{"id":100219,"date":"2026-02-15T20:28:00","date_gmt":"2026-02-15T20:28:00","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100219"},"modified":"2026-07-15T00:54:12","modified_gmt":"2026-07-15T00:54:12","slug":"section-moment-of-inertia","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/ar\/calculators\/section-moment-of-inertia\/","title":{"rendered":"\u0622\u0644\u0629 \u062d\u0633\u0627\u0628 \u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0642\u0633\u0645 \u062d\u0648\u0644 \u0627\u0644\u0645\u062d\u0648\u0631 \u0627\u0644\u0645\u0631\u0643\u0632\u064a x"},"content":{"rendered":"\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Centroidal x-Axis Section Properties\",\"description\":\"Calculate geometric area, centroidal x-axis second moment of area, elastic section modulus and radius of gyration for explicitly idealized sharp-corner sections.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/section-moment-of-inertia\/\",\"applicationCategory\":\"Engineering Calculator\",\"operatingSystem\":\"Any\",\"offers\":{\"@type\":\"Offer\",\"price\":\"0\",\"priceCurrency\":\"EUR\"},\"creator\":{\"@type\":\"Organization\",\"name\":\"Vibromera\",\"url\":\"https:\/\/vibromera.eu\/\"},\"dateModified\":\"2026-07-15\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true}<\/script>\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/vibromera.eu\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Calculators\",\"item\":\"https:\/\/vibromera.eu\/calculators\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Centroidal x-Axis Section Properties\",\"item\":\"https:\/\/vibromera.eu\/calculators\/section-moment-of-inertia\/\"}]}<\/script>\n\n<style>\n.vbm219{--ink:#17212b;--muted:#52606d;--line:#cfd8df;--soft:#f5f8fa;--blue:#075b89;--blue2:#073d5b;--warn:#8a5700;--warnbg:#fff8e6;--bad:#b42318;--badbg:#fff1f0;max-width:1000px;margin:0 auto;color:var(--ink);font:16px\/1.58 system-ui,-apple-system,\"Segoe UI\",sans-serif}.vbm219 *{box-sizing:border-box}.vbm219 h1,.vbm219 h2,.vbm219 h3{line-height:1.2;color:#102a3a}.vbm219 h1{margin:8px 0 14px;font-size:clamp(30px,5vw,46px)}.vbm219 h2{margin:0;font-size:25px}.vbm219 h3{margin:26px 0 8px;font-size:19px}.vbm219 p{margin:9px 0}.vbm219 a{color:var(--blue)}.vbm219-hero{padding:34px 28px;border:1px solid var(--line);border-radius:12px;background:linear-gradient(135deg,#eef7fb,#fff)}.vbm219-kicker{font-size:13px;font-weight:800;letter-spacing:.09em;text-transform:uppercase;color:var(--blue)}.vbm219-lead{max-width:860px;font-size:18px}.vbm219-badges{display:flex;flex-wrap:wrap;gap:8px;margin-top:18px}.vbm219-badge{padding:5px 10px;border:1px solid #a8c9da;border-radius:999px;background:#fff;color:var(--blue2);font-size:13px;font-weight:750}.vbm219-alert{margin-top:20px;padding:13px 15px;border-left:4px solid var(--warn);border-radius:5px;background:var(--warnbg)}.vbm219-card,.vbm219-section{margin-top:24px;border:1px solid var(--line);border-radius:10px;background:#fff;box-shadow:0 5px 18px rgba(24,44,58,.06);overflow:hidden}.vbm219-card-h{padding:20px 22px;border-bottom:1px solid var(--line);background:var(--soft)}.vbm219-body,.vbm219-section{padding:22px}.vbm219-grid{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:16px}.vbm219-field{min-width:0}.vbm219-field.full,.vbm219-fieldset,.vbm219-shape-fields{grid-column:1\/-1}.vbm219-shape-fields{display:none}.vbm219-shape-fields.active{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:16px;padding:16px;border:1px solid