5.2.1(3) allows a first-order analysis when the deformations do not increase the internal forces significantly. That may be assumed when:",{"type":25,"clause":26,"rows":27},"formula","EN 1993-1-1 §5.2.1(3)",[28,32],{"s":29,"f":30,"ref":31},"*α*_{cr}","= *F*_{cr} \u002F *F*_{Ed} ≥ 10 for elastic analysis, ≥ 15 for plastic analysis","(5.1)",{"note":33,"f":34},true,"*α*_{cr} is the factor by which the design loads would have to be increased to cause elastic instability in a global mode; *F*_{cr} follows from the initial elastic stiffnesses.",{"type":15,"text":36},"For portal frames with shallow roofs and for beam-and-column plane frames in buildings, §5.2.1(4)B gives an approximation per storey:",{"type":25,"clause":38,"rows":39},"EN 1993-1-1 §5.2.1(4)B",[40,43],{"s":29,"f":41,"ref":42},"= (*H*_{Ed} \u002F *V*_{Ed}) · (*h* \u002F *δ*_{H,Ed})","(5.2)",{"note":33,"f":44},"*H*_{Ed}: horizontal reaction at the bottom of the storey from the horizontal loads, including the equivalent forces of the imperfections. *V*_{Ed}: total vertical load at the bottom of the storey. *h*: storey height. *δ*_{H,Ed}: horizontal displacement of the top of the storey relative to its bottom, from a first-order analysis under those horizontal loads.",{"type":15,"text":46},"The approximation assumes a roof no steeper than 1:2 (26°) and no significant axial compression in the beams or rafters. The compression counts as significant when *λ̄* ≥ 0.3 √(*A f*_{y} \u002F *N*_{Ed}) (5.3), with *λ̄* the in-plane slenderness of the beam as if hinged at its ends. Outside that scope, *α*_{cr} comes from an elastic buckling analysis of the frame.",{"type":11,"id":48,"text":49},"imperfections","Imperfections: sway and bow",{"type":15,"text":51},"Real frames are never perfectly plumb and real members never perfectly straight. §5.3 covers this with equivalent geometric imperfections, which also stand in for residual stresses and small eccentricities. Two kinds matter for frames (FIG. 1).",{"type":53,"diagram":54,"caption":55,"clause":56},"figure","sway-imperfection","The global sway imperfection *φ* of a frame and the local bow imperfection *e*_{0} of a member, drawn exaggerated. Both may be replaced by equivalent horizontal forces.","EN 1993-1-1 §5.3.2",{"type":58,"text":59},"h3","Global sway imperfection",{"type":25,"clause":61,"rows":62},"EN 1993-1-1 §5.3.2(3)a",[63,67,70,73,76],{"s":64,"f":65,"ref":66},"*φ*","= *φ*_{0} · *α*_{h} · *α*_{m}","(5.5)",{"s":68,"f":69},"*φ*_{0}","= 1\u002F200, the basic value",{"s":71,"f":72},"*α*_{h}","= 2 \u002F √*h*, with 2\u002F3 ≤ *α*_{h} ≤ 1.0 and *h* the height of the structure in metres",{"s":74,"f":75},"*α*_{m}","= √(0.5 (1 + 1\u002F*m*))",{"note":33,"f":77},"*m* is the number of columns in a row, counting only the columns that carry at least 50 % of the average column load in the plane considered.",{"type":15,"text":79},"The sway may be modelled as an inclined geometry, or replaced by horizontal forces *φ* · *N*_{Ed} at the top of each column (§5.3.2(7)). It applies in every relevant horizontal direction, one direction at a time (§5.3.2(8)); in multi-storey frames the equivalent forces act at every floor and roof level (§5.3.2(9)). For building frames it may be ignored when *H*_{Ed} ≥ 0.15 *V*_{Ed} (5.7), that is, when the horizontal loads are already large compared with the vertical ones.",{"type":58,"text":81},"Local bow imperfection",{"type":15,"text":83},"The initial bow of a member, *e*_{0}\u002F*L*, depends on its buckling curve. The national annex may set its own values; Table 5.1 gives the recommended ones.",{"type":85,"caption":86,"head":87,"rows":94,"align":105,"compact":33},"table","Bow imperfection *e*_{0}\u002F*L*, recommended values (Table 5.1)",[88,89,90,91,92,93],"Buckling curve","a0","a","b","c","d",[95,102],[96,97,98,99,100,101],"Elastic analysis","1\u002F350","1\u002F300","1\u002F250","1\u002F200","1\u002F150",[103,98,99,100,101,104],"Plastic