[{"data":1,"prerenderedAt":4628},["ShallowReactive",2],{"kb-en-\u002Fkb\u002Ftheory\u002Fdirect-stiffness-method":3,"kb-nav-en":4244},{"page":4,"fallback":4234},{"id":5,"title":6,"body":7,"description":4232,"extension":4233,"faq":4234,"meta":4235,"navigation":4236,"ogImage":4237,"order":4238,"path":4239,"raw":4234,"robots":4237,"schemaOrg":4237,"seo":4240,"sitemap":4241,"stem":4242,"__hash__":4243},"content_en\u002Fkb\u002Ftheory\u002Fdirect-stiffness-method.md","The direct stiffness method",{"type":8,"value":9,"toc":4219},"minimark",[10,14,19,89,131,134,138,201,464,515,602,1110,1169,1173,1193,1305,1345,1349,1368,1632,1706,1709,1739,1855,1859,1900,2518,2591,3184,3191,3197,3203,3227,3231,3377,3396,3400,3461,3647,3720,3727,3731,4144,4147,4151,4216],[11,12,13],"p",{},"Every result in Elementarium comes out of one procedure. This page walks through it once,\nwith enough algebra to follow but not to implement, so that when a number surprises you\nthe surprise has somewhere to go.",[15,16,18],"h2",{"id":17},"the-idea","The idea",[11,20,21,22,26,51,52,69,70,88],{},"A structure is a set of elements joined at nodes. Each element resists the relative motion\nof its nodes with forces proportional to that motion (that is what \"linear elastic\"\nmeans), and the constant of proportionality is a matrix. Add up every element's\ncontribution at every node and you have the structure's ",[23,24,25],"strong",{},"stiffness matrix",[27,28,31],"span",{"className":29},[30],"katex",[27,32,36],{"className":33,"ariaHidden":35},[34],"katex-html","true",[27,37,40,45],{"className":38},[39],"base",[27,41],{"className":42,"style":44},[43],"strut","height:0.6861em;",[27,46,50],{"className":47},[48,49],"mord","mathbf","K",", which relates every nodal displacement ",[27,53,55],{"className":54},[30],[27,56,58],{"className":57,"ariaHidden":35},[34],[27,59,61,65],{"className":60},[39],[27,62],{"className":63,"style":64},[43],"height:0.4444em;",[27,66,68],{"className":67},[48,49],"u"," to every nodal force\n",[27,71,73],{"className":72},[30],[27,74,76],{"className":75,"ariaHidden":35},[34],[27,77,79,83],{"className":78},[39],[27,80],{"className":81,"style":82},[43],"height:0.6944em;",[27,84,87],{"className":85,"style":86},[48,49],"margin-right:0.109em;","f",":",[27,90,93],{"className":91},[92],"katex-display",[27,94,96],{"className":95},[30],[27,97,99,122],{"className":98,"ariaHidden":35},[34],[27,100,102,105,109,114,119],{"className":101},[39],[27,103],{"className":104,"style":44},[43],[27,106,108],{"className":107},[48,49],"Ku",[27,110],{"className":111,"style":113},[112],"mspace","margin-right:0.2778em;",[27,115,118],{"className":116},[117],"mrel","=",[27,120],{"className":121,"style":113},[112],[27,123,125,128],{"className":124},[39],[27,126],{"className":127,"style":82},[43],[27,129,87],{"className":130,"style":86},[48,49],[11,132,133],{},"Given the loads, solve for the displacements. Given the displacements, every internal\nforce follows. Everything else is bookkeeping.",[15,135,137],{"id":136},"_1-element-stiffness-in-local-axes","1. Element stiffness in local axes",[11,139,140,141,146,147,165,166,182,183,200],{},"Each element type has a stiffness matrix written in its own axes, derived from its\n",[142,143,145],"a",{"href":144},"\u002Fkb\u002Ftheory\u002Felement-formulations","formulation",". For a bar of length ",[27,148,150],{"className":149},[30],[27,151,153],{"className":152,"ariaHidden":35},[34],[27,154,156,160],{"className":155},[39],[27,157],{"className":158,"style":159},[43],"height:0.6833em;",[27,161,164],{"className":162},[48,163],"mathnormal","L",", area ",[27,167,169],{"className":168},[30],[27,170,172],{"className":171,"ariaHidden":35},[34],[27,173,175,178],{"className":174},[39],[27,176],{"className":177,"style":159},[43],[27,179,181],{"className":180},[48,163],"A"," and\nmodulus ",[27,184,186],{"className":185},[30],[27,187,189],{"className":188,"ariaHidden":35},[34],[27,190,192,195],{"className":191},[39],[27,193],{"className":194,"style":159},[43],[27,196,199],{"className":197,"style":198},[48,163],"margin-right:0.0576em;","E",", it is 2 × 2:",[27,202,204],{"className":203},[92],[27,205,207],{"className":206},[30],[27,208,210,229],{"className":209,"ariaHidden":35},[34],[27,211,213,216,220,223,226],{"className":212},[39],[27,214],{"className":215,"style":82},[43],[27,217,219],{"className":218},[48,49],"k",[27,221],{"className":222,"style":113},[112],[27,224,118],{"className":225},[117],[27,227],{"className":228,"style":113},[112],[27,230,232,236,320,324],{"className":231},[39],[27,233],{"className":234,"style":235},[43],"height:2.4em;vertical-align:-0.95em;",[27,237,239,244,316],{"className":238},[48],[27,240],{"className":241},[242,243],"mopen","nulldelimiter",[27,245,248],{"className":246},[247],"mfrac",[27,249,253,307],{"className":250},[251,252],"vlist-t","vlist-t2",[27,254,257,302],{"className":255},[256],"vlist-r",[27,258,262,276,287],{"className":259,"style":261},[260],"vlist","height:1.3603em;",[27,263,265,270],{"style":264},"top:-2.314em;",[27,266],{"className":267,"style":269},[268],"pstrut","height:3em;",[27,271,273],{"className":272},[48],[27,274,164],{"className":275},[48,163],[27,277,279,282],{"style":278},"top:-3.23em;",[27,280],{"className":281,"style":269},[268],[27,283],{"className":284,"style":286},[285],"frac-line","border-bottom-width:0.04em;",[27,288,290,293],{"style":289},"top:-3.677em;",[27,291],{"className":292,"style":269},[268],[27,294,296,299],{"className":295},[48],[27,297,199],{"className":298,"style":198},[48,163],[27,300,181],{"className":301},[48,163],[27,303,306],{"className":304},[305],"vlist-s","​",[27,308,310],{"className":309},[256],[27,311,314],{"className":312,"style":313},[260],"height:0.686em;",[27,315],{},[27,317],{"className":318},[319,243],"mclose",[27,321],{"className":322,"style":323},[112],"margin-right:0.1667em;",[27,325,328,339,457],{"className":326},[327],"minner",[27,329,333],{"className":330,"style":332},[242,331],"delimcenter","top:0em;",[27,334,338],{"className":335},[336,337],"delimsizing","size3","[",[27,340,342],{"className":341},[48],[27,343,346,401,406,409],{"className":344},[345],"mtable",[27,347,350],{"className":348},[349],"col-align-c",[27,351,353,392],{"className":352},[251,252],[27,354,356,389],{"className":355},[256],[27,357,360,373],{"className":358,"style":359},[260],"height:1.45em;",[27,361,363,366],{"style":362},"top:-3.61em;",[27,364],{"className":365,"style":269},[268],[27,367,369],{"className":368},[48],[27,370,372],{"className":371},[48],"1",[27,374,376,379],{"style":375},"top:-2.41em;",[27,377],{"className":378,"style":269},[268],[27,380,382,386],{"className":381},[48],[27,383,385],{"className":384},[48],"−",[27,387,372],{"className":388},[48],[27,390,306],{"className":391},[305],[27,393,395],{"className":394},[256],[27,396,399],{"className":397,"style":398},[260],"height:0.95em;",[27,400],{},[27,402],{"className":403,"style":405},[404],"arraycolsep","width:0.5em;",[27,407],{"className":408,"style":405},[404],[27,410,412],{"className":411},[349],[27,413,415,449],{"className":414},[251,252],[27,416,418,446],{"className":417},[256],[27,419,421,435],{"className":420,"style":359},[260],[27,422,423,426],{"style":362},[27,424],{"className":425,"style":269},[268],[27,427,429,432],{"className":428},[48],[27,430,385],{"className":431},[48],[27,433,372],{"className":434},[48],[27,436,437,440],{"style":375},[27,438],{"className":439,"style":269},[268],[27,441,443],{"className":442},[48],[27,444,372],{"className":445},[48],[27,447,306],{"className":448},[305],[27,450,452],{"className":451},[256],[27,453,455],{"className":454,"style":398},[260],[27,456],{},[27,458,460],{"className":459,"style":332},[319,331],[27,461,463],{"className":462},[336,337],"]",[11,465,466,467,484,485,514],{},"Push