var(--line);border-radius:8px;background:var(--soft)}.vbm219-shape-fields legend{padding:0 7px;font-weight:800}.vbm219 label{display:block;margin-bottom:6px;font-weight:750}.vbm219 input,.vbm219 select{width:100%;min-height:46px;padding:10px 12px;border:1.5px solid #9dacb7;border-radius:7px;background:#fff;color:var(--ink);font:inherit}.vbm219 input:focus,.vbm219 select:focus{outline:3px solid rgba(7,91,137,.2);border-color:var(--blue)}.vbm219-hint{display:block;margin-top:5px;color:var(--muted);font-size:13px}.vbm219-fieldset{padding:15px;border:1px solid var(--line);border-radius:8px}.vbm219-fieldset legend{padding:0 6px;font-weight:800}.vbm219-check{display:flex!important;align-items:flex-start;gap:9px;margin:8px 0!important;font-weight:500!important}.vbm219-check input{width:18px;min-height:18px;margin-top:4px;flex:0 0 auto}.vbm219-actions{display:flex;flex-wrap:wrap;gap:10px;margin-top:18px}.vbm219-btn{min-height:44px;padding:10px 18px;border:0;border-radius:7px;background:var(--blue);color:#fff;font:750 15px\/1.2 inherit;cursor:pointer}.vbm219-btn:hover{background:var(--blue2)}.vbm219-btn.secondary{border:1px solid #9dacb7;background:#fff;color:var(--ink)}.vbm219-error{display:none;margin-top:15px;padding:12px 14px;border-left:4px solid var(--bad);border-radius:5px;background:var(--badbg);color:#7a1b15}.vbm219-error.show{display:block}.vbm219-results{display:none}.vbm219-results.show{display:block}.vbm219-result-grid{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:12px}.vbm219-result{padding:15px;border:1px solid var(--line);border-radius:8px;background:var(--soft)}.vbm219-result.primary{grid-column:1\/-1;border:2px solid var(--blue);background:#edf7fc}.vbm219-result-label{color:var(--muted);font-size:12px;font-weight:800;text-transform:uppercase;letter-spacing:.05em}.vbm219-result-value{margin-top:4px;font:750 20px\/1.35 ui-monospace,\"Cascadia Mono\",monospace;overflow-wrap:anywhere}.vbm219-result.primary .vbm219-result-value{color:var(--blue2);font-size:27px}.vbm219-result-note{margin-top:4px;color:var(--muted);font-size:13px}.vbm219-boundary{margin-top:15px;padding:12px 14px;border-left:4px solid var(--warn);border-radius:5px;background:var(--warnbg)}.vbm219-equation{margin:12px 0;padding:13px;border:1px solid #b7ccd8;border-radius:7px;background:#f3f9fc;text-align:center;font:750 16px\/1.55 ui-monospace,\"Cascadia Mono\",monospace;overflow-wrap:anywhere}.vbm219-table-wrap{overflow-x:auto}.vbm219 table{width:100%;border-collapse:collapse;margin:12px 0;font-size:14px}.vbm219 th,.vbm219 td{padding:10px;border:1px solid var(--line);text-align:left;vertical-align:top}.vbm219 th{background:var(--soft)}.vbm219-limit{padding:14px;border-left:4px solid var(--warn);background:var(--warnbg)}.vbm219-footer{margin-top:24px;padding:16px;border-top:1px solid var(--line);color:var(--muted);font-size:13px}\n@media(max-width:680px){.vbm219-grid,.vbm219-shape-fields.active,.vbm219-result-grid{grid-template-columns:1fr}.vbm219-result.primary{grid-column:auto}.vbm219-hero,.vbm219-body,.vbm219-section{padding:18px}.vbm219-card-h{padding:17px}.vbm219 table{font-size:12px}.vbm219 th,.vbm219 td{padding:7px}}\n<\/style>\n\n<div class=\"vbm219\">\n  <header class=\"vbm219-hero\">\n    <div class=\"vbm219-kicker\">Explicit geometry \u00b7 centroidal horizontal axis<\/div>\n    <h1>Centroidal x-Axis Section Properties<\/h1>\n    <p class=\"vbm219-lead\">Calculate area <strong>A<\/strong>, centroidal horizontal-axis second moment of area <strong>I<sub>x<\/sub><\/strong>, elastic section modulus <strong>S<sub>x<\/sub><\/strong> and radius of gyration <strong>k<sub>x<\/sub><\/strong> for an idealized sharp-corner section.