{"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":"2026-03-05T15:43:38","modified_gmt":"2026-03-05T15:43:38","slug":"section-moment-of-inertia","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/pt_br\/calculators\/section-moment-of-inertia\/","title":{"rendered":"Calculadora do Momento de In\u00e9rcia da Se\u00e7\u00e3o | Ferramenta Online Gratuita"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Section Moment of Inertia Calculator\",\"alternateName\":[\"Cross-Section Properties Calculator\",\"Second Moment of Area Calculator\"],\"description\":\"Free online calculator for cross-section properties including moment of inertia, section modulus, area, and radius of gyration.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/section-moment-of-inertia\/\",\"applicationCategory\":\"Engineering Calculator\",\"operatingSystem\":\"Any (Web Browser)\",\"offers\":{\"@type\":\"Offer\",\"price\":\"0\",\"priceCurrency\":\"EUR\"},\"creator\":{\"@type\":\"Organization\",\"name\":\"Vibromera\",\"url\":\"https:\/\/vibromera.eu\/\"},\"datePublished\":\"2024-01-01\",\"dateModified\":\"2025-02-14\",\"inLanguage\":\"en\",\"isAccessibleForFree\":true,\"featureList\":[\"Solid circle\",\"Hollow circle (tube)\",\"Rectangle\",\"I-beam (symmetric)\",\"Channel section\",\"Auto-calculation\"],\"keywords\":\"moment of inertia, section modulus, cross section, I-beam, radius of gyration\"}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[\n{\"@type\":\"Question\",\"name\":\"What is the moment of inertia of a cross-section?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"The moment of inertia (second moment of area) I measures how the cross-section's area is distributed about the neutral axis. A larger I means more resistance to bending. Units are mm\u2074 or cm\u2074.\"}},\n{\"@type\":\"Question\",\"name\":\"What is section modulus?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Section modulus W (or S) equals I\/c, where c is the distance from the neutral axis to the extreme fiber. It directly relates bending moment to stress: \u03c3 = M\/W. A larger W means lower bending stress.\"}},\n{\"@type\":\"Question\",\"name\":\"What is the radius of gyration?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"The radius of gyration r = \u221a(I\/A) is a measure of how far the area is distributed from the neutral axis. It is used in column buckling calculations to determine the slenderness ratio \u03bb = L\/r.\"}},\n{\"@type\":\"Question\",\"name\":\"How do I calculate I for an I-beam?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"For a symmetric I-beam: I = [B\u00d7H\u00b3 \u2212 (B\u2212tw)\u00d7(H\u22122tf)\u00b3] \/ 12, where B is flange width, H is total height, tw is web thickness, and tf is flange thickness. This subtracts the hollow portions from the bounding rectangle.\"}},\n{\"@type\":\"Question\",\"name\":\"What is the difference between Ix and Iy?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Ix is the moment of inertia about the horizontal (x) axis, and Iy is about the vertical (y) axis. For bending in the vertical plane (most common), Ix is used. For lateral bending, Iy applies. This calculator computes the primary axis I.\"}}\n]}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[\n{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/vibromera.eu\/\"},\n{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Calculators\",\"item\":\"https:\/\/vibromera.eu\/calculators\/\"},\n{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Section Moment of Inertia\",\"item\":\"https:\/\/vibromera.eu\/calculators\/section-moment-of-inertia\/\"}\n]}\n<\/script>\n\n<link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\n\n<link rel=\"preconnect\" href=\"https:\/\/fonts.gstatic.com\" crossorigin>\n\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=DM+Sans:ital,opsz,wght@0,9..40,300;0,9..40,400;0,9..40,500;0,9..40,600;0,9..40,700;1,9..40,400&#038;family=JetBrains+Mono:wght@400;500;600&#038;family=Fraunces:opsz,wght@9..144,700;9..144,800&#038;display=swap\" rel=\"stylesheet\">\n\n<link rel=\"stylesheet\" href=\"https:\/\/cdn.jsdelivr.net\/npm\/katex@0.16.11\/dist\/katex.min.css\">\n\n<style>\n:root{--vc-bg:#f4f1ec;--vc-surface:#ffffff;--vc-surface-alt:#f8f6f2;--vc-ink:#1a1a1a;--vc-ink-secondary:#5a5650;--vc-ink-muted:#8a857e;--vc-accent:#c85a2a;--vc-accent-hover:#b04d22;--vc-accent-light:#fdf0ea;--vc-blue:#2a5c8c;--vc-blue-light:#eaf1f8;--vc-green:#2a7a4b;--vc-green-light:#eaf8ef;--vc-yellow:#a67c00;--vc-yellow-light:#fef9e8;--vc-red:#b91c1c;--vc-red-light:#fef2f2;--vc-border:#d9d4cc;--vc-border-light:#e8e4dd;--vc-shadow:0 1px 3px rgba(26,26,26,0.06),0 4px 12px rgba(26,26,26,0.04);--vc-shadow-lg:0 4px 12px rgba(26,26,26,0.08),0 16px 40px rgba(26,26,26,0.06);--vc-radius:8px;--vc-radius-lg:12px;--vc-font:'DM Sans',-apple-system,BlinkMacSystemFont,'Segoe UI',sans-serif;--vc-mono:'JetBrains Mono','Consolas','Monaco',monospace;--vc-display:'Fraunces',Georgia,serif}\n.vc-calculator{font-family:var(--vc-font);font-size:15px;line-height:1.65;color:var(--vc-ink);max-width:960px;margin:0 auto;padding:20px 16px 40px;-webkit-font-smoothing:antialiased}.vc-calculator *,.vc-calculator *::before,.vc-calculator *::after{box-sizing:border-box}.vc-header{text-align:center;padding:48px 24px 40px;position:relative}.vc-header::after{content:'';position:absolute;bottom:0;left:50%;transform:translateX(-50%);width:80px;height:3px;background:var(--vc-accent);border-radius:2px}.vc-header-eyebrow{font-family:var(--vc-mono);font-size:11px;font-weight:500;letter-spacing:0.15em;text-transform:uppercase;color:var(--vc-accent);margin:0 0 12px}.vc-header-title{font-family:var(--vc-display);font-size:clamp(24px,4vw,36px);font-weight:800;line-height:1.15;color:var(--vc-ink);margin:0 0 16px;letter-spacing:-0.02em}.vc-header-subtitle{font-size:15px;color:var(--vc-ink-secondary);margin:0 auto 20px;max-width:600px}.vc-badges{display:inline-flex;gap:8px;flex-wrap:wrap;justify-content:center}.vc-badge{font-family:var(--vc-mono);font-size:11px;font-weight:500;padding:4px 10px;border-radius:4px;background:var(--vc-surface);border:1px solid var(--vc-border);color:var(--vc-ink-secondary)}.vc-card{background:var(--vc-surface);border:1px solid var(--vc-border);border-radius:var(--vc-radius-lg);box-shadow:var(--vc-shadow);overflow:hidden;margin-top:32px}.vc-form{padding:24px}.vc-form-grid{display:grid;grid-template-columns:1fr 1fr;gap:18px}@media(max-width:600px){.vc-form-grid{grid-template-columns:1fr}}.vc-field{display:flex;flex-direction:column}.vc-field-full{grid-column:1\/-1}.vc-label{font-size:12px;font-weight:600;letter-spacing:0.04em;text-transform:uppercase;color:var(--vc-ink-secondary);margin-bottom:6px}.vc-label-hint{font-weight:400;text-transform:none;letter-spacing:0;color:var(--vc-ink-muted);font-size:11px}.vc-select,.vc-input{width:100%;padding:10px 12px;border:1.5px solid var(--vc-border);border-radius:6px;font-family:var(--vc-font);font-size:14px;color:var(--vc-ink);background:var(--vc-surface);transition:border-color 0.15s,box-shadow 0.15s}.vc-select{padding-right:36px;appearance:none;-webkit-appearance:none;cursor:pointer;background-image:url(\"data:image\/svg+xml,%3Csvg xmlns='http:\/\/www.w3.org\/2000\/svg' width='16' height='16' fill='none' stroke='%235a5650' stroke-width='2'%3E%3Cpath d='M4 6l4 4 4-4'\/%3E%3C\/svg%3E\");background-repeat:no-repeat;background-position:right 10px center}.vc-select:focus,.vc-input:focus{outline:none;border-color:var(--vc-accent);box-shadow:0 0 0 3px rgba(200,90,42,0.12)}.vc-select:hover,.vc-input:hover{border-color:var(--vc-ink-muted)}.vc-presets{display:flex;flex-wrap:wrap;gap:8px;margin-top:18px;padding-top:18px;border-top:1px solid var(--vc-border-light)}.vc-presets-label{font-family:var(--vc-mono);font-size:11px;font-weight:500;letter-spacing:0.1em;text-transform:uppercase;color:var(--vc-ink-muted);width:100%;margin-bottom:2px}.vc-preset-btn{padding:5px 12px;font-family:var(--vc-font);font-size:12px;font-weight:500;border:1px solid var(--vc-border);border-radius:6px;background:var(--vc-surface-alt);color:var(--vc-ink-secondary);cursor:pointer;transition:all 0.15s}.vc-preset-btn:hover{border-color:var(--vc-accent);color:var(--vc-accent);background:var(--vc-accent-light)}.vc-results{border-top:1px solid var(--vc-border-light);background:var(--vc-surface-alt);padding:0;max-height:0;overflow:hidden;transition:max-height 0.5s cubic-bezier(0.4,0,0.2,1),padding 0.3s}.vc-results.vc-visible{max-height:2000px;padding:24px}.vc-results-head{display:flex;align-items:center;justify-content:space-between;margin-bottom:18px}.vc-results-title{font-family:var(--vc-display);font-size:20px;font-weight:700;color:var(--vc-ink);margin:0}.vc-copy-btn{display:inline-flex;align-items:center;gap:5px;padding:6px 12px;border:1px solid var(--vc-border);border-radius:6px;background:var(--vc-surface);font-family:var(--vc-font);font-size:12px;font-weight:500;color:var(--vc-ink-secondary);cursor:pointer;transition:all 0.15s}.vc-copy-btn:hover{border-color:var(--vc-accent);color:var(--vc-accent)}.vc-copy-btn.vc-copied{background:var(--vc-green-light);border-color:var(--vc-green);color:var(--vc-green)}.vc-result-grid{display:grid;grid-template-columns:1fr 1fr;gap:12px;margin-bottom:20px}@media(max-width:500px){.vc-result-grid{grid-template-columns:1fr}}.vc-rcard{background:var(--vc-surface);border:1px solid var(--vc-border-light);border-radius:var(--vc-radius);padding:16px;transition:transform 0.2s}.vc-rcard:hover{transform:translateY(-1px)}.vc-rcard-primary{border-color:var(--vc-accent);border-width:2px;background:linear-gradient(135deg,#fdf6f2 0%,var(--vc-surface) 100%);grid-column:1\/-1}.vc-rcard-label{font-family:var(--vc-mono);font-size:10px;font-weight:500;letter-spacing:0.1em;text-transform:uppercase;color:var(--vc-ink-muted);margin-bottom:4px}.vc-rcard-value{font-family:var(--vc-mono);font-size:22px;font-weight:600;color:var(--vc-ink);line-height:1.2}.vc-rcard-primary .vc-rcard-value{font-size:30px;color:var(--vc-accent)}.vc-rcard-unit{font-size:14px;font-weight:400;color:var(--vc-ink-secondary)}.vc-section{margin-top:24px;border:1px solid var(--vc-border);border-radius:var(--vc-radius-lg);background:var(--vc-surface);box-shadow:var(--vc-shadow);overflow:hidden}.vc-section-toggle{width:100%;display:flex;align-items:center;justify-content:space-between;padding:18px 24px;border:none;background:transparent;cursor:pointer;font-family:var(--vc-font);text-align:left;transition:background 