[{"data":1,"prerenderedAt":8473},["ShallowReactive",2],{"kb-en-\u002Fkb\u002Fanalysis\u002Fverification":3,"kb-nav-en":8093},{"page":4,"fallback":8083},{"id":5,"title":6,"body":7,"description":8081,"extension":8082,"faq":8083,"meta":8084,"navigation":8085,"ogImage":8086,"order":8087,"path":8088,"raw":8083,"robots":8086,"schemaOrg":8086,"seo":8089,"sitemap":8090,"stem":8091,"__hash__":8092},"content_en\u002Fkb\u002Fanalysis\u002Fverification.md","Verifying the solver",{"type":8,"value":9,"toc":8064},"minimark",[10,14,23,28,225,228,621,625,632,1515,1519,1727,2407,2432,2437,2606,2610,2773,3526,3630,3638,3642,3645,3869,4005,4150,4153,4160,4321,4657,4739,4743,4750,4754,4757,5104,5175,5439,5598,6358,6365,6369,6424,6599,6636,6998,7132,7135,7184,7191,7194,7238,7242,7532,7879,7882,7910,7916,7920,7992,7996,8009,8013,8016,8056],[11,12,13],"p",{},"Do not take our word for it. Every element type in Elementarium is exercised against\nclosed-form solutions before anything merges, but that is our floor, not your evidence.\nThe only solver test that means anything to you is one you ran yourself, on a case whose\nanswer you already know.",[11,15,16,17,22],{},"Each case below is small enough to build in a few minutes, with the formula to compare\nagainst and the size of disagreement to expect from discretisation. Every beam, frame,\ncolumn and frequency case fits the free plan's 500 degrees of freedom; of the plate\nmeshes, 4 × 4 fits, 8 × 8 sits just under the cap once the edges are supported, and\n16 × 16 needs a student or Pro account. The\npage doubles as a guide to what each element assumes; see\n",[18,19,21],"a",{"href":20},"\u002Fkb\u002Ftheory\u002Felement-formulations","Element formulations"," for the formal version.",[24,25,27],"h2",{"id":26},"how-to-run-a-benchmark","How to run a benchmark",[29,30,31],"steps",{},[32,33,34,207,213,219],"ol",{},[35,36,37,41,42,111,112,158,159,206],"li",{},[38,39,40],"strong",{},"Build the case with round numbers:"," ",[43,44,47],"span",{"className":45},[46],"katex",[43,48,52,81],{"className":49,"ariaHidden":51},[50],"katex-html","true",[43,53,56,61,68,73,78],{"className":54},[55],"base",[43,57],{"className":58,"style":60},[59],"strut","height:0.6833em;",[43,62,67],{"className":63,"style":66},[64,65],"mord","mathnormal","margin-right:0.0576em;","E",[43,69],{"className":70,"style":72},[71],"mspace","margin-right:0.2778em;",[43,74,77],{"className":75},[76],"mrel","=",[43,79],{"className":80,"style":72},[71],[43,82,84,87,91,95,99,103],{"className":83},[55],[43,85],{"className":86,"style":60},[59],[43,88,90],{"className":89},[64],"210",[43,92],{"className":93,"style":94},[71],"margin-right:0.1667em;",[43,96,98],{"className":97},[64],"000",[43,100,102],{"className":101},[71]," ",[43,104,107],{"className":105},[64,106],"text",[43,108,110],{"className":109},[64],"MPa",", ",[43,113,115],{"className":114},[46],[43,116,118,137],{"className":117,"ariaHidden":51},[50],[43,119,121,124,128,131,134],{"className":120},[55],[43,122],{"className":123,"style":60},[59],[43,125,127],{"className":126},[64,65],"L",[43,129],{"className":130,"style":72},[71],[43,132,77],{"className":133},[76],[43,135],{"className":136,"style":72},[71],[43,138,140,144,148,151],{"className":139},[55],[43,141],{"className":142,"style":143},[59],"height:0.6444em;",[43,145,147],{"className":146},[64],"10",[43,149,102],{"className":150},[71],[43,152,154],{"className":153},[64,106],[43,155,157],{"className":156},[64],"m",",\n",[43,160,162],{"className":161},[46],[43,163,165,186],{"className":164,"ariaHidden":51},[50],[43,166,168,172,177,180,183],{"className":167},[55],[43,169],{"className":170,"style":171},[59],"height:0.4306em;",[43,173,176],{"className":174,"style":175},[64,65],"margin-right:0.0269em;","w",[43,178],{"className":179,"style":72},[71],[43,181,77],{"className":182},[76],[43,184],{"className":185,"style":72},[71],[43,187,189,193,196,199],{"className":188},[55],[43,190],{"className":191,"style":192},[59],"height:1em;vertical-align:-0.25em;",[43,194,147],{"className":195},[64],[43,197,102],{"className":198},[71],[43,200,202],{"className":201},[64,106],[43,203,205],{"className":204},[64],"kN\u002Fm",". Round numbers make a wrong factor of a thousand obvious.",[35,208,209,212],{},[38,210,211],{},"Solve, and read the quantity from the table",", not the diagram, so that you get all\nthe figures.",[35,214,215,218],{},[38,216,217],{},"Compute the closed form by hand and compare."," Record the percentage difference.",[35,220,221,224],{},[38,222,223],{},"If the difference is larger than this page says to expect, refine and repeat"," before\nconcluding anything. The tables below say which cases are exact and which converge.",[11,226,227],{},"Keep the project. A benchmark set you can re-run after every release is worth more than\nany release note.",[11,229,230,231,111,305,111,354,111,411,111,429,111,451,482,483,514,515,620],{},"For the record, the solver's own test suite runs the beam cases below on every change —\n",[43,232,234],{"className":233},[46],[43,235,237],{"className":236,"ariaHidden":51},[50],[43,238,240,244,248,251,293,297,300],{"className":239},[55],[43,241],{"className":242,"style":243},[59],"height:1.0641em;vertical-align:-0.25em;",[43,245,247],{"className":246},[64],"5",[43,249,176],{"className":250,"style":175},[64,65],[43,252,254,257],{"className":253},[64],[43,255,127],{"className":256},[64,65],[43,258,261],{"className":259},[260],"msupsub",[43,262,265],{"className":263},[264],"vlist-t",[43,266,269],{"className":267},[268],"vlist-r",[43,270,274],{"className":271,"style":273},[272],"vlist","height:0.8141em;",[43,275,277,282],{"style":276},"top:-3.063em;margin-right:0.05em;",[43,278],{"className":279,"style":281},[280],"pstrut","height:2.7em;",[43,283,289],{"className":284},[285,286,287,288],"sizing","reset-size6","size3","mtight",[43,290,292],{"className":291},[64,288],"4",[43,294,296],{"className":295},[64],"\u002F384",[43,298,67],{"className":299,"style":66},[64,65],[43,301,304],{"className":302,"style":303},[64,65],"margin-right:0.0785em;","I",[43,306,308],{"className":307},[46],[43,309,311],{"className":310,"ariaHidden":51},[50],[43,312,314,317,320,350],{"className":313},[55],[43,315],{"className":316,"style":243},[59],[43,318,176],{"className":319,"style":175},[64,65],[43,321,323,326],{"className":322},[64],[43,324,127],{"className":325},[64,65],[43,327,329],{"className":328},[260],[43,330,332],{"className":331},[264],[43,333,335],{"className":334},[268],[43,336,338],{"className":337,"style":273},[272],[43,339,340,343],{"style":276},[43,341],{"className":342,"style":281},[280],[43,344,346],{"className":345},[285,286,287,288],[43,347,349],{"className":348},[64,288],"2",[43,351,353],{"className":352},[64],"\u002F8",[43,355,357],{"className":356},[46],[43,358,360],{"className":359,"ariaHidden":51},[50],[43,361,363,366,371,401,405,408],{"className":362},[55],[43,364],{"className":365,"style":243},[59],[43,367,370],{"className":368,"style":369},[64,65],"margin-right:0.1389em;","P",[43,372,374,377],{"className":373},[64],[43,375,127],{"className":376},[64,65],[43,378,380],{"className":379},[260],[43,381,383],{"className":382},[264],[43,384,386],{"className":385},[268],[43,387,389],{"className":388,"style":273},[272],[43,390,391,394],{"style":276},[43,392],{"className":393,"style":281},[280],[43,395,397],{"className":396},[285,286,287,288],[43,398,400],{"className":399},[64,288],"3",[43,402,404],{"className":403},[64],"\u002F3",[43,406,67],{"className":407,"style":66},[64,65],[43,409,304],{"className":410,"style":303},[64,65],[43,412,414],{"className":413},[46],[43,415,417],{"className":416,"ariaHidden":51},[50],[43,418,420,423,426],{"className":419},[55],[43,421],{"className":422,"style":60},[59],[43,424,370],{"className":425,"style":369},[64,65],[43,427,127],{"className":428},[64,65],[43,430,432],{"className":431},[46],[43,433,435],{"className":434,"ariaHidden":51},[50],[43,436,438,441,444,447],{"className":437},[55],[43,439],{"className":440,"style":192},[59],[43,442,370],{"className":443,"style":369},[64,65],[43,445,127],{"className":446},[64,65],[43,448,450],{"className":449},[64],"\u002F4",[43,452,454],{"className":453},[46],[43,455,457],{"className":456,"ariaHidden":51},[50],[43,458,460,463,468,471,475,478],{"className":459},[55],[43,461],{"className":462,"style":192},[59],[43,464,467],{"className":465,"style":466},[64,65],"margin-right:0.109em;","N",[43,469,127],{"className":470},[64,65],[43,472,474],{"className":473},[64],"\u002F",[43,476,67],{"className":477,"style":66},[64,65],[43,479,481],{"className":480},[64,65],"A"," and ",[43,484,486],{"className":485},[46],[43,487,489],{"className":488,"ariaHidden":51},[50],[43,490,492,495,499,502,505,509],{"className":491},[55],[43,493],{"className":494,"style":192},[59],[43,496,498],{"className":497,"style":369},[64,65],"T",[43,500,127],{"className":501},[64,65],[43,503,474],{"className":504},[64],[43,506,508],{"className":507},[64,65],"G",[43,510,513],{"className":511,"style":512},[64,65],"margin-right:0.0962em;","J",", each to nine\ndecimal places — and drives the WebAssembly stability solver to Euler's load and the\nsecond-order amplifier ",[43,516,518],{"className":517},[46],[43,519,521,552],{"className":520,"ariaHidden":51},[50],[43,522,524,527,531,536,540,544,549],{"className":523},[55],[43,525],{"className":526,"style":192},[59],[43,528,530],{"className":529},[64],"1\u002F",[43,532,535],{"className":533},[534],"mopen","(",[43,537,539],{"className":538},[64],"1",[43,541],{"className":542,"style":543},[71],"margin-right:0.2222em;",[43,545,548],{"className":546},[547],"mbin","−",[43,550],{"className":551,"style":543},[71],[43,553,555,558,561,564,615],{"className":554},[55],[43,556],{"className":557,"style":192},[59],[43,559,467],{"className":560,"style":466},[64,65],[43,562,474],{"className":563},[64],[43,565,567,570],{"className":566},[64],[43,568,467],{"className":569,"style":466},[64,65],[43,571,573],{"className":572},[260],[43,574,577,606],{"className":575},[264,576],"vlist-t2",[43,578,580,601],{"className":579},[268],[43,581,584],{"className":582,"style":583},[272],"height:0.1514em;",[43,585,587,590],{"style":586},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[43,588],{"className":589,"style":281},[280],[43,591,593],{"className":592},[285,286,287,288],[43,594,596],{"className":595},[64,288],[43,597,600],{"className":598,"style":599},[64,65,288],"margin-right:0.0278em;","cr",[43,602,605],{"className":603},[604],"vlist-s","​",[43,607,609],{"className":608},[268],[43,610,613],{"className":611,"style":612},[272],"height:0.15em;",[43,614],{},[43,616,619],{"className":617},[618],"mclose",")",". That