<\/p>\n    <div class=\"vbm219-badges\"><span class=\"vbm219-badge\">area property, not mass inertia<\/span><span class=\"vbm219-badge\">one declared x-axis<\/span><span class=\"vbm219-badge\">no profile catalogue<\/span><span class=\"vbm219-badge\">no strength verdict<\/span><\/div>\n    <div class=\"vbm219-alert\"><strong>This calculator does not represent a real rolled, extruded, welded or machined profile unless that profile exactly matches the idealized dimensions.<\/strong> Fillets, corner radii, flange taper, weld metal, holes, coatings and tolerances are excluded.<\/div>\n  <\/header>\n\n  <section class=\"vbm219-card\" aria-labelledby=\"vbm219-input-title\">\n    <div class=\"vbm219-card-h\"><h2 id=\"vbm219-input-title\">Idealized geometry and record<\/h2><p>Choose one shape and one unit. Values start blank. Changing the unit clears every dimension and result; no hidden conversion is performed.<\/p><\/div>\n    <div class=\"vbm219-body\">\n      <form id=\"vbm219-form\" novalidate>\n        <div class=\"vbm219-grid\">\n          <div class=\"vbm219-field\"><label for=\"vbm219-shape\">Idealized cross-section<\/label><select id=\"vbm219-shape\" required><option value=\"\" selected>Select the geometry<\/option><option value=\"circle\">Solid circle<\/option><option value=\"annulus\">Concentric circular annulus<\/option><option value=\"rectangle\">Solid rectangle<\/option><option value=\"ibeam\">Double-symmetric sharp-corner I-section<\/option><option value=\"channel\">Equal-flange sharp-corner channel<\/option><\/select><\/div>\n          <div class=\"vbm219-field\"><label for=\"vbm219-unit\">Shared linear unit<\/label><select id=\"vbm219-unit\" required><option value=\"\" selected>Select the input unit<\/option><option value=\"mm\">millimetres (mm)<\/option><option value=\"in\">inches (in)<\/option><\/select><span class=\"vbm219-hint\">Outputs use the corresponding unit, unit\u00b2, unit\u00b3 and unit\u2074.<\/span><\/div>\n\n          <fieldset class=\"vbm219-shape-fields\" id=\"vbm219-fields-circle\" disabled><legend>Solid-circle dimensions<\/legend><div class=\"vbm219-field\"><label for=\"vbm219-d\">Diameter d<\/label><input id=\"vbm219-d\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><span class=\"vbm219-hint\">Positive complete decimal.<\/span><\/div><\/fieldset>\n          <fieldset class=\"vbm219-shape-fields\" id=\"vbm219-fields-annulus\" disabled><legend>Concentric-annulus dimensions<\/legend><div class=\"vbm219-field\"><label for=\"vbm219-D\">Outer diameter D<\/label><input id=\"vbm219-D\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><\/div><div class=\"vbm219-field\"><label for=\"vbm219-di\">Inner diameter d<sub>i<\/sub><\/label><input id=\"vbm219-di\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><span class=\"vbm219-hint\">Must be positive and smaller than D.<\/span><\/div><\/fieldset>\n          <fieldset class=\"vbm219-shape-fields\" id=\"vbm219-fields-rectangle\" disabled><legend>Rectangle dimensions<\/legend><div class=\"vbm219-field\"><label for=\"vbm219-b\">Width b, parallel to x<\/label><input id=\"vbm219-b\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><\/div><div class=\"vbm219-field\"><label for=\"vbm219-h\">Height h, perpendicular to x<\/label><input id=\"vbm219-h\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><\/div><\/fieldset>\n          <fieldset