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a/book/design_workflow/general_mechanics/cross_sectional_quantities.ipynb b/book/design_workflow/general_mechanics/cross_sectional_quantities.ipynb index 566dcd5..2b806d2 100644 --- a/book/design_workflow/general_mechanics/cross_sectional_quantities.ipynb +++ b/book/design_workflow/general_mechanics/cross_sectional_quantities.ipynb @@ -7,6 +7,173 @@ "# Cross sectional quantitites" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Definition section parameters\n", + "| Surface |Static moment of area| Moment of Inertia | Polar moment of inertia | Steiner's rule |\n", + "|--------------|---------------------|------------------------------|-----------------------------------------|----------------|\n", + "|$A=\\int_A\\,dA$|$S_y = \\int_A y\\, dA$|$I_{yy}=I_z=\\int_A y^2\\,dA$ |$I_p = \\int_A r^2\\, dA = I_{yy} + I_{zz}$|$I_{\\overline{yy}}=I_{yy(own)}+\\bar{y}_c^2 \\cdot A$| \n", + "| |$S_z = \\int_A z\\, dA$|$I_{yz}=I_{zy}=\\int_A y*z\\,dA$| |$I_{\\overline{yz}}=I_{\\overline{zy}}=I_{yz(own)}+\\bar{y}_c \\cdot \\bar{z}_c \\ A$|\n", + "| | |$I_{zz}=I_{y}=\\int_A z^2\\,dA$ | |$I_{\\overline{zz}}=I_{zz(own)}+\\bar{z}_c^2 \\cdot A$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Section parameters\n", + "\n", + "```{figure} Images/Rectangle shape.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "### Square\n", + "|Area, centre of gravity| Moments of intertia own | other |\n", + "|-----------------------|-------------------------|-----------------------------|\n", + "|$A=bh$ |$I_{yy}=\\frac{1}{12}*b^3*h$ |$I_{\\overline{yy}}=1/3b^3h$ | \n", + "|$\\bar{y}_C=\\frac{1}{2}b$ |$I_{zz}=\\frac{1}{12}bh^3$ |$I_{\\overline{zz}}=\\frac{1}{3}bh^3$ |\n", + "|$\\bar{z}_C=\\frac{1}{2}b$ |$I_{yz}=\\frac{1}{12}bh^3$ |$I_{\\overline{yz}}=\\frac{1}{4}b^2h^2$|\n", + "\n", + "```{figure} Images/Parallelogram shape.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "### Parallelogram\n", + "|Area, centre of gravity | Moments of intertia own | other |\n", + "|---------------------------|-------------------------------|-------------------------------|\n", + "|$A=bh$ |$I_{yy}=\\frac{1}{12}(a^2+b^2)$|$I_{\\overline{yy}}=1/3bh^3$ | \n", + "|$\\bar{y}_C=\\frac{1}{2}(a+b)$ |$I_{zz}=\\frac{1}{12}bh^3$ | |\n", + "|$\\bar{z_C}=\\frac{1}{2}h$ |$I_{yz}=\\frac{1}{12}abh^2$ | |\n", + "\n", + "```{figure} Images/Triangle shape.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "### Triangle\n", + "|Area, centre of gravity | Moments of intertia own | other |\n", + "|------------------------------|----------------------------------|---------------------------------|\n", + "|$A=\\frac{1}{2}bh$ |$I_{yy}=\\frac{1}{36}*(a^2-ab+b^2)$|$I_{\\overline{zz}}=\\frac{1}{3}bh^3$ | \n", + "|$\\bar{y}_C=\\frac{1}{3}(2a-b)$ |$I_{zz}=\\frac{1}{36}bh^3$ |$I_{\\overline{yz}}=\\frac{1}{8}(2a-b)bh^2$|\n", + "|$\\bar{z}_C=\\frac{2}{3}h$ |$I_{yz}=\\frac{1}{72}abh^2$ |$I_{\\overline{\\overline{zz}}}=\\frac{1}{12}bh^3$ |\n", + "\n", + "```{figure} Images/Trapezoid shape.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "### Trapezoid\n", + "|Area, centre of gravity | Moments of intertia own | other | \n", + "|-----------------------------------------|-----------------------------------------------|----------------------------------------------------|\n", + "|$A=\\frac{1}{2}(a+b)h$ |$I_{zz}=\\frac{1}{36}\\frac{a^2+4ab+b^2}{a+b}h^3$|$I_{\\overline{zz}}=\\frac{1}{3}bh^3$ | \n", + "|$\\bar{y}_C=\\frac{1}{3}\\frac{a+2b}{a+b}h$ | |$I_{\\overline{\\overline{zz}}}=\\frac{1}{12}(a+3b)h^3$|\n", + "|$\\bar{z}_C=\\frac{2}{3}h$ | | |\n", + "\n", + "```{figure} Images/Circle shape.