31:T6f1,{ const height = 300; const s = 20; const svg = d3.select(DOM.svg(width, height)); const scaleY = d3.scaleLinear().domain([0, 3]).range([height / 2 + 75, height / 2 - 75]); const line = d3.line().x(d => scaleX(d.x)).y(d => scaleY(d.y)); // undeformed supports svg.append("g") .attr("stroke", "black") .attr("stroke-width", 0.5) .attr("transform", `translate(${scaleX(X1.x)}, ${scaleY(X1.y)}) scale(1.5)`) .call(drawing.support); svg.append("g") .attr("stroke", "black") .attr("stroke-width", 0.5) .attr("transform", `translate(${scaleX(X2.x)}, ${scaleY(X2.y)}) scale(1.5) rotate(${dof == "u" ? 0 : -90})`) .call(drawing.slidingSupport); // undeformed truss svg.append("path") .attr("d", line([X1, X2])) .attr("stroke-width", 2) .attr("stroke", "black"); // undeformed nodes svg.append("g") .attr("class", "X1") .attr("stroke", "black") .attr("stroke-width", 0.5) .attr("transform", `translate(${scaleX(X1.x)}, ${scaleY(X1.y)}) scale(1.5)`) .call(drawing.node); svg.append("g") .attr("class", "X2") .attr("stroke", "black") .attr("stroke-width", 0.5) .attr("transform", `translate(${scaleX(X2.x)}, ${scaleY(X2.y)}) scale(1.5)`) .call(drawing.node); // external force svg.append("polyline") .attr("class", "f") .attr("stroke-width", 3) .attr("fill", "green") .attr("stroke", "green"); if (f == 0) { svg.select(".f") .attr("points", ""); } else if (dof == "u") { svg.select(".f") .call(arrow, scaleX(X2.x - f * s), scaleY(X2.y), scaleX(X2.x), scaleY(X2.y)); } else { svg.select(".f") .call(arrow, scaleX(X2.x), scaleY(X2.y - f * s), scaleX(X2.x), scaleY(X2.y)); } return svg.node(); }32:T405,lang({ de: md`Für dieses einfache System können wir die Residualkraft in Abhängigkeit von der Verschiebung ${tex`${dof}`} in einem Last-Verschiebungs-Diagramm darstellen (${svg``}, weiter unten). Die Residualkraft an der Stelle ${tex`${dof}`} ist im Last-Verschiebungs-Diagram als ${svg``} eingezeichnet.`, en: md`For this simple system we can plot the residual force w.r.t. the displacement ${tex`${dof}`} within a load-displacement-diagram (${svg``}, below). The residual force at ${tex`${dof}`} is visualized in the load-displacement-diagram as ${svg``}.`, })33:T446,{ const result = linearResult; const min = scaleX.invert(0) - X2.x; const max = scaleX.invert(width) - X2.x; const results = dof == "u" ? linspace(min, max, 100).map(d => linear(vec(X2.x + d, X2.y))) : linspace(min, max, 100).map(d => linear(vec(X2.x, X2.y + d))); const plots = []; if (showLoadDisplacement.includes("Linearized")) { plots.push( Plot.line(results.map(d => [d.d, result.r + result.k * (d.d - result.d)]), {stroke: "green"}), Plot.dot([[result.d - result.r / result.k, 0]], {stroke: "green", fill: "white"}), ); } plots.push( Plot.line([[result.d, 0], [result.d, result.r]], {stroke: "red"}), Plot.dot([[result.d, result.r]], {stroke: "red", fill: "white"}), ); return Plot.plot({ inset: 10, width, grid: true, x: { label: `Displacement ${dof} →`, }, y: { label: `↑ Residual r_${dof}`, domain: [-1, 1], }, marks: [ Plot.line(results.map(d => [d.d, d.r]), {stroke: "lightgray"}), Plot.ruleY([0], {stroke: "black"}), ...plots, ] }); }34:T49c,{ const result = nonlinearResult; const min = scaleX.invert(0) - X2.x; const max = scaleX.invert(width) - X2.x; const results = dof == "u" ? linspace(min, max, 100).map(d => nonlinear(vec(X2.x + d, X2.y))) : linspace(min, max, 100).map(d => nonlinear(vec(X2.x, X2.y + d))); const plots = []; if (showLoadDisplacement.includes("Linearized")) { plots.push( Plot.line(results.map(d => [d.d, result.r + result.k * (d.d - result.d)]), {stroke: "green"}), Plot.dot([[result.d - result.r / result.k, 0]], {stroke: "green", fill: "white"}), ); } plots.push( Plot.line([[result.d, 0], [result.d, result.r]], {stroke: "red"}), Plot.dot([[result.d, result.r]], {stroke: "red", fill: "white"}), ); const k0 = result.d - result.r / result.k; return Plot.plot({ inset: 10, width, grid: true, x: { label: `Displacement ${dof} →`, domain: [min, max], }, y: { label: `↑ Residual r_${dof}`, domain: [-0.2, 0.2], }, marks: [ Plot.line(results.map(d => [d.d, d.r]), {stroke: "lightgray"}), Plot.ruleY([0], {stroke: "black"}), ...plots, ] }); }35:T1015,generatePlot = function (model) { const svg = d3.select(DOM.svg(width, height)); // contour lines const color = d3.scaleLinear() .domain([-2, 0, 2]) .range(["blue", "black", "red"]) .interpolate(d3.interpolateRgb.gamma(2.2)); const isolines = svg.append("g") .attr("fill", "none") .attr("stroke-width", 0.5) .attr("stroke-opacity", 0.2) .attr("stroke-dasharray", [2, 