0.15s}.vc-section-toggle:hover{background:var(--vc-surface-alt)}.vc-section-toggle-text{display:flex;align-items:center;gap:10px}.vc-section-icon{font-size:20px;line-height:1}.vc-section-title{font-family:var(--vc-display);font-size:18px;font-weight:700;color:var(--vc-ink);margin:0}.vc-section-chevron{width:20px;height:20px;color:var(--vc-ink-muted);transition:transform 0.3s;flex-shrink:0}.vc-section.vc-open .vc-section-chevron{transform:rotate(180deg)}.vc-section-body{max-height:0;overflow:hidden;transition:max-height 0.4s cubic-bezier(0.4,0,0.2,1)}.vc-section.vc-open .vc-section-body{max-height:20000px}.vc-section-inner{padding:0 24px 24px;border-top:1px solid var(--vc-border-light)}.vc-theory h3{font-family:var(--vc-display);font-size:17px;font-weight:700;color:var(--vc-ink);margin:28px 0 10px}.vc-theory h3:first-child{margin-top:20px}.vc-theory p{font-size:14px;color:var(--vc-ink-secondary);margin:0 0 10px}.vc-theory ul,.vc-theory ol{font-size:14px;color:var(--vc-ink-secondary);margin:0 0 12px 18px;padding:0}.vc-theory li{margin-bottom:5px}.vc-formula-box{background:var(--vc-surface-alt);border:2px solid var(--vc-border);border-radius:6px;padding:14px 20px;font-family:var(--vc-mono);font-size:16px;font-weight:500;text-align:center;margin:14px 0;color:var(--vc-ink);overflow-x:auto}.vc-info-box{background:var(--vc-blue-light);border-left:3px solid var(--vc-blue);padding:14px 16px;border-radius:0 6px 6px 0;margin:14px 0}.vc-info-box p{margin:0;font-size:14px;color:var(--vc-ink-secondary)}.vc-warning-box{background:var(--vc-yellow-light);border-left:3px solid var(--vc-yellow);padding:14px 16px;border-radius:0 6px 6px 0;margin:14px 0}.vc-example-box{background:var(--vc-surface-alt);border:1px solid var(--vc-border-light);border-radius:6px;padding:16px;margin:14px 0}.vc-example-title{font-family:var(--vc-mono);font-size:13px;font-weight:600;color:var(--vc-accent);margin-bottom:8px}.vc-table{width:100%;border-collapse:collapse;margin:14px 0;font-size:13px}.vc-table th,.vc-table td{padding:10px 12px;border:1px solid var(--vc-border-light);text-align:left}.vc-table th{background:var(--vc-surface-alt);font-size:11px;font-weight:600;text-transform:uppercase;letter-spacing:0.05em;color:var(--vc-ink-secondary)}.vc-table tr:nth-child(even){background:var(--vc-surface-alt)}@media(max-width:600px){.vc-table{font-size:12px}.vc-table th,.vc-table td{padding:8px}}.vc-faq-list{display:flex;flex-direction:column;gap:8px;margin-top:20px}.vc-faq-item{border:1px solid var(--vc-border-light);border-radius:6px;overflow:hidden}.vc-faq-q{width:100%;display:flex;align-items:center;justify-content:space-between;gap:12px;padding:14px 16px;border:none;background:var(--vc-surface-alt);cursor:pointer;font-family:var(--vc-font);font-size:15px;font-weight:600;color:var(--vc-ink);text-align:left;transition:background 0.15s}.vc-faq-q:hover{background:#ede9e3}.vc-faq-chevron{width:16px;height:16px;color:var(--vc-ink-muted);flex-shrink:0;transition:transform 0.25s}.vc-faq-item.vc-open .vc-faq-chevron{transform:rotate(180deg)}.vc-faq-a{max-height:0;overflow:hidden;transition:max-height 0.35s}.vc-faq-item.vc-open .vc-faq-a{max-height:1200px}.vc-faq-a-inner{padding:14px 16px;border-top:1px solid var(--vc-border-light);font-size:14px;color:var(--vc-ink-secondary)}.vc-faq-a-inner p{margin:0 0 8px}.vc-faq-a-inner p:last-child{margin-bottom:0}.vc-faq-a-inner ul{margin:4px 0 8px 16px;padding:0}.vc-faq-a-inner li{margin-bottom:3px}.vc-related{display:flex;flex-wrap:wrap;gap:10px;margin-top:16px}.vc-related-link{display:inline-block;padding:8px 16px;font-family:var(--vc-font);font-size:13px;font-weight:600;background:var(--vc-surface-alt);color:var(--vc-ink-secondary);border:1px solid var(--vc-border);border-radius:var(--vc-radius);text-decoration:none;transition:all 0.15s}.vc-related-link:hover{border-color:var(--vc-accent);color:var(--vc-accent);background:var(--vc-accent-light)}.vc-promo{margin-top:24px;padding:20px;background:linear-gradient(135deg,var(--vc-accent-light),var(--vc-surface));border:1px solid var(--vc-border);border-radius:var(--vc-radius-lg);display:flex;align-items:center;gap:16px}.vc-promo-text{flex:1;font-size:14px;color:var(--vc-ink-secondary)}.vc-promo-text strong{color:var(--vc-ink)}.vc-promo-link{padding:8px 20px;font-size:13px;font-weight:700;color:#fff;background:var(--vc-accent);border-radius:var(--vc-radius);text-decoration:none;white-space:nowrap;transition:background 0.15s}.vc-promo-link:hover{background:var(--vc-accent-hover)}.vc-footer{text-align:center;padding:32px 16px;font-size:13px;color:var(--vc-ink-muted)}.vc-footer a{color:var(--vc-accent);text-decoration:none}.vc-footer a:hover{text-decoration:underline}.vc-footer-links{margin-top:8px;font-size:12px}.vc-footer-links a{margin:0 8px}@media print{.vc-section-body{max-height:none!important}.vc-faq-a{max-height:none!important}.vc-results{max-height:none!important;padding:24px!important}.vc-copy-btn,.vc-section-chevron,.vc-faq-chevron,.vc-presets,.vc-promo{display:none!important}}\n.vc-dim-fields{display:none}.vc-dim-fields.vc-active{display:contents}\n.vc-results.vc-visible{max-height:5000px}\n.vc-viz{margin:20px 0;background:var(--vc-surface);border:1px solid var(--vc-border-light);border-radius:var(--vc-radius);padding:16px;overflow:hidden}\n.vc-viz svg{width:100%;height:auto;display:block}\n.vc-viz-title{font-family:var(--vc-mono);font-size:11px;font-weight:500;letter-spacing:0.1em;text-transform:uppercase;color:var(--vc-ink-muted);margin-bottom:12px}\n<\/style>\n\n<div class=\"vc-calculator\" id=\"vc-section-inertia\">\n<header class=\"vc-header\">\n    <p class=\"vc-header-eyebrow\">Free Engineering Tool<\/p>\n    <h1 class=\"vc-header-title\">Section Moment of Inertia Calculator<\/h1>\n    <p class=\"vc-header-subtitle\">Calculate cross-section properties \u2014 moment of inertia I, section modulus W, area A, and radius of gyration r \u2014 for common shapes.