analysis","1\u002F100",[106,92,92,92,92,92],"l",{"type":15,"text":108},"In the global analysis of a frame that is sensitive to second-order effects, the bow has to be modelled only for compressed members with at least one moment-resisting joint and *λ̄* > 0.5 √(*A f*_{y} \u002F *N*_{Ed}) (5.8). In all other cases the member checks of §6.3 cover it.",{"type":11,"id":110,"text":111},"routes","Three ways to deal with second-order effects",{"type":15,"text":113},"Clause 5.2.2(3) allows three routes, depending on the frame and the analysis.",{"type":115,"ordered":33,"items":116},"list",[117,118,119],"**Everything in the global analysis.** Second-order effects, the sway and the member bow imperfections are all in the model. No member buckling check to §6.3 is needed (§5.2.2(7)a); the cross-sections are checked with the forces from the analysis.","**Partly in the global analysis, partly in member checks.** The analysis includes the global second-order effects and the sway imperfection. The members are then checked to §6.3 for what the analysis left out, using its end forces and moments, and the buckling length may be taken equal to the system length (§5.2.2(7)b).","**Member checks only.** For basic cases: a first-order analysis without imperfections, with members checked to §6.3 using buckling lengths from the global buckling mode of the frame (§5.2.2(8)).",{"type":15,"text":121},"Route 3 is the classic hand method. Its weak point is the buckling length: in a sway frame it depends on the stiffness of the beams and joints and on the distribution of the axial forces, so it changes from one load combination to the next. Routes 1 and 2 avoid that question.",{"type":11,"id":123,"text":124},"amplification","The amplification factor",{"type":15,"text":126},"For single-storey frames designed by elastic analysis there is a shortcut: §5.2.2(5)B allows the sway effects of a first-order analysis to be amplified instead of running a second-order analysis.",{"type":25,"clause":128,"rows":129},"EN 1993-1-1 §5.2.2(5)B",[130],{"s":131,"f":132,"ref":133},"","multiply *H*_{Ed} and *φ* · *V*_{Ed} by 1 \u002F (1 − 1\u002F*α*_{cr}), provided that *α*_{cr} ≥ 3.0","(5.4)",{"type":15,"text":135},"For *α*_{cr} \u003C 3.0 a more accurate second-order analysis applies. §5.2.2(6)B extends the method to multi-storey frames when all storeys have a similar distribution of vertical loads, horizontal loads and frame stiffness. FIG. 2 shows the factor.",{"type":53,"diagram":123,"caption":137,"clause":138},"The amplification factor of (5.4) against *α*_{cr}, with the limits of (5.1) and (5.4). The dots are the two cases of the worked example.","EN 1993-1-1 §5.2.1, §5.2.2",{"type":11,"id":140,"text":141},"example","Worked example: a portal frame",{"type":15,"text":143},"A single-storey frame with two columns, *h* = 6.0 m. Each column carries *N*_{Ed} = 120 kN, so *V*_{Ed} = 240 kN. The design wind load gives *H*_{Ed} = 12.0 kN at the top.",{"type":25,"title":145,"clause":56,"rows":146},"Sway imperfection",[147,149,151,153,156],{"s":71,"f":148},"= 2 \u002F √6.0 = 0.816, between 2\u002F3 and 1.0",{"s":74,"f":150},"= √(0.5 (1 + 1\u002F2)) = 0.866",{"s":64,"f":152,"ref":66,"res":33},"= 1\u002F200 · 0.816 · 0.866 = 0.00354 = **1\u002F283**",{"s":154,"f":155},"*φ N*_{Ed}","= 0.00354 · 120 = 0.42 kN per column, 0.85 kN in total",{"s":131,"f":157,"ref":158},"*H*_{Ed} = 12.0 kN \u003C 0.15 *V*_{Ed} = 36.0 kN, so the sway imperfection has to be included","(5.7)",{"type":15,"text":160},"A first-order analysis under the wind load plus the equivalent forces, 12.85 kN in total, gives a horizontal displacement of 20 mm at the top. The roof is shallow and the compression in the rafter small, so (5.2) applies.",{"type":25,"title":162,"clause":38,"rows":163},"Is first order enough?",[164],{"s":29,"f":165,"ref":42,"res":33},"= (12.85 \u002F 240) · (6000 \u002F 20) = **16.1** ≥ 10: first-order analysis is enough",{"type":15,"text":167},"With more flexible columns, the same loads