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