is the floor mentioned above; the cases\non this page are how you raise it on your own machine.",[24,622,624],{"id":623},"beams-exact-in-linear-static","Beams: exact in linear static",[11,626,627,628,631],{},"The beam element uses cubic interpolation, which is the exact solution for a prismatic\nmember under end loads and uniform load. These cases should agree to rounding with\n",[38,629,630],{},"one element per member"," and no refinement:",[633,634,635,651],"table",{},[636,637,638],"thead",{},[639,640,641,645,648],"tr",{},[642,643,644],"th",{},"Case",[642,646,647],{},"Quantity",[642,649,650],{},"Closed form",[652,653,654,727,792,823,866,928,999,1060,1133,1189,1250,1322,1355,1407,1453],"tbody",{},[639,655,656,675,678],{},[657,658,659,660],"td",{},"Simply supported, UDL ",[43,661,663],{"className":662},[46],[43,664,666],{"className":665,"ariaHidden":51},[50],[43,667,669,672],{"className":668},[55],[43,670],{"className":671,"style":171},[59],[43,673,176],{"className":674,"style":175},[64,65],[657,676,677],{},"Mid-span moment",[657,679,680],{},[43,681,683],{"className":682},[46],[43,684,686],{"className":685,"ariaHidden":51},[50],[43,687,689,692,695,724],{"className":688},[55],[43,690],{"className":691,"style":243},[59],[43,693,176],{"className":694,"style":175},[64,65],[43,696,698,701],{"className":697},[64],[43,699,127],{"className":700},[64,65],[43,702,704],{"className":703},[260],[43,705,707],{"className":706},[264],[43,708,710],{"className":709},[268],[43,711,713],{"className":712,"style":273},[272],[43,714,715,718],{"style":276},[43,716],{"className":717,"style":281},[280],[43,719,721],{"className":720},[285,286,287,288],[43,722,349],{"className":723},[64,288],[43,725,353],{"className":726},[64],[639,728,729,731,734],{},[657,730],{},[657,732,733],{},"Mid-span deflection",[657,735,736],{},[43,737,739],{"className":738},[46],[43,740,742],{"className":741,"ariaHidden":51},[50],[43,743,745,748,751,754,783,786,789],{"className":744},[55],[43,746],{"className":747,"style":243},[59],[43,749,247],{"className":750},[64],[43,752,176],{"className":753,"style":175},[64,65],[43,755,757,760],{"className":756},[64],[43,758,127],{"className":759},[64,65],[43,761,763],{"className":762},[260],[43,764,766],{"className":765},[264],[43,767,769],{"className":768},[268],[43,770,772],{"className":771,"style":273},[272],[43,773,774,777],{"style":276},[43,775],{"className":776,"style":281},[280],[43,778,780],{"className":779},[285,286,287,288],[43,781,292],{"className":782},[64,288],[43,784,296],{"className":785},[64],[43,787,67],{"className":788,"style":66},[64,65],[43,790,304],{"className":791,"style":303},[64,65],[639,793,794,796,799],{},[657,795],{},[657,797,798],{},"End shear",[657,800,801],{},[43,802,804],{"className":803},[46],[43,805,807],{"className":806,"ariaHidden":51},[50],[43,808,810,813,816,819],{"className":809},[55],[43,811],{"className":812,"style":192},[59],[43,814,176],{"className":815,"style":175},[64,65],[43,817,127],{"className":818},[64,65],[43,820,822],{"className":821},[64],"\u002F2",[639,824,825,843,846],{},[657,826,827,828],{},"Cantilever, tip load ",[43,829,831],{"className":830},[46],[43,832,834],{"className":833,"ariaHidden":51},[50],[43,835,837,840],{"className":836},[55],[43,838],{"className":839,"style":60},[59],[43,841,370],{"className":842,"style":369},[64,65],[657,844,845],{},"Root moment",[657,847,848],{},[43,849,851],{"className":850},[46],[43,852,854],{"className":853,"ariaHidden":51},[50],[43,855,857,860,863],{"className":856},[55],[43,858],{"className":859,"style":60},[59],[43,861,370],{"className":862,"style":369},[64,65],[43,864,127],{"className":865},[64,65],[639,867,868,870,873],{},[657,869],{},[657,871,872],{},"Tip deflection",[657,874,875],{},[43,876,878],{"className":877},[46],[43,879,881],{"className":880,"ariaHidden":51},[50],[43,882,884,887,890,919,922,925],{"className":883},[55],[43,885],{"className":886,"style":243},[59],[43,888,370],{"className":889,"style":369},[64,65],[43,891,893,896],{"className":892},[64],[43,894,127],{"className":895},[64,65],[43,897,899],{"className":898},[260],[43,900,902],{"className":901},[264],[43,903,905],{"className":904},[268],[43,906,908],{"className":907,"style":273},[272],[43,909,910,913],{"style":276},[43,911],{"className":912,"style":281},[280],[43,914,916],{"className":915},[285,286,287,288],[43,917,400],{"className":918},[64,288],[43,920,404],{"className":921},[64],[43,923,67],{"className":924,"style":66},[64,65],[43,926,304],{"className":927,"style":303},[64,65],[639,929,930,948,950],{},[657,931,932,933],{},"Cantilever, UDL ",[43,934,936],{"className":935},[46],[43,937,939],{"className":938,"ariaHidden":51},[50],[43,940,942,945],{"className":941},[55],[43,943],{"className":944,"style":171},[59],[43,946,176],{"className":947,"style":175},[64,65],[657,949,845],{},[657,951,952],{},[43,953,955],{"className":954},[46],[43,956,958],{"className":957,"ariaHidden":51},[50],[43,959,961,964,967,996],{"className":960},[55],[43,962],{"className":963,"style":243},[59],[43,965,176],{"className":966,"style":175},[64,65],[43,968,970,973],{"className":969},[64],[43,971,127],{"className":972},[64,65],[43,974,976],{"className":975},[260],[43,977,979],{"className":978},[264],[43,980,982],{"className":981},[268],[43,983,985],{"className":984,"style":273},[272],[43,986,987,990],{"style":276},[43,988],{"className":989,"style":281},[280],[43,991,993],{"className":992},[285,286,287,288],[43,994,349],{"className":995},[64,288],[43,997,822],{"className":998},[64],[639,1000,1001,1003,1005],{},[657,1002],{},[657,1004,872],{},[657,1006,1007],{},[43,1008,1010],{"className":1009},[46],[43,1011,1013],{"className":1012,"ariaHidden":51},[50],[43,1014,1016,1019,1022,1051,1054,1057],{"className":1015},[55],[43,1017],{"className":1018,"style":243},[59],[43,1020,176],{"className":1021,"style":175},[64,65],[43,1023,1025,1028],{"className":1024},[64],[43,1026,127],{"className":1027},[64,65],[43,1029,1031],{"className":1030},[260],[43,1032,1034],{"className":1033},[264],[43,1035,1037],{"className":1036},[268],[43,1038,1040],{"className":1039,"style":273},[272],[43,1041,1042,1045],{"style":276},[43,1043],{"className":1044,"style":281},[280],[43,1046,1048],{"className":1047},[285,286,287,288],[43,1049,292],{"className":1050},[64,288],[43,1052,353],{"className":1053},[64],[43,1055,67],{"className":1056,"style":66},[64,65],[43,1058,304],{"className":1059,"style":303},[64,65],[639,1061,1062,1080,1083],{},[657,1063,1064,1065],{},"Fixed 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3,2387,102],{"className":2388},[71],[43,2390,2392],{"className":2391},[64,106],[43,2393,2395],{"className":2394},[64],"mm",[43,2397,605],{"className":2398},[604],[43,2400,2402],{"className":2401},[268],[43,2403,2405],{"className":2404,"style":1802},[272],[43,2406],{},[11,2408,2409,2410,2431],{},"A point load at mid-span sits at a node if the member is split there, and the result is\nexact. Try it both ways, split and unsplit, and expect ",[43,2411,2413],{"className":2412},[46],[43,2414,2416],{"className":2415,"ariaHidden":51},[50],[43,2417,2419,2422,2425,2428],{"className":2418},[55],[43,2420],{"className":2421,"style":192},[59],[43,2423,370],{"className":2424,"style":369},[64,65],[43,2426,127],{"className":2427},[64,65],[43,2429,450],{"className":2430},[64]," to rounding from each. If\nthe unsplit member disagrees, that is worth reporting.",[2433,2434,2436],"h3",{"id":2435},"shear-deformation","Shear deformation",[11,2438,2439,2440,2548,2549,2605],{},"The beam includes shear deformation through its shear areas. On a slender beam the effect\nis small; on a stocky one it is not. Under a uniform load the shear term adds\n",[43,2441,2443],{"className":2442},[46],[43,2444,2446],{"className":2445,"ariaHidden":51},[50],[43,2447,2449,2452,2456,2459,2462,2465,2468,2471,2474,2516,2545],{"className":2448},[55],[43,2450],{"className":2451,"style":243},[59],[43,2453,2455],{"className":2454},[64],"9.6",[43,2457],{"className":2458,"style":94},[71],[43,2460,67],{"className":2461,"style":66},[64,65],[43,2463,304],{"className":2464,"style":303},[64,65],[43,2466,474],{"className":2467},[64],[43,2469,535],{"className":2470},[534],[43,2472,508],{"className":2473},[64,65],[43,2475,2477,2480],{"className":2476},[64],[43,2478,481],{"className":2479},[64,65],[43,2481,2483],{"className":2482},[260],[43,2484,2486,2508],{"className":2485},[264,576],[43,2487,2489,2505],{"className":2488},[268],[43,2490,2492],{"className":2491,"style":583},[272],[43,2493,2495,2498],{"style":2494},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[43,2496],{"className":2497,"style":281},[280],[43,2499,2501],{"className":2500},[285,286,287,288],[43,2502,2504],{"className":2503,"style":1650},[64,65,288],"v",[43,2506,605],{"className":2507},[604],[43,2509,2511],{"className":2510},[268],[43,2512,2514],{"className":2513,"style":612},[272],[43,2515],{},[43,2517,2519,2522],{"className":2518},[64],[43,2520,127],{"className":2521},[64,65],[43,2523,2525],{"className":2524},[260],[43,2526,2528],{"className":2527},[264],[43,2529,2531],{"className":2530},[268],[43,2532,2534],{"className":2533,"style":273},[272],[43,2535,2536,2539],{"style":276},[43,2537],{"className":2538,"style":281},[280],[43,2540,2542],{"className":2541},[285,286,287,288],[43,2543,349],{"className":2544},[64,288],[43,2546,619],{"className":2547},[618]," to the bending deflection ",[43,2550,2552],{"className":2551},[46],[43,2553,2555],{"className":2554,"ariaHidden":51},[50],[43,2556,2558,2561,2564,2567,2596,2599,2602],{"className":2557},[55],[43,2559],{"className":2560,"style":243},[59],[43,2562,247],{"className":2563},[64],[43,2565,176],{"className":2566,"style":175},[64,65],[43,2568,2570,2573],{"className":2569},[64],[43,2571,127],{"className":2572},[64,65],[43,2574,2576],{"className":2575},[260],[43,2577,2579],{"className":2578},[264],[43,2580,2582],{"className":2581},[268],[43,2583,2585],{"className":2584,"style":273},[272],[43,2586,2587,2590],{"style":276},[43,2588],{"className":2589,"style":281},[280],[43,2591,2593],{"className":2592},[285,286,287,288],[43,2594,292],{"className":2595},[64,288],[43,2597,296],{"className":2598},[64],[43,2600,67],{"className":2601,"style":66},[64,65],[43,2603,304],{"className":2604,"style":303},[64,65],": for an IPE 400 that is about\n1.4% at a 10 m span and about 34% at 2 m. That is the element being right. To compare with the pure bending formula, set the shear areas very