class=\"vbm219-shape-fields\" id=\"vbm219-fields-ibeam\" disabled><legend>Double-symmetric I-section dimensions<\/legend><div class=\"vbm219-field\"><label for=\"vbm219-iH\">Overall height H<\/label><input id=\"vbm219-iH\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><\/div><div class=\"vbm219-field\"><label for=\"vbm219-iB\">Overall flange width B<\/label><input id=\"vbm219-iB\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><\/div><div class=\"vbm219-field\"><label for=\"vbm219-itw\">Web thickness t<sub>w<\/sub><\/label><input id=\"vbm219-itw\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><span class=\"vbm219-hint\">Must be smaller than B.<\/span><\/div><div class=\"vbm219-field\"><label for=\"vbm219-itf\">Each flange thickness t<sub>f<\/sub><\/label><input id=\"vbm219-itf\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><span class=\"vbm219-hint\">2t<sub>f<\/sub> must be smaller than H.<\/span><\/div><\/fieldset>\n          <fieldset class=\"vbm219-shape-fields\" id=\"vbm219-fields-channel\" disabled><legend>Equal-flange channel dimensions<\/legend><div class=\"vbm219-field\"><label for=\"vbm219-cH\">Overall height H<\/label><input id=\"vbm219-cH\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><\/div><div class=\"vbm219-field\"><label for=\"vbm219-cB\">Overall flange width B<\/label><input id=\"vbm219-cB\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><\/div><div class=\"vbm219-field\"><label for=\"vbm219-ctw\">Web thickness t<sub>w<\/sub><\/label><input id=\"vbm219-ctw\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><span class=\"vbm219-hint\">Must be smaller than B.<\/span><\/div><div class=\"vbm219-field\"><label for=\"vbm219-ctf\">Each flange thickness t<sub>f<\/sub><\/label><input id=\"vbm219-ctf\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><span class=\"vbm219-hint\">2t<sub>f<\/sub> must be smaller than H.<\/span><\/div><\/fieldset>\n\n          <div class=\"vbm219-field full\"><label for=\"vbm219-geometry-record\">Geometry source or record ID<\/label><input id=\"vbm219-geometry-record\" autocomplete=\"off\" placeholder=\"drawing\/revision, measured geometry record or explicitly idealized study\"><span class=\"vbm219-hint\">Do not enter only a commercial profile name; this page has no catalogue lookup.<\/span><\/div>\n          <div class=\"vbm219-field full\"><label for=\"vbm219-purpose-record\">Axis and intended-use record<\/label><input id=\"vbm219-purpose-record\" autocomplete=\"off\" placeholder=\"centroidal horizontal x-axis; geometric comparison or downstream calculation boundary\"><\/div>\n          <fieldset class=\"vbm219-fieldset\"><legend>Required model confirmation<\/legend><label class=\"vbm219-check\"><input type=\"checkbox\" id=\"vbm219-c1\"><span>The entered section is intentionally idealized with sharp corners, constant listed thicknesses and the exact symmetry stated by the selected shape.<\/span><\/label><label class=\"vbm219-check\"><input type=\"checkbox\" id=\"vbm219-c2\"><span>I will not treat the geometric outputs as a catalogue property, bending capacity, buckling capacity, stress result or code-compliance verdict.