one end towards the other by ",[27,468,470],{"className":469},[30],[27,471,473],{"className":472,"ariaHidden":35},[34],[27,474,476,479],{"className":475},[39],[27,477],{"className":478,"style":82},[43],[27,480,483],{"className":481,"style":482},[48,163],"margin-right:0.0379em;","δ"," and the bar pushes back with ",[27,486,488],{"className":487},[30],[27,489,491],{"className":490,"ariaHidden":35},[34],[27,492,494,498,501,504,507,511],{"className":493},[39],[27,495],{"className":496,"style":497},[43],"height:1em;vertical-align:-0.25em;",[27,499,199],{"className":500,"style":198},[48,163],[27,502,181],{"className":503},[48,163],[27,505,483],{"className":506,"style":482},[48,163],[27,508,510],{"className":509},[48],"\u002F",[27,512,164],{"className":513},[48,163],". A\nbeam's matrix is 12 × 12 (six degrees of freedom at each of two nodes), and its bending\nterms come from cubic interpolation, which is why a prismatic beam under uniform load is\nexact without subdivision. A shell element's is 24 × 24 and is integrated numerically at\nGauss points.",[11,516,517,518,535,536,569,570,601],{},"The familiar beam terms are in there. The moment needed to rotate one end of a beam by\n",[27,519,521],{"className":520},[30],[27,522,524],{"className":523,"ariaHidden":35},[34],[27,525,527,530],{"className":526},[39],[27,528],{"className":529,"style":82},[43],[27,531,534],{"className":532,"style":533},[48,163],"margin-right:0.0278em;","θ"," while the far end is fixed is ",[27,537,539],{"className":538},[30],[27,540,542],{"className":541,"ariaHidden":35},[34],[27,543,545,548,552,555,560,563,566],{"className":544},[39],[27,546],{"className":547,"style":497},[43],[27,549,551],{"className":550},[48],"4",[27,553,199],{"className":554,"style":198},[48,163],[27,556,559],{"className":557,"style":558},[48,163],"margin-right:0.0785em;","I",[27,561,534],{"className":562,"style":533},[48,163],[27,564,510],{"className":565},[48],[27,567,164],{"className":568},[48,163],", and the moment it induces at the far\nend is ",[27,571,573],{"className":572},[30],[27,574,576],{"className":575,"ariaHidden":35},[34],[27,577,579,582,586,589,592,595,598],{"className":578},[39],[27,580],{"className":581,"style":497},[43],[27,583,585],{"className":584},[48],"2",[27,587,199],{"className":588,"style":198},[48,163],[27,590,559],{"className":591,"style":558},[48,163],[27,593,534],{"className":594,"style":533},[48,163],[27,596,510],{"className":597},[48],[27,599,164],{"className":600},[48,163],". With both ends held against translation, the end rotations and\nmoments are related by",[27,603,605],{"className":604},[92],[27,606,608],{"className":607},[30],[27,609,611,775],{"className":610,"ariaHidden":35},[34],[27,612,614,617,766,769,772],{"className":613},[39],[27,615],{"className":616,"style":235},[43],[27,618,620,626,760],{"className":619},[327],[27,621,623],{"className":622,"style":332},[242,331],[27,624,338],{"className":625},[336,337],[27,627,629],{"className":628},[48],[27,630,632],{"className":631},[345],[27,633,635],{"className":634},[349],[27,636,638,752],{"className":637},[251,252],[27,639,641,749],{"className":640},[256],[27,642,644,701],{"className":643,"style":359},[260],[27,645,646,649],{"style":362},[27,647],{"className":648,"style":269},[268],[27,650,652],{"className":651},[48],[27,653,655,659],{"className":654},[48],[27,656,658],{"className":657,"style":86},[48,163],"M",[27,660,663],{"className":661},[662],"msupsub",[27,664,666,692],{"className":665},[251,252],[27,667,669,689],{"className":668},[256],[27,670,673],{"className":671,"style":672},[260],"height:0.3011em;",[27,674,676,680],{"style":675},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[27,677],{"className":678,"style":679},[268],"height:2.7em;",[27,681,686],{"className":682},[683,684,337,685],"sizing","reset-size6","mtight",[27,687,372],{"className":688},[48,685],[27,690,306],{"className":691},[305],[27,693,695],{"className":694},[256],[27,696,699],{"className":697,"style":698},[260],"height:0.15em;",[27,700],{},[27,702,703,706],{"style":375},[27,704],{"className":705,"style":269},[268],[27,707,709],{"className":708},[48],[27,710,712,715],{"className":711},[48],[27,713,658],{"className":714,"style":86},[48,163],[27,716,718],{"className":717},[662],[27,719,721,741],{"className":720},[251,252],[27,722,724,738],{"className":723},[256],[27,725,727],{"className":726,"style":672},[260],[27,728,729,732],{"style":675},[27,730],{"className":731,"style":679},[268],[27,733,735],{"className":734},[683,684,337,685],[27,736,585],{"className":737},[48,685],[27,739,306],{"className":740},[305],[27,742,744],{"className":743},[256],[27,745,747],{"className":746,"style":698},[260],[27,748],{},[27,750,306],{"className":751},[305],[27,753,755],{"className":754},[256],[27,756,758],{"className":757,"style":398},[260],[27,759],{},[27,761,763],{"className":762,"style":332},[319,331],[27,764,463],{"className":765},[336,337],[27,767],{"className":768,"style":113},[112],[27,770,118],{"className":771},[117],[27,773],{"className":774,"style":113},[112],[27,776,778,781,846,849,966,969],{"className":777},[39],[27,779],{"className":780,"style":235},[43],[27,782,784,787,843],{"className":783},[48],[27,785],{"className":786},[242,243],[27,788,790],{"className":789},[247],[27,791,793,835],{"className":792},[251,252],[27,794,796,832],{"className":795},[256],[27,797,799,810,818],{"className":798,"style":261},[260],[27,800,801,804],{"style":264},[27,802],{"className":803,"style":269},[268],[27,805,807],{"className":806},[48],[27,808,164],{"className":809},[48,163],[27,811,812,815],{"style":278},[27,813],{"className":814,"style":269},[268],[27,816],{"className":817,"style":286},[285],[27,819,820,823],{"style":289},[27,821],{"className":822,"style":269},[268],[27,824,826,829],{"className":825},[48],[27,827,199],{"className":828,"style":198},[48,163],[27,830,559],{"className":831,"style":558},[48,163],[27,833,306],{"className":834},[305],[27,836,838],{"className":837},[256],[27,839,841],{"className":840,"style":313},[260],[27,842],{},[27,844],{"className":845},[319,243],[27,847],{"className":848,"style":323},[112],[27,850,852,858,960],{"className":851},[327],[27,853,855],{"className":854,"style":332},[242,331],[27,856,338],{"className":857},[336,337],[27,859,861],{"className":860},[48],[27,862,864,909,912,915],{"className":863},[345],[27,865,867],{"className":866},[349],[27,868,870,901],{"className":869},[251,252],[27,871,873,898],{"className":872},[256],[27,874,876,887],{"className":875,"style":359},[260],[27,877,878,881],{"style":362},[27,879],{"className":880,"style":269},[268],[27,882,884],{"className":883},[48],[27,885,551],{"className":886},[48],[27,888,889,892],{"style":375},[27,890],{"className":891,"style":269},[268],[27,893,895],{"className":894},[48],[27,896,585],{"className":897},[48],[27,899,306],{"className":900},[305],[27,902,904],{"className":903},[256],[27,905,907],{"className":906,"style":398},[260],[27,908],{},[27,910],{"className":911,"style":405},[404],[27,913],{"className":914,"style":405},[404],[27,916,918],{"className":917},[349],[27,919,921,952],{"className":920},[251,252],[27,922,924,949],{"className":923},[256],[27,925,927,938],{"className":926,"style":359},[260],[27,928,929,932],{"style":362},[27,930],{"className":931,"style":269},[268],[27,933,935],{"className":934},[48],[27,936,585],{"className":937},[48],[27,939,940,943],{"style":375},[27,941],{"className":942,"style":269},[268],[27,944,946],{"className":945},[48],[27,947,551],{"className":948},[48],[27,950,306],{"className":951},[305],[27,953,955],{"classNam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are