1]); const t = unit(G1); const n = perp(t); if (model === linear) { for (const eps of d3.ticks(-3, 3, 20)) { const l = (1 + eps) * L; isolines.append("line") .attr("x1", scaleX(X1.x + t.x * l + n.x * 10)) .attr("y1", scaleY(X1.y + t.y * l + n.y * 10)) .attr("x2", scaleX(X1.x + t.x * l - n.x * 10)) .attr("y2", scaleY(X1.y + t.y * l - n.y * 10)) .attr("stroke", color(eps)); } } if (model === nonlinear) { for (const eps of d3.ticks(-3, 3, 20)) { const l = Math.sqrt(1 + 2 * eps) * L; isolines.append("circle") .attr("cx", scaleX(X1.x)) .attr("cy", scaleY(X1.y)) .attr("r", scaleX(X1.x + l) - scaleX(X1.x)) .attr("stroke", color(eps)); } } // undeformed supports svg.append("g") .attr("stroke", "lightgray") .attr("stroke-width", 0.5) .attr("transform", `translate(${scaleX(X1.x)}, ${scaleY(X1.y)}) scale(1.5)`) .call(drawing.support); svg.append("g") .attr("stroke", "lightgray") .attr("stroke-width", 0.5) .attr("transform", `translate(${scaleX(X2.x)}, ${scaleY(X2.y)}) scale(1.5) rotate(${dof == "u" ? 0 : -90})`) .call(drawing.slidingSupport); // undeformed truss svg.append("path") .attr("d", line([X1, X2])) .attr("stroke-width", 2) .attr("stroke", "lightgray"); // undeformed nodes svg.append("g") .attr("class", "X1") .attr("stroke", "lightgray") .attr("stroke-width", 0.5) .attr("transform", `translate(${scaleX(X1.x)}, ${scaleY(X1.y)}) scale(1.5)`) .call(drawing.node); svg.append("g") .attr("class", "X2") .attr("stroke", "lightgray") .attr("stroke-width", 0.5) .attr("transform", `translate(${scaleX(X2.x)}, ${scaleY(X2.y)}) scale(1.5)`) .call(drawing.node); // deformed supports svg.append("g") .attr("class", "x1") .attr("stroke", "black") .attr("stroke-width", 0.5) .attr("transform", `translate(${scaleX(x1.x)}, ${scaleY(x1.y)}) scale(1.5)`) .call(drawing.support); svg.append("g") .attr("class", "x2") .attr("stroke", "black") .attr("stroke-width", 0.5) .call(drawing.slidingSupport); // deformed truss svg.append("path") .attr("class", "deformed") .attr("stroke-width", 2) .attr("stroke", "black"); // deformed residual force svg.append("polyline") .attr("class", "r") .attr("stroke-width", 3) .attr("fill", "red") .attr("stroke", "red"); // deformed internal force svg.append("polyline") .attr("class", "n") .attr("stroke-width", 3) .attr("fill", "purple") .attr("stroke", "purple"); // deformed external force svg.append("polyline") .attr("class", "f") .attr("stroke-width", 3) .attr("fill", "green") .attr("stroke", "green"); // deformed nodes svg.append("g") .attr("class", "x1") .attr("stroke", "black") .attr("stroke-width", 0.5) .attr("transform", `translate(${scaleX(x1.x)}, ${scaleY(x1.y)}) scale(1.5)`) .call(drawing.node); svg.append("g") .attr("class", "x2") .attr("stroke", "black") .attr("stroke-width", 0.5) .call(drawing.node); // hide border svg.append("rect") .attr("width", width) .attr("height", height) .attr("fill", "none") .attr("stroke-width", 3) .attr("stroke", "white"); // interaction const ondrag = (event) => { if (dof == "u") { mutable x2 = vec(scaleX.invert(clamp(0, width, event.x)), X2.y); } else { mutable x2 = vec(X2.x, scaleY.invert(clamp(0, height, event.y))); } }; svg.selectAll(".x2") .call(d3.drag().on("drag", ondrag)); return svg; }36:T6a5,updatePlot = function (svg, result) { svg.select(".deformed") .attr("d", line([x1, x2])); const s = 20; const _x2 = svg == linearPlot ? X2 : x2; if (Math.abs(result.r) < 1e-4) { svg.select(".r") .attr("points", ""); } else if (dof == "u") { svg.select(".r") .call(arrow, scaleX(_x2.x), scaleY(_x2.y), scaleX(_x2.x + result.r * s), scaleY(_x2.y)); } else { svg.select(".r") .call(arrow, scaleX(_x2.x), scaleY(_x2.y), scaleX(_x2.x), scaleY(_x2.y + result.r * s)); } const t = unit(G1); const n = perp(t); if (Math.abs(result.f) < 1e-4) { svg.select(".f") .attr("points", ""); } else if (dof == "u") { svg.select(".f") .call(arrow, scaleX(_x2.x - result.f * s + n.x * 0.1), scaleY(_x2.y + n.y * 0.1), scaleX(_x2.x + n.x * 0.1), scaleY(_x2.y + n.y * 0.1)); } else { svg.select(".f") .call(arrow, scaleX(_x2.x + n.x * 0.1), scaleY(_x2.y - result.f * s + n.y * 0.1), scaleX(_x2.x + n.x * 0.1), scaleY(_x2.y + n.y * 0.1)); } if (norm(result.n) < 1e-4) { svg.select(".n") .attr("points", ""); } else { svg.select(".n") .call(arrow, scaleX(_x2.x), scaleY(_x2.y), scaleX(_x2.x + result.n.x * s), scaleY(_x2.y + result.n.y * s)); } svg.select(".r") .attr("visibility", showPlot.includes("Residual") ? "visible" : "hidden"); svg.select(".f") .attr("visibility", showPlot.includes("External