{"id":66708,"date":"2026-10-13T08:41:14","date_gmt":"2026-10-13T05:41:14","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66708"},"modified":"2026-10-13T08:41:14","modified_gmt":"2026-10-13T05:41:14","slug":"beam-section-modulus","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/engineering\/beam-section-modulus\/","title":{"rendered":"Beam Section Modulus"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Find Section Properties for Beam Bending<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66708\" data-favorite-title=\"Section Modulus\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/engineering\/beam-section-modulus\/\" data-favorite-excerpt=\"Calculate the elastic section modulus and second moment of area for rectangle, circle, tube, or I\/H beam sections. Enter the dimensions, choose a shape, and get properties about the strong bending axis.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-civil\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/engineering\/beam-section-modulus\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66708\" data-post=\"66708\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Cross-section shape<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"crosssection_shape\"><option value=\"rect\" selected>Rectangle<\/option><option value=\"circle\">Solid circle<\/option><option value=\"tube\">Hollow circular tube<\/option><option value=\"isec\">I \/ H section<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Width (b)<\/label><div class=\"term\">b<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"width_b\" id=\"field-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"100\" min=\"0.01\" value=\"100\"\/><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-4\">Height (h)<\/label><div class=\"term\">h<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"height_h\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"200\" min=\"0.01\" value=\"200\"\/><div class=\"grid grid-cols-1\"><select id=\"field-5\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-4\">Depth measured parallel to the bending axis<\/small><\/div><div class=\"field has-term\" id=\"input-4\" hidden><label for=\"field-6\">Diameter (d)<\/label><div class=\"term\">d<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"diameter_d\" id=\"field-6\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"50\" min=\"0.01\" value=\"50\"\/><div class=\"grid grid-cols-1\"><select id=\"field-7\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><div class=\"field has-term\" id=\"input-5\" hidden><label for=\"field-8\">Outer diameter (D)<\/label><div class=\"term\">D<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"outer_diameter_d\" id=\"field-8\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"100\" min=\"0.01\" value=\"100\"\/><div class=\"grid grid-cols-1\"><select id=\"field-9\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><div class=\"field has-term\" id=\"input-6\" hidden><label for=\"field-10\">Inner diameter (d)<\/label><div class=\"term\">d<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"inner_diameter_d\" id=\"field-10\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"80\" min=\"0.01\" value=\"80\"\/><div class=\"grid grid-cols-1\"><select id=\"field-11\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><div class=\"field has-term\" id=\"input-7\" hidden><label for=\"field-12\">Total height (h)<\/label><div class=\"term\">h<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"total_height_h\" id=\"field-12\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"200\" min=\"0.01\" value=\"200\"\/><div class=\"grid grid-cols-1\"><select id=\"field-13\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><div class=\"field has-term\" id=\"input-8\" hidden><label for=\"field-14\">Flange width (b)<\/label><div class=\"term\">b<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"flange_width_b\" id=\"field-14\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"100\" min=\"0.01\" value=\"100\"\/><div class=\"grid grid-cols-1\"><select id=\"field-15\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><div class=\"field has-term\" id=\"input-9\" hidden><label for=\"field-16\">Flange thickness (tf)<\/label><div class=\"term\">tf<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"flange_thickness_tf\" id=\"field-16\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"10\" min=\"0.01\" value=\"10\"\/><div class=\"grid grid-cols-1\"><select id=\"field-17\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><div class=\"field has-term\" id=\"input-10\" hidden><label for=\"field-18\">Web thickness (tw)<\/label><div class=\"term\">tw<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"web_thickness_tw\" id=\"field-18\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"6\" min=\"0.01\" value=\"6\"\/><div class=\"grid grid-cols-1\"><select id=\"field-19\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66708\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Section Properties<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Second moment of area<\/span><span class=\"result-term\">Ixx<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Elastic section modulus<\/span><span class=\"result-term\">Sxx<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-3\"><span class=\"field-title\"><span>Formula and notes<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"66708\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Section Modulus<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ S_{xx} = \\frac{I_{xx}}{c} \\qquad \\sigma_{max} = \\frac{M}{S_{xx}} \\\\\\ I_{xx} = \\frac{b h^{3}}{12} \\ \\text{(rectangle)} \\qquad I_{xx} = \\frac{\\pi d^{4}}{64} \\ \\text{(solid circle)} \\\\\\ I_{xx} = \\frac{\\pi \\left(D^{4} - d^{4}\\right)}{64} \\ \\text{(hollow tube)} \\qquad I_{xx} = \\frac{b h^{3} - \\left(b - t_{w}\\right)\\left(h - 2 t_{f}\\right)^{3}}{12} \\ \\text{(I \/ H section)} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(S_{xx}\\) \u2014 elastic section modulus about the horizontal centroidal (x\u2013x) axis (mm\u00b3 or in\u00b3)<\/li>\n<li>\\(I_{xx}\\) \u2014 second moment of area about the same axis (mm\u2074 or in\u2074)<\/li>\n<li>\\(c\\) \u2014 distance from the neutral axis to the extreme fibre: \\(c = h\/2\\) or \\(c = D\/2\\)<\/li>\n<li>\\(b\\) \u2014 width (or flange width for an I-section)<\/li>\n<li>\\(h\\) \u2014 total height of the section<\/li>\n<li>\\(d\\) \u2014 diameter (solid circle) or inner diameter (tube)<\/li>\n<li>\\(D\\) \u2014 outer diameter of a tube<\/li>\n<li>\\(t_{f},\\ t_{w}\\) \u2014 flange and web thicknesses of an I \/ H section<\/li>\n<li>\\(M\\) \u2014 applied bending moment; \\(\\sigma_{max}\\) \u2014 maximum bending stress<\/li>\n<\/ul>\n<p>The section modulus measures how efficiently a cross-section resists bending: the larger \\(S_{xx}\\), the smaller the bending stress produced by a given moment.