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Pressure calculation and 2D Mohr's Circle examples.
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "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.
", "advice": "A closed, cylindrical, thin-walled pressure vessel has diameter $D = \\var{diameter}$ m and wall thickness $t = \\var{thickness}$ mm.
\nUsing $\\sigma_h = {p D \\over 2 t}$ and $\\sigma_a = {p D \\over 4 t}$, the von Mises stress is given by:
\n$\\sigma_V^2 = \\sigma_a^2 - \\sigma_a \\sigma_h + \\sigma_h^2 = \\left({p D \\over 4 t}\\right)^2 - \\left({p D \\over 4 t}\\right)\\left({p D \\over 2 t}\\right) +\\left({p D \\over 2 t}\\right)^2 = 3\\left({p D \\over 4 t}\\right)^2$
\ni.e.:
\n$\\sigma_V = \\sqrt{3}\\left({p D \\over 4 t}\\right)$
\nwhich can be rearranged to give pressure:
\n$p = \\sigma_V {4 t \\over D \\sqrt{3}} = \\sigma_V {4 \\times \\var{thickness} \\times 10^{-3} \\over \\var{diameter} \\times \\sqrt{3}} = \\var{siground(factor,3)}\\sigma_V$
\nwhere $\\sigma_a$ is the axial stress and $\\sigma_h$ is the hoop stress.
\nThe maximum pressure (such that $\\sigma_V < \\sigma_Y$) for:
\nYield stress of steel.
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", "templateType": "anything"}, "factor": {"name": "factor", "group": "Ungrouped variables", "definition": "(4*(thickness/1000))/(sqrt(3)*diameter)", "description": "pressure / yield stress
", "templateType": "anything"}}, "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": "\nA 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:
\n$\\sigma_V^2 = \\sigma_a^2 - \\sigma_a \\sigma_h + \\sigma_h^2$
\nwhere $\\sigma_a$ is the axial stress and $\\sigma_h$ is the hoop stress.
\nWhat is the maximum pressure (such that $\\sigma_V < \\sigma_Y$) for:
\nUsing Mohr's Circle to calculate principal stresses in plane stress 2D case.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "For components in plane stress, Mohr's circle provides a quick and easy method for determining the principal stresses and the maximum shear stress.
", "advice": "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 ($\\var{siground(sigmamean,3)},0$):
\nradius = $\\sqrt{(\\sigma_x-\\sigma_m)^2+\\tau_{xy}^2} =\\sqrt{(\\var{sigmax}-(\\var{siground(sigmamean,3)}))^2+(\\var{tauxy})^2} = \\var{siground(taumax,3)}$MPa
\nThe angle, $\\theta$, between the principal axes and the $xy-$axes is given by $\\tan(2\\theta)={\\tau_{xy} \\over \\sigma_x-\\sigma_m}$:
\n$\\theta={1 \\over 2} \\tan^{-1}\\left({\\var{tauxy} \\over \\var{siground(sigmax-sigmamean,3)}}\\right) = \\var{siground(theta,3)}^\\circ$.
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\nDetermine:
\nWhat is the angle, $\\theta$, between the principal axes and the $xy-$axes? [[4]] [Units: degrees, $0\\le\\theta<180$]
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