[{"data":1,"prerenderedAt":1258},["ShallowReactive",2],{"kb-en-\u002Fkb\u002Fanalysis\u002Fsecond-order":3,"kb-nav-en":865},{"page":4,"fallback":855},{"id":5,"title":6,"body":7,"description":853,"extension":854,"faq":855,"meta":856,"navigation":857,"ogImage":858,"order":859,"path":860,"raw":855,"robots":858,"schemaOrg":858,"seo":861,"sitemap":862,"stem":863,"__hash__":864},"content_en\u002Fkb\u002Fanalysis\u002Fsecond-order.md","Second-order analysis",{"type":8,"value":9,"toc":842},"minimark",[10,14,174,177,180,185,234,237,244,248,251,259,262,282,285,289,296,616,619,628,637,641,683,686,690,754,758,761,772,776],[11,12,13],"p",{},"Second-order analysis solves",[15,16,19],"span",{"className":17},[18],"katex-display",[15,20,23],{"className":21},[22],"katex",[15,24,28,162],{"className":25,"ariaHidden":27},[26],"katex-html","true",[15,29,32,37,143,147,150,154,159],{"className":30},[31],"base",[15,33],{"className":34,"style":36},[35],"strut","height:1.0361em;vertical-align:-0.2861em;",[15,38,41,48,55,60,65,68,126,130,134,139],{"className":39},[40],"minner",[15,42,47],{"className":43,"style":46},[44,45],"mopen","delimcenter","top:0em;","[",[15,49,54],{"className":50,"style":53},[51,52],"mord","mathnormal","margin-right:0.0715em;","K",[15,56],{"className":57,"style":59},[58],"mspace","margin-right:0.2222em;",[15,61,64],{"className":62},[63],"mbin","+",[15,66],{"className":67,"style":59},[58],[15,69,71,74],{"className":70},[51],[15,72,54],{"className":73,"style":53},[51,52],[15,75,78],{"className":76},[77],"msupsub",[15,79,83,117],{"className":80},[81,82],"vlist-t","vlist-t2",[15,84,87,112],{"className":85},[86],"vlist-r",[15,88,92],{"className":89,"style":91},[90],"vlist","height:0.1514em;",[15,93,95,100],{"style":94},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[15,96],{"className":97,"style":99},[98],"pstrut","height:2.7em;",[15,101,107],{"className":102},[103,104,105,106],"sizing","reset-size6","size3","mtight",[15,108,111],{"className":109,"style":110},[51,52,106],"margin-right:0.0359em;","g",[15,113,116],{"className":114},[115],"vlist-s","​",[15,118,120],{"className":119},[86],[15,121,124],{"className":122,"style":123},[90],"height:0.2861em;",[15,125],{},[15,127,129],{"className":128},[44],"(",[15,131,133],{"className":132},[51,52],"u",[15,135,138],{"className":136},[137],"mclose",")",[15,140,142],{"className":141,"style":46},[137,45],"]",[15,144],{"className":145,"style":146},[58],"margin-right:0.1667em;",[15,148,133],{"className":149},[51,52],[15,151],{"className":152,"style":153},[58],"margin-right:0.2778em;",[15,155,158],{"className":156},[157],"mrel","=",[15,160],{"className":161,"style":153},[58],[15,163,165,169],{"className":164},[31],[15,166],{"className":167,"style":168},[35],"height:0.8889em;vertical-align:-0.1944em;",[15,170,173],{"className":171,"style":172},[51,52],"margin-right:0.1076em;","f",[11,175,176],{},"by iteration. The structure is solved, the geometric stiffness of the deformed state is\nformed from the resulting axial forces, and the system is solved again, until the\ndisplacement increment between two steps falls below your tolerance.",[11,178,179],{},"The answer has the same shape a linear solve produces: displacements, internal forces,\nreactions and shell resultants. Everything downstream (viewers, tables, design checks,\nreport) reads it unchanged.",[181,182,184],"h2",{"id":183},"the-two-effects-and-why-they-need-different-meshes","The two effects, and why they need different meshes",[186,187,188,203],"table",{},[189,190,191],"thead",{},[192,193,194,197,200],"tr",{},[195,196],"th",{},[195,198,199],{},"What it is",[195,201,202],{},"What captures it",[204,205,206,221],"tbody",{},[192,207,208,215,218],{},[209,210,211],"td",{},[212,213,214],"strong",{},"P-Δ",[209,216,217],{},"The whole frame sways, and gravity acting through that sway adds moment",[209,219,220],{},"Node displacements alone — no refinement needed",[192,222,223,228,231],{},[209,224,225],{},[212,226,227],{},"P-δ",[209,229,230],{},"A member bows between its own two ends, and its axial force adds moment along the span",[209,232,233],{},"Interior nodes on the member — a single element has none",[235,236],"diagram-second-order",{},[11,238,239,240,243],{},"A member modelled as one element cannot bow, so it cannot show P-δ at all. Two elements\nshow