force") ? "visible" : "hidden"); svg.select(".n") .attr("visibility", showPlot.includes("Internal force") ? "visible" : "hidden"); svg.selectAll(".x2") .attr("transform", `translate(${scaleX(x2.x)}, ${scaleY(x2.y)}) scale(1.5) rotate(${dof == "v" ? -90 : 0})`); }37:[] 2f:{"title":"Linear and nonlinear analysis","cells":[{"id":0,"value":"# Linear and nonlinear analysis","pinned":false,"hidden":false,"mode":"md"},{"id":913,"value":"viewof lang = {\n const userLanguage = (navigator.language || navigator.userLanguage).split('-')[0];\n \n const list = [\"de\", \"en\"];\n\n const options = new Map(list.map(d => [d, l => l[d]]));\n \n const input = Inputs.radio(options, {value: options.get(list.includes(userLanguage) ? userLanguage : \"en\"), label: \"Language\"});\n\n return input;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":497,"value":"## Setup","pinned":false,"hidden":false,"mode":"md"},{"id":6,"value":"viewof dof = Inputs.radio([\"u\", \"v\"], {label: lang({de: \"Freiheitsgrad\", en: \"Degree of freedom\"}), value: \"v\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":14,"value":"viewof configuration = Inputs.radio([\"(5, 0)\", \"(4, 3)\"], {label: tex`\\mathbf{X}_2`, value: \"(4, 3)\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":138,"value":"viewof f = Inputs.range([-0.2, 0.2], {label: tex`F_\\text{${dof}}`, step: 0.01})","pinned":false,"hidden":false,"mode":"ojs"},{"id":485,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":815,"value":"lang({\n en: md`Vectors and matrices are denoted with bold letters. Quantities in the undeformed state are denoted with capital letters. Lowercase letters describe quantities in the deformed state.`,\n de: md`Vektoren und Matrizen werden mit fetten Buchstaben gekennzeichnet. Größen im unverformten Zustand werden mit Großbuchstaben bezeichnet. Kleinbuchstaben beschreiben Größen im verformten Zustand.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":74,"value":"---","pinned":false,"hidden":false,"mode":"md"},{"id":710,"value":"lang({\n de: md`## Lineare Analyse`,\n en: md`## Linear analysis`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":810,"value":"lang({\n en: md`The positions of the two nodes in the undeformed state are denoted ${tex`\\mathbf{X}_1`} and ${tex`\\mathbf{X}_2`}.`,\n de: md`Die Positionen der beiden Knoten im unverformten Zustand werden ${tex`\\mathbf{X}_1`} und ${tex`\\mathbf{X}_2`} bezeichnet.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":56,"value":"tex`\\mathbf{X}_1 = (X_1, Y_1) = (${X1.x}, ${X1.y})`","pinned":false,"hidden":false,"mode":"ojs"},{"id":61,"value":"tex`\\mathbf{X}_2 = (X_2, Y_2) = (${X2.x}, ${X2.y})`","pinned":false,"hidden":false,"mode":"ojs"},{"id":820,"value":"lang({\n de: md`Die unverformte Länge ergibt sich aus der Distanz der Knoten.`,\n en: md`The undeformed length results from the distance of the nodes.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":28,"value":"tex`L = \\lVert \\mathbf{X}_2 - \\mathbf{X}_1 \\rVert = \\sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2} = ${fmtTex(L)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":825,"value":"lang({\n de: md`Die Positionen der Knoten im verformten Zustand werden ${tex`\\mathbf{x}_1`} und ${tex`\\mathbf{x}_2`} bezeichnet.`,\n en: md`The positions of the nodes in the deformed state are denoted ${tex`\\mathbf{x}_1`} and ${tex`\\mathbf{x}_2`}.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":733,"value":"tex`\\mathbf{x}_1 = (x_1, y_1) = (${x1.x}, ${x1.y})`","pinned":false,"hidden":false,"mode":"ojs"},{"id":736,"value":"tex`\\mathbf{x}_2 = (x_2, y_2) = (X_2${dof == \"u\" ? \"+ u\" : \"\"}, Y_2 ${dof == \"u\" ? \"\" : \"+ v\"}) = (${fmtTex(x2.x)}, ${fmtTex(x2.y)})`","pinned":false,"hidden":false,"mode":"ojs"},{"id":793,"value":"lang({\n en: md`For the linear computation, the strain is linearized. The length of the deformed member is measured in the undeformed reference configuration. The deformed length depends on the displacement ${tex`${dof}`}.`,\n de: md`Für die lineare Berechnung wird die Dehnung linearisiert. Die Länge des verformten Stabes wird dazu in der unverformten Referenzkonfiguration gemessen. Die verformte Länge ist abhängig von der Verschiebung ${tex`${dof}`}.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":505,"value":"tex`l_{\\text{lin}}(${dof}) = (\\mathbf{x}_2 - \\mathbf{x}_1) \\cdot \\dfrac{(\\mathbf{X}_2 - \\mathbf{X}_1)}{L} = ${fmtTex(linearResult.l_lin)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":832,"value":"lang({\n de: md`Damit ergibt sich die linearisierte Dehnung und Spannung im Stab.