large.",[24,2607,2609],{"id":2608},"frames-exact-and-a-check-on-connectivity","Frames: exact, and a check on connectivity",[11,2611,2612,2613,2630,2631,2646,2647,2701,2702,2756,2757,2772],{},"A pinned-base portal with a uniform rafter load is the standard frame test. For a\nrectangular portal of height ",[43,2614,2616],{"className":2615},[46],[43,2617,2619],{"className":2618,"ariaHidden":51},[50],[43,2620,2622,2626],{"className":2621},[55],[43,2623],{"className":2624,"style":2625},[59],"height:0.6944em;",[43,2627,2629],{"className":2628},[64,65],"h",", span ",[43,2632,2634],{"className":2633},[46],[43,2635,2637],{"className":2636,"ariaHidden":51},[50],[43,2638,2640,2643],{"className":2639},[55],[43,2641],{"className":2642,"style":60},[59],[43,2644,127],{"className":2645},[64,65],", column ",[43,2648,2650],{"className":2649},[46],[43,2651,2653],{"className":2652,"ariaHidden":51},[50],[43,2654,2656,2660],{"className":2655},[55],[43,2657],{"className":2658,"style":2659},[59],"height:0.8333em;vertical-align:-0.15em;",[43,2661,2663,2666],{"className":2662},[64],[43,2664,304],{"className":2665,"style":303},[64,65],[43,2667,2669],{"className":2668},[260],[43,2670,2672,2693],{"className":2671},[264,576],[43,2673,2675,2690],{"className":2674},[268],[43,2676,2678],{"className":2677,"style":583},[272],[43,2679,2680,2683],{"style":1640},[43,2681],{"className":2682,"style":281},[280],[43,2684,2686],{"className":2685},[285,286,287,288],[43,2687,2689],{"className":2688},[64,65,288],"c",[43,2691,605],{"className":2692},[604],[43,2694,2696],{"className":2695},[268],[43,2697,2699],{"className":2698,"style":612},[272],[43,2700],{}," and rafter ",[43,2703,2705],{"className":2704},[46],[43,2706,2708],{"className":2707,"ariaHidden":51},[50],[43,2709,2711,2714],{"className":2710},[55],[43,2712],{"className":2713,"style":2659},[59],[43,2715,2717,2720],{"className":2716},[64],[43,2718,304],{"className":2719,"style":303},[64,65],[43,2721,2723],{"className":2722},[260],[43,2724,2726,2748],{"className":2725},[264,576],[43,2727,2729,2745],{"className":2728},[268],[43,2730,2733],{"className":2731,"style":2732},[272],"height:0.3361em;",[43,2734,2735,2738],{"style":1640},[43,2736],{"className":2737,"style":281},[280],[43,2739,2741],{"className":2740},[285,286,287,288],[43,2742,2744],{"className":2743},[64,65,288],"b",[43,2746,605],{"className":2747},[604],[43,2749,2751],{"className":2750},[268],[43,2752,2754],{"className":2753,"style":612},[272],[43,2755],{}," under rafter\nUDL ",[43,2758,2760],{"className":2759},[46],[43,2761,2763],{"className":2762,"ariaHidden":51},[50],[43,2764,2766,2769],{"className":2765},[55],[43,2767],{"className":2768,"style":171},[59],[43,2770,176],{"className":2771,"style":175},[64,65],":",[43,2774,2776],{"className":2775},[1730],[43,2777,2779],{"className":2778},[46],[43,2780,2782],{"className":2781,"ariaHidden":51},[50],[43,2783,2785,2789],{"className":2784},[55],[43,2786],{"className":2787,"style":2788},[59],"height:9.0672em;vertical-align:-4.2836em;",[43,2790,2792],{"className":2791},[64],[43,2793,2795,2960,3447,3452,3485],{"className":2794},[1750],[43,2796,2798],{"className":2797},[1754],[43,2799,2801,2951],{"className":2800},[264,576],[43,2802,2804,2948],{"className":2803},[268],[43,2805,2808,2822,2836,2892],{"className":2806,"style":2807},[272],"height:4.7836em;",[43,2809,2811,2814],{"style":2810},"top:-6.8477em;",[43,2812],{"className":2813,"style":1771},[280],[43,2815,2817],{"className":2816},[64],[43,2818,2821],{"className":2819,"style":2820},[64,65],"margin-right:0.0315em;","k",[43,2823,2825,2828],{"style":2824},"top:-4.1206em;",[43,2826],{"className":2827,"style":1771},[280],[43,2829,2831],{"className":2830},[64],[43,2832,2835],{"className":2833,"style":2834},[64,65],"margin-right:0.0813em;","H",[43,2837,2839,2842],{"style":2838},"top:-2.0446em;",[43,2840],{"className":2841,"style":1771},[280],[43,2843,2845],{"className":2844},[64],[43,2846,2848,2851],{"className":2847},[64],[43,2849,1778],{"className":2850,"style":466},[64,65],[43,2852,2854],{"className":2853},[260],[43,2855,2857,2884],{"className":2856},[264,576],[43,2858,2860,2881],{"className":2859},[268],[43,2861,2863],{"className":2862,"style":583},[272],[43,2864,2865,2868],{"style":586},[43,2866],{"className":2867,"style":281},[280],[43,2869,2871],{"className":2870},[285,286,287,288],[43,2872,2874],{"className":2873},[64,288],[43,2875,2877],{"className":2876},[64,106,288],[43,2878,2880],{"className":2879},[64,288],"eaves",[43,2882,605],{"className":2883},[604],[43,2885,2887],{"className":2886},[268],[43,2888,2890],{"className":2889,"style":612},[272],[43,2891],{},[43,2893,2895,2898],{"style":2894},"top:0.1065em;",[43,2896],{"className":2897,"style":1771},[280],[43,2899,2901],{"className":2900},[64],[43,2902,2904,2907],{"className":2903},[64],[43,2905,1778],{"className":2906,"style":466},[64,65],[43,2908,2910],{"className":2909},[260],[43,2911,2913,2940],{"className":2912},[264,576],[43,2914,2916,2937],{"className":2915},[268],[43,2917,2919],{"className":2918,"style":2732},[272],[43,2920,2921,2924],{"style":586},[43,2922],{"className":2923,"style":281},[280],[43,2925,2927],{"className":2926},[285,286,287,288],[43,2928,2930],{"className":2929},[64,288],[43,2931,2933],{"className":2932},[64,106,288],[43,2934,2936],{"className":2935},[64,288],"mid",[43,2938,605],{"className":2939},[604],[43,2941,2943],{"className":2942},[268],[43,2944,2946],{"className":2945,"style":612},[272],[43,2947],{},[43,2949,605],{"className":2950},[604],[43,2952,2954],{"className":2953},[268],[43,2955,2958],{"className":2956,"style":2957},[272],"height:4.2836em;",[43,2959],{},[43,2961,2963],{"className":2962},[1808],[43,2964,2966,3439],{"className":2965},[264,576],[43,2967,2969,3436],{"className":2968},[268],[43,2970,2972,3142,3284,3310],{"className":2971,"style":2807},[272],[43,2973,2974,2977],{"style":2810},[43,2975],{"className":2976,"style":1771},[280],[43,2978,2980,2983,2986,2989,2992],{"className":2979},[64],[43,2981],{"className":2982},[64],[43,2984],{"className":2985,"style":72},[71],[43,2987,77],{"className":2988},[76],[43,2990],{"className":2991,"style":72},[71],[43,2993,2995,2998,3139],{"className":2994},[64],[43,2996],{"className":2997},[534,1844],[43,2999,3001],{"className":3000},[1848],[43,3002,3004,3130],{"className":3003},[264,576],[43,3005,3007,3127],{"className":3006},[268],[43,3008,3011,3065,3073],{"className":3009,"style":3010},[272],"height:1.427em;",[43,3012,3013,3016],{"style":1861},[43,3014],{"className":3015,"style":1865},[280],[43,3017,3019,3059,3062],{"className":3018},[64],[43,3020,3022,3025],{"className":3021},[64],[43,3023,304],{"className":3024,"style":303},[64,65],[43,3026,3028],{"className":3027},[260],[43,3029,3031,3051],{"className":3030},[264,576],[43,3032,3034,3048],{"className":3033},[268],[43,3035,3037],{"className":3036,"style":583},[272],[43,3038,3039,3042],{"style":1640},[43,3040],{"className":3041,"style":281},[280],[43,3043,3045],{"className":3044},[285,286,287,288],[43,3046,2689],{"className":3047},[64,65,288],[43,3049,605],{"className":3050},[604],[43,3052,3054],{"className":3053},[268],[43,3055,3057],{"className":3056,"style":612},[272],[43,3058],{},[43,3060,474],{"className":3061},[64],[43,3063,2629],{"className":3064},[64,65],[43,3066,3067,3070],{"style":1875},[43,3068],{"className":3069,"style":1865},[280],[43,3071],{"className":3072,"style":1883},[1882],[43,3074,3075,3078],{"style":1886},[43,3076],{"className":3077,"style":1865},[280],[43,3079,3081,3121,3124],{"className":3080},[64],[43,3082,3084,3087],{"className":3083},[64],[43,3085,304],{"className":3086,"style":303},[64,65],[43,3088,3090],{"className":3089},[260],[43,3091,3093,3113],{"className":3092},[264,576],[43,3094,3096,3110],{"className":3095},[268],[43,3097,3099],{"className":3098,"style":2732},[272],[43,3100,3101,3104],{"style":1640},[43,3102],{"className":3103,"style":281},[280],[43,3105,3107],{"className":3106},[285,286,287,288],[43,3108,2744],{"className":3109},[64,65,288],[43,3111,605],{"className":3112},[604],[43,3114,3116],{"className":3115},[268],[43,3117,3119],{"className":3118,"style":612},[272],[43,3120],{},[43,3122,474],{"className":3123},[64],[43,3125,127],{"className":3126},[64,65],[43,3128,605],{"className":3129},[604],[43,3131,3133],{"className":3132},[268],[43,3134,3137],{"className":3135,"style":3136},[272],"height:0.936em;",[43,3138],{},[43,3140],{"className":3141},[618,1844],[43,3143,3144,3147],{"style":2824},[43,3145],{"className":3146,"style":1771},[280],[43,3148,3150,3153,3156,3159,3162],{"className":3149},[64],[43,3151],{"className":3152},[64],[43,3154],{"className":3155,"style":72},[71],[43,3157,77],{"className":3158},[76],[43,3160],{"className":3161,"style":72},[71],[43,3163,3165,3168,3281],{"className":3164},[64],[43,3166],{"className":3167},[534,1844],[43,3169,3171],{"className":3170},[1848],[43,3172,3174,3273],{"className":3173},[264,576],[43,3175,3177,3270],{"className":3176},[268],[43,3178,3180,3222,3230],{"className":3179,"style":1858},[272],[43,3181,3182,3185],{"style":1861},[43,3183],{"className":3184,"style":1865},[280],[43,3186,3188,3191,3194,3197,3200,3203,3206,3209,3213,3216,3219],{"className":3187},[64],[43,3189,292],{"className":3190},[64],[43,3192,2629],{"className":3193},[64,65],[43,3195],{"className":3196,"style":94},[71],[43,3198,535],{"className":3199},[534],[43,3201,349],{"className":3202},[64],[43,3204,2821],{"className":3205,"style":2820},[64,65],[43,3207],{"className":3208,"style":543},[71],[43,3210,3212],{"className":3211},[547],"+",[43,3214],{"className":3215,"style":543},[71],[43,3217,400],{"className":3218},[64],[43,3220,619],{"className":3221},[618],[43,3223,3224,3227],{"style":1875},[43,3225],{"className":3226,"style":1865},[280],[43,3228],{"className":3229,"style":1883},[1882],[43,3231,3232,3235],{"style":1886},[43,3233],{"className":3234,"style":1865},[280],[43,3236,3238,3241],{"className":3237},[64],[43,3239,176],{"className":3240,"style":175},[64,65],[43,3242,3244,3247],{"className":3243},[64],[43,3245,127],{"className":3246},[64,65],[43,3248,3250],{"className":3249},[260],[43,3251,3253],{"className":3252},[264],[43,3254,3256],{"className":3255},[268],[43,3257,3259],{"className":3258,"style":273},[272],[43,3260,3261,3264],{"style":276},[43,3262],{"className":3263,"style":281},[280],[43,3265,3267],{"className":3266},[285,286,287,288],[43,3268,349],{"className":3269},[64,288],[43,3271,605],{"className":3272},[604],[43,3274,3276],{"className":3275},[268],[43,3277,3279],{"className":3278,"style":3136},[272],[43,3280],{},[43,3282],{"className":3283},[618,1844],[43,3285,3286,3289],{"style":2838},[43,3287],{"