// Numbas version: exam_results_page_options
{"name": "2D Mohr's Circle", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "2D Mohr's Circle", "tags": [], "metadata": {"description": "<p>Using Mohr's Circle to calculate principal stresses in plane stress 2D case.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>For components in plane stress, Mohr's circle provides a quick and easy method for determining the principal stresses and the maximum shear stress.</p>", "advice": "<ol>\n<li>The mean stress, $\\sigma_m=(\\sigma_x+\\sigma_y)/2=\\var{siground(sigmamean,3)}$MPa.</li>\n</ol>\n<p>The mean stress determines the centre of Mohr's circle, and the radius can be found by specifying a coordinate on the circle, i.e.: ($\\sigma_x,\\tau_{xy}$) = ($\\var{sigmax},\\var{tauxy}$), and then using Pythagoras to determine the length of the radius from the centre of the circle at ($\\var{siground(sigmamean,3)},0$):</p>\n<p>radius = $\\sqrt{(\\sigma_x-\\sigma_m)^2+\\tau_{xy}^2} =\\sqrt{(\\var{sigmax}-(\\var{siground(sigmamean,3)}))^2+(\\var{tauxy})^2} = \\var{siground(taumax,3)}$MPa</p>\n<ol start=\"2\">\n<li>The maximum principal stress is the mean stress plus the radius: $\\var{siground(sigmamean+taumax,3)}$MPa.</li>\n<li>The minimum principal stress is the mean stress minus the radius: $\\var{siground(sigmamean-taumax,3)}$MPa.</li>\n<li>The maximum shear stress is just the radius: $\\var{siground(taumax,3)}$MPa.</li>\n</ol>\n<p>The angle, $\\theta$, between the principal axes and the $xy-$axes is given by $\\tan(2\\theta)={\\tau_{xy} \\over \\sigma_x-\\sigma_m}$:</p>\n<p>$\\theta={1 \\over 2} \\tan^{-1}\\left({\\var{tauxy} \\over \\var{siground(sigmax-sigmamean,3)}}\\right) = \\var{siground(theta,3)}^\\circ$.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"tauxy": {"name": "tauxy", "group": "Ungrouped variables", "definition": "random(-5..25#3)", "description": "<p>Shear stress in $xy$ plane.</p>", "templateType": "anything", "can_override": false}, "sigmax": {"name": "sigmax", "group": "Ungrouped variables", "definition": "random(-4..40#2)", "description": "<p>Normal stress in $x$ direction.</p>", "templateType": "anything", "can_override": false}, "theta": {"name": "theta", "group": "Ungrouped variables", "definition": "if(sigmax<sigmay,twotheta+180,if(tauxy<0,twotheta+360,twotheta))/2", "description": "<p>Angle to principal axes, adjusted for quadrant.</p>", "templateType": "anything", "can_override": false}, "taumax": {"name": "taumax", "group": "Ungrouped variables", "definition": "sqrt((sigmax-sigmamean)^2+tauxy^2)", "description": "<p>Maximum shear stress.</p>", "templateType": "anything", "can_override": false}, "twotheta": {"name": "twotheta", "group": "Ungrouped variables", "definition": "degrees(arctan(tauxy/(sigmax-sigmamean)))", "description": "<p>Angle to principal axes, doubled, not adjusted for quadrant.</p>", "templateType": "anything", "can_override": false}, "sigmay": {"name": "sigmay", "group": "Ungrouped variables", "definition": "random(-3..13#2)", "description": "<p>Normal stress in $y$ direction.</p>", "templateType": "anything", "can_override": false}, "sigmamean": {"name": "sigmamean", "group": "Ungrouped variables", "definition": "(sigmax+sigmay)/2", "description": "<p>Mean stress.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["sigmax", "sigmay", "sigmamean", "tauxy", "taumax", "twotheta", "theta"], "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>A sheet steel component is subject to stresses $\\sigma_x=\\var{sigmax}$ MPa, $\\sigma_y=\\var{sigmay}$ MPa and $\\tau_{xy}=\\var{tauxy}$ MPa.</p>\n<p>Determine:</p>\n<ol>\n<li>The mean stress, $\\sigma_m=$[[0]] [Units: MPa]</li>\n<li>The maximum principal stress, $\\sigma_1=$[[1]] [Units: MPa]</li>\n<li>The minimum principal stress, $\\sigma_2=$[[2]] [Units: MPa]</li>\n<li>The maximum shear stress, $\\tau_\\text{max}=$[[3]] [Units: MPa]</li>\n</ol>\n<p>What is the angle, $\\theta$, between the principal axes and the $xy-$axes? [[4]] [Units: degrees, $0\\le\\theta&lt;180$]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "sigmamean", "maxValue": "sigmamean", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "sigmamean+taumax", "maxValue": "sigmamean+taumax", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "sigmamean-taumax", "maxValue": "sigmamean-taumax", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "taumax", "maxValue": "taumax", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "theta", "maxValue": "theta", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}]}]}], "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}]}