`,\n en: md`This gives the linearized strain and stress in the member.`\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":516,"value":"tex`\\varepsilon_{\\text{lin}}(u) = \\dfrac{l_\\text{lin} - L}{L} = ${fmtTex(linearResult.epsilon)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":519,"value":"tex`\\sigma_\\text{lin}(u) = \\varepsilon_\\text{lin}(u) \\, E = ${fmtTex(linearResult.sigma)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":838,"value":"lang({\n de: md`Die innere Kraft ${tex`N`} wirkt in Richtung des unverformten Stabes.`,\n en: md`The internal force ${tex`N`} acts in the direction of the undeformed member.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":843,"value":"lang({\n de: md`Wir können als nächstes die virtuelle Arbeit berechnen.`,\n en: md`Next we can calculate the virtual work.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":527,"value":"tex`\\begin{aligned}\n\\text{d} W\n&=\n\\sigma_\\text{lin} \\, A \\, L \\, \\text{d} \\varepsilon_{\\text{lin}} + F_${dof} \\text{d} ${dof} \\\\\n&=\n(\\sigma_\\text{lin} \\, A \\, L \\, \\dfrac{\\partial \\varepsilon_\\text{lin}}{\\partial ${dof}} + F_${dof}) \\, \\text{d} ${dof}\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":847,"value":"lang({\n de: md`Für die Variation der Dehnung berechnen wir die partielle Ableitung nach der Verschiebung.`,\n en: md`For the variation of the strain, we calculate the partial derivative w.r.t. the displacement.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":522,"value":"tex`\\text{d} \\varepsilon_{\\text{lin}} =\n\\dfrac{\\partial \\varepsilon_\\text{lin}}{\\partial ${dof}} \\, \\text{d} ${dof} =\n\\dfrac{${dof == \"u\" ? \"X\" : \"Y\"}_2 - ${dof == \"u\" ? \"X\" : \"Y\"}_1}{L^2} \\, \\text{d} ${dof}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":852,"value":"lang({\n de: md`Daraus ergibt sich die Residualkraft, die am entsprechenden Freiheitsgrad wirkt.`,\n en: md`This results in the residual force acting on the corresponding degree of freedom.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":534,"value":"tex`r_${dof}(u) = \\dfrac{dW}{d${dof}} = \\sigma_\\text{lin} \\, A \\, L \\, \\dfrac{\\partial \\varepsilon_\\text{lin}}{\\partial ${dof}} - F_${dof} = ${fmtTex(linearResult.r)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1181,"value":"lang({\n de: md`Der Einfluss von ${tex`${dof}`} auf die Residualkraft ist linear.`,\n en: md`The influende of ${tex`${dof}`} on the residual force is linear.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1282,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":855,"value":"lang({\n de: md`Um das System in Gleichgewicht zu bringen, muss die Verschiebung so gewählt werden, dass die Residualkraft verschwindet. Wir verwenden hierfür das Newton-Raphson Verfahren.`,\n en: md`To bring the system into equilibrium, the displacement must be chosen so that the residual force vanishes. We use the Newton-Raphson method for this task.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":859,"value":"lang({\n de: md`Die Residualkraft wird dafür nach der Verschiebung abgeleitet. Man erhält die Steifigkeit des Systems für den entsprechenden Freiheitsgrad.`,\n en: md`The residual force is derived w.r.t. the displacement. From this we obtein the stiffness of the system for the corresponding degree of freedom.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":547,"value":"tex`\\text{d} r_${dof} =\n\\dfrac{\\partial r_${dof}}{\\partial ${dof}} \\text{d} ${dof} =\n\\dfrac{\\partial \\varepsilon_\\text{lin}}{\\partial ${dof}} \\, E \\, A \\, L \\, \\dfrac{\\partial \\varepsilon_\\text{lin}}{\\partial ${dof}} \\, \\text{d}${dof}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":559,"value":"tex`K_{${dof}${dof}}(u) =\n\\dfrac{\\text{d} r_${dof}}{\\text{d}${dof}} =\n\\dfrac{\\partial \\varepsilon_\\text{lin}}{\\partial ${dof}} \\, E \\, A \\, L \\, \\dfrac{\\partial \\varepsilon_\\text{lin}}{\\partial ${dof}} = ${fmtTex(linearResult.k)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1177,"value":"lang({\n de: md`Die Steifigkeit des linearen Systems ist konstant und nicht von ${dof} abhängig. Mithilfe der Steifigkeit, kann ${tex`r_${dof}`} an der Stelle ${tex`${dof}`} linearisiert werden (${svg``} im Last-Verschiebungs-Diagramm)`,\n en: md`The stiffness for the linear system is constant and does not depend on ${dof}. Using the stiffness, we can linearize ${tex`r_${dof}`} at ${tex`${dof}`} (${svg``} in the Load-Displacement-Diagram).