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Section Modulus \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a rectangular timber beam with a width of 100 mm and a height of 200 mm, loaded so that it bends about its strong axis.<\/p>\n<p>\\[ I_{xx} = \\frac{b h^{3}}{12} = \\frac{100 \\times 200^{3}}{12} = 66{,}666{,}667 \\ \\text{mm}^{4} \\approx 160.2 \\ \\text{in}^{4} \\]<\/p>\n<p>\\[ S_{xx} = \\frac{I_{xx}}{h\/2} = \\frac{66{,}666{,}667}{100} = 666{,}667 \\ \\text{mm}^{3} \\approx 40.68 \\ \\text{in}^{3} \\]<\/p>\n<p><strong>Result:<\/strong> \\(I_{xx} \\approx 6.67 \\times 10^{7}\\) mm\u2074 and \\(S_{xx} \\approx 6.67 \\times 10^{5}\\) mm\u00b3 (40.68 in\u00b3). A bending moment of 10 kN\u00b7m would therefore produce a maximum stress of \\(\\sigma = 10^{7} \/ 666{,}667 \\approx 15.0\\) N\/mm\u00b2 (MPa).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Select the <strong>Cross-section shape<\/strong>: Rectangle, Solid circle, Hollow circular tube, or I \/ H section. Only the dimension fields relevant to that shape are shown.<\/li>\n<li>Enter the dimensions in the visible fields \u2014 for example <strong>Width (b)<\/strong> and <strong>Height (h)<\/strong> for a rectangle, or <strong>Outer diameter (D)<\/strong> and <strong>Inner diameter (d)<\/strong> for a tube.<\/li>\n<li>Pick the unit (mm, cm, m, or in) for each dimension. The results follow your input system, with a conversion to the other system shown in brackets.<\/li>\n<li>Read the <strong>Second moment of area<\/strong> and the <strong>Elastic section modulus<\/strong> in the results panel, together with the exact formula used for your shape.<\/li>\n<\/ol>\n<h2>What is the section modulus?<\/h2>\n<p>When a beam bends, the material furthest from the neutral axis is stressed the most, while the material at the neutral axis carries almost no bending stress. The <strong>elastic section modulus<\/strong> \\(S\\) captures how much \"effective\" material a cross-section places far from that axis. It is defined as the second moment of area divided by the distance to the extreme fibre, \\(S = I\/c\\), and it links bending moment to stress through the flexure formula \\(\\sigma = M\/S\\).<\/p>\n<p>The practical consequence is simple: for the same applied moment, a larger section modulus means a lower maximum bending stress. This is why I-beams and H-sections are so common \u2014 by concentrating material in the flanges, far from the neutral axis, they achieve a high \\(S\\) with relatively little material. Doubling the depth of a rectangular beam, for instance, multiplies its section modulus by roughly four.<\/p>\n<p>All properties here are computed about the <strong>strong x\u2013x axis<\/strong> (the horizontal axis through the centroid), which is how beams are normally oriented in floor and roof applications. A section can also be loaded about its weak y\u2013y axis, in which case the section modulus will be considerably smaller \u2014 for a rectangle, \\(S_{yy} = h b^{2}\/6\\).<\/p>\n<p>To turn the result into a design check, combine it with the bending moment from your loading case. If you know the load on the beam, our calculators for a <a href=\"https:\/\/wpcalc.com\/en\/engineering\/simply-supported-beam-with-uniform-load\/\">uniformly loaded simply supported beam<\/a> or a <a href=\"https:\/\/wpcalc.com\/en\/engineering\/simply-supported-beam-with-center-point-load\/\">center point load<\/a> will give you the maximum moment; divide it by \\(S_{xx}\\) to get the peak stress and compare with your material's allowable stress.<\/p>\n<h2>Assumptions, limitations and practical tips<\/h2>\n<ul>\n<li><strong>Elastic behaviour only.<\/strong> The results assume the material stays below its yield point. For plastic design, the plastic section modulus (which is larger than the elastic one) must be used instead.<\/li>\n<li><strong>Homogeneous, symmetric sections.<\/strong> The formulas assume one material and a doubly symmetric shape, so the neutral axis passes through the centroid. Composite beams (e.g. concrete\u2013steel) need a transformed-section analysis.<\/li>\n<li><strong>Fillet radii and details are ignored.<\/strong> Real rolled I-sections have root radii and tapered flanges, so tabulated catalogue values differ from the idealized calculation by a few percent.<\/li>\n<li><strong>Orientation matters.<\/strong> Enter the height measured parallel to the bending direction. A 100 \u00d7 200 rectangle used flat (bending about the weak axis) has only a quarter of the section modulus of the same section stood upright.<\/li>\n<li><strong>Tube check.<\/strong> For a hollow tube the inner diameter must be smaller than the outer diameter; extremely thin walls may additionally be governed by local buckling rather than bending stress.<\/li>\n<li><strong>Bending is not the whole story.<\/strong> A beam can also fail by shear, deflection, or lateral-torsional buckling. Slender compression members should also be checked with the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/euler-buckling-load\/\">Euler buckling load calculator<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the elastic section modulus and second moment of area for rectangle, circle, tube, or I\/H beam sections. Enter the dimensions, choose a shape, and get properties about the strong bending axis.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[4],"tags":[97],"class_list":["post-66708","post","type-post","status-publish","format-standard","hentry","category-engineering","tag-beams"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66708","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=66708"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66708\/revisions"}],"predecessor-version":[{"id":66709,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66708\/revisions\/66709"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66708"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66708"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}