<\/p>\n    <div class=\"vc-badges\"><span class=\"vc-badge\">Circle<\/span><span class=\"vc-badge\">Rectangle<\/span><span class=\"vc-badge\">I-Beam<\/span><span class=\"vc-badge\">#165<\/span><\/div>\n<\/header>\n\n<div class=\"vc-card\">\n    <form class=\"vc-form\" id=\"vc-form\" autocomplete=\"off\">\n        <div class=\"vc-form-grid\">\n            <div class=\"vc-field vc-field-full\">\n                <label class=\"vc-label\" for=\"vc-shape\">Cross-Section Shape<\/label>\n                <select class=\"vc-select\" id=\"vc-shape\">\n                    <option value=\"circle\">Solid Circle<\/option>\n                    <option value=\"hollow_circle\">Hollow Circle (Tube)<\/option>\n                    <option value=\"rectangle\">Solid Rectangle<\/option>\n                    <option value=\"ibeam\" selected>I-Beam (Symmetric)<\/option>\n                    <option value=\"channel\">Channel (C-section)<\/option>\n                <\/select>\n            <\/div>\n            <!-- Circle -->\n            <div class=\"vc-dim-fields\" id=\"vc-dims-circle\">\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-d\">Diameter d <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-d\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 100\" value=\"100\"><\/div>\n            <\/div>\n            <!-- Hollow circle -->\n            <div class=\"vc-dim-fields\" id=\"vc-dims-hollow_circle\">\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-do\">Outer Diameter D <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-do\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 114.3\" value=\"114.3\"><\/div>\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-di\">Inner Diameter d <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-di\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 102.3\" value=\"102.3\"><\/div>\n            <\/div>\n            <!-- Rectangle -->\n            <div class=\"vc-dim-fields\" id=\"vc-dims-rectangle\">\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-bw\">Width b <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-bw\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 50\" value=\"50\"><\/div>\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-hw\">Height h <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-hw\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 100\" value=\"100\"><\/div>\n            <\/div>\n            <!-- I-beam -->\n            <div class=\"vc-dim-fields vc-active\" id=\"vc-dims-ibeam\">\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-H\">Total Height H <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-H\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 300\" value=\"300\"><\/div>\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-B\">Flange Width B <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-B\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 150\" value=\"150\"><\/div>\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-tw\">Web Thickness tw <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-tw\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 7.1\" value=\"7.1\"><\/div>\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-tf\">Flange Thickness tf <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-tf\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 10.7\" value=\"10.7\"><\/div>\n            <\/div>\n            <!-- Channel -->\n            <div class=\"vc-dim-fields\" id=\"vc-dims-channel\">\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-cH\">Total Height H <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-cH\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 200\" value=\"200\"><\/div>\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-cB\">Flange Width B <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-cB\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 75\" value=\"75\"><\/div>\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-ctw\">Web Thickness tw <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-ctw\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 6\" value=\"6\"><\/div>\n                <div class=\"vc-field\"><label class=\"vc-label\" for=\"vc-ctf\">Flange Thickness tf <span class=\"vc-label-hint\">(mm)<\/span><\/label><input class=\"vc-input\" type=\"number\" id=\"vc-ctf\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 10\" value=\"10\"><\/div>\n            <\/div>\n        <\/div>\n        <div