the ",[27,1114,1116],{"className":1115},[30],[27,1117,1119],{"className":1118,"ariaHidden":35},[34],[27,1120,1122,1125,1128,1131,1134,1137],{"className":1121},[39],[27,1123],{"className":1124,"style":497},[43],[27,1126,551],{"className":1127},[48],[27,1129,199],{"className":1130,"style":198},[48,163],[27,1132,559],{"className":1133,"style":558},[48,163],[27,1135,510],{"className":1136},[48],[27,1138,164],{"className":1139},[48,163]," and ",[27,1142,1144],{"className":1143},[30],[27,1145,1147],{"className":1146,"ariaHidden":35},[34],[27,1148,1150,1153,1156,1159,1162,1165],{"className":1149},[39],[27,1151],{"className":1152,"style":497},[43],[27,1154,585],{"className":1155},[48],[27,1157,199],{"className":1158,"style":198},[48,163],[27,1160,559],{"className":1161,"style":558},[48,163],[27,1163,510],{"className":1164},[48],[27,1166,164],{"className":1167},[48,163]," of moment distribution, which is the same method by hand.",[15,1170,1172],{"id":1171},"_2-transformation-to-global-axes","2. Transformation to global axes",[11,1174,1175,1176,1192],{},"The element's axes point along the element; the structure's do not. Each element matrix\nis rotated into the global system with a transformation matrix ",[27,1177,1179],{"className":1178},[30],[27,1180,1182],{"className":1181,"ariaHidden":35},[34],[27,1183,1185,1188],{"className":1184},[39],[27,1186],{"className":1187,"style":44},[43],[27,1189,1191],{"className":1190},[48,49],"T"," built from the\ndirection cosines of the member:",[27,1194,1196],{"className":1195},[92],[27,1197,1199],{"className":1198},[30],[27,1200,1202,1255],{"className":1201,"ariaHidden":35},[34],[27,1203,1205,1209,1246,1249,1252],{"className":1204},[39],[27,1206],{"className":1207,"style":1208},[43],"height:0.8312em;",[27,1210,1213],{"className":1211},[48,1212],"accent",[27,1214,1216],{"className":1215},[251],[27,1217,1219],{"className":1218},[256],[27,1220,1222,1231],{"className":1221,"style":1208},[260],[27,1223,1225,1228],{"style":1224},"top:-3em;",[27,1226],{"className":1227,"style":269},[268],[27,1229,219],{"className":1230},[48,49],[27,1232,1234,1237],{"style":1233},"top:-3.2634em;",[27,1235],{"className":1236,"style":269},[268],[27,1238,1242],{"className":1239,"style":1241},[1240],"accent-body","left:-0.25em;",[27,1243,1245],{"className":1244},[48],"ˉ",[27,1247],{"className":1248,"style":113},[112],[27,1250,118],{"className":1251},[117],[27,1253],{"className":1254,"style":113},[112],[27,1256,1258,1262,1293,1296,1299,1302],{"className":1257},[39],[27,1259],{"className":1260,"style":1261},[43],"height:0.8991em;",[27,1263,1265,1268],{"className":1264},[48],[27,1266,1191],{"className":1267},[48,49],[27,1269,1271],{"className":1270},[662],[27,1272,1274],{"className":1273},[251],[27,1275,1277],{"className":1276},[256],[27,1278,1280],{"className":1279,"style":1261},[260],[27,1281,1283,1286],{"style":1282},"top:-3.113em;margin-right:0.05em;",[27,1284],{"className":1285,"style":679},[268],[27,1287,1289],{"className":1288},[683,684,337,685],[27,1290,1191],{"className":1291},[48,1292,685],"mathsf",[27,1294],{"className":1295,"style":323},[112],[27,1297,219],{"className":1298},[48,49],[27,1300],{"className":1301,"style":323},[112],[27,1303,1191],{"className":1304},[48,49],[11,1306,1307,1308,1312,1313,1328,1329,1344],{},"This is where the ",[142,1309,1311],{"href":1310},"\u002Fkb\u002Ftheory\u002Fsign-conventions#local-member-axes","local axis convention","\nenters: a member drawn from A to B and one drawn from B to A have different ",[27,1314,1316],{"className":1315},[30],[27,1317,1319],{"className":1318,"ariaHidden":35},[34],[27,1320,1322,1325],{"className":1321},[39],[27,1323],{"className":1324,"style":44},[43],[27,1326,1191],{"className":1327},[48,49],",\nreport their internal forces with opposite signs, and contribute identically to\n",[27,1330,1332],{"className":1331},[30],[27,1333,1335],{"className":1334,"ariaHidden":35},[34],[27,1336,1338,1341],{"className":1337},[39],[27,1339],{"className":1340,"style":44},[43],[27,1342,50],{"className":1343},[48,49],".",[15,1346,1348],{"id":1347},"_3-assembly","3. Assembly",[11,1350,1351,1352,1367],{},"Each global element matrix is added into ",[27,1353,1355],{"className":1354},[30],[27,1356,1358],{"className":1357,"ariaHidden":35},[34],[27,1359,1361,1364],{"className":1360},[39],[27,1362],{"className":1363,"style":44},[43],[27,1365,50],{"className":1366},[48,49]," at the rows and columns belonging to\nits nodes' degrees of freedom:",[27,1369,1371],{"className":1370},[92],[27,1372,1374],{"className":1373},[30],[27,1375,1377,1395],{"className":1376,"ariaHidden":35},[34],[27,1378,1380,1383,1386,1389,1392],{"className":1379},[39],[27,1381],{"className":1382,"style":44},[43],[27,1384,50],{"className":1385},[48,49],[27,1387],{"className":1388,"style":113},[112],[27,1390,118],{"className":1391},[117],[27,1393],{"className":1394,"style":113},[112],[27,1396,1398,1402,1460,1463,1516,1519,1589,1592],{"className":1397},[39],[27,1399],{"className":1400,"style":1401},[43],"height:2.3em;vertical-align:-1.25em;",[27,1403,1407],{"className":1404},[1405,1406],"mop","op-limits",[27,1408,1410,1451],{"className":1409},[251,252],[27,1411,1413,1448],{"className":1412},[256],[27,1414,1417,1434],{"className":1415,"style":1416},[260],"height:1.05em;",[27,1418,1420,1424],{"style":1419},"top:-1.9em;margin-left:0em;",[27,1421],{"className":1422,"style":1423},[268],"height:3.05em;",[27,1425,1427],{"className":1426},[683,684,337,685],[27,1428,1430],{"className":1429},[48,685],[27,1431,1433],{"className":1432},[48,163,685],"e",[27,1435,1437,1440],{"style":1436},"top:-3.05em;",[27,1438],{"className":1439,"style":1423},[268],[27,1441,1442],{},[27,1443,1447],{"className":1444},[1405,1445,1446],"op-symbol","large-op","∑",[27,1449,306],{"className":1450},[305],[27,1452,1454],{"className":1453},[256],[27,1455,1458],{"className":1456,"style":1457},[260],"height:1.25em;",[27,1459],{},[27,1461],{"className":1462,"style":323},[112],[27,1464,1466,1469],{"className":1465},[48],[27,1467,181],{"className":1468},[48,49],[27,1470,1472],{"className":1471},[662],[27,1473,1475,1507],{"className":1474},[251,252],[27,1476,1478,1504],{"className":1477},[256],[27,1479,1481,1493],{"className":1480,"style":1261},[260],[27,1482,1484,1487],{"style":1483},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[27,1485],{"className":1486,"style":679},[268],[27,1488,1490],{"className":1489},[683,684,337,685],[27,1491,1433],{"className":1492},[48,163,685],[27,1494,1495,1498],{"style":1282},[27,1496],{"className":1497,"style":679},[268],[27,1499,1501],{"className":1500},[683,684,337,685],[27,1502,1191],{"className":1503},[48,1292,685],[27,1505,306],{"className":1506},[305],[27,1508,1510],{"className":1509},[256],[27,1511,1514],{"className":1512,"style":1513},[260],"height:0.247em;",[27,1515],{},[27,1517],{"className":1518,"style":323},[112],[27,1520,1522,1553],{"className":1521},[48],[27,1523,1525],{"className":1524},[48,1212],[27,1526,1528],{"className":1527},[251],[27,1529,1531],{"className":1530},[256],[27,1532,1534,1542],{"className":1533,"style