{
    "version" : "https://jsonfeed.org/version/1.1",
    "title" : "Marcin Woliński | Blog",
    "description": "engineering blog Marcin Wolinski — structural engineering and software.",
    "home_page_url" : "https://mwolinski.com/",
    "feed_url" : "https://mwolinski.com/index.feed",
    "icon" : "https://mwolinski.com/apple-touch-icon.png",
    "favicon" : "https://mwolinski.com/favicon.ico",
    "author" : {
        "name" : "M. Wolinski",
        "avatar": "https://mwolinski.com/images/avatar.jpg"
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    "items" : [
    {
        "title" : "Python In Excel",
        "date_published" : "2026-08-29T00:00:00Z",
        "date_modified" : "2026-09-04T18:22:58+02:00",
        "id" : "https://mwolinski.com/posts/python-in-excel/",
        "url" : "https://mwolinski.com/posts/python-in-excel/",
        "summary": "In 2024 Microsoft released the Python in Excel feature, yet I have never seen anyone use it - despite it being generally available. So I decided to try it myself, and it\u0026rsquo;s great. Let me show you how it can be used in daily Structural Engineer work.\nAnd let me be clear: I\u0026rsquo;m not a fan of Excel for structural calculations, but I can\u0026rsquo;t escape from reality.\nVisualisationThe best use case for me is visualization. Using matplotlib, I can visualize much more than standard built-in charts - shapes, dimensions, colors, finite element meshes, legends, and more.\n",
        "content_html" : "\u003cp\u003eIn 2024 Microsoft released the Python in Excel feature, yet I have never seen anyone use it - despite it being generally available. So I decided to try it myself, and it\u0026rsquo;s great. Let me show you how it can be used in daily Structural Engineer work.\u003c/p\u003e\n\u003cp\u003eAnd let me be clear: I\u0026rsquo;m not a fan of Excel for structural calculations, but I can\u0026rsquo;t escape from reality.\u003c/p\u003e\n\u003ch2 id=\"visualisation\" class=\"headerLink\"\u003e\n    \u003ca href=\"#visualisation\" class=\"header-mark\"\u003e\u003c/a\u003eVisualisation\u003c/h2\u003e\u003cp\u003eThe best use case for me is visualization. Using matplotlib, I can visualize much more than standard built-in charts - shapes, dimensions, colors, finite element meshes, legends, and more.\u003c/p\u003e\n\u003ch3 id=\"soil-profile-example\" class=\"headerLink\"\u003e\n    \u003ca href=\"#soil-profile-example\" class=\"header-mark\"\u003e\u003c/a\u003eSoil Profile Example\u003c/h3\u003e\u003cp\u003eA lot of the spreadsheets in my private library involve geotechnical calculations, and all of them have a soil profile. Previously it was just a table, but with Python in Excel I can dynamically plot the soil profile.\u003c/p\u003e\n\u003cfigure\u003e\u003cimg src=\"/posts/python-in-excel/soil-profile.png\"\n\t\t\talt=\"Plotted soil profile with layer colors and labels\"\u003e\u003cfigcaption\u003e\n\t\t\t\u003cp\u003eSoil profile plotted dynamically with Python in Excel.\u003c/p\u003e\n\t\t\u003c/figcaption\u003e\n\u003c/figure\u003e\n\n\u003ch3 id=\"strip-footing-example\" class=\"headerLink\"\u003e\n    \u003ca href=\"#strip-footing-example\" class=\"header-mark\"\u003e\u003c/a\u003eStrip Footing Example\u003c/h3\u003e\u003cp\u003eAnother real-life example is a spreadsheet for strip footings I wrote a while ago, but it always lacked good visuals, so I hesitated to use it for official printouts. Now I can plot a nice, dynamic figure and include it directly in the final report.\u003c/p\u003e\n\u003cfigure\u003e\u003cimg src=\"/posts/python-in-excel/foundation-pad.png\"\n\t\t\talt=\"Plotted strip footing plan with dimensions\"\u003e\u003cfigcaption\u003e\n\t\t\t\u003cp\u003eStrip footing geometry plotted with Python in Excel.\u003c/p\u003e\n\t\t\u003c/figcaption\u003e\n\u003c/figure\u003e\n\n\u003ch2 id=\"computation\" class=\"headerLink\"\u003e\n    \u003ca href=\"#computation\" class=\"header-mark\"\u003e\u003c/a\u003eComputation\u003c/h2\u003e\u003cp\u003eThis is serious stuff - you can use \u003cstrong\u003epandas\u003c/strong\u003e, \u003cstrong\u003enumpy\u003c/strong\u003e, and \u003cstrong\u003escipy\u003c/strong\u003e, which makes a lot of things possible: matrix math, statistics, and other scientific computation.\u003c/p\u003e\n\u003cp\u003eThe full list of libraries Excel is equipped with can be found in the \u003ca href=\"https://support.microsoft.com/en-us/excel/python/open-source-libraries-and-python-in-excel\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eofficial library list\u003c/a\u003e.\u003c/p\u003e\n\u003ch3 id=\"2d-reinforcement-for-in-plane-conditions\" class=\"headerLink\"\u003e\n    \u003ca href=\"#2d-reinforcement-for-in-plane-conditions\" class=\"header-mark\"\u003e\u003c/a\u003e2D Reinforcement for In-Plane Conditions\u003c/h3\u003e\u003cp\u003eI want to walk through computing tension reinforcement for in-plane stress, per Eurocode 2 Annex F.\u003c/p\u003e\n\u003cp\u003eLet\u0026rsquo;s assume we have FEA results from software that doesn\u0026rsquo;t support Eurocode 2 shell design, but can export nodes, elements, and corresponding stresses into Excel. Using Python in Excel, we can quickly and cleanly compute the reinforcement ratio for each element.