class=\"vc-presets\">\n            <div class=\"vc-presets-label\">Quick presets<\/div>\n            <button type=\"button\" class=\"vc-preset-btn\" data-shape=\"ibeam\" data-H=\"300\" data-B=\"150\" data-tw=\"7.1\" data-tf=\"10.7\">IPE 300<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-shape=\"ibeam\" data-H=\"200\" data-B=\"100\" data-tw=\"5.6\" data-tf=\"8.5\">IPE 200<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-shape=\"circle\" data-d=\"50\">Shaft \u220550<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-shape=\"hollow_circle\" data-do=\"114.3\" data-di=\"102.3\">Pipe DN100<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-shape=\"rectangle\" data-bw=\"50\" data-hw=\"100\">Rect 50\u00d7100<\/button>\n        <\/div>\n    <\/form>\n\n    <div class=\"vc-results\" id=\"vc-results\">\n        <div class=\"vc-results-head\"><h2 class=\"vc-results-title\">Results<\/h2>\n            <button type=\"button\" class=\"vc-copy-btn\" id=\"vc-copy-btn\"><svg width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><rect x=\"9\" y=\"9\" width=\"13\" height=\"13\" rx=\"2\"\/><path d=\"M5 15H4a2 2 0 01-2-2V4a2 2 0 012-2h9a2 2 0 012 2v1\"\/><\/svg> Copy<\/button>\n        <\/div>\n        <div class=\"vc-result-grid\">\n            <div class=\"vc-rcard vc-rcard-primary\"><div class=\"vc-rcard-label\">Moment of Inertia I<\/div><div class=\"vc-rcard-value\" id=\"vc-r-I\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Section Modulus W<\/div><div class=\"vc-rcard-value\" id=\"vc-r-W\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Cross-Section Area A<\/div><div class=\"vc-rcard-value\" id=\"vc-r-A\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Radius of Gyration r<\/div><div class=\"vc-rcard-value\" id=\"vc-r-r\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Distance to Extreme Fiber c<\/div><div class=\"vc-rcard-value\" id=\"vc-r-c\">\u2014<\/div><\/div>\n        <\/div>\n        <div class=\"vc-viz\" id=\"vc-section-viz\"><\/div>\n    <\/div>\n<\/div>\n\n<div class=\"vc-section vc-open\" id=\"vc-sec-theory\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"true\"><span class=\"vc-section-toggle-text\"><span class=\"vc-section-icon\">\ud83d\udcd8<\/span><span class=\"vc-section-title\">Theory &amp; Formulas<\/span><\/span><svg class=\"vc-section-chevron\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M6 9l6 6 6-6\"\/><\/svg><\/button>\n    <div class=\"vc-section-body\"><div class=\"vc-section-inner vc-theory\">\n        <h3>Solid Circle<\/h3>\n        <div class=\"vc-formula-box\" id=\"vc-katex-1\"><\/div>\n        <h3>Hollow Circle (Tube)<\/h3>\n        <div class=\"vc-formula-box\" id=\"vc-katex-2\"><\/div>\n        <h3>Rectangle<\/h3>\n        <div class=\"vc-formula-box\" id=\"vc-katex-3\"><\/div>\n        <h3>I-Beam (Symmetric)<\/h3>\n        <div class=\"vc-formula-box\" id=\"vc-katex-4\"><\/div>\n        <h3>General Relationships<\/h3>\n        <div class=\"vc-formula-box\" id=\"vc-katex-5\"><\/div>\n        <div class=\"vc-info-box\"><p><strong>\u2139\ufe0f<\/strong> Section modulus W = I\/c relates bending moment to stress: \u03c3 = M\/W. Radius of gyration r = \u221a(I\/A) is used in buckling analysis.<\/p><\/div>\n    <\/div><\/div>\n<\/div>\n\n<div class=\"vc-section\" id=\"vc-sec-related\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-toggle-text\"><span class=\"vc-section-icon\">\ud83d\udd17<\/span><span class=\"vc-section-title\">Related Calculators<\/span><\/span><svg class=\"vc-section-chevron\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M6 9l6 6 6-6\"\/><\/svg><\/button>\n    <div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-related\">\n        <a class=\"vc-related-link\" href=\"\/calculators\/beam-bending-calculator\/\">Beam Bending<\/a>\n        <a class=\"vc-related-link\" href=\"\/calculators\/stress-calculator\/\">Stress Calculator<\/a>\n        <a class=\"vc-related-link\" href=\"\/calculators\/euler-buckling-calculator\/\">Euler Buckling<\/a>\n        <a class=\"vc-related-link\" href=\"\/calculators\/safety-factor-calculator\/\">Safety Factor<\/a>\n    <\/div><\/div><\/div>\n<\/div>\n\n<div class=\"vc-section\" id=\"vc-sec-faq\">\n    <button type=\"button\" class=\"vc-section-toggle\" aria-expanded=\"false\"><span class=\"vc-section-toggle-text\"><span class=\"vc-section-icon\">\u2753<\/span><span class=\"vc-section-title\">Frequently Asked Questions<\/span><\/span><svg class=\"vc-section-chevron\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M6 9l6 6 6-6\"\/><\/svg><\/button>\n    <div class=\"vc-section-body\"><div class=\"vc-section-inner\"><div class=\"vc-faq-list\" id=\"vc-faq-list\"><\/div><\/div><\/div>\n<\/div>\n\n<div class=\"vc-promo\"><div class=\"vc-promo-text\"><strong>Vibromera \u2014 Portable Balancing &amp; Vibration Analysis<\/strong><br>Professional field balancing instruments and software. Used in 50+ countries.