<\/span><\/label><\/fieldset>\n        <\/div>\n        <div class=\"vbm219-actions\"><button class=\"vbm219-btn\" type=\"submit\">Calculate documented x-axis properties<\/button><button class=\"vbm219-btn secondary\" type=\"button\" id=\"vbm219-reset\">Clear<\/button><\/div>\n        <div class=\"vbm219-error\" id=\"vbm219-error\" role=\"alert\" tabindex=\"-1\"><\/div>\n      <\/form>\n    <\/div>\n  <\/section>\n\n  <section class=\"vbm219-card vbm219-results\" id=\"vbm219-results\" aria-labelledby=\"vbm219-results-title\" aria-live=\"polite\">\n    <div class=\"vbm219-card-h\"><h2 id=\"vbm219-results-title\">Geometric x-axis result<\/h2><p id=\"vbm219-result-context\"><\/p><\/div>\n    <div class=\"vbm219-body\">\n      <div class=\"vbm219-result-grid\">\n        <div class=\"vbm219-result primary\"><div class=\"vbm219-result-label\">Centroidal second moment of area I<sub>x<\/sub><\/div><div class=\"vbm219-result-value\" id=\"vbm219-r-I\">\u2014<\/div><div class=\"vbm219-result-note\">Length to the fourth power; not mass moment of inertia.<\/div><\/div>\n        <div class=\"vbm219-result\"><div class=\"vbm219-result-label\">Cross-sectional area A<\/div><div class=\"vbm219-result-value\" id=\"vbm219-r-A\">\u2014<\/div><\/div>\n        <div class=\"vbm219-result\"><div class=\"vbm219-result-label\">Elastic section modulus S<sub>x<\/sub>=I<sub>x<\/sub>\/c<\/div><div class=\"vbm219-result-value\" id=\"vbm219-r-S\">\u2014<\/div><div class=\"vbm219-result-note\">Geometric elastic section modulus about the declared x-axis.<\/div><\/div>\n        <div class=\"vbm219-result\"><div class=\"vbm219-result-label\">Radius of gyration k<sub>x<\/sub>=\u221a(I<sub>x<\/sub>\/A)<\/div><div class=\"vbm219-result-value\" id=\"vbm219-r-k\">\u2014<\/div><\/div>\n        <div class=\"vbm219-result\"><div class=\"vbm219-result-label\">Extreme-fibre distance c<\/div><div class=\"vbm219-result-value\" id=\"vbm219-r-c\">\u2014<\/div><div class=\"vbm219-result-note\">Equal top\/bottom distance for the supported symmetric-about-x shapes.<\/div><\/div>\n      <\/div>\n      <div class=\"vbm219-boundary\"><strong>No structural verdict is produced.<\/strong> A stress, deflection, fatigue, yielding or buckling calculation additionally needs a verified structural model, loads and combinations, material properties, supports\/effective length, imperfections, residual stress, local\/global stability, connection effects and the applicable design rules.<\/div>\n    <\/div>\n  <\/section>\n\n  <section class=\"vbm219-section\" aria-labelledby=\"vbm219-model-title\">\n    <h2 id=\"vbm219-model-title\">Equations, axes and model boundary<\/h2>\n    <p>The x-axis is horizontal and passes through the area centroid. The listed I-section is double-symmetric. The channel has equal top and bottom flanges, so its horizontal centroidal axis is at H\/2; its horizontal centroid coordinate, I<sub>y<\/sub>, product of inertia and principal axes are deliberately not calculated.<\/p>\n    <h3>Supported idealizations<\/h3>\n    <div class=\"vbm219-equation\">circle: A=\u03c0d\u00b2\/4; I\u2093=\u03c0d\u2074\/64<\/div>\n    <div class=\"vbm219-equation\">annulus: A=\u03c0(D\u00b2\u2212d\u1d62\u00b2)\/4; I\u2093=\u03c0(D\u2074\u2212d\u1d62\u2074)\/64<\/div>\n    <div class=\"vbm219-equation\">rectangle: A=bh; I\u2093=bh\u00b3\/12<\/div>\n    <div class=\"vbm219-equation\">sharp-corner I-section or equal-flange channel:<br>A=2Bt_f+t_w(H\u22122t_f); I\u2093=[BH\u00b3\u2212(B\u2212t_w)(H\u22122t_f)\u00b3]\/12<\/div>\n    <div class=\"vbm219-equation\">all shapes: c=d\/2, h\/2 or H\/2; S\u2093=I\u2093\/c; k\u2093=\u221a(I\u2093\/A)<\/div>\n    <p>For the I-section, the subtraction removes the two identical voids beside the web. For the equal-flange channel, it removes one rectangular void. Their I<sub>x<\/sub> expressions are identical because the removed total width and vertical extent are identical; their y-axis properties are not identical.