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "### Circle\n", + "|Area, centre of gravity | Moments of intertia own | other | \n", + "|------------------------|------------------------------|-----------------------------------------------------|\n", + "|$A=πR^2$ |$I_{yy}=I_{zz}=\\frac{1}{4}R^4$|$I_{\\overline{yy}}=I_{\\overline{zz}}=\\frac{5}{4}πR^4$| \n", + "| |$I_{yz}=0$ |$I_{\\overline{\\overline{yz}}}=πR^4$ |\n", + "| |$I_p=\\frac{1}{2}πR^4$ | |\n", + "\n", + "```{figure} Image/Thick walled ring shape.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "### Thick-walled ring\n", + "|Area, centre of gravity | Moments of intertia own | other | \n", + "|------------------------|-----------------------------------------|-----------------|\n", + "|$A=πR_0^2R_i^2$ |$I_{yy}=I_{zz}=\\frac{1}{4}π(R_0^4-R_i^4)$| | \n", + "| |$I_{yz}=0$ | |\n", + "| |$I_p=\\frac{1}{2}π(R_0^4-R_i^4)$ | |\n", + "\n", + "```{figure} Images/Thin walled ring shape.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "### Thin-walled ring\n", + "|Area, centre of gravity | Moments of intertia own | other | \n", + "|------------------------|------------------------------|--------------------------------------------|\n", + "|$A=2πRt$ |$I_{yy}=I_{zz}=πR^3t$ |$I_{\\overline{yy}}=I_{\\overline{zz}}=3πR^3t$| \n", + "| |$I_{yz}=0$ | |\n", + "| |$I_p=\\frac{1}{2}πR^3t$ | |\n", + "\n", + "```{figure} Images/Half circle shape.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "### Half circle\n", + "|Area, centre of gravity | Moments of intertia own | other | \n", + "|--------------------------|--------------------------------------|-----------------------------------------------------|\n", + "|$A=\\frac{1}{2}πR^2$ |$I_{yy}=\\frac{1}{8}πR^4$ |$I_{\\overline{yy}}=I_{\\overline{zz}}=\\frac{1}{8}πR^4$| \n", + "|$\\bar{y}_C=0$ |$I_{zz}=(\\frac{π}{8}-\\frac{8}{9π})R^4$|$I_{\\overline{\\overline{yz}}}=0$ |\n", + "|$\\bar{z}_C=\\frac{4}{3π}R$ |$I_{yz}=0$ | |\n", + "\n", + "```{figure} Images/Half thin walled ring shape.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "### Half thin-walled ring\n", + "|Area, centre of gravity | Moments of intertia own | other | \n", + "|------------------------|-------------------------------|------------------------------------------------------|\n", + "|$A=πRt$ |$I_{yy}=I_{zz}=\\frac{1}{2}R^3t$|$I_{\\overline{yy}}=I_{\\overline{zz}}=\\frac{1}{2}πR^3t$| \n", + "|$\\bar{y}_C=0$ |$I_{zz}=0$ |$I_{\\overline{yz}}=0$ |\n", + "|$\\bar{z}_C=0$ |$I_{yz}=0$ | |" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Centre of gravity for moment diagrams\n", + "\n", + "```{figure} Images/Centre of gravity 1.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "```{figure} Images/Centre of gravity 2.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "```{figure} Images/Centre of gravity 3.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```\n", + "\n", + "```{figure} Images/Centre of gravity 4.jpg\n", + "---\n", + "scale: 100%\n", + "---\n", + "```" + ] + }, { "cell_type": "markdown", "metadata": {},