`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":862,"value":"lang({\n de: md`In jeder Newton-Raphson Iteration wird die Verschiebung nach folgender Vorschrift angepasst:`,\n en: md`In each Newton-Raphson iteration the displacement is adjusted according to the following scheme:`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":863,"value":"tex`${dof}^{(i+1)} = ${dof}^{(i)} - K_{${dof}${dof}}\\left(${dof}^{(i)}\\right)^{-1} \\, r_${dof}\\left(${dof}^{(i)}\\right)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1296,"value":"lang({\n de: md`Das Ergebnis ist im Last-Verschiebungs-Diagramm als ${svg``} eingezeichnet.`,\n en: md`The solution is visualized in the load-displacement-diagram as ${svg``}.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":869,"value":"lang({\n de: md`Da die Residualkraft linear von der Verschiebung abhängt, löst der Algorithmus das System in einem einzigen Iterationsschritt. Die Lösung ist dabei eindeutig (sofern das System nicht kinematisch ist).`,\n en: md`Since the residual force depends linear on the displacement, the algorithm solves the system in a single iteration. The solution is unique (as long as the system is not kinematic).`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":893,"value":"lang({\n de: md`In der folgenden interaktiven Grafik, kann der Knoten mit der Maus verschoben werden. Die Kräfte und das Last-Verschiebungs-Diagramm werden angezeigt. Mit der Schaltfläche kann eine Newton-Iteration durchgeführt werden, um das System wieder in Gleichgewicht zu bringen.`,\n en: md`In the following interactive graphic, the node can be moved with the mouse. The forces and the load-displacement diagram are displayed. With the button, a Newton iteration can be performed to bring the system back to equilibrium.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":234,"value":"linearPlot.node()","pinned":false,"hidden":false,"mode":"ojs"},{"id":1114,"value":"viewof showPlot = Inputs.checkbox(new Map([[lang({en: \"Residual\", de: \"Residuum\"}), \"Residual\"], [lang({en: \"Internal Force\", de: \"Interne Kraft\"}), \"Internal force\"], [lang({en: \"External force\", de: \"Externe Kraft\"}), \"External force\"]]), {value: [\"Residual\", \"External force\"], format: x => html`${x[0]}`})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1220,"value":"Inputs.bind(Inputs.range([-0.2, 0.2], {label: tex`F_\\text{${dof}}`, step: 0.01}), viewof f)","pinned":false,"hidden":false,"mode":"ojs"},{"id":693,"value":"tex`${dof}^{(i)} = ${fmtTex(linearResult.d)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1338,"value":"tex`l_{\\text{lin}}(${dof}^{(i)}) = ${fmtTex(linearResult.l_lin)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":696,"value":"tex`r_${dof}(${dof}^{(i)}) = ${fmtTex(linearResult.r)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":690,"value":"tex`K_{${dof}${dof}}(${dof}^{(i)}) = ${fmtTex(linearResult.k)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":990,"value":"tex`${dof}^{(i + 1)} = ${fmtTex(linearResult.d)} - \\dfrac{${fmtTex(linearResult.r)}}{${fmtTex(linearResult.k)}} = ${fmtTex(linearResult.d - linearResult.r / linearResult.k)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":442,"value":"Inputs.button(linearResult.k < 1e-8 ? lang({en: \"Singular stiffness\", de: \"Steifigkeit singulär\"}) : lang({en: \"Perform Newton iteration for linear model\", de: \"Newton Iteration für das lineare System\"}), {reduce: () => {\n const result = linearResult;\n \n const d = result.d - result.r / result.k;\n \n mutable x2 = vec(X2.x + (dof == \"u\" ? d : 0), X2.y + (dof == \"v\" ? d : 0));\n}, disabled: linearResult.k < 1e-8})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1302,"value":"lang({\n de: md`**Last-Verschiebungs-Diagramm**`,\n en: md`**Load-Displacement-Diagram**`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":340,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":1133,"value":"viewof showLoadDisplacement = Inputs.checkbox(new Map([[lang({en: \"Linearized system\", de: \"Linearisiertes System\"}), \"Linearized\"]]), {format: x => html`${x[0]}`})","pinned":false,"hidden":false,"mode":"ojs"},{"id":715,"value":"lang({\n de: md`## Nichlineare Analyse`,\n en: md`## Noninear analysis`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":872,"value":"lang({\n de: md`Für die nichtlineare Analyse berechnen wir die Länge des Stabes im verformten Zustand.`,\n en: md`For the nonlinear analysis, we calculate the length of the member in the deformed state.