<\/div><a class=\"vc-promo-link\" href=\"https:\/\/vibromera.eu\/\" target=\"_blank\" rel=\"noopener\">Learn More<\/a><\/div>\n<footer class=\"vc-footer\"><p>\u00a9 2024\u20132025 <a href=\"https:\/\/vibromera.eu\/\">Vibromera<\/a> \u2014 Engineering Calculators<\/p><p style=\"font-size:12px;margin-top:4px;\">Cross-section property formulas. Last updated: February 2025<\/p><div class=\"vc-footer-links\"><a href=\"\/calculators\/engineering-calculators\/\">All Calculators<\/a><a href=\"https:\/\/vibromera.eu\/contacts\/\">Contact<\/a><\/div><\/footer>\n<\/div>\n\n<script src=\"https:\/\/cdn.jsdelivr.net\/npm\/katex@0.16.11\/dist\/katex.min.js\"><\/script>\n<script>\n(function(){\n'use strict';\nvar SLUG='section-moment-of-inertia';\nfunction $(id){return document.getElementById(id)}\nfunction fmt(n,d){if(isNaN(n)||!isFinite(n))return'\u2014';var f=n.toFixed(d===undefined?2:d);return f.replace(\/(\\.\\d*?)0+$\/,'$1').replace(\/\\.$\/,'')}\n\nfunction updateUI(){\n    var s=$('vc-shape').value;\n    document.querySelectorAll('.vc-dim-fields').forEach(function(el){el.classList.remove('vc-active')});\n    var t=$('vc-dims-'+s);if(t)t.classList.add('vc-active');\n}\n\nfunction calculate(){\n    updateUI();\n    var s=$('vc-shape').value;\n    var I=0,A=0,c=0;\n    var PI=Math.PI;\n\n    switch(s){\n        case'circle':{\n            var d=parseFloat($('vc-d').value);if(!d||d<=0){$('vc-results').classList.remove('vc-visible');return}\n            I=PI*Math.pow(d,4)\/64;A=PI*d*d\/4;c=d\/2;break;\n        }\n        case'hollow_circle':{\n            var D=parseFloat($('vc-do').value),di=parseFloat($('vc-di').value);\n            if(!D||D<=0||!di||di<=0||di>=D){$('vc-results').classList.remove('vc-visible');return}\n            I=PI*(Math.pow(D,4)-Math.pow(di,4))\/64;A=PI*(D*D-di*di)\/4;c=D\/2;break;\n        }\n        case'rectangle':{\n            var b=parseFloat($('vc-bw').value),h=parseFloat($('vc-hw').value);\n            if(!b||b<=0||!h||h<=0){$('vc-results').classList.remove('vc-visible');return}\n            I=b*Math.pow(h,3)\/12;A=b*h;c=h\/2;break;\n        }\n        case'ibeam':{\n            var H=parseFloat($('vc-H').value),B=parseFloat($('vc-B').value),tw=parseFloat($('vc-tw').value),tf=parseFloat($('vc-tf').value);\n            if(!H||!B||!tw||!tf||H<=0||B<=0||tw<=0||tf<=0||tf*2>=H||tw>=B){$('vc-results').classList.remove('vc-visible');return}\n            I=(B*Math.pow(H,3)-(B-tw)*Math.pow(H-2*tf,3))\/12;\n            A=2*B*tf+tw*(H-2*tf);c=H\/2;break;\n        }\n        case'channel':{\n            var cH=parseFloat($('vc-cH').value),cB=parseFloat($('vc-cB').value),ctw=parseFloat($('vc-ctw').value),ctf=parseFloat($('vc-ctf').value);\n            if(!cH||!cB||!ctw||!ctf||cH<=0||cB<=0||ctw<=0||ctf<=0){$('vc-results').classList.remove('vc-visible');return}\n            I=(cB*Math.pow(cH,3)-(cB-ctw)*Math.pow(cH-2*ctf,3))\/12;\n            A=2*cB*ctf+ctw*(cH-2*ctf);c=cH\/2;break;\n        }\n    }\n\n    var W=c>0?I\/c:0;\n    var r=A>0?Math.sqrt(I\/A):0;\n    var Icm=I\/1e4;\/\/ mm4 to cm4\n\n    $('vc-r-I').innerHTML=fmt(Icm,1)+' <span class=\"vc-rcard-unit\">cm\u2074<\/span>';\n    $('vc-r-W').innerHTML=fmt(W\/1e3,2)+' <span class=\"vc-rcard-unit\">cm\u00b3<\/span>';\n    $('vc-r-A').innerHTML=fmt(A,1)+' <span class=\"vc-rcard-unit\">mm\u00b2<\/span>';\n    $('vc-r-r').innerHTML=fmt(r,2)+' <span class=\"vc-rcard-unit\">mm<\/span>';\n    $('vc-r-c').innerHTML=fmt(c,2)+' <span class=\"vc-rcard-unit\">mm<\/span>';\n    $('vc-results').classList.add('vc-visible');\n\n    saveHistory({shape:s,I:Icm,W:W\/1e3,A:A,r:r,_summary:s+', I='+fmt(Icm,1)+' cm\u2074'});\n    drawSectionViz();\n}\n\nfunction drawSectionViz(){\n    var el=$('vc-section-viz');if(!el)return;\n    var shape=$('vc-shape').value;\n    var s='<svg viewBox=\"0 0 300 300\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"font-family:var(--vc-mono)\">';\n    var cx=150,cy=150;\n    function cross(){\n        s+='<line x1=\"'+(cx-7)+'\" y1=\"'+cy+'\" x2=\"'+(cx+7)+'\" y2=\"'+cy+'\" stroke=\"var(--vc-accent)\" stroke-width=\"1.5\"\/>';\n        s+='<line x1=\"'+cx+'\" y1=\"'+(cy-7)+'\" x2=\"'+cx+'\" y2=\"'+(cy+7)+'\" stroke=\"var(--vc-accent)\" stroke-width=\"1.5\"\/>';\n    }\n    function na(x1,x2){\n        s+='<line x1=\"'+x1+'\" y1=\"'+cy+'\" x2=\"'+x2+'\" y2=\"'+cy+'\" stroke=\"var(--vc-blue)\" stroke-width=\"1\" stroke-dasharray=\"5,3\"\/>';\n        s+='<text x=\"'+(x2+4)+'\" y=\"'+(cy+4)+'\" fill=\"var(--vc-blue)\" font-size=\"9\">N.A.<\/text>';\n    }\n    switch(shape){\n    case'circle':{\n        var d=parseFloat($('vc-d').value)||0;if(d<=0){el.innerHTML='';return;}\n        var sc=Math.min(220\/d,1.5),r=d\/2*sc;\n        s+='<circle cx=\"'+cx+'\" cy=\"'+cy+'\" r=\"'+r+'\" fill=\"var(--vc-accent-light)\" stroke=\"var(--vc-accent)\" stroke-width=\"2\"\/>';\n        na(cx-r-15,cx+r+15);\n        s+='<line x1=\"'+cx+'\" y1=\"'+(cy-r)+'\" x2=\"'+cx+'\" y2=\"'+(cy+r)+'\" stroke=\"var(--vc-ink)\" stroke-width=\"0.8\" stroke-dasharray=\"3,2\"\/>';\n        s+='<text x=\"'+(cx+8)+'\" y=\"'+(cy-r\/3)+'\" fill=\"var(--vc-ink)\" font-size=\"10\">d = '+fmt(d,1)+' mm<\/text>';\n        cross();break;}\n    case'hollow_circle':{\n        var D=parseFloat($('vc-do').value)||0,di=parseFloat($('vc-di').value)||0;\n        if(D<=0||di<=0||di>=D){el.innerHTML='';return;}\n        var sc=Math.min(220\/D,1.5),ro=D\/2*sc,ri=di\/2*sc;\n        s+='<circle cx=\"'+cx+'\" cy=\"'+cy+'\" r=\"'+ro+'\" fill=\"var(--vc-accent-light)\" stroke=\"var(--vc-accent)\" stroke-width=\"2\"\/>';\n        