className":3288,"style":1771},[280],[43,3290,3292,3295,3298,3301,3304,3307],{"className":3291},[64],[43,3293],{"className":3294},[64],[43,3296],{"className":3297,"style":72},[71],[43,3299,77],{"className":3300},[76],[43,3302],{"className":3303,"style":72},[71],[43,3305,2835],{"className":3306,"style":2834},[64,65],[43,3308,2629],{"className":3309},[64,65],[43,3311,3312,3315],{"style":2894},[43,3313],{"className":3314,"style":1771},[280],[43,3316,3318,3321,3324,3327,3330,3421,3424,3427,3430,3433],{"className":3317},[64],[43,3319],{"className":3320},[64],[43,3322],{"className":3323,"style":72},[71],[43,3325,77],{"className":3326},[76],[43,3328],{"className":3329,"style":72},[71],[43,3331,3333,3336,3418],{"className":3332},[64],[43,3334],{"className":3335},[534,1844],[43,3337,3339],{"className":3338},[1848],[43,3340,3342,3410],{"className":3341},[264,576],[43,3343,3345,3407],{"className":3344},[268],[43,3346,3348,3359,3367],{"className":3347,"style":1858},[272],[43,3349,3350,3353],{"style":1861},[43,3351],{"className":3352,"style":1865},[280],[43,3354,3356],{"className":3355},[64],[43,3357,1872],{"className":3358},[64],[43,3360,3361,3364],{"style":1875},[43,3362],{"className":3363,"style":1865},[280],[43,3365],{"className":3366,"style":1883},[1882],[43,3368,3369,3372],{"style":1886},[43,3370],{"className":3371,"style":1865},[280],[43,3373,3375,3378],{"className":3374},[64],[43,3376,176],{"className":3377,"style":175},[64,65],[43,3379,3381,3384],{"className":3380},[64],[43,3382,127],{"className":3383},[64,65],[43,3385,3387],{"className":3386},[260],[43,3388,3390],{"className":3389},[264],[43,3391,3393],{"className":3392},[268],[43,3394,3396],{"className":3395,"style":273},[272],[43,3397,3398,3401],{"style":276},[43,3399],{"className":3400,"style":281},[280],[43,3402,3404],{"className":3403},[285,286,287,288],[43,3405,349],{"className":3406},[64,288],[43,3408,605],{"className":3409},[604],[43,3411,3413],{"className":3412},[268],[43,3414,3416],{"className":3415,"style":1948},[272],[43,3417],{},[43,3419],{"className":3420},[618,1844],[43,3422],{"className":3423,"style":543},[71],[43,3425,548],{"className":3426},[547],[43,3428],{"className":3429,"style":543},[71],[43,3431,2835],{"className":3432,"style":2834},[64,65],[43,3434,2629],{"className":3435},[64,65],[43,3437,605],{"className":3438},[604],[43,3440,3442],{"className":3441},[268],[43,3443,3445],{"className":3444,"style":2957},[272],[43,3446],{},[43,3448],{"className":3449,"style":3451},[3450],"arraycolsep","width:1em;",[43,3453,3455],{"className":3454},[1754],[43,3456,3458,3476],{"className":3457},[264,576],[43,3459,3461,3473],{"className":3460},[268],[43,3462,3465],{"className":3463,"style":3464},[272],"height:2.1206em;",[43,3466,3467,3470],{"style":2824},[43,3468],{"className":3469,"style":1771},[280],[43,3471],{"className":3472},[64],[43,3474,605],{"className":3475},[604],[43,3477,3479],{"className":3478},[268],[43,3480,3483],{"className":3481,"style":3482},[272],"height:0.6065em;",[43,3484],{},[43,3486,3488],{"className":3487},[1808],[43,3489,3491,3518],{"className":3490},[264,576],[43,3492,3494,3515],{"className":3493},[268],[43,3495,3497],{"className":3496,"style":3464},[272],[43,3498,3499,3502],{"style":2824},[43,3500],{"className":3501,"style":1771},[280],[43,3503,3505,3508],{"className":3504},[64],[43,3506],{"className":3507},[64],[43,3509,3511],{"className":3510},[64,106],[43,3512,3514],{"className":3513},[64],"horizontal base reaction",[43,3516,605],{"className":3517},[604],[43,3519,3521],{"className":3520},[268],[43,3522,3524],{"className":3523,"style":3482},[272],[43,3525],{},[11,3527,3528,3529,3576,3577,3624,3625,3629],{},"The mid-span moment must come out below ",[43,3530,3532],{"className":3531},[46],[43,3533,3535],{"className":3534,"ariaHidden":51},[50],[43,3536,3538,3541,3544,3573],{"className":3537},[55],[43,3539],{"className":3540,"style":243},[59],[43,3542,176],{"className":3543,"style":175},[64,65],[43,3545,3547,3550],{"className":3546},[64],[43,3548,127],{"className":3549},[64,65],[43,3551,3553],{"className":3552},[260],[43,3554,3556],{"className":3555},[264],[43,3557,3559],{"className":3558},[268],[43,3560,3562],{"className":3561,"style":273},[272],[43,3563,3564,3567],{"style":276},[43,3565],{"className":3566,"style":281},[280],[43,3568,3570],{"className":3569},[285,286,287,288],[43,3571,349],{"className":3572},[64,288],[43,3574,353],{"className":3575},[64]," and the eaves moment must be the\ndifference. If the rafter shows ",[43,3578,3580],{"className":3579},[46],[43,3581,3583],{"className":3582,"ariaHidden":51},[50],[43,3584,3586,3589,3592,3621],{"className":3585},[55],[43,3587],{"className":3588,"style":243},[59],[43,3590,176],{"className":3591,"style":175},[64,65],[43,3593,3595,3598],{"className":3594},[64],[43,3596,127],{"className":3597},[64,65],[43,3599,3601],{"className":3600},[260],[43,3602,3604],{"className":3603},[264],[43,3605,3607],{"className":3606},[268],[43,3608,3610],{"className":3609,"style":273},[272],[43,3611,3612,3615],{"style":276},[43,3613],{"className":3614,"style":281},[280],[43,3616,3618],{"className":3617},[285,286,287,288],[43,3619,349],{"className":3620},[64,288],[43,3622,353],{"className":3623},[64]," exactly, the eaves are not connected — see\n",[18,3626,3628],{"href":3627},"\u002Fkb\u002Fmodeling\u002Ffirst-model","Your first model",".",[11,3631,3632,3633,3637],{},"The ",[18,3634,3636],{"href":3635},"\u002Fkb\u002Fgetting-started\u002Fquickstart","quickstart"," builds this frame.",[24,3639,3641],{"id":3640},"columns-buckling-converges-with-subdivision","Columns: buckling converges with subdivision",[11,3643,3644],{},"Linear buckling uses a geometric stiffness matrix derived from the same cubic shape\nfunctions, and for that problem the cubic is an approximation. The Euler load",[43,3646,3648],{"className":3647},[1730],[43,3649,3651],{"className":3650},[46],[43,3652,3654,3713],{"className":3653,"ariaHidden":51},[50],[43,3655,3657,3660,3704,3707,3710],{"className":3656},[55],[43,3658],{"className":3659,"style":2659},[59],[43,3661,3663,3666],{"className":3662},[64],[43,3664,370],{"className":3665,"style":369},[64,65],[43,3667,3669],{"className":3668},[260],[43,3670,3672,3696],{"className":3671},[264,576],[43,3673,3675,3693],{"className":3674},[268],[43,3676,3678],{"className":3677,"style":583},[272],[43,3679,3681,3684],{"style":3680},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[43,3682],{"className":3683,"style":281},[280],[43,3685,3687],{"className":3686},[285,286,287,288],[43,3688,3690],{"className":3689},[64,288],[43,3691,600],{"className":3692,"style":599},[64,65,288],[43,3694,605],{"className":3695},[604],[43,3697,3699],{"className":3698},[268],[43,3700,3702],{"className":3701,"style":612},[272],[43,3703],{},[43,3705],{"className":3706,"style":72},[71],[43,3708,77],{"className":3709},[76],[43,3711],{"className":3712,"style":72},[71],[43,3714,3716,3720],{"className":3715},[55],[43,3717],{"className":3718,"style":3719},[59],"height:2.4241em;vertical-align:-0.933em;",[43,3721,3723,3726,3866],{"className":3722},[64],[43,3724],{"className":3725},[534,1844],[43,3727,3729],{"className":3728},[1848],[43,3730,3732,3857],{"className":3731},[264,576],[43,3733,3735,3854],{"className":3734},[268],[43,3736,3738,3802,3810],{"className":3737,"style":1858},[272],[43,3739,3740,3743],{"style":1861},[43,3741],{"className":3742,"style":1865},[280],[43,3744,3746],{"className":3745},[64],[43,3747,3749,3752],{"className":3748},[64],[43,3750,127],{"className":3751},[64,65],[43,3753,3755],{"className":3754},[260],[43,3756,3758,3793],{"className":3757},[264,576],[43,3759,3761,3790],{"className":3760},[268],[43,3762,3764,3779],{"className":3763,"style":2069},[272],[43,3765,3767,3770],{"style":3766},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[43,3768],{"className":3769,"style":281},[280],[43,3771,3773],{"className":3772},[285,286,287,288],[43,3774,3776],{"className":3775},[64,288],[43,3777,600],{"className":3778,"style":599},[64,65,288],[43,3780,3781,3784],{"style":2072},[43,3782],{"className":3783,"style":281},[280],[43,3785,3787],{"className":3786},[285,286,287,288],[43,3788,349],{"className":3789},[64,288],[43,3791,605],{"className":3792},[604],[43,3794,3796],{"className":3795},[268],[43,3797,3800],{"className":3798,"style":3799},[272],"height:0.247em;",[43,3801],{},[43,3803,3804,3807],{"style":1875},[43,3805],{"className":3806,"style":1865},[280],[43,3808],{"className":3809,"style":1883},[1882],[43,3811,3812,3815],{"style":1886},[43,3813],{"className":3814,"style":1865},[280],[43,3816,3818,3848,3851],{"className":3817},[64],[43,3819,3821,3825],{"className":3820},[64],[43,3822,3824],{"className":3823,"style":1650},[64,65],"π",[43,3826,3828],{"className":3827},[260],[43,3829,3831],{"className":3830},[264],[43,3832,3834],{"className":3833},[268],[43,3835,3837],{"className":3836,"style":273},[272],[43,3838,3839,3842],{"style":276},[43,3840],{"className":3841,"style":281},[280],[43,3843,3845],{"className":3844},[285,286,287,288],[43,3846,349],{"className":3847},[64,288],[43,3849,67],{"className":3850,"style":66},[64,65],[43,3852,304],{"className":3853,"style":303},[64,65],[43,3855,605],{"className":3856},[604],[43,3858,3860],{"className":3859},[268],[43,3861,3864],{"className":3862,"style":3863},[272],"height:0.933em;",[43,3865],{},[43,3867],{"className":3868},[618,1844],[11,3870,3871,3872,3945,3946,3964,3965,3984,3985,4004],{},"with ",[43,3873,3875],{"className":3874},[46],[43,3876,3878,3936],{"className":3877,"ariaHidden":51},[50],[43,3879,3881,3884,3927,3930,3933],{"className":3880},[55],[43,3882],{"className":3883,"style":2659},[59],[43,3885,3887,3890],{"className":3886},[64],[43,3888,127],{"className":3889},[64,65],[43,3891,3893],{"className":3892},[260],[43,3894,3896,3919],{"className":3895},[264,576],[43,3897,3899,3916],{"className":3898},[268],[43,3900,3902],{"className":3901,"style":583},[272],[43,3903,3904,3907],{"style":2494},[43,3905],{"className":3906,"style":281},[280],[43,3908,3910],{"className":3909},[285,286,287,288],[43,3911,3913],{"className":3912},[64,288],[43,3914,600],{"className":3915,"style":599},[64,65,288],[43,3917,605],{"className":3918},[604],[43,3920,3922],{"className":3921},[268],[43,3923,3925],{"className":3924,"style":612},[272],[43,3926],{},[43,3928],{"className":3929,"style":72},[71],[43,3931,77],{"className":3932},[76],[43,3934],{"className":3935,"style":72},[71],[43,3937,3939,3942],{"className":3938},[55],[43,3940],{"className":3941,"style":60},[59],[43,3943,127],{"className":3944},[64,65]," for a pin-ended