<\/p>\n    <div class=\"vbm219-limit\"><strong>Excluded geometry:<\/strong> root\/toe radii, flange taper, unequal flanges, lips, holes, cut-outs, welds, compound or transformed sections, corrosion allowance, dimensional tolerances and non-concentric tubes. Use a verified drawing-based composite-area or CAD\/section-property calculation for those cases.<\/div>\n    <h3>Why the former profile presets were removed<\/h3>\n    <p>The former page labelled four-number sharp-corner approximations as \u201cIPE 200\u201d, \u201cIPE 300\u201d and \u201cPipe DN100\u201d. Actual catalogue\/standard profiles can include radii, taper and schedule-dependent wall dimensions. The page neither cited a controlled profile table nor matched those details, so the labels could be mistaken for certified catalogue properties. Exact commercial-profile properties remain <strong>NEEDS_LICENSED_SOURCE<\/strong> or require a current manufacturer catalogue.<\/p>\n    <h3>Exact numerical checks<\/h3>\n    <div class=\"vbm219-table-wrap\"><table><thead><tr><th>Inputs<\/th><th>A<\/th><th>I<sub>x<\/sub><\/th><th>S<sub>x<\/sub><\/th><th>k<sub>x<\/sub><\/th><th>c<\/th><\/tr><\/thead><tbody>\n      <tr><td>circle d=100 mm<\/td><td>7853.981633974 mm\u00b2<\/td><td>4908738.521234052 mm\u2074<\/td><td>98174.770424681 mm\u00b3<\/td><td>25 mm<\/td><td>50 mm<\/td><\/tr>\n      <tr><td>rectangle b=50 mm; h=100 mm<\/td><td>5000 mm\u00b2<\/td><td>4166666.666666667 mm\u2074<\/td><td>83333.333333333 mm\u00b3<\/td><td>28.867513459 mm<\/td><td>50 mm<\/td><\/tr>\n    <\/tbody><\/table><\/div>\n    <h3>Sources and classification<\/h3>\n    <div class=\"vbm219-table-wrap\"><table><thead><tr><th>Source<\/th><th>Use on this page<\/th><\/tr><\/thead><tbody>\n      <tr><td><a href=\"https:\/\/mechanicsmap.psu.edu\/websites\/centroidtables\/centroids2D\/centroids2D.html\" target=\"_blank\" rel=\"noopener\">Penn State Mechanics Map \u2014 centroids and area moments for 2D shapes<\/a><\/td><td>Centroidal rectangle and circle area\/second-moment formulas.<\/td><\/tr>\n      <tr><td><a href=\"https:\/\/engineeringstatics.org\/Chapter_10-moment-of-inertia-of-composite-shapes.html\" target=\"_blank\" rel=\"noopener\">Engineering Statics \u2014 composite shapes<\/a><\/td><td>Addition\/subtraction of simple areas and centroidal moments; basis for the idealized I\/channel derivation.<\/td><\/tr>\n      <tr><td><a href=\"https:\/\/engineeringstatics.org\/radius-of-gyration-sec.html\" target=\"_blank\" rel=\"noopener\">Engineering Statics \u2014 radius of gyration<\/a><\/td><td>Definition k<sub>x<\/sub>=\u221a(I<sub>x<\/sub>\/A), with I and k referred to the same axis.<\/td><\/tr>\n      <tr><td><a href=\"https:\/\/mechref.engr.illinois.edu\/sol\/bending.html\" target=\"_blank\" rel=\"noopener\">University of Illinois Mechanics Reference \u2014 bending<\/a><\/td><td>Elastic section modulus S=I\/c and the conditions attached to elastic flexure; this calculator returns geometry only.<\/td><\/tr>\n    <\/tbody><\/table><\/div>\n    <p><strong>Accessed:<\/strong> 15 July 2026. These are general geometric\/mechanics relationships, not formulas attributed to an ISO profile standard.<\/p>\n    <div class=\"vbm219-limit\"><strong>Independent review required.<\/strong> For a real product, compare against the current controlled drawing or manufacturer\/standard section-property table. Do not infer compliance from numerical agreement with this idealization.