![](\"resources/question-resources/gsm1_v7hUrSp.jpg\"/)
u
", "name": "u2", "group": "Ungrouped variables", "definition": "-{a23}*{k4}/({a22}*{a33}-{a23}*{a32})", "templateType": "anything"}}, "tags": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "variable_groups": [], "statement": "Determine the global stiffness matrix for the spring system below, where
\nk1 = \\(\\var{k1}N/mm\\), k2 = \\(\\var{k2}N/mm\\) , k3 = \\(\\var{k3}N/mm\\).
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First set up the size of the answer matrix (choose the correct number of rows and columns in the boxes) and then input the entries.
\n\\(\\mathbf A=\\) [[0]]
", "showFeedbackIcon": true, "gaps": [{"tolerance": 0, "correctAnswer": "matrix([\n [a11,a12,a13,a14],\n [a21,a22,a23,a24],\n [a31,a32,a33,a34],\n [a41,a42,a43,a44]\n])", "variableReplacements": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": "4", "correctAnswerFractions": false, "numRows": "1", "allowFractions": false, "markPerCell": true, "numColumns": "1", "type": "matrix", "showFeedbackIcon": true, "allowResize": true, "showCorrectAnswer": true}], "showCorrectAnswer": true}, {"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "prompt": "A force of \\(\\var{k4}N\\) is applied at node 3 in a positive direction, i.e. F3 = \\(\\var{k4}N\\)and \\(u_1=u_4=0\\)
\nCalculate the displacements at nodes 2 and 3.
\nGive your answers correct to 3 decimal places.
\n\\(u_2=\\) [[0]]mm
\n\\(u_3=\\) [[1]]mm
", "showFeedbackIcon": true, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "scripts": {}, "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "marks": 1, "precisionType": "dp", "precisionPartialCredit": 0, "maxValue": "{u2}", "minValue": "{u2}", "type": "numberentry", "strictPrecision": false, "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "mustBeReducedPC": 0, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "scripts": {}, "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "marks": 1, "precisionType": "dp", "precisionPartialCredit": 0, "maxValue": "{u3}", "minValue": "{u3}", "type": "numberentry", "strictPrecision": false, "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "mustBeReducedPC": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true}], "advice": "The global stiffness matrix is given by:
\n\\(\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}&\\var{a14}\\\\ \\var{a21}&\\var{a22}&\\var{a23}&\\var{a24}\\\\ \\var{a31}&\\var{a32}&\\var{a33}&\\var{a34}\\\\\\var{a41}&\\var{a42}&\\var{a43}&\\var{a44} \\end{pmatrix}\\)
\n(b)
\n\\(\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}&\\var{a14}\\\\ \\var{a21}&\\var{a22}&\\var{a23}&\\var{a24}\\\\ \\var{a31}&\\var{a32}&\\var{a33}&\\var{a34}\\\\\\var{a41}&\\var{a42}&\\var{a43}&\\var{a44} \\end{pmatrix}\\begin{pmatrix} u_1\\\\u_2\\\\u_3\\\\u_4\\end{pmatrix}=\\begin{pmatrix} F_1\\\\F_2\\\\F_3\\\\F_4\\end{pmatrix}\\)
\n\\(\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}&\\var{a14}\\\\ \\var{a21}&\\var{a22}&\\var{a23}&\\var{a24}\\\\ \\var{a31}&\\var{a32}&\\var{a33}&\\var{a34}\\\\\\var{a41}&\\var{a42}&\\var{a43}&\\var{a44} \\end{pmatrix}\\begin{pmatrix} 0\\\\u_2\\\\u_3\\\\0\\end{pmatrix}=\\begin{pmatrix} F_1\\\\0\\\\\\var{k3}\\\\F_4\\end{pmatrix}\\)
\n\\(As\\,u_1=0\\,and\\,u_4=0\\,we \\,can\\, use\\, the\\, reduced \\,central\\,matrix\\, below \\,to\\, solve\\, for\\, u_2\\,and\\,u_3\\)
\n\\(\\begin{pmatrix}\\var{a22}&\\var{a23}\\\\\\var{a32}&\\var{a33}\\end{pmatrix}\\begin{pmatrix} u_2\\\\u_3\\end{pmatrix}=\\begin{pmatrix} 0\\\\\\var{k3}\\end{pmatrix}\\)
\nusing the inverse matrix method:
\n\\(\\begin{pmatrix} u_2\\\\u_3\\end{pmatrix}=\\frac{1}{\\var{det}}\\begin{pmatrix}\\var{a33}&-\\var{a23}\\\\-\\var{a32}&\\var{a22}\\end{pmatrix}\\begin{pmatrix} 0\\\\\\var{k3}\\end{pmatrix}\\)
\n\\(\\begin{pmatrix} u_2\\\\u_3\\end{pmatrix}=\\frac{1}{\\var{det}}\\begin{pmatrix} \\simplify{-{a23}*{k6}}\\\\\\simplify{{a33}*{k}}\\end{pmatrix}\\)
\n\\(\\begin{pmatrix} u_2\\\\u_3\\end{pmatrix}=\\begin{pmatrix} \\var{u2}\\\\\\var{u3}\\end{pmatrix}\\)
\n", "name": "FEM #2", "ungrouped_variables": ["k1", "k2", "k3", "a11", "a12", "a13", "a14", "a21", "a22", "a24", "a23", "a31", "a32", "a33", "a34", "a41", "a42", "a43", "a44", "k4", "u3", "u2", "det"], "preamble": {"css": "", "js": ""}, "extensions": [], "rulesets": {}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}