":1208},[260],[27,1535,1536,1539],{"style":1224},[27,1537],{"className":1538,"style":269},[268],[27,1540,219],{"className":1541},[48,49],[27,1543,1544,1547],{"style":1233},[27,1545],{"className":1546,"style":269},[268],[27,1548,1550],{"className":1549,"style":1241},[1240],[27,1551,1245],{"className":1552},[48],[27,1554,1556],{"className":1555},[662],[27,1557,1559,1581],{"className":1558},[251,252],[27,1560,1562,1578],{"className":1561},[256],[27,1563,1566],{"className":1564,"style":1565},[260],"height:0.1514em;",[27,1567,1569,1572],{"style":1568},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[27,1570],{"className":1571,"style":679},[268],[27,1573,1575],{"className":1574},[683,684,337,685],[27,1576,1433],{"className":1577},[48,163,685],[27,1579,306],{"className":1580},[305],[27,1582,1584],{"className":1583},[256],[27,1585,1587],{"className":1586,"style":698},[260],[27,1588],{},[27,1590],{"className":1591,"style":323},[112],[27,1593,1595,1598],{"className":1594},[48],[27,1596,181],{"className":1597},[48,49],[27,1599,1601],{"className":1600},[662],[27,1602,1604,1624],{"className":1603},[251,252],[27,1605,1607,1621],{"className":1606},[256],[27,1608,1610],{"className":1609,"style":1565},[260],[27,1611,1612,1615],{"style":1568},[27,1613],{"className":1614,"style":679},[268],[27,1616,1618],{"className":1617},[683,684,337,685],[27,1619,1433],{"className":1620},[48,163,685],[27,1622,306],{"className":1623},[305],[27,1625,1627],{"className":1626},[256],[27,1628,1630],{"className":1629,"style":698},[260],[27,1631],{},[11,1633,1634,1635,1688,1689,1705],{},"where ",[27,1636,1638],{"className":1637},[30],[27,1639,1641],{"className":1640,"ariaHidden":35},[34],[27,1642,1644,1648],{"className":1643},[39],[27,1645],{"className":1646,"style":1647},[43],"height:0.8361em;vertical-align:-0.15em;",[27,1649,1651,1654],{"className":1650},[48],[27,1652,181],{"className":1653},[48,49],[27,1655,1657],{"className":1656},[662],[27,1658,1660,1680],{"className":1659},[251,252],[27,1661,1663,1677],{"className":1662},[256],[27,1664,1666],{"className":1665,"style":1565},[260],[27,1667,1668,1671],{"style":1568},[27,1669],{"className":1670,"style":679},[268],[27,1672,1674],{"className":1673},[683,684,337,685],[27,1675,1433],{"className":1676},[48,163,685],[27,1678,306],{"className":1679},[305],[27,1681,1683],{"className":1682},[256],[27,1684,1686],{"className":1685,"style":698},[260],[27,1687],{}," only picks out element ",[27,1690,1692],{"className":1691},[30],[27,1693,1695],{"className":1694,"ariaHidden":35},[34],[27,1696,1698,1702],{"className":1697},[39],[27,1699],{"className":1700,"style":1701},[43],"height:0.4306em;",[27,1703,1433],{"className":1704},[48,163],"'s degrees of freedom from the global list.\nA node shared by three members receives three contributions in the same rows. That is what\n\"connected\" means numerically, and it is why two members whose ends are 3 mm apart are not\nconnected: they have no rows in common.",[1707,1708],"diagram-stiffness",{},[11,1710,1711,1726,1727,1730,1731,1734,1735,1344],{},[27,1712,1714],{"className":1713},[30],[27,1715,1717],{"className":1716,"ariaHidden":35},[34],[27,1718,1720,1723],{"className":1719},[39],[27,1721],{"className":1722,"style":44},[43],[27,1724,50],{"className":1725},[48,49]," is ",[23,1728,1729],{},"sparse",": a node is only coupled to the nodes it shares an element with,\nso nearly every entry is zero. It is also ",[23,1732,1733],{},"symmetric",", by Maxwell's reciprocal theorem.\nThe solvers exploit both; see ",[142,1736,1738],{"href":1737},"\u002Fkb\u002Fanalysis\u002Fsolvers","Choosing a linear solver",[11,1740,1741,1742,1757,1758,1761,1762,1779,1780,1802,1803,1854],{},"Loads are assembled into ",[27,1743,1745],{"className":1744},[30],[27,1746,1748],{"className":1747,"ariaHidden":35},[34],[27,1749,1751,1754],{"className":1750},[39],[27,1752],{"className":1753,"style":82},[43],[27,1755,87],{"className":1756,"style":86},[48,49]," the same way. A nodal load goes straight in. A\ndistributed load on a member is converted to ",[23,1759,1760],{},"equivalent nodal forces",": for a uniform\n",[27,1763,1765],{"className":1764},[30],[27,1766,1768],{"className":1767,"ariaHidden":35},[34],[27,1769,1771,1774],{"className":1770},[39],[27,1772],{"className":1773,"style":1701},[43],[27,1775,1778],{"className":1776,"style":1777},[48,163],"margin-right:0.0269em;","w"," on a beam, ",[27,1781,1783],{"className":1782},[30],[27,1784,1786],{"className":1785,"ariaHidden":35},[34],[27,1787,1789,1792,1795,1798],{"className":1788},[39],[27,1790],{"className":1791,"style":497},[43],[27,1793,1778],{"className":1794,"style":1777},[48,163],[27,1796,164],{"className":1797},[48,163],[27,1799,1801],{"className":1800},[48],"\u002F2"," at each end and fixed-end moments ",[27,1804,1806],{"className":1805},[30],[27,1807,1809],{"className":1808,"ariaHidden":35},[34],[27,1810,1812,1816,1819,1850],{"className":1811},[39],[27,1813],{"className":1814,"style":1815},[43],"height:1.0641em;vertical-align:-0.25em;",[27,1817,1778],{"className":1818,"style":1777},[48,163],[27,1820,1822,1825],{"className":1821},[48],[27,1823,164],{"className":1824},[48,163],[27,1826,1828],{"className":1827},[662],[27,1829,1831],{"className":1830},[251],[27,1832,1834],{"className":1833},[256],[27,1835,1838],{"className":1836,"style":1837},[260],"height:0.8141em;",[27,1839,1841,1844],{"style":1840},"top:-3.063em;margin-right:0.05em;",[27,1842],{"className":1843,"style":679},[268],[27,1845,1847],{"className":1846},[683,684,337,685],[27,1848,585],{"className":1849},[48,685],[27,1851,1853],{"className":1852},[48],"\u002F12",". The element's own\nend-force treatment later corrects for these, so the moment along the member is exact.",[15,1856,1858],{"id":1857},"_4-constraints","4. Constraints",[11,1860,1861,1864,1865,1882,1883,1899],{},[23,1862,1863],{},"Supports"," remove degrees of freedom. A restrained one has a prescribed displacement\n(zero, or a settlement), so its row and column leave the system and its reaction is\nrecovered afterwards from the equilibrium of that row. Splitting the degrees of freedom\ninto free (",[27,1866,1868],{"className":1867},[30],[27,1869,1871],{"className":1870,"ariaHidden":35},[34],[27,1872,1874,1878],{"className":1873},[39],[27,1875],{"className":1876,"style":1877},[43],"height:0.8889em;vertical-align:-0.1944em;",[27,1879,87],{"className":1880,"style":1881},[48,163],"margin-right:0.1076em;",") and supported (",[27,1884,1886],{"className":1885},[30],[27,1887,1889],{"className":1888,"ariaHidden":35},[34],[27,1890,1892,1895],{"className":1891},[39],[27,1893],{"className":1894,"style":1701},[43],[27,1896,1898],{"className":1897},[48,163],"s","):",[27,1901,1903],{"className":1902},[92],[27,1904,1906],{"className":1905},[30],[27,1907,1909,2356],{"className":1908,"ariaHidden":35},[34],[27,1910,1912,1915,2204,2207,2347,2350,2353],{"className":1911},[39],[27,1913],{"className":1914,"style":235},[43],[27,1916,1918,1924,2198],{"className":1917},[327],[27,1919,1921],{"className":1920,"style":332},[242,331],[27,1922,338],{"className":1923},[336,337],[27,1925,1927],{"className":1926},[48],[27,1928,1930,2063,2066,2069],{"className":1929},[345],[27,1931,1933],{"className":1932},[349],[27,1934,1936,2055],{"className":1935},[251,252],[27,1937,1939,2052],{"className":1938},[256],[27,1940,1942,1998],{"className":1941,"style":359},[260],[27,1943,1944,1947],{"style":362},[27,1945],{"className":1946,"style":269},[268],[27,1948,1950],{"className":1949},[48],[27,1951,1953,1956],{"className":1952},[48],[27,1954,50],{"className":1955},[48,49],[27,1957,1959],{"className":1958},[662],[27,1960,1962,1989],{"className":1961},[251,252],[27,1963,1965,1986],{"className":1964},[256],[27,1966,1969],{"className":1967,"style":1968},[260],"height:0.3361em;",[27,1970,1971,1974],{"style":1568},[27,1972],{"className":1973,"style":679},[268],[27,1975,1977],{"className":1976},[683,684,337,685],[27,1978,1980,1983],{"className":1979},[48,685],[27,1981,87],{"className":1982,"style":1881},[48,163,685],[27,1984,87],{"className":1985,"style":1881},[48,163,685],[27,1987,306],{"className":1988},[305],[27,1990,1992],{"className":1991},[256],[27,1993,1996],{"className":1994,"style":1995},[260],"height:0.2861em;",[27,1997],{},[27,1999,2000,2003],{"style":375},[27,2001],{"className":2002,"style":269},[268],[27,2004,2006],{"className":2005},[48],[27,2007,2009,2012],{"className":2008},[48],[27,2010,50],{"className":2011},[48,49],[27,2013,2015],{"className":2014},[662],[27,2016,2018,2044],{"className":2017},[251,252],[27,2019,2021,2041],{"className":2020},[256],[27,2022,2024],{"className":2023,"style":1968},[260],[27,2025,2026,2029],{"style":1568},[27,2027],{"className":2028,"style":679},[268],[27,2030,2032],{"className":2031},[683,684,337,685],[27,2033,2035,2038],{"className":2034},[48,685],[27,2036,1898],{"className":2037},[48,163,685],[27,2039,87],{"className":2040,"style":1881},[48,163,685],[27,2042,306],{"className":2043},[305],[27,2045,2047],{"className":2046},[256],[27,2048,2050],{"className":2049,"style":1995},[260],[27,2051],{},[27,2053,306],{"className":2054},[305],[27,2056,2058],{"className":2057},[256],[27,2059,2061],{"className":2060,"style":398},[260],[27,2062],{},[27,2064],{"className":2065,"style":405},[404],[27,2067],{"className":2068,"style":405},[404],[27,2070,2072],{"className":2071},[349],[27,2073,2075,2190],{"className":2074},[251,252],[27,2076,2078,2187],{"className":2077},[256],[27,2079,2081,2135],{"className":2080,"style":359},[260],[27,2082,2083,2086],{"style":362},[27,2084],{"className":2085,"style":269},[268],[27,2087,2089],{"className":2088},[48],[27,2090,2092,2095],{"className":2091},[48],[27,2093,50],{"className":2094},[48,49],[27,2096,2098],{"className":2097},[662],[27,2099,2101,2127],{"className":2100},[251,252],[27,2102,2104,2124],{"className":2103},[256],[27,2105,2107],{"className":2106,"style":1968},[260],[27,2108,2109,2112],{"style":1568},[27,2110],{"className":2111,"style":679},[268],[27,2113,2115],{"className":2114},[683,684,337,685],[27,2116,2118,2121],{"className":2117},[48,685],[27,2119,87],{"className":2120,"style":1881},[48,163,685],[27,2122,1898],{"className":2123},[48,163,685],[27,2125,306],{"className":2126},[305],[27,2128,2130],{"className":2129},[256],[27,2131,2133],{"className":2132,"style":1995},[260],[27,2134],{},[27,2136,2137,2140],{"style":375},[27,2138],{"className":2139,"style":269},[268],[27,2141,2143],{"className":2142},[48],[27,2144,2146,2149],{"className":2145},[48],[27,2147,50],{"className":2148},[48,49],[27,2150,2152],{"className":2151},[662],[27,2153,2155,2179],{"className":2154},[251,252],[27,2156,2158,2176],{"className":2157},[256],[27,2159,2161],{"className":2160,"style":1565},[260],[27,2162,2163,2166],{"style":1568},[27,2164],{"className":2165,"style":679},[268],[27,2167,2169],{"className":2168},[683,684,337,685],[27,2170,2172],{"className":2171},[48,685],[27,2173,2175],{"className":2174},[48,163,685],"ss",[27,2177,306],{"className":2178},[305],[27,2180,2182],{"className":2181},[256],[27,2183,2185],{"className":2184,"style":698},[260],[27,2186],{},[27,2188,306],{"className":2189},[305],[27,2191,2193],{"className":2192},[256],[27,2194,2196],{"className":2195,"style":398},[260],[27,2197],{},[27,2199,2201],{"className":2200,"style":332},[319,331],[27,2202,463],{"className":2203},[336,337],[27,2205],{"className":2206,"style":323},[112],[27,2208,2210,2216,2341],{"className":2209},[327],[27,2211,2213],{"class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me":2914},[256],[27,2916,2918],{"className":2917,"style":698},[260],[27,2919],{},[27,2921,2922,2925],{"style":2727},[27,2923],{"className":2924,"style":269},[268],[27,2926,2928,2931,2934,2937,2940,2986,2989,3029,3032,3035,3038,3081,3084,3124,3127,3130,3133],{"className":2927},[48],[27,2929],{"className":2930},[48],[27,2932],{"className":2933,"style":113},[112],[27,2935,118],{"className":2936},[117],[27,2938],{"className":2939,"style":113},[112],[27,2941,2943,2946],{"className":2942},[48],[27,2944,50],{"className":2945},[48,49],[27,2947,2949],{"className":2948},[662],[27,2950,2952,2978],{"className":2951},[251,252],[27,2953,2955,2975],{"className":2954},[256],[27,2956,2958],{"className":2957,"style":1968},[260],[27,2959,2960,2963],{"style":1568},[27,2961],{"className":2962,"style":679},[268],[27,2964,2966],{"className":2965},[683,684,337,685],[27,2967,2969,2972],{"className":2968},[48,685],[27,2970,1898],{"className":2971},[48,163,685],[27,2973,87],{"className":2974,"style":1881},[48,163,685],[27,2976,306],{"className":2977},[305],[27,2979,2981],{"className":2980},[256],[27,2982,2984],{"className":2983,"style":1995},[260],[27,2985],{},[27,2987],{"className":2988,"style":323},[112],[27,2990,2992,2995],{"className":2991},[48],[27,2993,68],{"className":2994},[48,49],[27,2996,2998],{"className":2997},[662],[27,2999,3001,3021],{"className":3000},[251,252],[27,3002,3004,3018],{"className":3003},[256],[27,3005,3007],{"className":3006,"style":1968},[260],[27,3008,3009,3012],{"style":1568},[27,3010],{"className":3011,"style":679},[268],[27,3013,3015],{"className":3014},[683,684,337,685],[27,3016,87],{"className":3017,"style":1881},[48,163,685],[27,3019,306],{"className":3020},[305],[27,3022,3024],{"className":3023},[256],[27,3025,3027],{"className":3026,"style":1995},[260],[27,3028],{},[27,3030],{"className":3031,"style":2488},[112],[27,3033,2493],{"className":3034},[2492],[27,3036],{"className":3037,"style":2488},[112],[27,3039,3041,3044],{"className":3040},[48],[27,3042,50],{"className":3043},[48,49],[27,3045,3047],{"className":3046},[662],[27,3048,3050,3073],{"className":3049},[251,252],[27,3051,3053,3070],{"className":3052},[256],[27,3054,3056],{"className":3055,"style":1565},[260],[27,3057,3058,3061],{"style":1568},[27,3059],{"className":3060,"style":679},[268],[27,3062,3064],{"className":3063},[683,684,337,685],[27,3065,3067],{"className":3066},[48,685],[27,3068,2175],{"className":3069},[48,163,685],[27,3071,306],{"className":3072},[305],[27,3074,3076],{"className":3075},[256],[27,3077,3079],{"className":3078,"style":698},[260],[27,3080],{},[27,3082],{"className":3083,"style":323},[112],[27,3085,3087,3090],{"className":3086},[48],[27,3088,68],{"className":3089},[48,49],[27,3091,3093],{"className":3092},[662],[27,3094,3096,3116],{"className":3095},[251,252],[27,3097,3099,3113],{"className":3098},[256],[27,3100,3102],{"className":3101,"style":1565},[260],[27,3103,3104,3107],{"style":1568},[27,3105],{"className":3106,"style":679},[268],[27,3108,3110],{"className":3109},[683,684,337,685],[27,3111,1898],{"className":3112},[48,163,685],[27,3114,306],{"className":3115},[305],[27,3117,3119],{"className":3118},[256],[27,3120,3122],{"className":3121,"style":698},[260],[27,3123],{},[27,3125],{"className":3126,"style":2488},[112],[27,3128,385],{"className":3129},[2492],[27,3131],{"className":3132,"style":2488},[112],[27,3134,3136,3139],{"className":3135},[48],[27,3137,87],{"className":3138,"style":86},[48,49],[27,3140,3142],{"className":3141},[662],[27,3143,3145,3165],{"className":3144},[251,252],[27,3146,3148,3162],{"className":3147},[256],[27,3149,3151],{"className":3150,"style":1565},[260],[27,3152,3153,3156],{"style":675},[27,3154],{"className":3155,"style":679},[268],[27,3157,3159],{"className":3158},[683,684,337,685],[27,3160,1898],{"className":3161},[48,163,685],[27,3163,306],{"className":3164},[305],[27,3166,3168],{"className":3167},[256],[27,3169,3171],{"className":3170,"style":698},[260],[27,3172],{},[27,3174,306],{"className":3175},[305],[27,3177,3179],{"className":3178},[256],[27,3180,3182],{"className":3181,"style":2746},[260],[27,3183],{},[11,3185,3186,3187,3190],{},"A ",[23,3188,3189],{},"spring"," does not remove the degree of freedom; it adds its stiffness to the diagonal.",[11,3192,3193,3196],{},[23,3194,3195],{},"Releases"," modify an element before assembly. A full release decouples that degree of\nfreedom from the node, so the member carries no moment across that end; a partial release\nputs a finite stiffness between the two, which is the semi-rigid connection.",[11,3198,3199,3202],{},[23,3200,3201],{},"Rigid arms"," are constraints that tie one node's degrees of freedom to another's by a\nkinematic relation, applied at assembly instead of as a very stiff element. That is why\nthey do not damage conditioning the way a stiff element does.",[11,3204,3205,3206,3221,3222,3226],{},"After this step, ",[27,3207,3209],{"className":3208},[30],[27,3210,3212],{"className":3211,"ariaHidden":35},[34],[27,3213,3215,3218],{"className":3214},[39],[27,3216],{"className":3217,"style":44},[43],[27,3219,50],{"className":3220},[48,49]," is singular if and only if the structure can move without\nresistance: a ",[142,3223,3225],{"href":3224},"\u002Fkb\u002Ftroubleshooting\u002Fsingular-stiffness-matrix","mechanism",". Elementarium\nchecks for that before solving, and names the free motion.",[15,3228,3230],{"id":3229},"_5-solution","5. Solution",[11,3232,3233,3266,3267,3282,3283,3328,3329,3374,3375,1344],{},[27,3234,3236],{"className":3235},[30],[27,3237,3239,3257],{"className":3238,"ariaHidden":35},[34],[27,3240,3242,3245,3248,3251,3254],{"className":3241},[39],[27,3243],{"className":3244,"style":44},[43],[27,3246,108],{"className":3247},[48,49],[27,3249],{"className":3250,"style":113},[112],[27,3252,118],{"className":3253},[117],[27,3255],{"className":3256,"style":113},[112],[27,3258,3260,3263],{"className":3259},[39],[27,3261],{"className":3262,"style":82},[43],[27,3264,87],{"className":3265,"style":86},[48,49]," is solved once per load state. With a direct solver\n",[27,3268,3270],{"className":3269},[30],[27,3271,3273],{"className":3272,"ariaHidden":35},[34],[27,3274,3276,3279],{"className":3275},[39],[27,3277],{"className":3278,"style":44},[43],[27,3280,50],{"className":3281},[48,49]," is factorised, as ",[27,3284,3286],{"className":3285},[30],[27,3287,3289],{"className":3288,"ariaHidden":35},[34],[27,3290,3292,3296,3299],{"className":3291},[39],[27,3293],{"className":3294,"style":3295},[43],"height:0.8491em;",[27,3297,164],{"className":3298},[48,49],[27,3300,3302,3305],{"className":3301},[48],[27,3303,164],{"className":3304},[48,49],[27,3306,3308],{"className":3307},[662],[27,3309,3311],{"className":3310},[251],[27,3312,3314],{"className":3313},[256],[27,3315,3317],{"className":3316,"style":3295},[260],[27,3318,3319,3322],{"style":1840},[27,3320],{"className":3321,"style":679},[268],[27,3323,3325],{"className":3324},[683,684,337,685],[27,3326,1191],{"className":3327},[48,1292,685]," or\n",[27,3330,3332],{"className":3331},[30],[27,3333,3335],{"className":3334,"ariaHidden":35},[34],[27,3336,3338,3341,3345],{"className":3337},[39],[27,3339],{"className":3340,"style":3295},[43],[27,3342,3344],{"className":3343},[48,49],"LD",[27,3346,3348,3351],{"className":3347},[48],[27,3349,164],{"className":3350},[48,49],[27,3352,3354],{"className":3353},[662],[27,3355,3357],{"className":3356},[251],[27,3358,3360],{"className":3359},[256],[27,3361,3363],{"className":3362,"style":3295},[260],[27,3364,3365,3368],{"style":1840},[27,3366],{"className":3367,"style":679},[268],[27,3369,3371],{"className":3370},[683,684,337,685],[27,3372,1191],{"className":3373},[48,1292,685],", and each load case or combination is a cheap\nresolve against the factors. That is why fifty combinations cost barely more than one.\nWith an iterative solver the system is converged separately for each right-hand side. See\n",[142,3376,1738],{"href":1737},[11,3378,3379,3380,3395],{},"The result is ",[27,3381,3383],{"className":3382},[30],[27,3384,3386],{"className":3385,"ariaHidden":35},[34],[27,3387,3389,3392],{"className":3388},[39],[27,3390],{"className":3391,"style":64},[43],[27,3393,68],{"className":3394},[48,49],": every translation and rotation at every node, for that load\nstate.",[15,3397,3399],{"id":3398},"_6-recovery","6. Recovery",[11,3401,3402,3403,3460],{},"Internal forces come from the displacements, element by element: transform the element's\nnodal displacements back to its local axes, multiply by its local stiffness, and add back\nthe fixed-end forces ",[27,3404,3406],{"className":3405},[30],[27,3407,3409],{"className":3408,"ariaHidden":35},[34],[27,3410,3412,3416],{"className":3411},[39],[27,3413],{"className":3414,"style":3415},[43],"height:0.8444em;vertical-align:-0.15em;",[27,3417,3419,3422],{"className":3418},[48],[27,3420,87],{"className":3421,"style":86},[48,49],[27,3423,3425],{"className":3424},[662],[27,3426,3428,3452],{"className":3427},[251,252],[27,3429,3431,3449],{"className":3430},[256],[27,3432,3434],{"className":3433,"style":672},[260],[27,3435,3436,3439],{"style":675},[27,3437],{"className":3438,"style":679},[268],[27,3440,3442],{"className":3441},[683,684,337,685],[27,3443,3445],{"className":3444},[48,685],[27,3446,3448],{"className":3447},[48,685],"0",[27,3450,306],{"className":3451},[305],[27,3453,3455],{"className":3454},[256],[27,3456,3458],{"className":3457,"style":698},[260],[27,3459],{}," from any load along its length:",[27,3462,3464],{"className":3463},[92],[27,3465,3467],{"className":3466},[30],[27,3468,3470,3528,3598],{"className":3469,"ariaHidden":35},[34],[27,3471,3473,3476,3519,3522,3525],{"className":3472},[39],[27,3474],{"className":3475,"style":3415},[43],[27,3477,3479,3482],{"className":3478},[48],[27,3480,87],{"className":3481,"style":86},[48,49],[27,3483,3485],{"className":3484},[662],[27,3486,3488,3511],{"className":3487},[251,252],[27,3489,3491,3508],{"className":3490},[256],[27,3492,3494],{"className":3493,"style":1565},[260],[27,3495,3496,3499],{"style":675},[27,3497],{"className":3498,"style":679},[268],[27,3500,3502],{"className":3501},[683,684,337,685],[27,3503,3505],{"className":3504},[48,685],[27,3506,1433],{"className":3507},[48,163,685],[27,3509,306],{"className":3510},[305],[27,3512,3514],{"className":3513},[256],[27,3515,3517],{"className":3516,"style":698},[260],[27,3518],{},[27,3520],{"className":3521,"style":113},[112],[27,3523,118],{"className":3524},[117],[27,3526],{"className":3527,"style":113},[112],[27,3529,3531,3534,3537,3540,3543,3546,3589,3592,3595],{"className":3530},[39],[27,3532],{"className":3533,"style":3415},[43],[27,3535,219],{"className":3536},[48,49],[27,3538],{"className":3539,"style":323},[112],[27,3541,1191],{"className":3542},[48,49],[27,3544],{"className":3545,"style":323},[112],[27,3547,3549,3552],{"className":3548},[48],[27,3550,68],{"className":3551},[48,49],[27,3553,3555],{"className":3554},[662],[27,3556,3558,3581],{"className":3557},[251,252