it; four is where it is worth trusting. A second-order setup therefore carries its\nown ",[212,241,242],{},"elements per member",", defaulting to four, and raises the analysis mesh to match,\nincluding members you have set a coarser count on yourself.",[181,245,247],{"id":246},"results-do-not-superpose","Results do not superpose",[11,249,250],{},"This is the one thing that genuinely behaves differently from a linear analysis, and it\nchanges how the work is organised.",[252,253,256],"callout",{"type":254,"title":255},"warning","No factored sums",[11,257,258],{},"The amplification one load causes depends on every other load acting with it. A factored\nsum of load-case results, which is exactly what a linear combination is, is not the\nsecond-order answer for that combination. Elementarium does not form one.",[11,260,261],{},"Instead:",[263,264,265,273,279],"ul",{},[266,267,268,269,272],"li",{},"A second-order setup ",[212,270,271],{},"solves each combination as its own run",".",[266,274,275,276,272],{},"A combination that expands to several permutations (an envelope, an EN 1990 rule with\neach variable action leading in turn) is solved ",[212,277,278],{},"once per permutation",[266,280,281],{},"Results read the direct answer rather than any sum that could have been formed from\nload cases.",[11,283,284],{},"The practical consequence: a second-order run over forty combinations costs forty solves,\nnot one solve per load case and forty sums. Point the setup at the combinations that\nmatter rather than at everything.",[181,286,288],{"id":287},"geometric-imperfections-en-1993-1-1-53","Geometric imperfections — EN 1993-1-1 §5.3",[11,290,291,292,295],{},"A second-order analysis of a perfectly straight, perfectly plumb frame understates the\neffects. Real capacity is limited by how far the structure is already out of line before\nthe load arrives. §5.3 covers this in three parts, and all three are available as an\n",[212,293,294],{},"initial deformation"," of the model rather than as equivalent horizontal forces:",[186,297,298,311],{},[189,299,300],{},[192,301,302,305,308],{},[195,303,304],{},"Clause",[195,306,307],{},"Imperfection",[195,309,310],{},"Applied as",[204,312,313,507,605],{},[192,314,315,318,488],{},[209,316,317],{},"§5.3.2(3)a",[209,319,320,321],{},"Global sway ",[15,322,324],{"className":323},[22],[15,325,327,346],{"className":326,"ariaHidden":27},[26],[15,328,330,333,337,340,343],{"className":329},[31],[15,331],{"className":332,"style":168},[35],[15,334,336],{"className":335},[51,52],"ϕ",[15,338],{"className":339,"style":153},[58],[15,341,158],{"className":342},[157],[15,344],{"className":345,"style":153},[58],[15,347,349,352,396,399,444,447],{"className":348},[31],[15,350],{"className":351,"style":168},[35],[15,353,355,358],{"className":354},[51],[15,356,336],{"className":357},[51,52],[15,359,361],{"className":360},[77],[15,362,364,387],{"className":363},[81,82],[15,365,367,384],{"className":366},[86],[15,368,371],{"className":369,"style":370},[90],"height:0.3011em;",[15,372,374,377],{"style":373},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[15,375],{"className":376,"style":99},[98],[15,378,380],{"className":379},[103,104,105,106],[15,381,383],{"className":382},[51,106],"0",[15,385,116],{"className":386},[115],[15,388,390],{"className":389},[86],[15,391,394],{"className":392,"style":393},[90],"height:0.15em;",[15,395],{},[15,397],{"className":398,"style":146},[58],[15,400,402,407],{"className":401},[51],[15,403,406],{"className":404,"style":405},[51,52],"margin-right:0.0037em;","α",[15,408,410],{"className":409},[77],[15,411,413,436],{"className":412},[81,82],[15,414,416,433],{"className":415},[86],[15,417,420],{"className":418,"style":419},[90],"height:0.3361em;",[15,421,423,426],{"style":422},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[15,424],{"className":425,"style":99},[98],[15,427,429],{"className":428},[103,104,105,106],[15,430,432],{"className":431},[51,52,106],"h",[15,434,116],{"className":435},[115],[15,437,439],{"className":438},[86],[15,440,442],{"className":441,"style":393},[90],[15,443],{},[15,445],{"className":446,"style":146},[58],[15,448,450,453],{"className":449},[51],[15,451,406],{"className":452,"style":405},[51,52],[15,454,456],{"className":455},[77],[15,457,459,480],{"className":458},[81,82],[15,460,462,477],{"className":461},[86],[15,463,465],{"className":464,"style":91},[90],[15,466,467,470],{"style":422},[15,468],{"className":469,"style":99},[98],[15,471,473],{"className":472},[103,104,105,106],[15,474,476],{"className":475},[51,52,106],"m",[15,478,116],{"className":479},[115],[15,481,483],{"className":482},[86],[15,484,486],{"className":485,"style":393},[90],[15,487],{},[209,489,490,491,506],{},"Every