\u003c/p\u003e\n\u003cp\u003eThe standard Eurocode math:\u003c/p\u003e\n\u003ch4 id=\"tensile-forces-f_tdx-f_tdz\" class=\"headerLink\"\u003e\n    \u003ca href=\"#tensile-forces-f_tdx-f_tdz\" class=\"header-mark\"\u003e\u003c/a\u003eTensile Forces ($f_{tdx}, f_{tdz}$)\u003c/h4\u003e\u003cp\u003eGiven stress components $\\sigma_{xx}, \\sigma_{zz}, \\sigma_{xz}$:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eCase 1: Full Compression\u003c/strong\u003e ($\\sigma_{xx} \u0026lt; 0$, $\\sigma_{zz} \\le 0$, and $\\sigma_{xx}\\sigma_{zz} \u0026gt; \\sigma_{xz}^2$)\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ef_{tdx} = 0, \\quad f_{tdz} = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eCase 2: Pure Tensile / Low Compression\u003c/strong\u003e ($\\sigma_{xx} \\ge -|\\sigma_{xz}|$ and $\\sigma_{zz} \\ge -|\\sigma_{xz}|$)\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eσ\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∣\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eσ\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi mathvariant=\"normal\"\u003e∣\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eσ\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∣\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eσ\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi mathvariant=\"normal\"\u003e∣\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ef_{tdx} = \\sigma_{xx} + |\\sigma_{xz}|, \\quad f_{tdz} = \\sigma_{zz} + |\\sigma_{xz}|\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eσ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003exx\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∣\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eσ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∣\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eσ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ezz\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∣\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eσ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∣\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eCase 3: High Compression in X\u003c/strong\u003e ($\\sigma_{zz} \\ge \\sigma_{xx}$ and $\\sigma_{xx} \\le -|\\sigma_{xz}|$)\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eσ\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmsubsup\u003e\u003cmi\u003eσ\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msubsup\u003e\u003cmsub\u003e\u003cmi\u003eσ\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ef_{tdx} = 0, \\quad f_{tdz} = \\sigma_{zz} + \\frac{\\sigma_{xz}^2}{\\sigma_{xx}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eσ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ezz\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.3271em;vertical-align:-0.836em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eσ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003exx\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eσ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.247em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.836em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eCase 4: High Compression in Z\u003c/strong\u003e ($\\sigma_{xx} \u0026gt; \\sigma_{zz}$ and $\\sigma_{zz} \\le -|\\sigma_{xz}|$)\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eσ\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmsubsup\u003e\u003cmi\u003eσ\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msubsup\u003e\u003cmsub\u003e\u003cmi\u003eσ\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mfrac\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ef_{tdx} = \\sigma_{xx} + \\frac{\\sigma_{xz}^2}{\\sigma_{zz}}, \\quad f_{tdz} = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eσ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003exx\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.3271em;vertical-align:-0.836em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eσ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ezz\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eσ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.247em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.836em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eDefault / Fallback:\u003c/strong\u003e\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ef_{tdx} = 0, \\quad f_{tdz} = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"reinforcement-ratios--areas\" class=\"headerLink\"\u003e\n    \u003ca href=\"#reinforcement-ratios--areas\" class=\"header-mark\"\u003e\u003c/a\u003eReinforcement Ratios \u0026amp; Areas\u003c/h4\u003e\u003cp\u003eUsing yield strength $f_{yd}$ and thickness $th$:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eReinforcement Ratios:\u003c/strong\u003e\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mfrac\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003erc_x = \\frac{f_{tdx}}{f_{yd}}, \\quad rc_z = \\frac{f_{tdz}}{f_{yd}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.3435em;vertical-align:-0.9721em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9721em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.3435em;vertical-align:-0.9721em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9721em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eReinforcement