// Numbas version: finer_feedback_settings
{"name": "Thin-walled pressure vessel", "extensions": [], "custom_part_types": [], "resources": [["question-resources/ThinWalledCylinder-2.png", "/srv/numbas/media/question-resources/ThinWalledCylinder-2.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Thin-walled pressure vessel", "tags": [], "metadata": {"description": "<p>Determine maximum pressure in a closed thin-walled cylindrical pressure vessel before yield.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>A closed, cylindrical, thin-walled pressure vessel can be considered as a biaxial stress case with the hoop stress and axial stress as principal stresses.</p>", "advice": "<p>A closed, cylindrical, thin-walled pressure vessel has diameter $D = \\var{diameter}$ m and wall thickness $t = \\var{thickness}$ mm.</p>\n<p>Using $\\sigma_h = {p D \\over 2 t}$ and $\\sigma_a = {p D \\over 4 t}$, the von Mises stress is given by:</p>\n<p style=\"text-align: center;\">$\\sigma_V^2 = \\sigma_a^2 - \\sigma_a \\sigma_h + \\sigma_h^2 = \\left({p D \\over 4 t}\\right)^2 - \\left({p D \\over 4 t}\\right)\\left({p D \\over 2 t}\\right) +\\left({p D \\over 2 t}\\right)^2 = 3\\left({p D \\over 4 t}\\right)^2$</p>\n<p>i.e.:</p>\n<p style=\"text-align: center;\">$\\sigma_V = \\sqrt{3}\\left({p D \\over 4 t}\\right)$</p>\n<p>which can be rearranged to give pressure:</p>\n<p style=\"text-align: center;\">$p = \\sigma_V {4 t \\over D \\sqrt{3}} = \\sigma_V {4 \\times \\var{thickness} \\times 10^{-3} \\over \\var{diameter} \\times \\sqrt{3}} = \\var{siground(factor,3)}\\sigma_V$</p>\n<p>where $\\sigma_a$ is the axial stress and $\\sigma_h$ is the hoop stress.</p>\n<p>The maximum pressure (such that $\\sigma_V &lt; \\sigma_Y$) for:</p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>a steel ($\\sigma_Y=\\var{sYFe}$ MPa) pressure vessel is $\\var{siground(factor*sYFe,3)}$MPa.</li>\n<li>an aluminium ($\\sigma_Y=\\var{sYAl}$ MPa) pressure vessel is $\\var{siground(factor*sYAl,3)}$MPa.</li>\n</ol>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"sYFe": {"name": "sYFe", "group": "Ungrouped variables", "definition": "random(300..400#10)", "description": "<p>Yield stress of steel.</p>", "templateType": "anything", "can_override": false}, "thickness": {"name": "thickness", "group": "Ungrouped variables", "definition": "random(0.5..10#0.1)", "description": "<p>Wall thickness of thin-walled pressure vessel.</p>", "templateType": "anything", "can_override": false}, "sYAl": {"name": "sYAl", "group": "Ungrouped variables", "definition": "random(200..280#10)", "description": "<p>Yield stress of aluminium.</p>", "templateType": "anything", "can_override": false}, "diameter": {"name": "diameter", "group": "Ungrouped variables", "definition": "random(0.9..2.2#0.1)", "description": "<p>Diameter of thin-walled pressure vessel.</p>", "templateType": "anything", "can_override": false}, "factor": {"name": "factor", "group": "Ungrouped variables", "definition": "(4*(thickness/1000))/(sqrt(3)*diameter)", "description": "<p>pressure / yield stress</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["diameter", "thickness", "sYFe", "sYAl", "factor"], "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 style=\"text-align: center;\"><img src=\"resources/question-resources/ThinWalledCylinder-2.png\"/></p>\n<p>A closed, cylindrical, thin-walled pressure vessel has diameter $D = \\var{diameter}$ m and wall thickness $t = \\var{thickness}$ mm. The von Mises stress is given by:</p>\n<p style=\"text-align: center;\">$\\sigma_V^2 = \\sigma_a^2 - \\sigma_a \\sigma_h + \\sigma_h^2$</p>\n<p>where $\\sigma_a$ is the axial stress and $\\sigma_h$ is the hoop stress.</p>\n<p>What is the maximum pressure (such that $\\sigma_V &lt; \\sigma_Y$) for:</p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>a steel ($\\sigma_Y=\\var{sYFe}$ MPa) pressure vessel? [[0]] [Units: MPa]</li>\n<li>an aluminium ($\\sigma_Y=\\var{sYAl}$ MPa) pressure vessel?&nbsp;[[1]] [Units: MPa]</li>\n</ol>", "gaps": [{"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": "factor*sYFe", "maxValue": "factor*sYFe", "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": "factor*sYAl", "maxValue": "factor*sYAl", "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", "type": "question", "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/"}]}