// Numbas version: exam_results_page_options
{"name": "FEM#1", "extensions": [], "custom_part_types": [], "resources": [["question-resources/3_Springs_RWtq3Yl.jpg", "/srv/numbas/media/question-resources/3_Springs_RWtq3Yl.jpg"], ["question-resources/gsm1.jpg", "/srv/numbas/media/question-resources/gsm1.jpg"], ["question-resources/gsm1_v7hUrSp.jpg", "/srv/numbas/media/question-resources/gsm1_v7hUrSp.jpg"], ["question-resources/gsm2.jpg", "/srv/numbas/media/question-resources/gsm2.jpg"], ["question-resources/gsm3.jpg", "/srv/numbas/media/question-resources/gsm3.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "extensions": [], "variable_groups": [], "ungrouped_variables": ["k4", "k2", "k3", "a11", "a12", "a13", "a21", "a22", "a23", "a31", "a32", "a33", "k1", "k5", "a14", "a24", "a34", "a41", "a42", "a43", "a44", "k6", "u2", "u3", "det"], "parts": [{"prompt": "<p style=\"padding-left: 60px;\"><img src=\"resources/question-resources/gsm3.jpg\"/></p>\n<p>First set up&nbsp;the size of the answer matrix (choose the correct number of rows and columns in the boxes) and then input the entries.</p>\n<p style=\"padding-left: 60px;\">\\(\\mathbf A=\\)&nbsp;[[0]]&nbsp;</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "scripts": {}, "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"tolerance": 0, "markPerCell": true, "numRows": "1", "allowResize": true, "variableReplacements": [], "correctAnswerFractions": false, "marks": "4", "allowFractions": false, "showFeedbackIcon": true, "correctAnswer": "matrix([\n    [a11,a12,a13,a14],\n    [a21,a22,a23,a24],\n    [a31,a32,a33,a34],\n    [a41,a42,a43,a44]\n])", "scripts": {}, "type": "matrix", "showCorrectAnswer": true, "numColumns": "1", "variableReplacementStrategy": "originalfirst"}], "marks": 0}, {"prompt": "<p>A force of \\(\\var{k6}N\\) is applied at node 3 in a positive direction, i.e. <em>F<sub>3</sub></em><sub> </sub>=&nbsp;\\(\\var{k6}\\)N.</p>\n<p>Calculate the displacements at nodes 2 and 3.</p>\n<p>Give your answers correct to 3 decimal places.</p>\n<p>\\(u_2=\\)&nbsp;[[0]]mm</p>\n<p>\\(u_3=\\)&nbsp;[[1]]mm</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "scripts": {}, "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"variableReplacements": [], "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "maxValue": "{u2}", "showPrecisionHint": false, "marks": 1, "mustBeReduced": false, "precisionPartialCredit": 0, "scripts": {}, "minValue": "{u2}", "showFeedbackIcon": true, "strictPrecision": false, "mustBeReducedPC": 0, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precision": "3", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "precisionMessage": "You have not given your answer to the correct precision."}, {"variableReplacements": [], "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "maxValue": "{u3}", "showPrecisionHint": false, "marks": 1, "mustBeReduced": false, "precisionPartialCredit": 0, "scripts": {}, "minValue": "{u3}", "showFeedbackIcon": true, "strictPrecision": false, "mustBeReducedPC": 0, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "precision": "3", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "precisionMessage": "You have not given your answer to the correct precision."}], "marks": 0}], "tags": [], "preamble": {"css": "", "js": ""}, "variables": {"k3": {"definition": "random(140..290#10)", "group": "Ungrouped variables", "description": "", "name": "k3", "templateType": "randrange"}, "a43": {"definition": "-k4", "group": "Ungrouped variables", "description": "", "name": "a43", "templateType": "anything"}, "a24": {"definition": "0", "group": "Ungrouped variables", "description": "", "name": "a24", "templateType": "anything"}, "k2": {"definition": "random(150..210#10)", "group": "Ungrouped variables", "description": "", "name": "k2", "templateType": "randrange"}, "a32": {"definition": "-k3", "group": "Ungrouped variables", "description": "", "name": "a32", "templateType": "anything"}, "u2": {"definition": "-{a23}*{k6}/({a22}*{a33}-{a23}*{a32})", "group": "Ungrouped variables", "description": "", "name": "u2", "templateType": "anything"}, "det": {"definition": "{a22}*{a33}-{a23}*{a32}", "group": "Ungrouped variables", "description": "", "name": "det", "templateType": "anything"}, "a44": {"definition": "k1+k4+k5", "group": "Ungrouped variables", "description": "", "name": "a44", "templateType": "anything"}, "a14": {"definition": "-k1-k5", "group": "Ungrouped variables", "description": "", "name": "a14", "templateType": "anything"}, "k5": {"definition": "k1", "group": "Ungrouped variables", "description": "", "name": "k5", "templateType": "anything"}, "a11": {"definition": "k1+k2+k5", "group": "Ungrouped