Areas:\u003c/strong\u003e\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003eh\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmi\u003eh\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003earc_x = rc_x \\cdot th, \\quad arc_z = rc_z \\cdot th\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5945em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e⋅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eh\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5945em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e⋅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eh\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThis is transformed into the following Python code:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers  tw:group tw:my-2\"\u003e\n  \u003cdiv class=\"\n    \n    tw:flex \n    tw:flex-row\n    tw:flex-1 \n    tw:justify-between \n    tw:w-full tw:bg-bgColor-secondary\n    \"\u003e      \n    \u003cbutton \n      class=\"\n        code-block-button\n        tw:mx-2 \n        tw:flex\n        tw:flex-row\n        tw:flex-1\"\n      aria-hidden=\"true\"\u003e\n          \u003cspan class=\"tw:group-[.is-open]:rotate-90 tw:transition-[transform] tw:duration-500 tw:ease-in-out tw:print:hidden! tw:w-min tw:h-min tw:my-1 tw:mx-1\"\u003e\u003csvg class=\"icon\"\n    xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n          \u003cspan class=\"tw:select-none tw:my-1! tw:block\"\u003epython\u003c/span\u003e\n      \u003c/button\u003e\n\n   \u003cdiv class=\"tw:flex\"\u003e\n      \u003cbutton \n        class=\"\n          line-number-button\n          tw:mx-2 \n          tw:hidden \n          tw:group-[.is-open]:block \n          tw:group-[.show-line-numbers]:text-fgColor-link \n          tw:print:hidden!\" \n        title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n    xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n      \u003cbutton \n        class=\"\n          wrap-code-button\n          tw:select-none \n          tw:mx-2 \n          tw:hidden \n          tw:group-[.is-open]:block \n          tw:group-[.is-wrap]:text-fgColor-link \n          tw:print:hidden!\" \n        title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n    xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n      \n      \u003cbutton \n        class=\"\n          copy-code-button\n          tw:select-none\n          tw:mx-2 \n          tw:hidden\n          tw:group-[.is-open]:block\n          tw:hover:text-fgColor-link \n          tw:print:hidden!\"\n        title=\"Copy code\"\u003e\n          \u003cspan class=\"copy-icon tw:block\"\u003e\u003csvg class=\"icon\"\n    xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n          \u003cspan class=\"check-icon tw:hidden\"\u003e\u003csvg class=\"icon\"\n    xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n      \u003c/button\u003e\n        \n      \u003cbutton \n        class=\"\n          tw:select-none \n          tw:mx-2 \n          tw:block \n          tw:group-[.is-open]:hidden \n          tw:print:hidden!\" \n        disabled\n        aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n    xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n    \u003c/div\u003e\n  \u003c/div\u003e\n  \u003cpre style=\"counter-reset: codeblock;\" class=\"tw:block tw:m-0 tw:p-0\"\u003e\u003ccode \n    id=\"codeblock-id-1\" \n    class=\"\n      chroma \n      tw:block! \n      tw:p-0\n      tw:m-0\n      tw:transition-[max-height] \n      tw:duration-500 \n      tw:ease-in-out \n      tw:group-[.is-closed]:max-h-0! \n      tw:group-[.is-wrap]:text-wrap\n      tw:overflow-y-hidden\n      tw:overflow-x-auto\n      tw:scrollbar-thin\n      \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"kn\"\u003eimport\u003c/span\u003e \u003cspan class=\"nn\"\u003epandas\u003c/span\u003e \u003cspan class=\"k\"\u003eas\u003c/span\u003e \u003cspan class=\"nn\"\u003epd\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"n\"\u003estress\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003exl\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;Stress[#All]\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003eheaders\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"kc\"\u003eTrue\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"n\"\u003estress\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003estress\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003estress\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;layer\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e \u003cspan class=\"o\"\u003e==\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;mid\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003efor\u003c/span\u003e \u003cspan class=\"n\"\u003ecol\u003c/span\u003e \u003cspan