give a displacement of 40 mm.",{"type":25,"title":169,"clause":138,"rows":170},"The same frame, twice as flexible",[171,173,175],{"s":29,"f":172,"ref":42},"= (12.85 \u002F 240) · (6000 \u002F 40) = 8.03 \u003C 10: second-order effects count",{"s":131,"f":174,"ref":133,"res":33},"*α*_{cr} ≥ 3, so amplify by 1 \u002F (1 − 1\u002F8.03) = **1.142**",{"note":33,"f":176},"The wind load and the equivalent forces *φ N*_{Ed} are multiplied by 1.142, and the frame is checked with the resulting forces and moments.",{"type":15,"text":178},"In both cases the members are then checked to §6.3 for flexural and lateral-torsional buckling, with the end moments from the analysis. Because the analysis contains the global second-order effects and the sway imperfection, the buckling lengths may be taken equal to the system lengths.",{"type":11,"id":180,"text":181},"software","How XFEM4U and XFrame2D handle it",{"type":15,"text":183},"In [XFEM4U](product:xfem4u) and [XFrame2D](product:xframe2d), a Eurocode check for steel or timber always runs a geometrically non-linear (second-order) analysis, with the global sway imperfection of §5.3.2(3)a included automatically. XFEM4U applies it by shifting the node coordinates, once in the positive and once in the negative direction, so every ULS combination is calculated twice; XFrame2D applies the tilt in both directions of its plane. In both programs you enter *m*, the number of columns in a row, in the calculation settings.",{"type":15,"text":185},"The member bow imperfections of §5.3.2(3)b are not in the analysis. That is route 2: the members are checked to §6.3, and because the analysis contains the global second-order effects and the sway imperfection, the buckling length may be taken equal to the member length (§5.2.2(7)b). You do not have to derive buckling lengths from a buckling mode, and you do not have to classify the frame as sway or non-sway: if it is sensitive to second-order effects, that follows from the analysis.",{"type":15,"text":187},"Two limits. The second-order analysis applies to the beams; plates in XFEM4U are calculated geometrically linear. And the member checks still matter: a geometrically non-linear analysis without bow imperfections does not replace the buckling checks of §6.3.",{"type":189,"tone":190,"text":191},"callout","key",[192],"Check *α*_{cr} first. From 10 upwards (elastic analysis), first order is enough; between 3 and 10, amplify the sway effects or analyse second order; below 3, only a proper second-order analysis will do. Whatever the route, the members still need the §6.3 checks for the effects that the analysis leaves out.",{"type":11,"id":194,"text":195},"mistakes","Common mistakes",{"type":115,"items":197},[198,199,200,201,202],"**One direction only.** The sway imperfection applies in each relevant horizontal direction, one at a time (§5.3.2(8)). A frame that is safe leaning one way may not be safe leaning the other.","**No equivalent forces in *α*_{cr}.** *H*_{Ed} in (5.2) includes the equivalent forces of the imperfections, and *δ*_{H,Ed} has to come from the same loads.","**(5.2) outside its scope.** It is an approximation for shallow roofs and beams without significant compression. For other frames, *α*_{cr} comes from an elastic buckling analysis.","**System length after a first-order analysis.** The shortcut of §5.2.2(7)b only holds when the analysis includes the global second-order effects and the sway imperfection.","**Amplifying below *α*_{cr} = 3.** Then (5.4) no longer applies and a more accurate second-order analysis is required.",{"type":204,"items":205},"sources",[206,207],"EN 1993-1-1:2005, Eurocode 3: Design of steel structures, Part 1-1 (in the Netherlands NEN-EN 1993-1-1+C2:2011 with its national annex): §5.2.1, §5.2.2, §5.3.2, §5.3.4, Table 5.1, §6.3.","The [XFEM4U manual](manual:xfem4u) and the [XFrame2D manual](manual:xframe2d): Calculation settings, and References, Background, Buckling.",1792702121175]