column, ",[43,3947,3949],{"className":3948},[46],[43,3950,3952],{"className":3951,"ariaHidden":51},[50],[43,3953,3955,3958,3961],{"className":3954},[55],[43,3956],{"className":3957,"style":60},[59],[43,3959,349],{"className":3960},[64],[43,3962,127],{"className":3963},[64,65]," for a cantilever, ",[43,3966,3968],{"className":3967},[46],[43,3969,3971],{"className":3970,"ariaHidden":51},[50],[43,3972,3974,3977,3981],{"className":3973},[55],[43,3975],{"className":3976,"style":60},[59],[43,3978,3980],{"className":3979},[64],"0.5",[43,3982,127],{"className":3983},[64,65]," fixed–fixed and\n",[43,3986,3988],{"className":3987},[46],[43,3989,3991],{"className":3990,"ariaHidden":51},[50],[43,3992,3994,3997,4001],{"className":3993},[55],[43,3995],{"className":3996,"style":60},[59],[43,3998,4000],{"className":3999},[64],"0.7",[43,4002,127],{"className":4003},[64,65]," fixed–pinned, is reproduced to a precision set by the number of elements:",[633,4006,4007,4073],{},[636,4008,4009],{},[639,4010,4011,4014],{},[642,4012,4013],{},"Elements per member",[642,4015,4016,4017,4072],{},"Error in ",[43,4018,4020],{"className":4019},[46],[43,4021,4023],{"className":4022,"ariaHidden":51},[50],[43,4024,4026,4029],{"className":4025},[55],[43,4027],{"className":4028,"style":2659},[59],[43,4030,4032,4035],{"className":4031},[64],[43,4033,370],{"className":4034,"style":369},[64,65],[43,4036,4038],{"className":4037},[260],[43,4039,4041,4064],{"className":4040},[264,576],[43,4042,4044,4061],{"className":4043},[268],[43,4045,4047],{"className":4046,"style":583},[272],[43,4048,4049,4052],{"style":3680},[43,4050],{"className":4051,"style":281},[280],[43,4053,4055],{"className":4054},[285,286,287,288],[43,4056,4058],{"className":4057},[64,288],[43,4059,600],{"className":4060,"style":599},[64,65,288],[43,4062,605],{"className":4063},[604],[43,4065,4067],{"className":4066},[268],[43,4068,4070],{"className":4069,"style":612},[272],[43,4071],{},", pin-ended",[652,4074,4075,4136,4143],{},[639,4076,4077,4079],{},[657,4078,539],{},[657,4080,4081,4082],{},"about +22% — the single element gives ",[43,4083,4085],{"className":4084},[46],[43,4086,4088],{"className":4087,"ariaHidden":51},[50],[43,4089,4091,4094,4098,4101,4104,4107],{"className":4090},[55],[43,4092],{"className":4093,"style":243},[59],[43,4095,4097],{"className":4096},[64],"12",[43,4099,67],{"className":4100,"style":66},[64,65],[43,4102,304],{"className":4103,"style":303},[64,65],[43,4105,474],{"className":4106},[64],[43,4108,4110,4113],{"className":4109},[64],[43,4111,127],{"className":4112},[64,65],[43,4114,4116],{"className":4115},[260],[43,4117,4119],{"className":4118},[264],[43,4120,4122],{"className":4121},[268],[43,4123,4125],{"className":4124,"style":273},[272],[43,4126,4127,4130],{"style":276},[43,4128],{"className":4129,"style":281},[280],[43,4131,4133],{"className":4132},[285,286,287,288],[43,4134,349],{"className":4135},[64,288],[639,4137,4138,4140],{},[657,4139,349],{},[657,4141,4142],{},"under 1%",[639,4144,4145,4147],{},[657,4146,292],{},[657,4148,4149],{},"under 0.1%",[4151,4152],"diagram-buckling",{},[11,4154,3632,4155,4159],{},[18,4156,4158],{"href":4157},"\u002Fkb\u002Fanalysis\u002Fbuckling","stability"," setup carries its own minimum subdivision for this\nreason, so the default run is already on the right side of the table. Compare a run at the\nsetup's minimum with one at twice that; the difference should be inside the figures above.\nIf the first mode is not the one you expect (a cantilever's first mode has no inflection\npoint and a fixed–fixed column's has two), leave the Sturm-sequence completeness check on and\nit will tell you whether a mode was skipped.",[11,4161,4162,4163,4205,4206,4320],{},"Worked: pin-ended, ",[43,4164,4166],{"className":4165},[46],[43,4167,4169,4187],{"className":4168,"ariaHidden":51},[50],[43,4170,4172,4175,4178,4181,4184],{"className":4171},[55],[43,4173],{"className":4174,"style":60},[59],[43,4176,127],{"className":4177},[64,65],[43,4179],{"className":4180,"style":72},[71],[43,4182,77],{"className":4183},[76],[43,4185],{"className":4186,"style":72},[71],[43,4188,4190,4193,4196,4199],{"className":4189},[55],[43,4191],{"className":4192,"style":143},[59],[43,4194,2082],{"className":4195},[64],[43,4197,102],{"className":4198},[71],[43,4200,4202],{"className":4201},[64,106],[43,4203,157],{"className":4204},[64],", HEA 300 (",[43,4207,4209],{"className":4208},[46],[43,4210,4212,4269],{"className":4211,"ariaHidden":51},[50],[43,4213,4215,4218,4260,4263,4266],{"className":4214},[55],[43,4216],{"className":4217,"style":2659},[59],[43,4219,4221,4224],{"className":4220},[64],[43,4222,304],{"className":4223,"style":303},[64,65],[43,4225,4227],{"className":4226},[260],[43,4228,4230,4252],{"className":4229},[264,576],[43,4231,4233,4249],{"className":4232},[268],[43,4234,4236],{"className":4235,"style":583},[272],[43,4237,4238,4241],{"style":1640},[43,4239],{"className":4240,"style":281},[280],[43,4242,4244],{"className":4243},[285,286,287,288],[43,4245,4248],{"className":4246,"style":4247},[64,65,288],"margin-right:0.044em;","z",[43,4250,605],{"className":4251},[604],[43,4253,4255],{"className":4254},[268],[43,4256,4258],{"className":4257,"style":612},[272],[43,4259],{},[43,4261],{"className":4262,"style":72},[71],[43,4264,77],{"className":4265},[76],[43,4267],{"className":4268,"style":72},[71],[43,4270,4272,4275,4278,4281,4285,4288],{"className":4271},[55],[43,4273],{"className":4274,"style":273},[59],[43,4276,2082],{"className":4277},[64],[43,4279],{"className":4280,"style":94},[71],[43,4282,4284],{"className":4283},[64],"310",[43,4286,102],{"className":4287},[71],[43,4289,4291,4297],{"className":4290},[64],[43,4292,4294],{"className":4293},[64,106],[43,4295,1702],{"className":4296},[64],[43,4298,4300],{"className":4299},[260],[43,4301,4303],{"className":4302},[264],[43,4304,4306],{"className":4305},[268],[43,4307,4309],{"className":4308,"style":273},[272],[43,4310,4311,4314],{"style":276},[43,4312],{"className":4313,"style":281},[280],[43,4315,4317],{"className":4316},[285,286,287,288],[43,4318,292],{"className":4319},[64,288],", weak axis):",[43,4322,4324],{"className":4323},[1730],[43,4325,4327],{"className":4326},[46],[43,4328,4330,4388,4631],{"className":4329,"ariaHidden":51},[50],[43,4331,4333,4336,4379,4382,4385],{"className":4332},[55],[43,4334],{"className":4335,"style":2659},[59],[43,4337,4339,4342],{"className":4338},[64],[43,4340,370],{"className":4341,"style":369},[64,65],[43,4343,4345],{"className":4344},[260],[43,4346,4348,4371],{"className":4347},[264,576],[43,4349,4351,4368],{"className":4350},[268],[43,4352,4354],{"className":4353,"style":583},[272],[43,4355,4356,4359],{"style":3680},[43,4357],{"className":4358,"style":281},[280],[43,4360,4362],{"className":4361},[285,286,287,288],[43,4363,4365],{"className":4364},[64,288],[43,4366,600],{"className":4367,"style":599},[64,65,288],[43,4369,605],{"className":4370},[604],[43,4372,4374],{"className":4373},[268],[43,4375,4377],{"className":4376,"style":612},[272],[43,4378],{},[43,4380],{"className":4381,"style":72},[71],[43,4383,77],{"className":4384},[76],[43,4386],{"className":4387,"style":72},[71],[43,4389,4391,4395,4622,4625,4628],{"className":4390},[55],[43,4392],{"className":4393,"style":4394},[59],"height:2.1771em;vertical-align:-0.686em;",[43,4396,4398,4401,4619],{"className":4397},[64],[43,4399],{"className":4400},[534,1844],[43,4402,4404],{"className":4403},[1848],[43,4405,4407,4611],{"className":4406},[264,576],[43,4408,4410,4608],{"className":4409},[268],[43,4411,4413,4450,4458],{"className":4412,"style":1858},[272],[43,4414,4415,4418],{"style":1861},[43,4416],{"className":4417,"style":1865},[280],[43,4419,4421],{"className":4420},[64],[43,4422,4424,4427],{"className":4423},[64],[43,4425,2082],{"className":4426},[64],[43,4428,4430],{"className":4429},[260],[43,4431,4433],{"className":4432},[264],[43,4434,4436],{"className":4435},[268],[43,4437,4439],{"className":4438,"style":2069},[272],[43,4440,4441,4444],{"style":2072},[43,4442],{"className":4443,"style":281},[280],[43,4445,4447],{"className":4446},[285,286,287,288],[43,4448,349],{"className":4449},[64,288],[43,4451,4452,4455],{"style":1875},[43,4453],{"className":4454,"style":1865},[280],[43,4456],{"className":4457,"style":1883},[1882],[43,4459,4460,4463],{"style":1886},[43,4461],{"className":4462,"style":1865},[280],[43,4464,4466,4495,4498,4501,4504,4507,4510,4513,4516,4519,4548,4551,4554,4557,4561,4564,4567,4570,4573],{"className":4465},[64],[43,4467,4469,4472],{"className":4468},[64],[43,4470,3824],{"className":4471,"style":1650},[64,65],[43,4473,4475],{"className":4474},[260],[43,4476,4478],{"className":4477},[264],[43,4479,4481],{"className":4480},[268],[43,4482,4484],{"className":4483,"style":273},[272],[43,4485,4486,4489],{"style":276},[43,4487],{"className":4488,"style":281},[280],[43,4490,4492],{"className":4491},[285,286,287,288],[43,4493,349],{"className":4494},[64,288],[43,4496],{"className":4497,"style":543},[71],[43,4499,1902],{"className":4500},[547],[43,4502],{"className":4503,"style":543},[71],[43,4505,90],{"className":4506},[64],[43,4508],{"className":4509,"style":543},[71],[43,4511,1902],{"className":4512},[547],[43,4514],{"className":4515,"style":543},[71],[43,4517,539],{"className":4518},[64],[43,4520,4522,4525],{"className":4521},[64],[43,4523,1915],{"className":4524},[64],[43,4526,4528],{"className":4527},[260],[43,4529,4531],{"className":4530},[264],[43,4532,4534],{"className":4533},[268],[43,4535,4537],{"className":4536,"style":273},[272],[43,4538,4539,4542],{"style":276},[43,4540],{"className":4541,"style":281},[280],[43,4543,4545],{"className":4544},[285,286,287,288],[43,4546,2082],{"className":4547},[64,288],[43,4549],{"className":4550,"style":543},[71],[43,4552,1902],{"className":4553},[547],[43,4555],{"className":4556,"style":543},[71],[43,4558,4560],{"className":4559},[64],"6.31",[43,4562],{"className":4563,"style":543},[71],[43,4565,1902],{"className":4566},[547],[43,4568],{"className":4569,"style":543},[71],[43,4571,539],{"className":4572},[64],[43,4574,4576,4579],{"className":4575},[64],[43,4577,1915],{"className":4578},[64],[43,4580,4582],{"className":4581},[260],[43,4583,4585],{"className":4584},[264],[43,4586,4588],{"className":4587},[268],[43,4589,4591],{"className":4590,"style":273},[272],[43,4592,4593,4596],{"style":276},[43,4594],{"className":4595,"style":281},[280],[43,4597,4599],{"className":4598},[285,286,287,288],[43,4600,4602,4605],{"className":4601},[64,288],[43,4603,548],{"className":4604},[64,288],[43,4606,247],{"className":4607},[64,288],[43,4609,605],{"className":4610},[604],[43,4612,4614],{"className":4613},[268],[43,4615,4617],{"className":4616,"style":1948},[272],[43,4618],{},[43,4620],{"className":4621},[618,1844],[43,4623],{"className":4624,"style":72},[71],[43,4626,77],{"className":4627},[76],[43,4629],{"className":4630,"style":72},[71],[43,4632,4634,4637,4640,4643,4647,4650],{"className":4633},[55],[43,4635],{"className":4636,"style":2625},[59],[43,4638,400],{"className":4639},[64],[43,4641],{"className":4642,"style":94},[71],[43,4644,4646],{"className":4645},[64],"633",[43,4648,102],{"className":4649},[71],[43,4651,4653],{"className":4652},[64,106],[43,4654,4656],{"className":4655},[64],"kN",[11,4658,4659,4660,3629],{},"Apply