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":749,"value":"tex`l(${dof}) = \\lVert \\mathbf{x}_2 - \\mathbf{x}_1 \\rVert = \\sqrt{(X_2${dof == \"u\" ? `+ ${dof}` : \"\"} - X_1)^2 + (Y_2${dof == \"v\" ? `+ ${dof}` : \"\"} - Y_1)^2} = ${fmtTex(nonlinearResult.l)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":875,"value":"lang({\n de: md`Wir verwenden das Green-Lagrange Dehnungsmaß und erhalten die 2. Piola-Kirchhoff Spannung.`,\n en: md`We use the Green-Lagrange strain measure and obtain Piola-Kirchhoff stresses of 2nd kind.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":753,"value":"tex`\\varepsilon_{\\text{GL}}(u) = \\dfrac{l(u)^2 - L^2}{2 L^2} = ${fmtTex(nonlinearResult.epsilon)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":759,"value":"tex`\\sigma_\\text{PK2}(u) = \\varepsilon_\\text{GL}(u) \\, E = ${fmtTex(nonlinearResult.sigma)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":884,"value":"lang({\n de: md`Die virtuelle Arbeit, Residualkraft und Steifigkeit berechnen wir analog zum linearen Fall.`,\n en: md`We calculate the virtual work, residual force and stiffness analogously to the linear case.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":881,"value":"tex`\n\\text{d} W\n=\n\\sigma \\, A \\, L \\, \\text{d} \\varepsilon_{\\text{GL}} + F_${dof} \\text{d} ${dof}\n=\n(\\sigma \\, A \\, L \\, \\dfrac{\\partial \\varepsilon_\\text{GL}}{\\partial ${dof}} + F_${dof}) \\, \\text{d} ${dof}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":764,"value":"tex`\n\\text{d} \\varepsilon_{\\text{GL}}\n= \\dfrac{\\partial \\varepsilon_\\text{GL}}{\\partial ${dof}} \\, \\text{d} ${dof}\n= \\dfrac{${dof == \"u\" ? \"X\" : \"Y\"}_2 + ${dof} - ${dof == \"u\" ? \"X\" : \"Y\"}_1}{L^2} \\, \\text{d} ${dof}\n= ${fmtTex(nonlinearResult.d_epsilon)}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1164,"value":"lang({\n de: md`Für den nichtlinearen Fall benötigen wir auch die zweite Variation der Dehnung und die Variation der Spannung.`,\n en: md`For the nonlinear case we also need the second variation of the strain and the variation of the stress.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1052,"value":"tex`\n\\text{d}^2 \\varepsilon_{\\text{GL}}\n= \\dfrac{\\partial^2 \\varepsilon_\\text{GL}}{\\partial ${dof}^2} \\, \\text{d} ${dof}\n= \\dfrac{1}{L^2} \\, \\text{d} ${dof}\n= ${fmtTex(nonlinearResult.d2_epsilon)}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1058,"value":"tex`\n\\text{d} \\sigma_{\\text{PK2}}\n= \\dfrac{\\partial \\sigma_\\text{PK2}}{\\partial ${dof}} \\, \\text{d} ${dof}\n= \\dfrac{\\partial \\varepsilon_\\text{GL}}{\\partial ${dof}} \\, E \\, \\text{d} ${dof}\n= ${fmtTex(nonlinearResult.d_sigma)}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1170,"value":"lang({\n de: md`Der Einfluss von ${dof} auf die Residualkraft ist nicht mehr linear.`,\n en: md`The influende of ${dof} on the residual force is not linear.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":778,"value":"tex`\nr_${dof}(u)\n= \\dfrac{dW}{d${dof}}\n= \\sigma_\\text{PK2} \\, A \\, L \\, \\dfrac{\\partial \\varepsilon_\\text{GL}}{\\partial ${dof}} - F_${dof}\n= ${fmtTex(nonlinearResult.r)}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1174,"value":"lang({\n de: md`Die Steifigkeit ist nicht mehr konstant.`,\n en: md`The stiffness is not constant.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1027,"value":"tex`\\text{d} r_${dof}\n= \\dfrac{\\partial r_${dof}}{\\partial ${dof}} \\text{d} ${dof}\n= \\dfrac{\\partial \\sigma_\\text{PK2}}{\\partial ${dof}} \\text{d}${dof} \\, A \\, L \\, \\dfrac{\\partial \\varepsilon_\\text{GL}}{\\partial ${dof}} + \\sigma_\\text{PK2} \\, AL \\, \\dfrac{\\partial^2 \\varepsilon_\\text{GL}}{\\partial ${dof}^2} \\text{d}u\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":783,"value":"tex`\nK_{${dof}${dof}}(u)\n= \\dfrac{\\text{d} r_${dof}}{\\text{d}${dof}}\n= \\dfrac{\\partial \\varepsilon_\\text{GL}}{\\partial ${dof}} \\, E \\, A \\, L \\, \\dfrac{\\partial \\varepsilon_\\text{GL}}{\\partial ${dof}} +\n\\sigma_{\\text{PK2}} \\, A \\, L \\, \\dfrac{\\partial \\varepsilon^2_\\text{GL}}{\\partial ${dof}^2}\n= ${fmtTex(nonlinearResult.k)}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":903,"value":"lang({\n de: md`Wir können wieder Newton-Iterationen durchführen, um das System in Gleichgewicht zu bringen. Da das System nichtlinear ist, benötigen wir dafür i.d.R. mehrere Iterationen. Man sieht außerdem, dass es mehrere Lösungen geben kann.