node moves horizontally by ",[15,492,494],{"className":493},[22],[15,495,497],{"className":496,"ariaHidden":27},[26],[15,498,500,503],{"className":499},[31],[15,501],{"className":502,"style":168},[35],[15,504,336],{"className":505},[51,52]," × its height above the base",[192,508,509,512,602],{},[209,510,511],{},"§5.3.2(3)b",[209,513,514,515,601],{},"Member bow ",[15,516,518],{"className":517},[22],[15,519,521,578],{"className":520,"ariaHidden":27},[26],[15,522,524,528,569,572,575],{"className":523},[31],[15,525],{"className":526,"style":527},[35],"height:0.5806em;vertical-align:-0.15em;",[15,529,531,535],{"className":530},[51],[15,532,534],{"className":533},[51,52],"e",[15,536,538],{"className":537},[77],[15,539,541,561],{"className":540},[81,82],[15,542,544,558],{"className":543},[86],[15,545,547],{"className":546,"style":370},[90],[15,548,549,552],{"style":373},[15,550],{"className":551,"style":99},[98],[15,553,555],{"className":554},[103,104,105,106],[15,556,383],{"className":557},[51,106],[15,559,116],{"className":560},[115],[15,562,564],{"className":563},[86],[15,565,567],{"className":566,"style":393},[90],[15,568],{},[15,570],{"className":571,"style":153},[58],[15,573,158],{"className":574},[157],[15,576],{"className":577,"style":153},[58],[15,579,581,585,589,593],{"className":580},[31],[15,582],{"className":583,"style":584},[35],"height:1em;vertical-align:-0.25em;",[15,586,588],{"className":587},[51,52],"L",[15,590,592],{"className":591},[51],"\u002F",[15,594,597],{"className":595},[51,596],"text",[15,598,600],{"className":599},[51],"ratio",", Table 5.1",[209,603,604],{},"A half sine along each member, about its weak or its strong axis",[192,606,607,610,613],{},[209,608,609],{},"§5.3.2(11)",[209,611,612],{},"The elastic critical buckling shape, scaled",[209,614,615],{},"Replaces the other two — that is what \"a unique global and local imperfection\" means",[11,617,618],{},"Deforming the geometry rather than applying notional forces is deliberate: the P-Δ moment then falls out of the second-order solve\nitself, for every load state, with no per-storey force bookkeeping and no assumption that\nthe structure is a stack of floors.",[252,620,622],{"type":621},"note",[11,623,624,627],{},[212,625,626],{},"Displacements come back measured from the imperfect shape."," Internal forces, which are\nwhat a design check reads, are exact.",[11,629,630,631,636],{},"The mode-shaped imperfection needs a ",[632,633,635],"a",{"href":634},"\u002Fkb\u002Fanalysis\u002Fbuckling","linear buckling"," result to\nscale, so run one first.",[181,638,640],{"id":639},"when-the-iteration-stops","When the iteration stops",[186,642,643,653],{},[189,644,645],{},[192,646,647,650],{},[195,648,649],{},"Outcome",[195,651,652],{},"Meaning",[204,654,655,663,675],{},[192,656,657,660],{},[209,658,659],{},"Converged",[209,661,662],{},"The displacement increment fell below the tolerance. This is the answer.",[192,664,665,668],{},[209,666,667],{},"Instability detected",[209,669,670,671,272],{},"The tangent stiffness stopped being positive definite — the structure has passed its critical load ",[672,673,674],"em",{},"under this very load state",[192,676,677,680],{},[209,678,679],{},"Iteration limit reached",[209,681,682],{},"Neither of the above happened within the iteration budget. Not an answer; treat it as a failure to converge.",[11,684,685],{},"Instability detection watches the factorisation for negative pivots and is on by default.