class=\"ow\"\u003ein\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;sxx\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;syy\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;szz\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;sxy\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;syz\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;szx\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]:\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003estress\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ecol\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003epd\u003c/span\u003e\u003cspan class=\"o\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003eto_numeric\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003estress\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ecol\u003c/span\u003e\u003cspan class=\"p\"\u003e],\u003c/span\u003e \u003cspan class=\"n\"\u003eerrors\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;coerce\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"n\"\u003efyd\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mf\"\u003e434.8e6\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"n\"\u003eth\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mf\"\u003e0.2\u003c/span\u003e  \u003cspan class=\"c1\"\u003e# Thickness and fy can be a dynamic input\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003edef\u003c/span\u003e \u003cspan class=\"nf\"\u003ecalculate_rc_ratio\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003estress_tensor\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003efyd\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003eth\u003c/span\u003e\u003cspan class=\"p\"\u003e):\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003estress_tensor\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"mi\"\u003e0\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003estress_tensor\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"mi\"\u003e2\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003esxz\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003estress_tensor\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"mi\"\u003e4\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003eif\u003c/span\u003e \u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e \u003cspan class=\"ow\"\u003eand\u003c/span\u003e \u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e \u003cspan class=\"ow\"\u003eand\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"o\"\u003e*\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"o\"\u003e**\u003c/span\u003e\u003cspan class=\"mi\"\u003e2\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdx\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdz\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003eelif\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;=\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"ow\"\u003eand\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;=\u003c/span\u003e \u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"mi\"\u003e1\u003c/span\u003e \u003cspan class=\"o\"\u003e*\u003c/span\u003e \u003cspan class=\"nb\"\u003eabs\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"p\"\u003e):\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdx\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"o\"\u003e+\u003c/span\u003e \u003cspan class=\"nb\"\u003eabs\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdz\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"o\"\u003e+\u003c/span\u003e \u003cspan class=\"nb\"\u003eabs\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003eelif\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;=\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"ow\"\u003eand\u003c/span\u003e \u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"mi\"\u003e1\u003c/span\u003e \u003cspan class=\"o\"\u003e*\u003c/span\u003e \u003cspan class=\"nb\"\u003eabs\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;=\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdx\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdz\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"o\"\u003e+\u003c/span\u003e \u003cspan class=\"nb\"\u003eabs\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e \u003cspan class=\"o\"\u003e**\u003c/span\u003e \u003cspan class=\"mi\"\u003e2\u003c/span\u003e \u003cspan class=\"o\"\u003e/\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003eelif\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"ow\"\u003eand\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"mi\"\u003e1\u003c/span\u003e \u003cspan class=\"o\"\u003e*\u003c/span\u003e \u003cspan class=\"nb\"\u003eabs\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"p\"\u003e):\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdx\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"o\"\u003e+\u003c/span\u003e \u003cspan class=\"nb\"\u003eabs\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdz\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"o\"\u003e+\u003c/span\u003e \u003cspan class=\"nb\"\u003eabs\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003eelif\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e \u003cspan class=\"ow\"\u003eand\u003c/span\u003e \u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"mi\"\u003e1\u003c/span\u003e \u003cspan class=\"o\"\u003e*\u003c/span\u003e \u003cspan