`,\n en: md`We can perform Newton iterations to bring the system into equilibrium. Since the system is nonlinear, we usually need several iterations to do this. We can also see that there can be several solutions.`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":335,"value":"nonlinearPlot.node()","pinned":false,"hidden":false,"mode":"ojs"},{"id":1212,"value":"Inputs.bind(Inputs.checkbox(new Map([[lang({en: \"Residual\", de: \"Residuum\"}), \"Residual\"], [lang({en: \"Internal Force\", de: \"Interne Kraft\"}), \"Internal force\"], [lang({en: \"External force\", de: \"Externe Kraft\"}), \"External force\"]]), {value: [\"Residual\", \"External force\"], format: x => html`${x[0]}`}), viewof showPlot)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1232,"value":"Inputs.bind(Inputs.range([-0.2, 0.2], {label: tex`F_\\text{${dof}}`, step: 0.01}), viewof f)","pinned":false,"hidden":false,"mode":"ojs"},{"id":411,"value":"tex`${dof}^{(i)} = ${fmtTex(nonlinearResult.d)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1357,"value":"tex`l(${dof}^{(i)}) = ${fmtTex(nonlinearResult.l)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":406,"value":"tex`r_${dof}(${dof}^{(i)}) = ${fmtTex(nonlinearResult.r)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":686,"value":"tex`K_{${dof}${dof}}(${dof}^{(i)}) = ${fmtTex(nonlinearResult.k)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":993,"value":"tex`${dof}^{(i + 1)} = ${fmtTex(nonlinearResult.d)} - \\dfrac{${fmtTex(nonlinearResult.r)}}{${fmtTex(nonlinearResult.k)}} = ${fmtTex(nonlinearResult.d - nonlinearResult.r / nonlinearResult.k)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":428,"value":"Inputs.button(Math.abs(nonlinearResult.k) < 1e-8 ? lang({en: \"Singular stiffness\", de: \"Steifigkeit singulär\"}) : lang({en: \"Perform Newton iteration for nonlinear model\", de: \"Newton Iteration für das nichtlineare System\"}), {reduce: () => {\n const result = nonlinearResult;\n \n const d = result.d - result.r / result.k;\n \n mutable x2 = vec(X2.x + (dof == \"u\" ? d : 0), X2.y + (dof == \"v\" ? d : 0));\n}, disabled: Math.abs(nonlinearResult.k) < 1e-8})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1311,"value":"lang({\n de: md`**Last-Verschiebungs-Diagramm**`,\n en: md`**Load-Displacement-Diagram**`,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":394,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":1197,"value":"Inputs.bind(Inputs.checkbox(new Map([[lang({en: \"Linearized system\", de: \"Linearisiertes System\"}), \"Linearized\"]]), {format: x => html`${x[0]}`}), viewof showLoadDisplacement)","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"---\n## Appendix\n\n*Not relevant...*","pinned":false,"hidden":false,"mode":"md"},{"id":124,"value":"E = 0.8","pinned":false,"hidden":false,"mode":"ojs"},{"id":127,"value":"A = 0.5","pinned":false,"hidden":false,"mode":"ojs"},{"id":66,"value":"X1 = vec(0, 0)","pinned":false,"hidden":false,"mode":"ojs"},{"id":10,"value":"X2 = configuration == \"(5, 0)\" ? vec(5, 0) : vec(4, 3)","pinned":false,"hidden":false,"mode":"ojs"},{"id":113,"value":"G1 = sub(X2, X1)","pinned":false,"hidden":false,"mode":"ojs"},{"id":79,"value":"L = dist(X1, X2)","pinned":false,"hidden":false,"mode":"ojs"},{"id":87,"value":"x1 = vec(0, 0)","pinned":false,"hidden":false,"mode":"ojs"},{"id":90,"value":"mutable x2 = vec(L, 0)","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"l = dist(x1, x2)","pinned":false,"hidden":false,"mode":"ojs"},{"id":189,"value":"---","pinned":false,"hidden":false,"mode":"md"},{"id":99,"value":"linear = function (x2) {\n const u = x2.x - X2.x;\n const v = x2.y - X2.y;\n\n const d = dof == \"u\" ? u : v;\n\n const g1 = sub(x2, x1);\n \n const l_lin = dot(g1, G1) / L;\n\n const dX = dof == \"u\" ? G1.x : G1.y;\n\n const epsilon = (l_lin - L) / L;\n \n const d_epsilon = dX / L**2;\n\n const sigma = E * epsilon;\n \n const d_sigma = E * d_epsilon;\n \n const r = sigma * A * d_epsilon * L - f;\n const k = d_sigma * A * d_epsilon * L;\n\n const rnorm = Math.abs(r);\n const detk = k;\n\n const n = scale(sigma * A, unit(sub(X1, X2)));\n \n return {u, v, d, f, epsilon, d_epsilon, sigma, r, k, rnorm, detk, n, l_lin, L};\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":151,"value":"nonlinear = function (x2) {\n const u = x2.x - X2.x;\n const v = x2.y - X2.y;\n\n const d = dof == \"u\" ? u : v;\n\n const g1 = sub(x2, x1);\n \n const l = norm(g1);\n\n const dx = dof == \"u\" ? g1.x : g1.y;\n\n const epsilon = (l**2 - L**2) / (2 * L**2);\n const d_epsilon = dx / L**2;\n const d2_epsilon = 1 / L**2;\n\n const sigma = E * epsilon;\n const d_sigma = E * d_epsilon;\n \n const r = sigma * A * d_epsilon * L - f;\n const k = d_sigma * A * d_epsilon * L + sigma * A * d2_epsilon * L;\n\n const rnorm = Math.abs(r);\n const detk = k;\n\n const n = scale(sigma * A * l / L, unit(sub(x1, x2)));\n \n return {u, v, d, f, epsilon, d_epsilon, d2_epsilon, sigma, d_sigma, r, k, rnorm, detk, n, l, L};\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":164,"value":"linearResult = {\n const result = linear(x2);\n updatePlot(linearPlot, result);\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":159,"value":"nonlinearResult = {\n const result = nonlinear(x2);\n updatePlot(nonlinearPlot, result);\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":193,"value":"---","pinned":false,"hidden":false,"mode":"md"},{"id":282,"value":"{\n if (dof == \"u\") {\n mutable x2 = vec(mutable x2.x, X2.y);\n } else {\n mutable x2 = vec(X2.x, mutable x2.y);\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":199,"value":"height = 500","pinned":false,"hidden":false,"mode":"ojs"},{"id":205,"value":"scaleX = d3.scaleLinear().domain([0, 4]).range([width / 2 - 100, width / 2 + 100])","pinned":false,"hidden":false,"mode":"ojs"},{"id":206,"value":"scaleY = d3.scaleLinear().domain([0, 3]).range([height / 2 + 75, height / 2 - 75])","pinned":false,"hidden":false,"mode":"ojs"},{"id":218,"value":"line = d3.line().x(d => scaleX(d.x)).y(d => scaleY(d.y))","pinned":false,"hidden":false,"mode":"ojs"},{"id":197,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":237,"value":"linearPlot = generatePlot(linear)","pinned":false,"hidden":false,"mode":"ojs"},{"id":239,"value":"nonlinearPlot = generatePlot(nonlinear)","pinned":false,"hidden":false,"mode":"ojs"},{"id":243,"value":"$36","pinned":false,"hidden":false,"mode":"ojs"},{"id":245,"value":"drawing = {\n const nodeSvg = `\n `;\n \n const supportSvg = `\n \n \n \n \n \n \n \n \n \n `;\n \n const slidingSupportSvg = `\n \n `;\n \n return {\n node: (selection) => selection.html(nodeSvg),\n support: (selection) => selection.html(supportSvg),\n slidingSupport: (selection) => selection.html(slidingSupportSvg),\n };\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":83,"value":"---","pinned":false,"hidden":false,"mode":"md"},{"id":423,"value":"within = function (min, max, value) {\n console.log(min , max, value)\n if (value < min) {\n return false;\n }\n if (value > max) {\n return false;\n }\n return true;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":258,"value":"clamp = function (min, max, value) {\n if (value < min) {\n return min;\n }\n if (value > max) {\n return max;\n }\n return value;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":262,"value":"lerp = function (a, b, t) {\n return a * (1 - t) + b * t;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":347,"value":"linspace = function (a, b, n) {\n return d3.range(n).map(d => lerp(a, b, d / (n - 1)));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":24,"value":"vec = function (x, y) {\n return {x, y};\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":34,"value":"sub = function (u, v) {\n const x = u.x - v.x;\n const y = u.y - v.y;\n return {x, y};\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":460,"value":"scale = function (a, v) {\n const x = a * v.x;\n const y = a * v.y;\n return {x, y};\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":110,"value":"dot = function (u, v) {\n return u.x * v.x + u.y * v.y;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":33,"value":"dist = function (u, v) {\n return norm(sub(u, v));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":40,"value":"norm = function (v) {\n return Math.sqrt(v.x**2 + v.y**2);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":290,"value":"unit = function (v) {\n const l = norm(v);\n return vec(v.x / l, v.y / l);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":295,"value":"perp = function (v) {\n return vec(-v.y, v.x);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":46,"value":"fmt = d3.format(\".4g\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":45,"value":"fmtTex = function (value) {\n return fmt(value).replace(/e([+-]\\d+)/, \" \\\\cdot 10^{$1}\")\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":323,"value":"import {arrow} from \"@oberbichler/arrow\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W37","stdlib":"1","titling":"auto"}