\nWithout it the iteration either wanders towards a large deflected shape that looks like a\nresult, or fails to converge for a reason nobody can read off the numbers.",[181,687,689],{"id":688},"choosing-between-α_cr-and-a-second-order-run","Choosing between α_cr and a second-order run",[11,691,692,695,696,753],{},[632,693,694],{"href":634},"Linear buckling"," gives you ",[15,697,699],{"className":698},[22],[15,700,702],{"className":701,"ariaHidden":27},[26],[15,703,705,708],{"className":704},[31],[15,706],{"className":707,"style":527},[35],[15,709,711,714],{"className":710},[51],[15,712,406],{"className":713,"style":405},[51,52],[15,715,717],{"className":716},[77],[15,718,720,745],{"className":719},[81,82],[15,721,723,742],{"className":722},[86],[15,724,726],{"className":725,"style":91},[90],[15,727,728,731],{"style":422},[15,729],{"className":730,"style":99},[98],[15,732,734],{"className":733},[103,104,105,106],[15,735,737],{"className":736},[51,106],[15,738,741],{"className":739,"style":740},[51,52,106],"margin-right:0.0278em;","cr",[15,743,116],{"className":744},[115],[15,746,748],{"className":747},[86],[15,749,751],{"className":750,"style":393},[90],[15,752],{}," cheaply, and\nEN 1993-1-1 lets you neglect second-order effects above the threshold. Below it the code\npermits two options: the amplification rules applied by hand, or this analysis. Where the\nframe is close to the threshold, or the amplifier is large, running it is faster than\njustifying the shortcut.",[181,755,757],{"id":756},"what-is-still-linear-here","What is still linear here",[11,759,760],{},"The geometry is not. Everything else is:",[263,762,763,766,769],{},[266,764,765],{},"Material stays elastic — no yielding, no cracking.",[266,767,768],{},"Supports do not gap, lift off or change stiffness with load.",[266,770,771],{},"Sections stay plane and prismatic between nodes.",[181,773,775],{"id":774},"related","Related",[11,777,778,780,781,836,837,841],{},[632,779,694],{"href":634}," supplies ",[15,782,784],{"className":783},[22],[15,785,787],{"className":786,"ariaHidden":27},[26],[15,788,790,793],{"className":789},[31],[15,791],{"className":792,"style":527},[35],[15,794,796,799],{"className":795},[51],[15,797,406],{"className":798,"style":405},[51,52],[15,800,802],{"className":801},[77],[15,803,805,828],{"className":804},[81,82],[15,806,808,825],{"className":807},[86],[15,809,811],{"className":810,"style":91},[90],[15,812,813,816],{"style":422},[15,814],{"className":815,"style":99},[98],[15,817,819],{"className":818},[103,104,105,106],[15,820,822],{"className":821},[51,106],[15,823,741],{"className":824,"style":740},[51,52,106],[15,826,116],{"className":827},[115],[15,829,831],{"className":830},[86],[15,832,834],{"className":833,"style":393},[90],[15,835],{}," and the mode shape this\nanalysis can use as an imperfection; ",[632,838,840],{"href":839},"\u002Fkb\u002Fanalysis\u002Flinear-static","Linear static"," explains\nthe superposition this analysis gives up.",{"title":843,"searchDepth":844,"depth":844,"links":845},"",2,[846,847,848,849,850,851,852],{"id":183,"depth":844,"text":184},{"id":246,"depth":844,"text":247},{"id":287,"depth":844,"text":288},{"id":639,"depth":844,"text":640},{"id":688,"depth":844,"text":689},{"id":756,"depth":844,"text":757},{"id":774,"depth":844,"text":775},"P-Δ and P-δ solved by iteration, geometric imperfections to EN 1993-1-1 §5.3, and why a second-order result cannot be superposed.","md",false,{},true,null,4,"\u002Fkb\u002Fanalysis\u002Fsecond-order",{"title":6,"description":853},{"loc":860},"kb\u002Fanalysis\u002Fsecond-order","evElRYE-fxsYbokc2HU8IEbA3DAoh7_sDrZv9jtNLXU",{"sections":866,"extras":1174,"flat":1187},[867,902,945,964,999,1024,1051,1082,1109,1136,1151],{"path":868,"slug":869,"title":870,"description":871,"icon":872,"articles":873},"\u002Fkb\u002Fgetting-started","getting-started","Getting started","From an empty tab to a solved, checked structure — the three guides that get you productive in Elementarium.","lucide:rocket",[874,878,882,886,890,894,898],{"path":875,"title":876,"description":877,"translated":857},"\u002Fkb\u002Fgetting-started\u002Fkeyboard-shortcuts","Keyboard shortcuts","Every key binding in the workspace, the pointer gestures in the viewports, what the number fields accept, and what the command palette understands.",{"path":879,"title":880,"description":881,"translated":857},"\u002Fkb\u002Fgetting-started\u002Fquickstart","Quickstart","Build a portal frame, load it, solve it and read the bending moment diagram — about ten minutes from an empty tab.",{"path":883,"title":884,"description":885,"translated":857},"\u002Fkb\u002Fgetting-started\u002Fscope-and-limits","Scope and limits","What Elementarium models, solves