<\/div>\n    <div class=\"vbm219-footer\">Revision: scientific audit 2026-07-15 \u00b7 English source page \u00b7 Explicit centroidal x-axis only \u00b7 General geometry, non-normative.<\/div>\n  <\/section>\n<\/div>\n\n<script>\n(function(){\n  'use strict';\n  function el(id){return document.getElementById(id)}\n  var form=el('vbm219-form');\n  var shapeIds=['circle','annulus','rectangle','ibeam','channel'];\n  var numericIds=['vbm219-d','vbm219-D','vbm219-di','vbm219-b','vbm219-h','vbm219-iH','vbm219-iB','vbm219-itw','vbm219-itf','vbm219-cH','vbm219-cB','vbm219-ctw','vbm219-ctf'];\n  function clearError(){var box=el('vbm219-error');box.textContent='';box.classList.remove('show')}\n  function invalidate(){clearError();el('vbm219-results').classList.remove('show')}\n  function showError(err){invalidate();var box=el('vbm219-error');box.textContent=err.message||String(err);box.classList.add('show');box.focus()}\n  function strictPositive(raw,name){var s=String(raw==null?'':raw).trim();if(!\/^(?:\\d+(?:[.,]\\d*)?|[.,]\\d+)$\/.test(s))throw new Error(name+' must be one complete positive decimal number.');var n=Number(s.replace(',','.'));if(!Number.isFinite(n)||n<=0)throw new Error(name+' must be finite and greater than zero.');return n}\n  function finite(n,name){if(!Number.isFinite(n)||n<=0)throw new Error(name+' is outside the finite positive calculation range.');return n}\n  function calculate(x){\n    if(!x||shapeIds.indexOf(x.shape)<0)throw new Error('Select one supported idealized cross-section.');if(['mm','in'].indexOf(x.unit)<0)throw new Error('Select the shared linear unit.');\n    var A,I,c,label;\n    if(x.shape==='circle'){\n      var d=strictPositive(x.d,'Diameter d');A=Math.PI*d*d\/4;I=Math.PI*Math.pow(d,4)\/64;c=d\/2;label='Solid circle';\n    }else if(x.shape==='annulus'){\n      var D=strictPositive(x.D,'Outer diameter D'),di=strictPositive(x.di,'Inner diameter di');if(!(di<D))throw new Error('Inner diameter di must be smaller than outer diameter D.');A=Math.PI*(D*D-di*di)\/4;I=Math.PI*(Math.pow(D,4)-Math.pow(di,4))\/64;c=D\/2;label='Concentric circular annulus';\n    }else if(x.shape==='rectangle'){\n      var b=strictPositive(x.b,'Width b'),h=strictPositive(x.h,'Height h');A=b*h;I=b*Math.pow(h,3)\/12;c=h\/2;label='Solid rectangle';\n    }else{\n      var H=strictPositive(x.H,'Overall height H'),B=strictPositive(x.B,'Overall flange width B'),tw=strictPositive(x.tw,'Web thickness tw'),tf=strictPositive(x.tf,'Flange thickness tf');if(!(tw<B))throw new Error('Web thickness tw must be smaller than overall flange width B.');if(!(2*tf<H))throw new Error('Twice the flange thickness 2tf must be smaller than overall height H.');A=2*B*tf+tw*(H-2*tf);I=(B*Math.pow(H,3)-(B-tw)*Math.pow(H-2*tf,3))\/12;c=H\/2;label=x.shape==='ibeam'?'Double-symmetric sharp-corner I-section':'Equal-flange sharp-corner channel';\n    }\n    A=finite(A,'Area A');I=finite(I,'Second moment Ix');c=finite(c,'Extreme-fibre distance c');var S=finite(I\/c,'Elastic section modulus Sx'),k=finite(Math.sqrt(I\/A),'Radius of gyration kx');return {shape:x.shape,label:label,unit:x.unit,A:A,I:I,c:c,S:S,k:k};\n  }\n  function requiredText(id,name){var s=el(id).value.trim();if(!s)throw new Error(name+' is required.');return s}\n  function format(n){if(Object.is(n,-0))n=0;var s=n.toPrecision(15);if(\/e\/i.test(s))return s.replace(\/\\.0+e\/i,'e').replace(\/(\\.\\d*?[1-9])0+e\/i,'$1e');return s.replace(\/(\\.\\d*?