a 1 000 kN axial load as the reference case and expect ",[43,4661,4663],{"className":4662},[46],[43,4664,4666,4729],{"className":4665,"ariaHidden":51},[50],[43,4667,4669,4673,4719,4722,4726],{"className":4668},[55],[43,4670],{"className":4671,"style":4672},[59],"height:0.6331em;vertical-align:-0.15em;",[43,4674,4676,4681],{"className":4675},[64],[43,4677,4680],{"className":4678,"style":4679},[64,65],"margin-right:0.0037em;","α",[43,4682,4684],{"className":4683},[260],[43,4685,4687,4711],{"className":4686},[264,576],[43,4688,4690,4708],{"className":4689},[268],[43,4691,4693],{"className":4692,"style":583},[272],[43,4694,4696,4699],{"style":4695},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[43,4697],{"className":4698,"style":281},[280],[43,4700,4702],{"className":4701},[285,286,287,288],[43,4703,4705],{"className":4704},[64,288],[43,4706,600],{"className":4707,"style":599},[64,65,288],[43,4709,605],{"className":4710},[604],[43,4712,4714],{"className":4713},[268],[43,4715,4717],{"className":4716,"style":612},[272],[43,4718],{},[43,4720],{"className":4721,"style":72},[71],[43,4723,4725],{"className":4724},[76],"≈",[43,4727],{"className":4728,"style":72},[71],[43,4730,4732,4735],{"className":4731},[55],[43,4733],{"className":4734,"style":143},[59],[43,4736,4738],{"className":4737},[64],"3.63",[24,4740,4742],{"id":4741},"frequencies-converge-from-above","Frequencies: converge from above",[11,4744,4745,4746,4749],{},"Modal analysis has the same character. With a consistent mass matrix the frequencies\nconverge to the exact values ",[38,4747,4748],{},"from above","; with a lumped matrix, from below.",[4751,4752],"diagram-convergence",{"kind":4753},"modal",[11,4755,4756],{},"The closed form for a uniform beam 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The classical case is a simply supported square plate, side ",[43,6373,6375],{"className":6374},[46],[43,6376,6378],{"className":6377,"ariaHidden":51},[50],[43,6379,6381,6384],{"className":6380},[55],[43,6382],{"className":6383,"style":171},[59],[43,6385,18],{"className":6386},[64,65],",\nthickness ",[43,6389,6391],{"className":6390},[46],[43,6392,6394],{"className":6393,"ariaHidden":51},[50],[43,6395,6397,6401],{"className":6396},[55],[43,6398],{"className":6399,"style":6400},[59],"height:0.6151em;",[43,6402,6404],{"className":6403},[64,65],"t",", under uniform pressure ",[43,6407,6409],{"className":6408},[46],[43,6410,6412],{"className":6411,"ariaHidden":51},[50],[43,6413,6415,6419],{"className":6414},[55],[43,6416],{"className":6417,"style":6418},[59],"height:0.625em;vertical-align:-0.1944em;",[43,6420,6422],{"className":6421,"style":1650},[64,65],"q",", 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",[43,6603,6605],{"className":6604},[46],[43,6606,6608,6626],{"className":6607,"ariaHidden":51},[50],[43,6609,6611,6614,6617,6620,6623],{"className":6610},[55],[43,6612],{"className":6613,"style":171},[59],[43,6615,6510],{"className":6616,"style":6509},[64,65],[43,6618],{"className":6619,"style":72},[71],[43,6621,77],{"className":6622},[76],[43,6624],{"className":6625,"style":72},[71],[43,6627,6629,6632],{"className":6628},[55],[43,6630],{"className":6631,"style":143},[59],[43,6633,6635],{"className":6634},[64],"0.3",[43,6637,6639],{"className":6638},[1730],[43,6640,6642],{"className":6641},[46],[43,6643,6645],{"className":6644,"ariaHidden":51},[50],[43,6646,6648,6652],{"className":6647},[55],[43,6649],{"className":6650,"style":6651},[59],"height:3.7012em;vertical-align:-1.6006em;",[43,6653,6655],{"className":6654},[64],[43,6656,6658,6796],{"className":6657},[1750],[43,6659,6661],{"className":6660},[1754],[43,6662,6664,6787],{"className":6663},[264,576],[43,6665,6667,6784],{"className":6666},[268],[43,6668,6671,6729],{"className":6669,"style":6670},[272],"height:2.1006em;",[43,6672,6674,6677],{"style":6673},"top:-4.1006em;",[43,6675],{"className":6676,"style":1771},[280],[43,6678,6680],{"className":6679},[64],[43,6681,6683,6686],{"className":6682},[64],[43,6684,176],{"className":6685,"style":175},[64,65],[43,6687,6689],{"className":6688},[260],[43,6690,6692,6721],{"className":6691},[264,576],[43,6693,6695,6718],{"className":6694},[268],[43,6696,6699],{"className":6697,"style":6698},[272],"height:0.2806em;",[43,6700,6702,6705],{"style":6701},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[43,6703],{"className":6704,"style":281},[280],[43,6706,6708],{"className":6707},[285,286,287,288],[43,6709,6711],{"className":6710},[64,288],[43,6712,6714],{"className":6713},[64,106,288],[43,6715,6717],{"className":6716},[64,288],"centre",[43,6719,605],{"className":6720},[604],[43,6722,6724],{"className":6723},[268],[43,6725,6727],{"className":6726,"style":612},[272],[43,6728],{},[43,6730,6732,6735],{"style":6731},"top:-2.2505em;",[43,6733],{"className":6734,"style":1771},[280],[43,6736,6738],{"className":6737},[64],[43,6739,6741,6744],{"className":6740},[64],[43,6742,157],{"className":6743},[64,65],[43,6745,6747],{"className":6746},[260],[43,6748,6750,6776],{"className":6749},[264,576],[43,6751,6753,6773],{"className":6752},[268],[43,6754,6756],{"className":6755,"style":6698},[272],[43,6757,6758,6761],{"style":2494},[43,6759],{"className":6760,"style":281},[280],[43,6762,6764],{"className":6763},[285,286,287,288],[43,6765,6767],{"className":6766},[64,288],[43,6768,6770],{"className":6769},[64,106,288],[43,6771,6717],{"className":6772},[64,288],[43,6774,605],{"className":6775},[604],[43,6777,6779],{"className":6778},[268],[43,6780,6782],{"className":6781,"style":612},[272],[43,6783],{},[43,6785,605],{"className":6786},[604],[43,6788,6790],{"className":6789},[268],[43,6791,6794],{"className":6792,"style":6793},[272],"height:1.6006em;",[43,6795],{},[43,6797,6799],{"className":6798},[1808],[43,6800,6802,6990],{"className":6801},[264,576],[43,6803,6805,6987],{"className":6804},[268],[43,6806,6808,6926],{"className":6807,"style":6670},[272],[43,6809,6810,6813],{"style":6673},[43,6811],{"className":6812,"style":1771},[280],[43,6814,6816,6819,6822,6825,6828,6832,6835],{"className":6815},[64],[43,6817],{"className":6818},[64],[43,6820],{"className":6821,"style":72},[71],[43,6823,77],{"className":6824},[76],[43,6826],{"className":6827,"style":72},[71],[43,6829,6831],{"className":6830},[64],"0.00406",[43,6833],{"className":6834,"style":94},[71],[43,6836,6838,6841,6923],{"className":6837},[64],[43,6839],{"className":6840},[534,1844],[43,6842,6844],{"className":6843},[1848],[43,6845,6847,6915],{"className":6846},[264,576],[43,6848,6850,6912],{"className":6849},[268],[43,6851,6853,6864,6872],{"className":6852,"style":1858},[272],[43,6854,6855,6858],{"style":1861},[43,6856],{"className":6857,"style":1865},[280],[43,6859,6861],{"className":6860},[64],[43,6862,6442],{"className":6863,"style":599},[64,65],[43,6865,6866,6869],{"style":1875},[43,6867],{"className":6868,"style":1865},[280],[43,6870],{"className":6871,"style":1883},[1882],[43,6873,6874,6877],{"style":1886},[43,6875],{"className":6876,"style":1865},[280],[43,6878,6880,6883],{"className":6879},[64],[43,6881,6422],{"className":6882,"style":1650},[64,65],[43,6884,6886,6889],{"className":6885},[64],[43,6887,18],{"className":6888},[64,65],[43,6890,6892],{"className":6891},[260],[43,6893,6895],{"className":6894},[264],[43,6896,6898],{"className":6897},[268],[43,6899,6901],{"className":6900,"style":273},[272],[43,6902,6903,6906],{"style":276},[43,6904],{"className":6905,"style":281},[280],[43,6907,6909],{"className":6908},[285,286,287,288],[43,6910,292],{"className":6911},[64,288],[43,6913,605],{"className":6914},[604],[43,6916,6918],{"className":6917},[268],[43,6919,6921],{"className":6920,"style":1948},[272],[43,6922],{},[43,6924],{"className":6925},[618,1844],[43,6927,6928,6931],{"style":6731},[43,6929],{"className":6930,"style":1771},[280],[43,6932,6934,6937,6940,6943,6946,6950,6953,6956],{"className":6933},[64],[43,6935],{"className":6936},[64],[43,6938],{"className":6939,"style":72},[71],[43,6941,77],{"className":6942},[76],[43,6944],{"className":6945,"style":72},[71],[43,6947,6949],{"className":6948},[64],"0.0479",[43,6951],{"className":6952,"style":94},[71],[43,6954,6422],{"className":6955,"style":1650},[64,65],[43,6957,6959,6962],{"className":6958},[64],[43,6960,18],{"className":6961},[64,65],[43,6963,6965],{"className":6964},[260],[43,6966,6968],{"className":6967},[264],[43,6969,6971],{"className":6970},[268],[43,6972,6975],{"className":6973,"style":6974},[272],"height:0.8641em;",[43,6976,6978,6981],{"style":6977},"top:-3.113em;margin-right:0.05em;",[43,6979],{"className":6980,"style":281},[280],[43,6982,6984],{"className":6983},[285,286,287,288],[43,6985,349],{"className":6986},[64,288],[43,6988,605],{"className":6989},[604],[43,6991,6993],{"className":6992},[268],[43,6994,6996],{"className":6995,"style":6793},[272],[43,6997],{},[11,6999,7000,7001,7076,7077,3629],{},"For a clamped square plate, ",[43,7002,7004],{"className":7003},[46],[43,7005,7007,7025],{"className":7006,"ariaHidden":51},[50],[43,7008,7010,7013,7016,7019,7022],{"className":7009},[55],[43,7011],{"className":7012,"style":171},[59],[43,7014,176],{"className":7015,"style":175},[64,65],[43,7017],{"className":7018,"style":72},[71],[43,7020,77],{"className":7021},[76],[43,7023],{"className":7024,"style":72},[71],[43,7026,7028,7031,7035,7038,7041,7070,7073],{"className":7027},[55],[43,7029],{"className":7030,"style":243},[59],[43,7032,7034],{"className":7033},[64],"0.00126",[43,7036],{"className":7037,"style":94},[71],[43,7039,6422],{"className":7040,"style":1650},[64,65],[43,7042,7044,7047],{"className":7043},[64],[43,7045,18],{"className":7046},[64,65],[43,7048,7050],{"className":7049},[260],[43,7051,7053],{"className":7052},[264],[43,7054,7056],{"className":7055},[268],[43,7057,7059],{"className":7058,"style":273},[272],[43,7060,7061,7064],{"style":276},[43,7062],{"className":7063,"style":281},[280],[43,7065,7067],{"className":7066},[285,286,287,288],[43,7068,292],{"className":7069},[64,288],[43,7071,474],{"className":7072},[64],[43,7074,6442],{"className":7075,"style":599},[64,65]," at