variables", "description": "", "name": "a11", "templateType": "anything"}, "a23": {"definition": "-k3", "group": "Ungrouped variables", "description": "", "name": "a23", "templateType": "anything"}, "a12": {"definition": "-k2", "group": "Ungrouped variables", "description": "", "name": "a12", "templateType": "anything"}, "a33": {"definition": "k4+k3", "group": "Ungrouped variables", "description": "", "name": "a33", "templateType": "anything"}, "a31": {"definition": "0", "group": "Ungrouped variables", "description": "", "name": "a31", "templateType": "anything"}, "a21": {"definition": "-k2", "group": "Ungrouped variables", "description": "", "name": "a21", "templateType": "anything"}, "k4": {"definition": "random(120..250#10)", "group": "Ungrouped variables", "description": "", "name": "k4", "templateType": "randrange"}, "k1": {"definition": "random(250..400#25)", "group": "Ungrouped variables", "description": "", "name": "k1", "templateType": "randrange"}, "a22": {"definition": "k2+k3", "group": "Ungrouped variables", "description": "", "name": "a22", "templateType": "anything"}, "a41": {"definition": "-k1-k5", "group": "Ungrouped variables", "description": "", "name": "a41", "templateType": "anything"}, "u3": {"definition": "{a22}*{k6}/({a22}*{a33}-{a23}*{a32})", "group": "Ungrouped variables", "description": "", "name": "u3", "templateType": "anything"}, "k6": {"definition": "random(100..170#10)", "group": "Ungrouped variables", "description": "", "name": "k6", "templateType": "randrange"}, "a42": {"definition": "0", "group": "Ungrouped variables", "description": "", "name": "a42", "templateType": "anything"}, "a34": {"definition": "-k4", "group": "Ungrouped variables", "description": "", "name": "a34", "templateType": "anything"}, "a13": {"definition": "0", "group": "Ungrouped variables", "description": "", "name": "a13", "templateType": "anything"}}, "statement": "<p>Determine the global stiffness matrix for the spring system below, where</p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; k<sub>1</sub> =&nbsp;\\(\\var{k1}N/mm\\),&nbsp;&nbsp;&nbsp; k<sub>2</sub> = \\(\\var{k2}N/mm\\),&nbsp; &nbsp;k<sub>3</sub> = \\(\\var{k3}N/mm\\),&nbsp; &nbsp; k<sub>4</sub>&nbsp;= \\(\\var{k4}N/mm\\),&nbsp; k<sub>5</sub>&nbsp;&nbsp;= \\(\\var{k5}N/mm\\) and&nbsp; \\(u_1=u_4=0\\).</p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "name": "FEM#1", "advice": "<p>The global stiffness matrix is given by:</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}&amp;\\var{a14}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{a23}&amp;\\var{a24}\\\\ \\var{a31}&amp;\\var{a32}&amp;\\var{a33}&amp;\\var{a34}\\\\\\var{a41}&amp;\\var{a42}&amp;\\var{a43}&amp;\\var{a44}&nbsp;\\end{pmatrix}\\)</p>\n<p>(b)</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}&amp;\\var{a14}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{a23}&amp;\\var{a24}\\\\ \\var{a31}&amp;\\var{a32}&amp;\\var{a33}&amp;\\var{a34}\\\\\\var{a41}&amp;\\var{a42}&amp;\\var{a43}&amp;\\var{a44}&nbsp;\\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}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{pmatrix} \\var{a11}&amp;\\var{a12}&amp;\\var{a13}&amp;\\var{a14}\\\\ \\var{a21}&amp;\\var{a22}&amp;\\var{a23}&amp;\\var{a24}\\\\ \\var{a31}&amp;\\var{a32}&amp;\\var{a33}&amp;\\var{a34}\\\\\\var{a41}&amp;\\var{a42}&amp;\\var{a43}&amp;\\var{a44}&nbsp;\\end{pmatrix}\\begin{pmatrix} 0\\\\u_2\\\\u_3\\\\0\\end{pmatrix}=\\begin{pmatrix} F_1\\\\0\\\\\\var{k6}\\\\F_4\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(As\\,u_1=0\\,and\\,u_4=0\\,we \\,can\\, use\\, the\\, reduced \\,central\\,matrix\\, below \\,to\\, solve\\, for\\, u_2\\,and\\,u_3\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\begin{pmatrix}\\var{a22}&amp;\\var{a23}\\\\\\var{a32}&amp;\\var{a33}\\end{pmatrix}\\begin{pmatrix} u_2\\\\u_3\\end{pmatrix}=\\begin{pmatrix} 0\\\\\\var{k6}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">using the inverse matrix method:</p>\n<p style=\"padding-left: 60px;\">\\(\\begin{pmatrix} u_2\\\\u_3\\end{pmatrix}=\\frac{1}{\\var{det}}\\begin{pmatrix}\\var{a33}&amp;\\simplify{-{a23}}\\\\\\simplify{-{a32}}&amp;\\var{a22}\\end{pmatrix}\\begin{pmatrix} 0\\\\\\var{k6}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 60px;\">\\(\\begin{pmatrix} u_2\\\\u_3\\end{pmatrix}=\\frac{1}{\\var{det}}\\begin{pmatrix} \\simplify{-{a23}*{k6}}\\\\\\simplify{{a22}*{k6}}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 60px;\">\\(\\begin{pmatrix} u_2\\\\u_3\\end{pmatrix}=\\begin{pmatrix} \\var{u2}\\\\\\var{u3}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 60px;\"></p>", "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/"}]}