class=\"nb\"\u003eabs\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;=\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdx\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003esxx\u003c/span\u003e \u003cspan class=\"o\"\u003e+\u003c/span\u003e \u003cspan class=\"nb\"\u003eabs\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003esxz\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e \u003cspan class=\"o\"\u003e**\u003c/span\u003e \u003cspan class=\"mi\"\u003e2\u003c/span\u003e \u003cspan class=\"o\"\u003e/\u003c/span\u003e \u003cspan class=\"n\"\u003eszz\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdz\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003eelse\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdx\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ef_tdz\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003erc_x\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003ef_tdx\u003c/span\u003e \u003cspan class=\"o\"\u003e/\u003c/span\u003e \u003cspan class=\"n\"\u003efyd\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003erc_z\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003ef_tdz\u003c/span\u003e \u003cspan class=\"o\"\u003e/\u003c/span\u003e \u003cspan class=\"n\"\u003efyd\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003earc_x\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003erc_x\u003c/span\u003e \u003cspan class=\"o\"\u003e*\u003c/span\u003e \u003cspan class=\"n\"\u003eth\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003earc_z\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003erc_z\u003c/span\u003e \u003cspan class=\"o\"\u003e*\u003c/span\u003e \u003cspan class=\"n\"\u003eth\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003ereturn\u003c/span\u003e \u003cspan class=\"n\"\u003erc_x\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003erc_z\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003earc_x\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003earc_z\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"n\"\u003eout\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003estress\u003c/span\u003e\u003cspan class=\"p\"\u003e[[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;element_id\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;local_index\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;node_id\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]]\u003c/span\u003e\u003cspan class=\"o\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003ereset_index\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003edrop\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"kc\"\u003eTrue\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"n\"\u003eres\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003estress\u003c/span\u003e\u003cspan class=\"o\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003eapply\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003elambda\u003c/span\u003e \u003cspan class=\"n\"\u003er\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e \u003cspan class=\"n\"\u003ecalculate_rc_ratio\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003er\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;sxx\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e],\u003c/span\u003e \u003cspan class=\"n\"\u003er\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;syy\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e],\u003c/span\u003e \u003cspan class=\"n\"\u003er\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;szz\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e],\u003c/span\u003e \u003cspan class=\"n\"\u003er\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;sxy\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e],\u003c/span\u003e \u003cspan class=\"n\"\u003er\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;szx\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e],\u003c/span\u003e \u003cspan class=\"n\"\u003er\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;syz\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]],\u003c/span\u003e \u003cspan class=\"n\"\u003efyd\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003eth\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"p\"\u003e),\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003eaxis\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"mi\"\u003e1\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003eresult_type\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;expand\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"n\"\u003eres\u003c/span\u003e\u003cspan class=\"o\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003ecolumns\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;rc_x\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;rc_z\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;arc_x\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;arc_z\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"n\"\u003epd\u003c/span\u003e\u003cspan class=\"o\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003econcat\u003c/span\u003e\u003cspan class=\"p\"\u003e([\u003c/span\u003e\u003cspan class=\"n\"\u003eout\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003eres\u003c/span\u003e\u003cspan class=\"p\"\u003e],\u003c/span\u003e \u003cspan class=\"n\"\u003eaxis\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"mi\"\u003e1\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result gets printed into cells, and Python can then be used again for visualisation.