// Numbas version: exam_results_page_options
{"name": "2D stress in a bar", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Truss-4.png", "/srv/numbas/media/question-resources/Truss-4.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "2D stress in a bar", "tags": [], "metadata": {"description": "<p>Calculate direct and shear stress in an axially loaded bar in the XY plane.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Axial stress in a bar is a straightforward calculation, but $\\sigma_x$, $\\sigma_y$ and $\\tau_{xy}$ depend on the choice of $x$ and $y$ axes and the orientation of the bar.</p>", "advice": "<p>The bar has a solid square cross-section (10mm&times;10mm), so the cross-sectional area is 100mm$^2$.</p>\n<ol>\n<li>The horizontal section area is 100mm$^2/\\sin(\\var{theta})=\\var{siground(AH,3)}$mm$^2$.</li>\n<li>The vertical section area is 100mm$^2/\\cos(\\var{theta})=\\var{siground(AV,3)}$mm$^2$.</li>\n<li>The axial stress is $\\var{force}$kN / 100mm$^2=\\var{SA}$MPa.</li>\n<li>The direct stress in the $x$ direction is the force component in the $x$ direction ($F\\cos(\\theta)$) divided by the vertical section area, i.e.: $\\sigma_x= \\var{force}$kN $\\times \\cos^2(\\var{theta}) \\div 100$mm$^2 = \\var{siground(SX,3)}$MPa.</li>\n<li>The direct stress in the $y$ direction is the force component in the $y$ direction ($F\\sin(\\theta)$) divided by the&nbsp;horizontal section area, i.e.: $\\sigma_y= \\var{force}$kN $\\times \\sin^2(\\var{theta}) \\div 100$mm$^2 = \\var{siground(SY,3)}$MPa.</li>\n<li>The shear stress in the $xy$ plane&nbsp;is the force component in the $x$ direction ($F\\cos(\\theta)$) divided by the&nbsp;horizontal section area, or equivalently the force component in the $y$&nbsp;direction ($F\\sin(\\theta)$)&nbsp;divided by the vertical section area, i.e.:? $\\tau_{xy}= \\var{force}$kN $\\times \\sin(\\var{theta})\\cos(\\var{theta}) \\div 100$mm$^2 = \\var{siground(TXY,3)}$MPa.</li>\n</ol>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"theta": {"name": "theta", "group": "Ungrouped variables", "definition": "random(5..355#10)", "description": "<p>Angle of bar to x-axis.</p>", "templateType": "anything", "can_override": false}, "force": {"name": "force", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "<p>Applied axial load.</p>", "templateType": "anything", "can_override": false}, "AH": {"name": "AH", "group": "Ungrouped variables", "definition": "100/abs(sin(radians(theta)))", "description": "<p>Area of horizontal section.</p>", "templateType": "anything", "can_override": false}, "AV": {"name": "AV", "group": "Ungrouped variables", "definition": "100/abs(cos(radians(theta)))", "description": "<p>Area of vertical section</p>", "templateType": "anything", "can_override": false}, "SA": {"name": "SA", "group": "Ungrouped variables", "definition": "1000*force/100", "description": "<p>Axial stress.</p>", "templateType": "anything", "can_override": false}, "SX": {"name": "SX", "group": "Ungrouped variables", "definition": "1000*(force/100)*(cos(radians(theta)))^2", "description": "<p>Direct stress (sxx).</p>", "templateType": "anything", "can_override": false}, "SY": {"name": "SY", "group": "Ungrouped variables", "definition": "1000*(force/100)*(sin(radians(theta)))^2", "description": "<p>Direct stress (syy).</p>", "templateType": "anything", "can_override": false}, "TXY": {"name": "TXY", "group": "Ungrouped variables", "definition": "1000*(force/100)*sin(radians(theta))*cos(radians(theta))", "description": "<p>Shear stress (txy).</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["theta", "force", "AH", "AV", "SA", "SX", "SY", "TXY"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The figure below shows a bar with axial load $F$, at angle $\\theta$ to the $x$ axis.</p>\n<p><img src=\"resources/question-resources/Truss-4.png\" style=\"width: 50%;\" alt=\"Axially loaded bar\"/></p>\n<p>The bar has a solid square cross-section (10mm&times;10mm). If the axial load is $F=\\var{force}$ kN, and the angle is $\\theta=\\var{theta}^\\circ$:</p>\n<ol>\n<li>What is the horizontal section area? [[0]] [Units: mm$^2$]</li>\n<li>What is the vertical section area? [[1]] [Units: mm$^2$]</li>\n<li>What is the axial stress? [[2]] [Units: MPa]</li>\n<li>What is the direct stress in the $x$ direction? $\\sigma_x=$[[3]] [Units: MPa]</li>\n<li>What is the direct stress in the $y$ direction? $\\sigma_y=$[[4]] [Units: MPa]</li>\n<li>What is the shear stress in the $xy$ plane? $\\tau_{xy}=$[[5]] [Units: MPa]</li>\n</ol>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "AH", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "AH", "maxValue": "AH", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct 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