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Pressure calculation and 2D Mohr's Circle examples.

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Determine maximum pressure in a closed thin-walled cylindrical pressure vessel before yield.

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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.

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Using $\\sigma_h = {p D \\over 2 t}$ and $\\sigma_a = {p D \\over 4 t}$, the von Mises stress is given by:

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$\\sigma_V^2 = \\sigma_a^2 - \\sigma_a \\sigma_h + \\sigma_h^2 = \\left({p D \\over 4 t}\\right)^2 - \\left({p D \\over 4 t}\\right)\\left({p D \\over 2 t}\\right) +\\left({p D \\over 2 t}\\right)^2 = 3\\left({p D \\over 4 t}\\right)^2$

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i.e.:

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$\\sigma_V = \\sqrt{3}\\left({p D \\over 4 t}\\right)$

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which can be rearranged to give pressure:

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$p = \\sigma_V {4 t \\over D \\sqrt{3}} = \\sigma_V {4 \\times \\var{thickness} \\times 10^{-3} \\over \\var{diameter} \\times \\sqrt{3}} = \\var{siground(factor,3)}\\sigma_V$

\n

where $\\sigma_a$ is the axial stress and $\\sigma_h$ is the hoop stress.

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The maximum pressure (such that $\\sigma_V < \\sigma_Y$) for:

\n
    \n
  1. a steel ($\\sigma_Y=\\var{sYFe}$ MPa) pressure vessel is $\\var{siground(factor*sYFe,3)}$MPa.
  2. \n
  3. an aluminium ($\\sigma_Y=\\var{sYAl}$ MPa) pressure vessel is $\\var{siground(factor*sYAl,3)}$MPa.
  4. \n
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Yield stress of steel.

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Wall thickness of thin-walled pressure vessel.

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Yield stress of aluminium.

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Diameter of thin-walled pressure vessel.

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pressure / yield stress

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\n

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:

\n

$\\sigma_V^2 = \\sigma_a^2 - \\sigma_a \\sigma_h + \\sigma_h^2$

\n

where $\\sigma_a$ is the axial stress and $\\sigma_h$ is the hoop stress.

\n

What is the maximum pressure (such that $\\sigma_V < \\sigma_Y$) for:

\n
    \n
  1. a steel ($\\sigma_Y=\\var{sYFe}$ MPa) pressure vessel? [[0]] [Units: MPa]
  2. \n
  3. an aluminium ($\\sigma_Y=\\var{sYAl}$ MPa) pressure vessel? [[1]] [Units: MPa]
  4. \n
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Using 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.

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    \n
  1. The mean stress, $\\sigma_m=(\\sigma_x+\\sigma_y)/2=\\var{siground(sigmamean,3)}$MPa.
  2. \n
\n

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 ($\\var{siground(sigmamean,3)},0$):

\n

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

\n
    \n
  1. The maximum principal stress is the mean stress plus the radius: $\\var{siground(sigmamean+taumax,3)}$MPa.
  2. \n
  3. The minimum principal stress is the mean stress minus the radius: $\\var{siground(sigmamean-taumax,3)}$MPa.
  4. \n
  5. The maximum shear stress is just the radius: $\\var{siground(taumax,3)}$MPa.
  6. \n
\n

The 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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Shear stress in $xy$ plane.

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Normal stress in $x$ direction.

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Maximum shear stress.

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Angle to principal axes, doubled, not adjusted for quadrant.

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Normal stress in $y$ direction.

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Mean stress.

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A sheet steel component is subject to stresses $\\sigma_x=\\var{sigmax}$ MPa, $\\sigma_y=\\var{sigmay}$ MPa and $\\tau_{xy}=\\var{tauxy}$ MPa.

\n

Determine:

\n
    \n
  1. The mean stress, $\\sigma_m=$[[0]] [Units: MPa]
  2. \n
  3. The maximum principal stress, $\\sigma_1=$[[1]] [Units: MPa]
  4. \n
  5. The minimum principal stress, $\\sigma_2=$[[2]] [Units: MPa]
  6. \n
  7. The maximum shear stress, $\\tau_\\text{max}=$[[3]] [Units: MPa]
  8. \n
\n

What is the angle, $\\theta$, between the principal axes and the $xy-$axes? [[4]] [Units: degrees, $0\\le\\theta<180$]

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Practice for thin-walled pressure vessel and 2D principal stresses

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