[1-9])0+$\/,'$1').replace(\/\\.0+$\/,'')}\n  function syncShape(){var selected=el('vbm219-shape').value;shapeIds.forEach(function(s){var fs=el('vbm219-fields-'+s),active=s===selected;fs.classList.toggle('active',active);fs.disabled=!active})}\n  function readModel(){\n    var shape=el('vbm219-shape').value,unit=el('vbm219-unit').value;if(shapeIds.indexOf(shape)<0)throw new Error('Select one supported idealized cross-section.');if(!unit)throw new Error('Select the shared linear unit.');\n    var pre=shape==='ibeam'?'vbm219-i':shape==='channel'?'vbm219-c':'';\n    var r=calculate({shape:shape,unit:unit,d:el('vbm219-d').value,D:el('vbm219-D').value,di:el('vbm219-di').value,b:el('vbm219-b').value,h:el('vbm219-h').value,H:pre?el(pre+'H').value:'',B:pre?el(pre+'B').value:'',tw:pre?el(pre+'tw').value:'',tf:pre?el(pre+'tf').value:''});\n    var source=requiredText('vbm219-geometry-record','Geometry source or record ID'),purpose=requiredText('vbm219-purpose-record','Axis and intended-use record');if(!el('vbm219-c1').checked||!el('vbm219-c2').checked)throw new Error('Both model-boundary confirmations are required.');\n    r.source=source;r.purpose=purpose;return r;\n  }\n  function render(r){el('vbm219-result-context').textContent=r.label+' \u00b7 '+r.source+' \u00b7 '+r.purpose;el('vbm219-r-I').textContent=format(r.I)+' '+r.unit+'\u2074';el('vbm219-r-A').textContent=format(r.A)+' '+r.unit+'\u00b2';el('vbm219-r-S').textContent=format(r.S)+' '+r.unit+'\u00b3';el('vbm219-r-k').textContent=format(r.k)+' '+r.unit;el('vbm219-r-c').textContent=format(r.c)+' '+r.unit;el('vbm219-results').classList.add('show')}\n  form.addEventListener('submit',function(e){e.preventDefault();try{clearError();render(readModel())}catch(err){showError(err)}});\n  form.addEventListener('keydown',function(e){if(e.key==='Enter'&&e.target.tagName!=='BUTTON'){e.preventDefault();form.requestSubmit()}});\n  form.addEventListener('input',invalidate);\n  el('vbm219-shape').addEventListener('change',function(){syncShape();invalidate()});\n  el('vbm219-unit').addEventListener('change',function(){numericIds.forEach(function(id){el(id).value=''});invalidate()});\n  el('vbm219-reset').addEventListener('click',function(){form.reset();numericIds.forEach(function(id){el(id).value=''});syncShape();invalidate();el('vbm219-shape').focus()});\n  syncShape();\n})();\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>Calculate idealized centroidal x-axis area, second moment of area, elastic section modulus and radius of gyration with explicit geometry and scope.<\/p>","protected":false},"featured_media":0,"template":"","meta":{"ai_generated_summary":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-100219","calculator","type-calculator","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/ar\/wp-json\/wp\/v2\/calculator\/100219","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/ar\/wp-json\/wp\/v2\/calculator"}],"about":[{"href":"https:\/\/vibromera.eu\/ar\/wp-json\/wp\/v2\/types\/calculator"}],"version-history":[{"count":5,"href":"https:\/\/vibromera.eu\/ar\/wp-json\/wp\/v2\/calculator\/100219\/revisions"}],"predecessor-version":[{"id":102564,"href":"https:\/\/vibromera.eu\/ar\/wp-json\/wp\/v2\/calculator\/100219\/revisions\/102564"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/ar\/wp-json\/wp\/v2\/media?parent=100219"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/ar\/wp-json\/wp\/v2\/categories?post=100219"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/ar\/wp-json\/wp\/v2\/tags?post=100219"}],"curies":[{"name":"\u0648\u0648\u0631\u062f\u0628\u0631\u064a\u0633","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}