and checks today, and what it does not — on one page, before you commit a job to it.",{"path":887,"title":888,"description":889,"translated":857},"\u002Fkb\u002Fgetting-started\u002Fsystem-requirements","System requirements","Which browsers and devices run Elementarium, how threading affects solve speed, and where the practical ceiling on model size comes from.",{"path":891,"title":892,"description":893,"translated":857},"\u002Fkb\u002Fgetting-started\u002Finterface-tour","Tour of the workspace","The ribbon, properties panel, dockable viewports, table dock, command palette and status bar — what each part of the editor is for.",{"path":895,"title":896,"description":897,"translated":857},"\u002Fkb\u002Fgetting-started\u002Funits","Units","How units work in Elementarium — SI inside, your unit system on screen — and the handful of conversions that cause most wrong answers in structural models.",{"path":899,"title":900,"description":901,"translated":857},"\u002Fkb\u002Fgetting-started\u002Ffirst-analysis","Your first analysis","Load cases, combinations and a linear static solve — then how to decide whether the answer is trustworthy.",{"path":903,"slug":904,"title":905,"description":906,"icon":907,"articles":908},"\u002Fkb\u002Fmodeling","modeling","Modeling","Bars, beams, 3D frames, slabs and walls. Geometry, sections, supports and releases that behave like the real structure.","lucide:frame",[909,913,917,921,925,929,933,937,941],{"path":910,"title":911,"description":912,"translated":857},"\u002Fkb\u002Fmodeling\u002Fbars-beams-and-trusses","Bars, beams and trusses","When a member should be a bar and when a beam, how to model trusses and bracing, and what to do about members that only work in tension.",{"path":914,"title":915,"description":916,"translated":857},"\u002Fkb\u002Fmodeling\u002Fdrawings-and-snapping","Drawings & snapping","Importing a DXF as a reference underlay, choosing its units and plane, and snapping model geometry onto it accurately.",{"path":918,"title":919,"description":920,"translated":857},"\u002Fkb\u002Fmodeling\u002Felastic-foundations","Elastic foundations","Grade beams, strip footings and walls on ground as line supports on springs — choosing a subgrade modulus, and where the model stops being true.",{"path":922,"title":923,"description":924,"translated":857},"\u002Fkb\u002Fmodeling\u002Fplane-frames-in-3d","Plane frames in a 3D solver","When to use the 2D Frame and 2D Truss model types, why a flat structure in a 3D model is a mechanism, and how to restrain it.",{"path":926,"title":927,"description":928,"translated":857},"\u002Fkb\u002Fmodeling\u002Fsections-and-materials","Sections & materials","What a section and a material contribute to the stiffness matrix and the design checks, and the values worth checking twice.",{"path":930,"title":931,"description":932,"translated":857},"\u002Fkb\u002Fmodeling\u002Fsurfaces","Slabs, walls and shells","Modelling surfaces in Elementarium — outlines, openings and thickness, how a beam or a wall attaches to a slab, local axes, and what a surface result means.",{"path":934,"title":935,"description":936,"translated":857},"\u002Fkb\u002Fmodeling\u002Fstructure-generators","Structure generators","Thirty-five parametric structures — trusses, frames, arches, towers, domes and shells — generated from a few dimensions, with supports and loads.",{"path":938,"title":939,"description":940,"translated":857},"\u002Fkb\u002Fmodeling\u002Fsupports-and-releases","Supports & releases","Pins, fixities, springs and end releases — what each one does to the stiffness matrix, and when each is right.",{"path":942,"title":943,"description":944,"translated":857},"\u002Fkb\u002Fmodeling\u002Ffirst-model","Your first model","The modeling loop end to end — geometry, sections, supports and releases — on a structure small enough to check by hand.",{"path":946,"slug":947,"title":948,"description":949,"icon":950,"articles":951},"\u002Fkb\u002Fmeshing","meshing","Meshing","Mesh density, local refinement and the quality checks that tell you when a mesh is fine enough to trust.","lucide:grid-3x3",[952,956,960],{"path":953,"title":954,"description":955,"translated":857},"\u002Fkb\u002Fmeshing\u002Felement-size","Choosing element size","Starting mesh 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