s+='<circle cx=\"'+cx+'\" cy=\"'+cy+'\" r=\"'+ri+'\" fill=\"var(--vc-surface)\" stroke=\"var(--vc-accent)\" stroke-width=\"1.5\"\/>';\n        na(cx-ro-15,cx+ro+15);\n        s+='<line x1=\"'+(cx-6)+'\" y1=\"'+(cy-ro)+'\" x2=\"'+(cx-6)+'\" y2=\"'+(cy+ro)+'\" stroke=\"var(--vc-ink)\" stroke-width=\"0.8\"\/>';\n        s+='<text x=\"'+(cx-18)+'\" y=\"'+(cy-ro\/2)+'\" fill=\"var(--vc-ink)\" font-size=\"9\" text-anchor=\"end\">D = '+fmt(D,1)+'<\/text>';\n        s+='<line x1=\"'+(cx+6)+'\" y1=\"'+(cy-ri)+'\" x2=\"'+(cx+6)+'\" y2=\"'+(cy+ri)+'\" stroke=\"var(--vc-ink-muted)\" stroke-width=\"0.8\"\/>';\n        s+='<text x=\"'+(cx+12)+'\" y=\"'+(cy-ri\/3)+'\" fill=\"var(--vc-ink-muted)\" font-size=\"9\">d = '+fmt(di,1)+'<\/text>';\n        cross();break;}\n    case'rectangle':{\n        var b=parseFloat($('vc-bw').value)||0,h=parseFloat($('vc-hw').value)||0;\n        if(b<=0||h<=0){el.innerHTML='';return;}\n        var sc=Math.min(220\/Math.max(b,h),1.5),rw=b*sc,rh=h*sc;\n        s+='<rect x=\"'+(cx-rw\/2)+'\" y=\"'+(cy-rh\/2)+'\" width=\"'+rw+'\" height=\"'+rh+'\" fill=\"var(--vc-accent-light)\" stroke=\"var(--vc-accent)\" stroke-width=\"2\" rx=\"1\"\/>';\n        na(cx-rw\/2-15,cx+rw\/2+15);\n        s+='<text x=\"'+cx+'\" y=\"'+(cy+rh\/2+16)+'\" text-anchor=\"middle\" fill=\"var(--vc-ink)\" font-size=\"10\">b = '+fmt(b,1)+' mm<\/text>';\n        s+='<text x=\"'+(cx+rw\/2+10)+'\" y=\"'+cy+'\" fill=\"var(--vc-ink)\" font-size=\"10\">h = '+fmt(h,1)+'<\/text>';\n        cross();break;}\n    case'ibeam':{\n        var H=parseFloat($('vc-H').value)||0,B=parseFloat($('vc-B').value)||0;\n        var tw=parseFloat($('vc-tw').value)||0,tf=parseFloat($('vc-tf').value)||0;\n        if(H<=0||B<=0||tw<=0||tf<=0){el.innerHTML='';return;}\n        var sc=Math.min(220\/Math.max(H,B),1.5),sH=H*sc,sB=B*sc,sTw=Math.max(tw*sc,3),sTf=Math.max(tf*sc,3);\n        var top=cy-sH\/2;\n        var p='M'+(cx-sB\/2)+','+top+' L'+(cx+sB\/2)+','+top+' L'+(cx+sB\/2)+','+(top+sTf)+' L'+(cx+sTw\/2)+','+(top+sTf)+' L'+(cx+sTw\/2)+','+(top+sH-sTf)+' L'+(cx+sB\/2)+','+(top+sH-sTf)+' L'+(cx+sB\/2)+','+(top+sH)+' L'+(cx-sB\/2)+','+(top+sH)+' L'+(cx-sB\/2)+','+(top+sH-sTf)+' L'+(cx-sTw\/2)+','+(top+sH-sTf)+' L'+(cx-sTw\/2)+','+(top+sTf)+' L'+(cx-sB\/2)+','+(top+sTf)+' Z';\n        s+='<path d=\"'+p+'\" fill=\"var(--vc-accent-light)\" stroke=\"var(--vc-accent)\" stroke-width=\"2\"\/>';\n        na(cx-sB\/2-15,cx+sB\/2+15);\n        s+='<text x=\"'+cx+'\" y=\"'+(top+sH+16)+'\" text-anchor=\"middle\" fill=\"var(--vc-ink)\" font-size=\"9\">B = '+fmt(B,1)+'<\/text>';\n        s+='<text x=\"'+(cx+sB\/2+10)+'\" y=\"'+(cy+4)+'\" fill=\"var(--vc-ink)\" font-size=\"9\">H = '+fmt(H,1)+'<\/text>';\n        s+='<text x=\"'+(cx+sTw\/2+5)+'\" y=\"'+(cy-12)+'\" fill=\"var(--vc-ink-muted)\" font-size=\"8\">tw='+fmt(tw,1)+'<\/text>';\n        s+='<text x=\"'+(cx+sB\/4)+'\" y=\"'+(top+sTf\/2+4)+'\" fill=\"var(--vc-ink-muted)\" font-size=\"8\">tf='+fmt(tf,1)+'<\/text>';\n        cross();break;}\n    case'channel':{\n        var cH=parseFloat($('vc-cH').value)||0,cB=parseFloat($('vc-cB').value)||0;\n        var ctw=parseFloat($('vc-ctw').value)||0,ctf=parseFloat($('vc-ctf').value)||0;\n        if(cH<=0||cB<=0||ctw<=0||ctf<=0){el.innerHTML='';return;}\n        var sc=Math.min(220\/Math.max(cH,cB),1.5),sH=cH*sc,sB=cB*sc,sTw=Math.max(ctw*sc,3),sTf=Math.max(ctf*sc,3);\n        var top=cy-sH\/2,left=cx-sB\/2;\n        var p='M'+left+','+top+' L'+(left+sB)+','+top+' L'+(left+sB)+','+(top+sTf)+' L'+(left+sTw)+','+(top+sTf)+' L'+(left+sTw)+','+(top+sH-sTf)+' L'+(left+sB)+','+(top+sH-sTf)+' L'+(left+sB)+','+(top+sH)+' L'+left+','+(top+sH)+' Z';\n        s+='<path d=\"'+p+'\" fill=\"var(--vc-accent-light)\" stroke=\"var(--vc-accent)\" stroke-width=\"2\"\/>';\n        na(left-15,left+sB+15);\n        s+='<text x=\"'+cx+'\" y=\"'+(top+sH+16)+'\" text-anchor=\"middle\" fill=\"var(--vc-ink)\" font-size=\"9\">B = '+fmt(cB,1)+'<\/text>';\n        s+='<text x=\"'+(left-8)+'\" y=\"'+(cy+4)+'\" text-anchor=\"end\" fill=\"var(--vc-ink)\" font-size=\"9\">H = '+fmt(cH,1)+'<\/text>';\n        s+='<text x=\"'+(left+sTw+4)+'\" y=\"'+(cy-10)+'\" fill=\"var(--vc-ink-muted)\" font-size=\"8\">tw='+fmt(ctw,1)+'<\/text>';\n        s+='<text x=\"'+(left+sB\/2)+'\" y=\"'+(top+sTf\/2+4)+'\" text-anchor=\"middle\" fill=\"var(--vc-ink-muted)\" font-size=\"8\">tf='+fmt(ctf,1)+'<\/text>';\n        cross();break;}\n    }\n    s+='<\/svg>';\n    el.innerHTML='<div class=\"vc-viz-title\">Cross-Section<\/div>'+s;\n}\n\ndocument.querySelectorAll('.vc-preset-btn').forEach(function(btn){\n    btn.addEventListener('click',function(e){e.preventDefault();\n        var s=this.getAttribute('data-shape');$('vc-shape').value=s;\n        ['d','do','di','bw','hw','H','B','tw','tf','cH','cB','ctw','ctf'].forEach(function(k){\n            var v=btn.getAttribute('data-'+k);if(v){var el=$('vc-'+k);if(el)el.value=v;}\n        });\n        calculate();\n    });\n});\n\n$('vc-form').addEventListener('input',calculate);\n$('vc-form').addEventListener('change',calculate);\n\n$('vc-copy-btn').addEventListener('click',function(){\n    var text='Section Properties\\n\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\nI: '+$('vc-r-I').textContent+'\\nW: '+$('vc-r-W').textContent+'\\nA: '+$('vc-r-A').textContent+'\\nr: '+$('vc-r-r').textContent+'\\nc: '+$('vc-r-c').textContent+'\\n\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\nvibromera.eu';\n    if(navigator.clipboard){navigator.clipboard.writeText(text).then(function(){var b=$('vc-copy-btn');b.classList.add('vc-copied');b.innerHTML='<svg width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><polyline points=\"20 6 9 17 4 12\"\/><\/svg> Copied!';setTimeout(function(){b.classList.remove('vc-copied');b.innerHTML='<svg width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><rect x=\"9\" y=\"9\" width=\"13\" height=\"13\" rx=\"2\"\/><path d=\"M5 15H4a2 2 0 01-2-2V4a2 2 0 012-2h9a2 2 0 012 2v1\"\/><\/svg> Copy';},2000)})}\n});\n\ndocument.querySelectorAll('.vc-section-toggle').forEach(function(btn){btn.addEventListener('click',function(){var s=this.closest('.vc-section');s.classList.toggle('vc-open');this.setAttribute('aria-expanded',s.classList.contains('vc-open'))})});\n\nvar faqData=[\n    {q:'What is the moment of inertia of a cross-section?',a:'<p>The moment of inertia (second moment of area) I measures how the cross-section\\'s area is distributed about the neutral axis. A larger I means more resistance to bending. Units are mm\u2074 or cm\u2074.<\/p>'},\n    {q:'What is section modulus?',a:'<p>Section modulus W = I\/c relates bending moment directly to stress: \u03c3 = M\/W. A larger section modulus means lower stress for the same bending moment.<\/p>'},\n    {q:'What is the radius of gyration?',a:'<p>Radius of gyration r = \u221a(I\/A) indicates how far the cross-section\\'s area is spread from the centroid. It\\'s essential in column buckling: slenderness ratio \u03bb = KL\/r.<\/p>'},\n    {q:'How do I calculate I for an I-beam?',a:'<p>For a symmetric I-beam: I = [B\u00b7H\u00b3 \u2212 (B\u2212tw)\u00b7(H\u22122tf)\u00b3] \/ 12. This treats the section as a solid rectangle minus the two hollow rectangular portions beside the web.<\/p>'},\n    {q:'What is the difference between Ix and Iy?',a:'<p>Ix is the moment of inertia about the horizontal axis (strong axis), Iy about the vertical axis (weak axis). For bending in the vertical plane (most common), Ix governs. This calculator computes the strong-axis I.<\/p>'}\n];\nvar fl=$('vc-faq-list');faqData.forEach(function(f){fl.innerHTML+='<div class=\"vc-faq-item\"><button type=\"button\" class=\"vc-faq-q\"><span>'+f.q+'<\/span><svg class=\"vc-faq-chevron\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M6 9l6 6 6-6\"\/><\/svg><\/button><div class=\"vc-faq-a\"><div class=\"vc-faq-a-inner\">'+f.a+'<\/div><\/div><\/div>'});\nfl.addEventListener('click',function(e){var b=e.target.closest('.vc-faq-q');if(!b)return;b.closest('.vc-faq-item').classList.toggle('vc-open')});\n\nfunction saveHistory(entry){var key='vc_history_'+SLUG;var arr=[];try{arr=JSON.parse(localStorage.getItem(key))||[]}catch(e){}entry._ts=new Date().toISOString();arr.unshift(entry);if(arr.length>10)arr=arr.slice(0,10);try{localStorage.setItem(key,JSON.stringify(arr))}catch(e){}}\n\nfunction renderFormulas(){try{\n    katex.render('I=\\\\frac{\\\\pi d^4}{64},\\\\quad W=\\\\frac{\\\\pi d^3}{32},\\\\quad A=\\\\frac{\\\\pi d^2}{4}',$('vc-katex-1'),{displayMode:true,throwOnError:false});\n    katex.render('I=\\\\frac{\\\\pi(D^4-d^4)}{64},\\\\quad A=\\\\frac{\\\\pi(D^2-d^2)}{4}',$('vc-katex-2'),{displayMode:true,throwOnError:false});\n    katex.render('I=\\\\frac{bh^3}{12},\\\\quad W=\\\\frac{bh^2}{6},\\\\quad A=bh',$('vc-katex-3'),{displayMode:true,throwOnError:false});\n    katex.render('I=\\\\frac{BH^3-(B-t_w)(H-2t_f)^3}{12}',$('vc-katex-4'),{displayMode:true,throwOnError:false});\n    katex.render('W=\\\\frac{I}{c},\\\\quad r=\\\\sqrt{\\\\frac{I}{A}}',$('vc-katex-5'),{displayMode:true,throwOnError:false});\n}catch(e){}}\n\nfunction init(){renderFormulas();calculate()}\nif(document.readyState==='loading'){document.addEventListener('DOMContentLoaded',init)}else{init()}\n})();\n<\/script>\n","protected":false},"excerpt":{"rendered":"<p>Calculadora online gratuita para propriedades de se\u00e7\u00f5es transversais. Calcule o momento de in\u00e9rcia I, o m\u00f3dulo de se\u00e7\u00e3o W, a \u00e1rea e o raio de gira\u00e7\u00e3o para se\u00e7\u00f5es circulares, retangulares, em I, em U e ocas.<\/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\/pt_br\/wp-json\/wp\/v2\/calculator\/100219","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/pt_br\/wp-json\/wp\/v2\/calculator"}],"about":[{"href":"https:\/\/vibromera.eu\/pt_br\/wp-json\/wp\/v2\/types\/calculator"}],"version-history":[{"count":2,"href":"https:\/\/vibromera.eu\/pt_br\/wp-json\/wp\/v2\/calculator\/100219\/revisions"}],"predecessor-version":[{"id":100777,"href":"https:\/\/vibromera.eu\/pt_br\/wp-json\/wp\/v2\/calculator\/100219\/revisions\/100777"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/pt_br\/wp-json\/wp\/v2\/media?parent=100219"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/pt_br\/wp-json\/wp\/v2\/categories?post=100219"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/pt_br\/wp-json\/wp\/v2\/tags?post=100219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}