the centre and the moment at the\nmid-side of an edge is ",[43,7078,7080],{"className":7079},[46],[43,7081,7083],{"className":7082,"ariaHidden":51},[50],[43,7084,7086,7090,7093,7097,7100,7103],{"className":7085},[55],[43,7087],{"className":7088,"style":7089},[59],"height:1.0085em;vertical-align:-0.1944em;",[43,7091,548],{"className":7092},[64],[43,7094,7096],{"className":7095},[64],"0.0513",[43,7098],{"className":7099,"style":94},[71],[43,7101,6422],{"className":7102,"style":1650},[64,65],[43,7104,7106,7109],{"className":7105},[64],[43,7107,18],{"className":7108},[64,65],[43,7110,7112],{"className":7111},[260],[43,7113,7115],{"className":7114},[264],[43,7116,7118],{"className":7117},[268],[43,7119,7121],{"className":7120,"style":273},[272],[43,7122,7123,7126],{"style":276},[43,7124],{"className":7125,"style":281},[280],[43,7127,7129],{"className":7128},[285,286,287,288],[43,7130,349],{"className":7131},[64,288],[11,7133,7134],{},"Expected behaviour on the shell element:",[633,7136,7137,7150],{},[636,7138,7139],{},[639,7140,7141,7144,7147],{},[642,7142,7143],{},"Mesh",[642,7145,7146],{},"Centre deflection",[642,7148,7149],{},"Centre moment",[652,7151,7152,7163,7174],{},[639,7153,7154,7157,7160],{},[657,7155,7156],{},"4 × 4",[657,7158,7159],{},"Within a few percent, low",[657,7161,7162],{},"Several percent low",[639,7164,7165,7168,7171],{},[657,7166,7167],{},"8 × 8",[657,7169,7170],{},"Within about 1%",[657,7172,7173],{},"Within 1–2%",[639,7175,7176,7179,7182],{},[657,7177,7178],{},"16 × 16",[657,7180,7181],{},"Converged",[657,7183,7181],{},[11,7185,7186,7187,3629],{},"Deflection converges from below (the coarse mesh is stiff) and the moment follows. Two\nrefinements agreeing within a percent is the convergence argument; a result that changes\nby more than that between the last two meshes is not yet a number to quote. See\n",[18,7188,7190],{"href":7189},"\u002Fkb\u002Fmeshing\u002Felement-size#convergence","Choosing element size",[4751,7192],{"kind":7193},"mesh",[11,7195,7196,7197,7237],{},"A thin plate, ",[43,7198,7200],{"className":7199},[46],[43,7201,7203,7227],{"className":7202,"ariaHidden":51},[50],[43,7204,7206,7209,7212,7215,7218,7221,7224],{"className":7205},[55],[43,7207],{"className":7208,"style":192},[59],[43,7210,18],{"className":7211},[64,65],[43,7213,474],{"className":7214},[64],[43,7216,6404],{"className":7217},[64,65],[43,7219],{"className":7220,"style":72},[71],[43,7222,77],{"className":7223},[76],[43,7225],{"className":7226,"style":72},[71],[43,7228,7230,7233],{"className":7229},[55],[43,7231],{"className":7232,"style":143},[59],[43,7234,7236],{"className":7235},[64],"100"," or more, is the test that exposes shear locking in a\ndisplacement-based Reissner–Mindlin element. The shell element does not lock. A plate\nthat comes out far too stiff as it gets thin is what a locking element does, and it is\nwhat this test is for.",[24,7239,7241],{"id":7240},"walls-bending-in-their-own-plane","Walls: bending in their own plane",[11,7243,7244,7245,7287,7288,7330,7331,111,7374,111,7418,7452,7453,7495,7496,7531],{},"The same element carries a wall's in-plane action, and the test that exposes a weak\nmembrane is a deep cantilever bent in its own plane. Take a wall ",[43,7246,7248],{"className":7247},[46],[43,7249,7251,7269],{"className":7250,"ariaHidden":51},[50],[43,7252,7254,7257,7260,7263,7266],{"className":7253},[55],[43,7255],{"className":7256,"style":60},[59],[43,7258,127],{"className":7259},[64,65],[43,7261],{"className":7262,"style":72},[71],[43,7264,77],{"className":7265},[76],[43,7267],{"className":7268,"style":72},[71],[43,7270,7272,7275,7278,7281],{"className":7271},[55],[43,7273],{"className":7274,"style":143},[59],[43,7276,2082],{"className":7277},[64],[43,7279,102],{"className":7280},[71],[43,7282,7284],{"className":7283},[64,106],[43,7285,157],{"className":7286},[64]," long and\n",[43,7289,7291],{"className":7290},[46],[43,7292,7294,7312],{"className":7293,"ariaHidden":51},[50],[43,7295,7297,7300,7303,7306,7309],{"className":7296},[55],[43,7298],{"className":7299,"style":60},[59],[43,7301,2835],{"className":7302,"style":2834},[64,65],[43,7304],{"className":7305,"style":72},[71],[43,7307,77],{"className":7308},[76],[43,7310],{"className":7311,"style":72},[71],[43,7313,7315,7318,7321,7324],{"className":7314},[55],[43,7316],{"className":7317,"style":143},[59],[43,7319,539],{"className":7320},[64],[43,7322,102],{"className":7323},[71],[43,7325,7327],{"className":7326},[64,106],[43,7328,157],{"className":7329},[64]," deep, ",[43,7332,7334],{"className":7333},[46],[43,7335,7337,7355],{"className":7336,"ariaHidden":51},[50],[43,7338,7340,7343,7346,7349,7352],{"className":7339},[55],[43,7341],{"className":7342,"style":6400},[59],[43,7344,6404],{"className":7345},[64,65],[43,7347],{"className":7348,"style":72},[71],[43,7350,77],{"className":7351},[76],[43,7353],{"className":7354,"style":72},[71],[43,7356,7358,7361,7365,7368],{"className":7357},[55],[43,7359],{"className":7360,"style":143},[59],[43,7362,7364],{"className":7363},[64],"200",[43,7366,102],{"className":7367},[71],[43,7369,7371],{"className":7370},[64,106],[43,7372,2395],{"className":7373},[64],[43,7375,7377],{"className":7376},[46],[43,7378,7380,7398],{"className":7379,"ariaHidden":51},[50],[43,7381,7383,7386,7389,7392,7395],{"className":7382},[55],[43,7384],{"className":7385,"style":60},[59],[43,7387,67],{"className":7388,"style":66},[64,65],[43,7390],{"className":7391,"style":72},[71],[43,7393,77],{"className":7394},[76],[43,7396],{"className":7397,"style":72},[71],[43,7399,7401,7404,7408,7411],{"className":7400},[55],[43,7402],{"className":7403,"style":60},[59],[43,7405,7407],{"className":7406},[64],"30",[43,7409,102],{"className":7410},[71],[43,7412,7414],{"className":7413},[64,106],[43,7415,7417],{"className":7416},[64],"GPa",[43,7419,7421],{"className":7420},[46],[43,7422,7424,7442],{"className":7423,"ariaHidden":51},[50],[43,7425,7427,7430,7433,7436,7439],{"className":7426},[55],[43,7428],{"className":7429,"style":171},[59],[43,7431,6510],{"className":7432,"style":6509},[64,65],[43,7434],{"className":7435,"style":72},[71],[43,7437,77],{"className":7438},[76],[43,7440],{"className":7441,"style":72},[71],[43,7443,7445,7448],{"className":7444},[55],[43,7446],{"className":7447,"style":143},[59],[43,7449,7451],{"className":7450},[64],"0.2",", clamped\nalong one short edge, with ",[43,7454,7456],{"className":7455},[46],[43,7457,7459,7477],{"className":7458,"ariaHidden":51},[50],[43,7460,7462,7465,7468,7471,7474],{"className":7461},[55],[43,7463],{"className":7464,"style":60},[59],[43,7466,370],{"className":7467,"style":369},[64,65],[43,7469],{"className":7470,"style":72},[71],[43,7472,77],{"className":7473},[76],[43,7475],{"className":7476,"style":72},[71],[43,7478,7480,7483,7486,7489],{"className":7479},[55],[43,7481],{"className":7482,"style":2625},[59],[43,7484,7236],{"className":7485},[64],[43,7487,102],{"className":7488},[71],[43,7490,7492],{"className":7491},[64,106],[43,7493,4656],{"className":7494},[64]," of shear spread over the other. With shear\ndeformation included (",[43,7497,7499],{"className":7498},[46],[43,7500,7502,7521],{"className":7501,"ariaHidden":51},[50],[43,7503,7505,7508,7512,7515,7518],{"className":7504},[55],[43,7506],{"className":7507,"style":171},[59],[43,7509,7511],{"className":7510},[64,65],"κ",[43,7513],{"className":7514,"style":72},[71],[43,7516,77],{"className":7517},[76],[43,7519],{"className":7520,"style":72},[71],[43,7522,7524,7527],{"className":7523},[55],[43,7525],{"className":7526,"style":192},[59],[43,7528,7530],{"className":7529},[64],"5\u002F6","):",[43,7533,7535],{"className":7534},[1730],[43,7536,7538],{"className":7537},[46],[43,7539,7541],{"className":7540,"ariaHidden":51},[50],[43,7542,7544,7548],{"className":7543},[55],[43,7545],{"className":7546,"style":7547},[59],"height:3.6771em;vertical-align:-1.5886em;",[43,7549,7551],{"className":7550},[64],[43,7552,7554,7600],{"className":7553},[1750],[43,7555,7557],{"className":7556},[1754],[43,7558,7560,7591],{"className":7559},[264,576],[43,7561,7563,7588],{"className":7562},[268],[43,7564,7567,7579],{"className":7565,"style":7566},[272],"height:2.0886em;",[43,7568,7570,7573],{"style":7569},"top:-4.0886em;",[43,7571],{"className":7572,"style":1771},[280],[43,7574,7576],{"className":7575},[64],[43,7577,1792],{"className":7578,"style":1791},[64,65],[43,7580,7582,7585],{"style":7581},"top:-2.2626em;",[43,7583],{"className":7584,"style":1771},[280],[43,7586],{"className":7587},[64],[43,7589,605],{"className":7590},[604],[43,7592,7594],{"className":7593},[268],[43,7595,7598],{"className":7596,"style":7597},[272],"height:1.5886em;",[43,7599],{},[43,7601,7603],{"className":7602},[1808],[43,7604,7606,7871],{"className":7605},[264,576],[43,7607,7609,7868],{"className":7608},[268],[43,7610,7612,7809],{"className":7611,"style":7566},[272],[43,7613,7614,7617],{"style":7569},[43,7615],{"className":7616,"style":1771},[280],[43,7618,7620,7623,7626,7629,7632,7729,7732,7735,7738],{"className":7619},[64],[43,7621],{"className":7622},[64],[43,7624],{"className":7625,"style":72},[71],[43,7627,77],{"className":7628},[76],[43,7630],{"className":7631,"style":72},[71],[43,7633,7635,7638,7726],{"className":7634},[64],[43,7636],{"className":7637},[534,1844],[43,7639,7641],{"className":7640},[1848],[43,7642,7644,7718],{"className":7643},[264,576],[43,7645,7647,7715],{"className":7646},[268],[43,7648,7650,7667,7675],{"className":7649,"style":1858},[272],[43,7651,7652,7655],{"style":1861},[43,7653],{"className":7654,"style":1865},[280],[43,7656,7658,7661,7664],{"className":7657},[64],[43,7659,400],{"className":7660},[64],[43,7662,67],{"className":7663,"style":66},[64,65],[43,7665,304],{"className":7666,"style":303},[64,65],[43,7668,7669,7672],{"style":1875},[43,7670],{"className":7671,"style":1865},[280],[43,7673],{"className":7674,"style":1883},[1882],[43,7676,7677,7680],{"style":1886},[43,7678],{"className":7679,"style":1865},[280],[43,7681,7683,7686],{"className":7682},[64],[43,7684,370],{"className":7685,"style":369},[64,65],[43,7687,7689,7692],{"className":7688},[64],[43,7690,127],{"className":7691},[64,65],[43,7693,7695],{"className":7694},[260],[43,7696,7698],{"className":7697},[264],[43,7699,7701],{"className":7700},[268],[43,7702,7704],{"className":7703,"style":273},[272],[43,7705,7706,7709],{"style":276},[43,7707],{"className":7708,"style":281},