// Numbas version: exam_results_page_options
{"name": "3D Stress - General case and von Mises calculation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "3D Stress - General case and von Mises calculation", "tags": [], "metadata": {"description": "<p>Calculate invariants and von Mises stress for a general 3D stress state.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>The principal stresses and maximum shear stress, and the von Mises stress, can all be determined from the 3D stress matrix and its invariants, i.e., if:</p>\n<p>\\[\\sigma=\\begin{pmatrix}\\sigma_x &amp; \\tau_{xy} &amp; \\tau_{zx} \\\\ \\tau_{xy} &amp; \\sigma_y &amp; \\tau_{yz} \\\\ \\tau_{zx} &amp; \\tau_{yz} &amp; \\sigma_z\\end{pmatrix}\\]</p>\n<p>then the three invariants are:</p>\n<ol>\n<li>The sum of the diagonal elements (the 'trace'): $I_1 = \\sigma_x + \\sigma_y + \\sigma_z$ [Units: Pa].</li>\n<li>An expression that is a sum of sub-determinants: $I_2 = \\sigma_x \\sigma_y + \\sigma_y \\sigma_z + \\sigma_z \\sigma_x - \\tau_{xy}^2&nbsp;- \\tau_{yz}^2 - \\tau_{zx}^2$&nbsp;[Units: Pa$^2$].</li>\n<li>The determinant: $I_3 = \\sigma_x \\sigma_y \\sigma_z&nbsp;+ 2 \\tau_{xy} \\tau_{yz} \\tau_{zx} - \\sigma_x \\tau_{yz}^2&nbsp;-\\sigma_y \\tau_{zx}^2 - \\sigma_z \\tau_{xy}^2$ [Units: Pa$^3$].</li>\n</ol>\n<p>From these we can easily calculate:</p>\n<ol>\n<li>The mean ('hydrostatic') stress, which is the sum of the diagonal elements divided by three: $\\sigma_\\text{mean} = I_1 / 3$ [Units: Pa].</li>\n<li>The second deviatoric stress invariant: $J_2 = I_2 - I_1^2 / 3$ [Units: Pa$^2$].</li>\n<li>The von Mises ('equivalent') stress: $\\sigma_V = \\sqrt{-3J_2}$ [Units: Pa].</li>\n</ol>\n<p>The principal stresses can be found by solving the eigenvalue/eigenvector matrix problem (the eigenvalues are the principal stresses), or by finding the three roots (the three roots - $\\lambda_1$, $\\lambda_2$, $\\lambda_3$ - are the principal stresses) of the equation:</p>\n<p>\\[\\lambda^3 - I_1 \\lambda^2 + I_2 \\lambda - I_3 = 0\\]</p>\n<p><strong><em>Note A.</em></strong> If the three principal stresses [Units: Pa] are different (which is usually but not always the case):</p>\n<ol>\n<li>The <strong>maximum</strong> principal stress, $\\sigma_\\text{max}$, is the most tensile (most positive or least negative) of the principal stresses.</li>\n<li>The <strong>minimum</strong> principal stress, $\\sigma_\\text{min}$, is the most compressive (most negative or least positive) of the principal stresses.</li>\n<li>The <strong>middle</strong> principal stress, $\\sigma_\\text{middle}$, is in between the other two, so that $\\sigma_\\text{min} \\le \\sigma_\\text{middle} \\le \\sigma_\\text{max}$.</li>\n</ol>\n<p><em><strong>Note B:</strong></em> 1 MPa = $10^6$ Pa = $10^6$ N/m$^2$ = 1 N/mm$^2$</p>\n<p><em><strong>Note C:</strong></em> 1 MPa$^2$ = $10^{12}$ Pa$^2$, etc.</p>", "advice": "<p>Calculate the invariants:</p>\n<ol>\n<li>$I_1=\\sigma_x + \\sigma_y + \\sigma_z = \\var{sigmax} +\\var{sigmay} +\\var{sigmaz} = \\var{I1}$MPa.</li>\n<li>$I_2=\\sigma_x \\sigma_y + \\sigma_y \\sigma_z + \\sigma_z \\sigma_x - \\tau_{xy}^2&nbsp;- \\tau_{yz}^2 - \\tau_{zx}^2 = (\\var{sigmax}) \\times (\\var{sigmay}) + (\\var{sigmay}) \\times (\\var{sigmaz}) + (\\var{sigmaz}) \\times (\\var{sigmax}) - (\\var{tauxy})^2&nbsp;- (\\var{tauyz})^2 - (\\var{tauzx})^2 = \\var{I2}$MPa$^2$.</li>\n<li>$I_3=\\sigma_x \\sigma_y \\sigma_z&nbsp;+ 2 \\tau_{xy} \\tau_{yz} \\tau_{zx} - \\sigma_x \\tau_{yz}^2&nbsp;-\\sigma_y \\tau_{zx}^2 - \\sigma_z \\tau_{xy}^2$ = $(\\var{sigmax}) \\times (\\var{sigmay}) \\times (\\var{sigmaz}) + 2 \\times (\\var{tauxy}) \\times (\\var{tauyz}) \\times (\\var{tauzx}) - (\\var{sigmax}) \\times (\\var{tauyz})^2&nbsp;- (\\var{sigmay}) \\times (\\var{tauzx})^2 - (\\var{sigmaz}) \\times (\\var{tauxy})^2 = \\var{I3}$MPa$^3$.