],[27,3559,3561,3578],{"className":3560},[256],[27,3562,3564],{"className":3563,"style":1565},[260],[27,3565,3566,3569],{"style":1568},[27,3567],{"className":3568,"style":679},[268],[27,3570,3572],{"className":3571},[683,684,337,685],[27,3573,3575],{"className":3574},[48,685],[27,3576,1433],{"className":3577},[48,163,685],[27,3579,306],{"className":3580},[305],[27,3582,3584],{"className":3583},[256],[27,3585,3587],{"className":3586,"style":698},[260],[27,3588],{},[27,3590],{"className":3591,"style":2488},[112],[27,3593,2493],{"className":3594},[2492],[27,3596],{"className":3597,"style":2488},[112],[27,3599,3601,3604],{"className":3600},[39],[27,3602],{"className":3603,"style":3415},[43],[27,3605,3607,3610],{"className":3606},[48],[27,3608,87],{"className":3609,"style":86},[48,49],[27,3611,3613],{"className":3612},[662],[27,3614,3616,3639],{"className":3615},[251,252],[27,3617,3619,3636],{"className":3618},[256],[27,3620,3622],{"className":3621,"style":672},[260],[27,3623,3624,3627],{"style":675},[27,3625],{"className":3626,"style":679},[268],[27,3628,3630],{"className":3629},[683,684,337,685],[27,3631,3633],{"className":3632},[48,685],[27,3634,3448],{"className":3635},[48,685],[27,3637,306],{"className":3638},[305],[27,3640,3642],{"className":3641},[256],[27,3643,3645],{"className":3644,"style":698},[260],[27,3646],{},[11,3648,3649,3650,3666,3667,3666,3683,1140,3699,3714,3715,3719],{},"For a beam that gives ",[27,3651,3653],{"className":3652},[30],[27,3654,3656],{"className":3655,"ariaHidden":35},[34],[27,3657,3659,3662],{"className":3658},[39],[27,3660],{"className":3661,"style":159},[43],[27,3663,3665],{"className":3664,"style":86},[48,163],"N",", ",[27,3668,3670],{"className":3669},[30],[27,3671,3673],{"className":3672,"ariaHidden":35},[34],[27,3674,3676,3679],{"className":3675},[39],[27,3677],{"className":3678,"style":159},[43],[27,3680,3682],{"className":3681,"style":2488},[48,163],"V",[27,3684,3686],{"className":3685},[30],[27,3687,3689],{"className":3688,"ariaHidden":35},[34],[27,3690,3692,3695],{"className":3691},[39],[27,3693],{"className":3694,"style":159},[43],[27,3696,1191],{"className":3697,"style":3698},[48,163],"margin-right:0.1389em;",[27,3700,3702],{"className":3701},[30],[27,3703,3705],{"className":3704,"ariaHidden":35},[34],[27,3706,3708,3711],{"className":3707},[39],[27,3709],{"className":3710,"style":159},[43],[27,3712,658],{"className":3713,"style":86},[48,163]," at the ends. Between the ends they follow by\nstatics from the end forces and the load on the member, which is why the moment under a\nuniform load comes out parabolic although the shape function is cubic. For a shell, stress resultants are evaluated at the Gauss\npoints and carried to the corners by the\n",[142,3716,3718],{"href":3717},"\u002Fkb\u002Ftheory\u002Felement-formulations#integration-and-result-recovery","recovery method"," the\nanalysis setup chooses.",[11,3721,3722,3723,1344],{},"Reactions are the forces at the restrained degrees of freedom, and their sum must equal\nthe applied load: the check described in\n",[142,3724,3726],{"href":3725},"\u002Fkb\u002Fgetting-started\u002Ffirst-analysis","Your first analysis",[15,3728,3730],{"id":3729},"where-the-other-analyses-depart","Where the other analyses depart",[3732,3733,3734,3747],"table",{},[3735,3736,3737],"thead",{},[3738,3739,3740,3744],"tr",{},[3741,3742,3743],"th",{},"Analysis",[3741,3745,3746],{},"What changes",[3748,3749,3750,3881,4056],"tbody",{},[3738,3751,3752,3759],{},[3753,3754,3755],"td",{},[142,3756,3758],{"href":3757},"\u002Fkb\u002Fanalysis\u002Fmodal","Modal",[3753,3760,3761,3762,3777,3778,3880],{},"A mass matrix ",[27,3763,3765],{"className":3764},[30],[27,3766,3768],{"className":3767,"ariaHidden":35},[34],[27,3769,3771,3774],{"className":3770},[39],[27,3772],{"className":3773,"style":44},[43],[27,3775,658],{"className":3776},[48,49]," is assembled the same way; 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That is why a modelling error in step 3, a member that does not\nconnect, appears in every analysis type at once.",[15,4148,4150],{"id":4149},"what-the-method-assumes","What the method assumes",[4152,4153,4154,4193,4199,4205],"ul",{},[4155,4156,4157,4160,4161,4176,4177,4192],"li",{},[23,4158,4159],{},"Linearity."," ",[27,4162,4164],{"className":4163},[30],[27,4165,4167],{"className":4166,"ariaHidden":35},[34],[27,4168,4170,4173],{"className":4169},[39],[27,4171],{"className":4172,"style":44},[43],[27,4174,50],{"className":4175},[48,49]," does not depend on ",[27,4178,4180],{"className":4179},[30],[27,4181,4183],{"className":4182,"ariaHidden":35},[34],[27,4184,4186,4189],{"className":4185},[39],[27,4187],{"className":4188,"style":64},[43],[27,4190,68],{"className":4191},[48,49],"; the material stays elastic\nand the geometry stays where it was drawn. The exceptions are the three rows above.",[4155,4194,4195,4198],{},[23,4196,4197],{},"Small displacements."," Equilibrium is written on the undeformed shape, except in\nsecond-order analysis.",[4155,4200,4201,4204],{},[23,4202,4203],{},"Compatibility at nodes only."," Elements are joined at their nodes and nowhere else.\nTwo surfaces that overlap without sharing nodes pass through each other.",[4155,4206,4207,4210,4211,4215],{},[23,4208,4209],{},"The element's shape functions."," A beam is exact; a shell is an approximation that\nconverges with refinement. ",[142,4212,4214],{"href":4213},"\u002Fkb\u002Fmeshing","Meshing"," is the consequence.",[11,4217,4218],{},"The method is the one taught in every structural analysis course and implemented in every\nfinite element package.",{"title":4220,"searchDepth":4221,"depth":4221,"links":4222},"",2,[4223,4224,4225,4226,4227,4228,4229,4230,4231],{"id":17,"depth":4221,"text":18},{"id":136,"depth":4221,"text":137},{"id":1171,"depth":4221,"text":1172},{"id":1347,"depth":4221,"text":1348},{"id":1857,"depth":4221,"text":1858},{"id":3229,"depth":4221,"text":3230},{"id":3398,"depth":4221,"text":3399},{"id":3729,"depth":4221,"text":3730},{"id":4149,"depth":4221,"text":4150},"How a finite element solver turns a model into K u = f and back — stiffness, transformation, assembly, constraints, solution and recovery.","md",false,{},true,null,4,"\u002Fkb\u002Ftheory\u002Fdirect-stiffness-method",{"title":6,"description":4232},{"loc":4239},"kb\u002Ftheory\u002Fdirect-stiffness-method","kT_p03ADx4RJu1-Wlr_GVpKwy4Q4DIy1JoGJDJZo0YY",{"sections":4245,"extras":4544,"flat":4557},[4246,4279,4322,4339,4374,4399,4426,4457,4484,4507,4521],{"path":4247,"slug":4248,"title":4249,"description":4250,"icon":4251,"articles":4252},"\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",[4253,4257,4261,4265,4269,4273,4277],{"path":4254,"title":4255,"description":4256,"translated":4236},"\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":4258,"title":4259,"description":4260,"translated":4236},"\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":4262,"title":4263,"description":4264,"translated":4236},"\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":4266,"title":4267,"description":4268,"translated":4236},"\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":4270,"title":4271,"description":4272,"translated":4236},"\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":4274,"title":4275,"description":4276,"translated":4236},"\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":3725,"title":3726,"description":4278,"translated":4236},"Load cases, combinations and a linear static solve — then how to decide whether