[280],[43,7710,7712],{"className":7711},[285,286,287,288],[43,7713,400],{"className":7714},[64,288],[43,7716,605],{"className":7717},[604],[43,7719,7721],{"className":7720},[268],[43,7722,7724],{"className":7723,"style":1948},[272],[43,7725],{},[43,7727],{"className":7728},[618,1844],[43,7730],{"className":7731,"style":543},[71],[43,7733,3212],{"className":7734},[547],[43,7736],{"className":7737,"style":543},[71],[43,7739,7741,7744,7806],{"className":7740},[64],[43,7742],{"className":7743},[534,1844],[43,7745,7747],{"className":7746},[1848],[43,7748,7750,7798],{"className":7749},[264,576],[43,7751,7753,7795],{"className":7752},[268],[43,7754,7756,7773,7781],{"className":7755,"style":4992},[272],[43,7757,7758,7761],{"style":1861},[43,7759],{"className":7760,"style":1865},[280],[43,7762,7764,7767,7770],{"className":7763},[64],[43,7765,7511],{"className":7766},[64,65],[43,7768,508],{"className":7769},[64,65],[43,7771,481],{"className":7772},[64,65],[43,7774,7775,7778],{"style":1875},[43,7776],{"className":7777,"style":1865},[280],[43,7779],{"className":7780,"style":1883},[1882],[43,7782,7783,7786],{"style":1886},[43,7784],{"className":7785,"style":1865},[280],[43,7787,7789,7792],{"className":7788},[64],[43,7790,370],{"className":7791,"style":369},[64,65],[43,7793,127],{"className":7794},[64,65],[43,7796,605],{"className":7797},[604],[43,7799,7801],{"className":7800},[268],[43,7802,7804],{"className":7803,"style":1948},[272],[43,7805],{},[43,7807],{"className":7808},[618,1844],[43,7810,7811,7814],{"style":7581},[43,7812],{"className":7813,"style":1771},[280],[43,7815,7817,7820,7823,7826,7829,7833,7836,7839,7842,7846,7849,7852,7855,7859,7862],{"className":7816},[64],[43,7818],{"className":7819},[64],[43,7821],{"className":7822,"style":72},[71],[43,7824,77],{"className":7825},[76],[43,7827],{"className":7828,"style":72},[71],[43,7830,7832],{"className":7831},[64],"14.40",[43,7834],{"className":7835,"style":543},[71],[43,7837,3212],{"className":7838},[547],[43,7840],{"className":7841,"style":543},[71],[43,7843,7845],{"className":7844},[64],"0.29",[43,7847],{"className":7848,"style":72},[71],[43,7850,77],{"className":7851},[76],[43,7853],{"className":7854,"style":72},[71],[43,7856,7858],{"className":7857},[64],"14.69",[43,7860,102],{"className":7861},[71],[43,7863,7865],{"className":7864},[64,106],[43,7866,2395],{"className":7867},[64],[43,7869,605],{"className":7870},[604],[43,7872,7874],{"className":7873},[268],[43,7875,7877],{"className":7876,"style":7597},[272],[43,7878],{},[11,7880,7881],{},"A fine plane-stress mesh converges to 14.67 mm; the 0.15 % between the two is the clamped\nroot, which beam theory does not see. Read the in-plane deflection at the free end as the\naverage over its nodes.",[633,7883,7884,7892],{},[636,7885,7886],{},[639,7887,7888,7890],{},[642,7889,7143],{},[642,7891,872],{},[652,7893,7894,7902],{},[639,7895,7896,7899],{},[657,7897,7898],{},"6 × 1, one element deep",[657,7900,7901],{},"About 1 % low",[639,7903,7904,7907],{},[657,7905,7906],{},"Finer meshes",[657,7908,7909],{},"Converge to 14.67 mm",[11,7911,7912,7913,3629],{},"A plain four-node membrane reaches about 68 % on the 6 × 1 mesh. If a wall comes out that\nstiff, the membrane is locking, and this is the case that shows it. Lean the interior\nelement edges and the result drops — to about 80 % when they lean by 40 % of the element\nwidth — because no four-node membrane that passes the patch test is exact on a trapezoid.\nSee ",[18,7914,21],{"href":7915},"\u002Fkb\u002Ftheory\u002Felement-formulations#walls-and-deep-beams-the-membrane",[24,7917,7919],{"id":7918},"reactions-the-universal-check","Reactions: the universal check",[7921,7922,7925,7928,7985],"callout",{"title":7923,"type":7924},"Reactions balance the loads","check",[11,7926,7927],{},"Every case above, and every real model, satisfies the same identity: the sum of reactions\nequals the sum of applied loads, per direction, per load case.",[43,7929,7931],{"className":7930},[1730],[43,7932,7934],{"className":7933},[46],[43,7935,7937,7969],{"className":7936,"ariaHidden":51},[50],[43,7938,7940,7944,7952,7955,7960,7963,7966],{"className":7939},[55],[43,7941],{"className":7942,"style":7943},[59],"height:1.6em;vertical-align:-0.55em;",[43,7945,7951],{"className":7946,"style":7950},[7947,7948,7949],"mop","op-symbol","large-op","position:relative;top:0em;","∑",[43,7953],{"className":7954,"style":94},[71],[43,7956,7959],{"className":7957,"style":7958},[64,65],"margin-right:0.0077em;","R",[43,7961],{"className":7962,"style":72},[71],[43,7964,77],{"className":7965},[76],[43,7967],{"className":7968,"style":72},[71],[43,7970,7972,7975,7978,7981],{"className":7971},[55],[43,7973],{"className":7974,"style":7943},[59],[43,7976,7951],{"className":7977,"style":7950},[7947,7948,7949],[43,7979],{"className":7980,"style":94},[71],[43,7982,7984],{"className":7983,"style":369},[64,65],"F",[11,7986,7987,7988,3629],{},"Read the reactions table and add it up. A mismatch beyond rounding is a load applied to\nsomething not connected to the supports — see\n",[18,7989,7991],{"href":7990},"\u002Fkb\u002Fgetting-started\u002Ffirst-analysis","Your first analysis",[24,7993,7995],{"id":7994},"cross-checking-in-another-solver","Cross-checking in another solver",[11,7997,7998,7999,8003,8004,8008],{},"Beyond closed-form cases, the real test of a real model is to run it somewhere else.\nOn Pro, Elementarium exports a ",[18,8000,8002],{"href":8001},"\u002Fkb\u002Freports\u002Fexchange","SAF workbook"," that SCIA Engineer,\nRFEM and others read (and an ",[18,8005,8007],{"href":8006},"\u002Fkb\u002Freports\u002Fexchange#oofem-an-input-deck-for-independent-verification","OOFEM input deck",",\nif OOFEM is what you have). Put the\ndisplacements side by side. Two independent implementations agreeing is the evidence a\nsceptical reviewer accepts.",[24,8010,8012],{"id":8011},"when-they-disagree","When they disagree",[11,8014,8015],{},"Work through, in order:",[29,8017,8018],{},[32,8019,8020,8029,8035,8041,8046],{},[35,8021,8022,8025,8026,3629],{},[38,8023,8024],{},"Units"," — the factor is 1 000 or 10⁶. ",[18,8027,8024],{"href":8028},"\u002Fkb\u002Fgetting-started\u002Funits",[35,8030,8031,8034],{},[38,8032,8033],{},"Supports"," — the closed form assumes a support condition; the model has the one you\ngave it. Pinned is translations only.",[35,8036,8037,8040],{},[38,8038,8039],{},"Discretisation"," — buckling, modal and plate cases converge; check the tables above\nbefore concluding.",[35,8042,8043,8045],{},[38,8044,2436],{}," — a stocky beam is more flexible than pure bending says,\ncorrectly: about a third more for an IPE 400 over 2 m.",[35,8047,8048,8051,8052,3629],{},[38,8049,8050],{},"Sign convention"," — a negative where you expected a positive is usually the local axis\ndirection. ",[18,8053,8055],{"href":8054},"\u002Fkb\u002Ftheory\u002Fsign-conventions","Sign conventions",[11,8057,8058,8059,8063],{},"If none of those explains it, that is a bug or a limitation worth reporting. Send the\nproject and the hand calculation to\n",[18,8060,8062],{"href":8061},"mailto:hello@elementarium.app","hello@elementarium.app","; during the beta it reaches the\nperson who wrote the solver, and either the solver or this page gets corrected.",{"title":8065,"searchDepth":8066,"depth":8066,"links":8067},"",2,[8068,8069,8073,8074,8075,8076,8077,8078,8079,8080],{"id":26,"depth":8066,"text":27},{"id":623,"depth":8066,"text":624,"children":8070},[8071],{"id":2435,"depth":8072,"text":2436},3,{"id":2608,"depth":8066,"text":2609},{"id":3640,"depth":8066,"text":3641},{"id":4741,"depth":8066,"text":4742},{"id":6367,"depth":8066,"text":6368},{"id":7240,"depth":8066,"text":7241},{"id":7918,"depth":8066,"text":7919},{"id":7994,"depth":8066,"text":7995},{"id":8011,"depth":8066,"text":8012},"Closed-form cases you can build in ten minutes to test the solver — beams, frames, columns, plates, frequencies — with the expected error of each.","md",false,{},true,null,6,"\u002Fkb\u002Fanalysis\u002Fverification",{"title":6,"description":8081},{"loc":8088},"kb\u002Fanalysis\u002Fverification","RRsrMi-9Aqp3Yayq0L9DBTG2mYmyakwc73DfD7NT2eY",{"sections":8094,"extras":8389,"flat":8402},[8095,8125,8166,8184,8219,8244,8271,8301,8328,8351,8366],{"path":8096,"slug":8097,"title":8098,"description":8099,"icon":8100,"articles":8101},"\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",[8102,8106,8109,8113,8117,8121,8123],{"path":8103,"title":8104,"description":8105,"translated":8085},"\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":3635,"title":8107,"description":8108,"translated":8085},"Quickstart","Build a portal frame, load it, solve it and read the bending moment diagram — about ten minutes from an empty tab.",{"path":8110,"title":8111,"description":8112,"translated":8085},"\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":8114,"title":8115,"description":8116,"translated":8085},"\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":8118,"title":8119,"description":8120,"translated":8085},"\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":8028,"title":8024,"description":8122,"translated":8085},"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":7990,"title":7991,"description":8124,"translated":8085},"Load cases, combinations and a linear static solve — then how to decide whether the answer is trustworthy.",{"path":8126,"slug":8127,"title":8128,"description":8129,"icon":8130,"articles":8131},"\u002Fkb\u002Fmodeling","modeling","Modeling","Bars, beams, 3D frames, slabs and walls. Geometry, sections, supports and releases that behave like the real structure.","lucide:frame",[8132,8136,8140,8144,8148,8152,8156,8160,8164],{"path":8133,"title":8134,"description":8135,"translated":8085},"\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":8137,"title":8138,"description":8139,"translated":8085},"\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":8141,"title":8142,"description":8143,"translated":8085},"\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":8145,"title":8146,"description":8147,"translated":8085},"\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":8149,"title":8150,"description":8151,"translated":8085},"\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":8153,"title":8154,"description":8155,"translated":8085},"\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":8157,"title":8158,"description":8159,"translated":8085},"\u002Fkb\u002Fmodeling\u002Fstructure-generators","Structure generators","Thirty-five parametric 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