</li>\n</ol>\n<p>and thus calculate:</p>\n<ol>\n<li>The mean&nbsp;('hydrostatic') stress: $\\sigma_\\text{mean} = I_1 / 3 = \\var{I1} / 3 = \\var{siground(sigmamean,3)}$MPa.</li>\n<li>The second deviatoric stress invariant: $J_2 = I_2 - I_1^2 / 3 = \\var{I2} - (\\var{I1})^2 / 3 = \\var{siground(J2,3)}$MPa$^2$.</li>\n<li>The von Mises stress: $\\sigma_V = \\sqrt{-3 J_2} = \\sqrt{-3 \\times (\\var{siground(J2,3)})} = \\var{siground(sigmav,3)}$MPa.</li>\n</ol>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"tauyz": {"name": "tauyz", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "<p>Shear stress in $yz$ plane.</p>", "templateType": "anything"}, "tauzx": {"name": "tauzx", "group": "Ungrouped variables", "definition": "random(5..15)", "description": "<p>Shear stress&nbsp;in $zx$ plane.</p>", "templateType": "anything"}, "I1": {"name": "I1", "group": "Ungrouped variables", "definition": "sigmax+sigmay+sigmaz", "description": "<p>First invariant.</p>", "templateType": "anything"}, "sigmaz": {"name": "sigmaz", "group": "Ungrouped variables", "definition": "-random(-16..17#3)", "description": "<p>Normal stress in $z$ direction</p>", "templateType": "anything"}, "sigmay": {"name": "sigmay", "group": "Ungrouped variables", "definition": "random(-18..15#3)", "description": "<p>Normal stress in $y$ direction.</p>", "templateType": "anything"}, "sigmamean": {"name": "sigmamean", "group": "Ungrouped variables", "definition": "I1/3", "description": "<p>Mean stress.</p>", "templateType": "anything"}, "sigmax": {"name": "sigmax", "group": "Ungrouped variables", "definition": "random(-17..16#3)", "description": "<p>Normal stress in $x$ direction.</p>", "templateType": "anything"}, "tauxy": {"name": "tauxy", "group": "Ungrouped variables", "definition": "random(-15..-5)", "description": "<p>Shear stress in $xy$ plane.</p>", "templateType": "anything"}, "I2": {"name": "I2", "group": "Ungrouped variables", "definition": "sigmax*sigmay+sigmay*sigmaz+sigmaz*sigmax-tauzx^2-tauxy^2-tauyz^2", "description": "<p>Second invariant.</p>", "templateType": "anything"}, "J2": {"name": "J2", "group": "Ungrouped variables", "definition": "I2-I1^2/3", "description": "<p>Second deviatoric invariant.</p>", "templateType": "anything"}, "I3": {"name": "I3", "group": "Ungrouped variables", "definition": "sigmax*sigmay*sigmaz+2*tauxy*tauyz*tauzx-sigmax*tauyz^2-sigmay*tauzx^2-sigmaz*tauxy^2", "description": "<p>Third invariant.</p>", "templateType": "anything"}, "sigmav": {"name": "sigmav", "group": "Ungrouped variables", "definition": "sqrt(-3*J2)", "description": "<p>von Mises stress.</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["sigmax", "sigmay", "sigmaz", "tauxy", "tauyz", "tauzx", "I1", "I2", "I3", "J2", "sigmav", "sigmamean"], "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 stress at a particular point in a component has been calculated as:</p>\n<p>\\[\\sigma=\\begin{pmatrix} \\var{sigmax} &amp; \\var{tauxy} &amp; \\var{tauzx} \\\\ \\var{tauxy} &amp; \\var{sigmay} &amp; \\var{tauyz} \\\\ \\var{tauzx} &amp; \\var{tauyz} &amp; \\var{sigmaz} \\end{pmatrix} \\text{[Units: MPa]}\\]</p>\n<p>Calculate the invariants:</p>\n<ol>\n<li>$I_1=$[[0]] [Units: MPa].</li>\n<li>$I_2=$[[1]] [Units: MPa$^2$].</li>\n<li>$I_3=$[[2]] [Units: MPa$^3$].</li>\n</ol>\n<p>and thus calculate:</p>\n<ol>\n<li>The mean stress: $\\sigma_\\text{mean}=$[[3]] [Units: MPa].</li>\n<li>The second deviatoric stress invariant: $J_2=$[[4]] [Units: MPa$^2$]</li>\n<li>The von Mises stress: $\\sigma_V=$[[5]] [Units: MPa].</li>\n</ol>", "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": "I1", "maxValue": "I1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, 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