the answer is trustworthy.",{"path":4280,"slug":4281,"title":4282,"description":4283,"icon":4284,"articles":4285},"\u002Fkb\u002Fmodeling","modeling","Modeling","Bars, beams, 3D frames, slabs and walls. Geometry, sections, supports and releases that behave like the real structure.","lucide:frame",[4286,4290,4294,4298,4302,4306,4310,4314,4318],{"path":4287,"title":4288,"description":4289,"translated":4236},"\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":4291,"title":4292,"description":4293,"translated":4236},"\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":4295,"title":4296,"description":4297,"translated":4236},"\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":4299,"title":4300,"description":4301,"translated":4236},"\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":4303,"title":4304,"description":4305,"translated":4236},"\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":4307,"title":4308,"description":4309,"translated":4236},"\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":4311,"title":4312,"description":4313,"translated":4236},"\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":4315,"title":4316,"description":4317,"translated":4236},"\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":4319,"title":4320,"description":4321,"translated":4236},"\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":4213,"slug":4323,"title":4214,"description":4324,"icon":4325,"articles":4326},"meshing","Mesh density, local refinement and the quality checks that tell you when a mesh is fine enough to trust.","lucide:grid-3x3",[4327,4331,4335],{"path":4328,"title":4329,"description":4330,"translated":4236},"\u002Fkb\u002Fmeshing\u002Felement-size","Choosing element size","Starting mesh densities for slabs, walls and rafts, and the convergence test that decides whether a mesh is fine enough.",{"path":4332,"title":4333,"description":4334,"translated":4236},"\u002Fkb\u002Fmeshing\u002Frefinement","Local refinement","Where to spend elements, where not to bother, and how to refine without wrecking element shape.",{"path":4336,"title":4337,"description":4338,"translated":4236},"\u002Fkb\u002Fmeshing\u002Fquality-checks","Mesh quality","Aspect ratio, skew and warping — what degrades a result, what the mesher reports, and how to tell whether the elements you are worried about matter.",{"path":4340,"slug":4341,"title":4342,"description":4343,"icon":4344,"articles":4345},"\u002Fkb\u002Floads","loads","Loads & combinations","Build load cases, generate code combinations automatically and find out which one actually governs.","lucide:arrow-down-to-line",[4346,4350,4354,4358,4362,4366,4370],{"path":4347,"title":4348,"description":4349,"translated":4236},"\u002Fkb\u002Floads\u002Fasce-7-combinations","ASCE 7 combinations","LRFD and ASD combinations to ASCE 7-22 — how the generator maps your load groups onto D, L, Lr, S, W and E, and pairing them with the AISC check.",{"path":4351,"title":4352,"description":4353,"translated":4236},"\u002Fkb\u002Floads\u002Fcombinations","Automatic combinations","Generating EN 1990 and ASCE 7 combinations, understanding the equations behind them, and reading the generated set critically.",{"path":4355,"title":4356,"description":4357,"translated":4236},"\u002Fkb\u002Floads\u002Fenvelopes","Envelopes","Extracting the governing value across dozens of combinations without reading every one — and the traps in doing so.",{"path":4359,"title":4360,"description":4361,"translated":4236},"\u002Fkb\u002Floads\u002Fload-cases","Load cases","Actions, self-weight, imposed loads and load arrangements — how to build cases that stay separable.",{"path":4363,"title":4364,"description":4365,"translated":4236},"\u002Fkb\u002Floads\u002Fload-panels","Load panels","Areas that hand a floor, roof or wall pressure to the members around them as line loads, one way or two way, without meshing a slab.",{"path":4367,"title":4368,"description":4369,"translated":4236},"\u002Fkb\u002Floads\u002Ftemperature-loads","Temperature loads","Uniform temperature change and through-depth gradients — what each does to a free and a restrained structure, and the coefficients to use.",{"path":4371,"title":4372,"description":4373,"translated":4236},"\u002Fkb\u002Floads\u002Fwind-and-snow","Wind and snow","Wind to EN 1991-1-4 and snow to EN 1991-1-3, generated on a building envelope — site inputs, recognised roof shapes, the cases produced, and the limits.",{"path":4375,"slug":4376,"title":4377,"description":4378,"icon":4379,"articles":4380},"\u002Fkb\u002Fanalysis","analysis","Analysis types","Linear static, modal, linear buckling and second-order — what each analysis assumes, what it returns and how to read the result.","lucide:activity",[4381,4383,4385,4389,4392,4395],{"path":1737,"title":1738,"description":4382,"translated":4236},"Conjugate gradients with Jacobi, incomplete Cholesky or multigrid, or a direct factorisation — what each does, how to watch it, and when the choice matters.",{"path":3887,"title":3888,"description":4384,"translated":4236},"Critical load factors and buckling shapes — what α_cr means, why it is an upper bound, and how to use it for second-order effects.",{"path":4386,"title":4387,"description":4388,"translated":4236},"\u002Fkb\u002Fanalysis\u002Flinear-static","Linear static analysis","The workhorse analysis — what K u = f actually assumes, what it returns, and where linearity stops being a reasonable model.",{"path":3757,"title":4390,"description":4391,"translated":4236},"Modal analysis","Natural frequencies and mode shapes — mass modelling, participation factors and what the numbers are good for.",{"path":4062,"title":4393,"description":4394,"translated":4236},"Second-order analysis","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.",{"path":4396,"title":4397,"description":4398,"translated":4236},"\u002Fkb\u002Fanalysis\u002Fverification","Verifying the solver","Closed-form cases you can build in ten minutes to test the solver — beams, frames, columns, plates, frequencies — with the expected error of each.",{"path":4400,"slug":4401,"title":4402,"description":4403,"icon":4404,"articles":4405},"\u002Fkb\u002Fdesign-checks","design-checks","Design checks","Utilisation ratios, governing cases and the clause behind every number a check reports.","lucide:clipboard-check",[4406,4410,4414,4418,4422],{"path":4407,"title":4408,"description":4409,"translated":4236},"\u002Fkb\u002Fdesign-checks\u002Fnational-annexes","National annexes","Where nationally determined parameters change the answer, how to select the right annex, and what to state in the report.",{"path":4411,"title":4412,"description":4413,"translated":4236},"\u002Fkb\u002Fdesign-checks\u002Fsection-classification","Section classification","How EN 1993-1-1 Table 5.2 classifies a steel section, why the class depends on the combination, and what the check does with shapes it cannot classify.",{"path":4415,"title":4416,"description":4417,"translated":4236},"\u002Fkb\u002Fdesign-checks\u002Fserviceability","Serviceability checks","Deflection, drift and floor frequency checked member by member at SLS, to the national annexes, EN 1990:2023, IBC, AISC or EN 1995-1-1.",{"path":4419,"title":4420,"description":4421,"translated":4236},"\u002Fkb\u002Fdesign-checks\u002Faisc","Steel — AISC 360","Member strength to Chapters D to H, LRFD or ASD — compression buckling, lateral-torsional buckling and every C_b option written 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