\u003c/p\u003e\n\u003cfigure\u003e\u003cimg src=\"/posts/python-in-excel/reinforcement-results.png\"\n\t\t\talt=\"Screenshot of reinforcement ratio plot in Excel\"\u003e\u003cfigcaption\u003e\n\t\t\t\u003cp\u003eReinforcement ratio results visualized in Excel via Python.\u003c/p\u003e\n\t\t\u003c/figcaption\u003e\n\u003c/figure\u003e\n\n\u003cp\u003eThese are just a few examples from my private library. In a similar manner, I\u0026rsquo;ve used this for visualization in commercial work as well - for example, in a soil subgrade calculator built from FE data exported from calculation software.\u003c/p\u003e\n\u003ch2 id=\"you-dont-need-to-know-how-to-code\" class=\"headerLink\"\u003e\n    \u003ca href=\"#you-dont-need-to-know-how-to-code\" class=\"header-mark\"\u003e\u003c/a\u003eYou Don\u0026rsquo;t Need to Know How to Code\u003c/h2\u003e\u003cp\u003eJust ask AI to build the code for you - remember to explicitly mention \u0026ldquo;Python in Excel\u0026rdquo; in your prompt.\u003c/p\u003e\n\u003cp\u003eI\u0026rsquo;ve tested both \u003cstrong\u003eCopilot in Excel\u003c/strong\u003e and \u003cstrong\u003eClaude in Excel\u003c/strong\u003e, and both work great. You can also use AI models outside of Excel, but the built-in ones have full access to your data, which saves time.\u003c/p\u003e\n\u003ch2 id=\"why-this-is-useful\" class=\"headerLink\"\u003e\n    \u003ca href=\"#why-this-is-useful\" class=\"header-mark\"\u003e\u003c/a\u003eWhy This Is Useful\u003c/h2\u003e\u003cp\u003eNot everyone knows Python, and people hesitate to use things that look complicated. The Excel interface helps bridge that gap - you can build small things that make a real difference for your team, and make your reports look good.\u003c/p\u003e\n\u003ch2 id=\"how-to-start\" class=\"headerLink\"\u003e\n    \u003ca href=\"#how-to-start\" class=\"header-mark\"\u003e\u003c/a\u003eHow to Start\u003c/h2\u003e\u003cp\u003eGo to \u0026ldquo;Formulas\u0026rdquo; on the top ribbon and check if you can see the Python tools. If yes, you\u0026rsquo;re good to go - just prompt an AI. Results go into a cell, preceded by \u003cstrong\u003e=PY\u003c/strong\u003e, then use \u003cstrong\u003eCtrl+Enter\u003c/strong\u003e to commit the code into the cell.\u003c/p\u003e\n\u003ch2 id=\"limitations\" class=\"headerLink\"\u003e\n    \u003ca href=\"#limitations\" class=\"header-mark\"\u003e\u003c/a\u003eLimitations\u003c/h2\u003e\u003cp\u003eThe solution isn\u0026rsquo;t perfect. Limitations I\u0026rsquo;ve found so far:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eExcel has a limit of 8,192 characters per formula, and the code is treated as a formula - so keep code light, avoid unnecessary comments, and use one-liners where possible.\u003c/li\u003e\n\u003cli\u003eThere are some computation limits; I wouldn\u0026rsquo;t recommend it for very heavy data, but test the limit yourself.\u003c/li\u003e\n\u003cli\u003eExcel sometimes doesn\u0026rsquo;t cooperate and displays \u003ccode\u003e#CONNECT!\u003c/code\u003e - hitting \u003cstrong\u003eReset Runtime\u003c/strong\u003e usually helps.\u003c/li\u003e\n\u003cli\u003eError messages are fairly vague, but AI usually handles working through them.\u003c/li\u003e\n\u003cli\u003eYou\u0026rsquo;re restricted to the preinstalled libraries - it\u0026rsquo;s a closed box, with no option to install additional packages or to import/export files outside of the Excel file itself.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eSee the \u003ca href=\"https://support.microsoft.com/en-us/excel/python/introduction-to-python-in-excel\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eofficial documentation\u003c/a\u003e for this feature.\u003c/p\u003e\n",
        "language": "en"
    },
    {
        "title" : "Welcome",
        "date_published" : "2026-08-20T00:00:00Z",
        "date_modified" : "2026-09-03T20:38:28+02:00",
        "id" : "https://mwolinski.com/posts/welcome/",
        "url" : "https://mwolinski.com/posts/welcome/",
        "summary": "Welcome to my site. I\u0026rsquo;m a structural engineer, and here I\u0026rsquo;ll be writing about structural engineering, software, and the overlap between the two — automation for tedious calculations, tools like Python in Excel, and anything else that speeds up daily engineering work.\nThanks for stopping by. Feel free to connect with me on LinkedIn if you\u0026rsquo;d like to chat.\n",
        "content_html" : "\u003cp\u003eWelcome to my site. I\u0026rsquo;m a structural engineer, and here I\u0026rsquo;ll be writing about structural engineering, software, and the overlap between the two — automation for tedious calculations, tools like Python in Excel, and anything else that speeds up daily engineering work.\u003c/p\u003e\n\u003cp\u003eThanks for stopping by. Feel free to connect with me on LinkedIn if you\u0026rsquo;d like to chat.\u003c/p\u003e\n",
        "language": "en"
    }
    ]
}