{"id":100257,"date":"2026-02-15T20:30:09","date_gmt":"2026-02-15T20:30:09","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100257"},"modified":"2026-02-15T20:30:09","modified_gmt":"2026-02-15T20:30:09","slug":"thermal-stress-calculator","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/bn\/calculators\/thermal-stress-calculator\/","title":{"rendered":"\u09a4\u09be\u09aa\u09c0\u09af\u09bc \u099a\u09be\u09aa \u0995\u09cd\u09af\u09be\u09b2\u0995\u09c1\u09b2\u09c7\u099f\u09b0 | \u03c3 = E\u00b7\u03b1\u00b7\u0394T | \u09ac\u09bf\u09a8\u09be\u09ae\u09c2\u09b2\u09cd\u09af\u09c7 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09b8\u09b0\u099e\u09cd\u099c\u09be\u09ae"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Thermal Stress Calculator\",\"alternateName\":[\"Temperature Stress Calculator\",\"Constrained Expansion Stress\"],\"description\":\"Free online thermal stress calculator. \u03c3 = E \u00d7 \u03b1 \u00d7 \u0394T for fully constrained members.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/thermal-stress-calculator\/\",\"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\":[\"Material presets with CTE values\",\"Thermal stress calculation\",\"Thermal strain\",\"Force calculation\",\"Auto-calculation\"],\"keywords\":\"thermal stress, CTE, coefficient thermal expansion, constrained expansion, temperature stress\"}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[\n{\"@type\":\"Question\",\"name\":\"What is thermal stress?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Thermal stress develops when a material is heated or cooled but cannot freely expand or contract. For a fully constrained member: \u03c3 = E \u00d7 \u03b1 \u00d7 \u0394T, where E is elastic modulus, \u03b1 is the coefficient of thermal expansion (CTE), and \u0394T is the temperature change.\"}},\n{\"@type\":\"Question\",\"name\":\"What is the coefficient of thermal expansion (CTE)?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"CTE (\u03b1) measures how much a material expands per degree of temperature change. Steel: ~12\u00d710\u207b\u2076\/\u00b0C, Aluminum: ~23\u00d710\u207b\u2076\/\u00b0C, Copper: ~17\u00d710\u207b\u2076\/\u00b0C, Stainless steel: ~17\u00d710\u207b\u2076\/\u00b0C.\"}},\n{\"@type\":\"Question\",\"name\":\"When does thermal stress occur?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Thermal stress occurs when expansion is constrained: pipes fixed at both ends, rails welded continuously, dissimilar materials bonded together, thick sections with temperature gradients, or any structure restrained from free movement.\"}},\n{\"@type\":\"Question\",\"name\":\"How can thermal stress be reduced?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Use expansion joints or loops in piping, allow sliding supports, choose materials with lower CTE, reduce \u0394T with insulation, or use flexible connections. In bimetallic systems, match CTE values as closely as possible.\"}},\n{\"@type\":\"Question\",\"name\":\"Is thermal stress always compressive when heating?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Yes, for a fully constrained member heated uniformly. The material tries to expand but cannot, so internal compressive stress develops. Cooling causes tensile stress. The sign depends on whether expansion or contraction is restrained.\"}}\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\":\"Thermal Stress Calculator\",\"item\":\"https:\/\/vibromera.eu\/calculators\/thermal-stress-calculator\/\"}\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{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-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)}.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-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<\/style>\n\n<div class=\"vc-calculator\" id=\"vc-thermal-stress\">\n<header class=\"vc-header\">\n    <p class=\"vc-header-eyebrow\">Free Engineering Tool<\/p>\n    <h1 class=\"vc-header-title\">Thermal Stress Calculator<\/h1>\n    <p class=\"vc-header-subtitle\">Calculate thermal stress \u03c3 = E \u00d7 \u03b1 \u00d7 \u0394T for a fully constrained member subjected to temperature change.<\/p>\n    <div class=\"vc-badges\"><span class=\"vc-badge\">\u03c3 = E\u00b7\u03b1\u00b7\u0394T<\/span><span class=\"vc-badge\">Constrained Member<\/span><span class=\"vc-badge\">#169<\/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-mat\">Material Preset<\/label>\n                <select class=\"vc-select\" id=\"vc-mat\">\n                    <option value=\"custom\">Custom values<\/option>\n                    <option value=\"steel\" selected>Carbon Steel (E=200, \u03b1=12)<\/option>\n                    <option value=\"ss\">Stainless Steel 304 (E=193, \u03b1=17.3)<\/option>\n                    <option value=\"al\">Aluminum 6061 (E=69, \u03b1=23.6)<\/option>\n                    <option value=\"cu\">Copper (E=117, \u03b1=16.6)<\/option>\n                    <option value=\"ti\">Titanium (E=114, \u03b1=8.6)<\/option>\n                    <option value=\"brass\">Brass (E=100, \u03b1=19)<\/option>\n                <\/select>\n            <\/div>\n            <div class=\"vc-field\">\n                <label class=\"vc-label\" for=\"vc-E\">Elastic Modulus E <span class=\"vc-label-hint\">(GPa)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-E\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 200\" value=\"200\">\n            <\/div>\n            <div class=\"vc-field\">\n                <label class=\"vc-label\" for=\"vc-alpha\">CTE \u03b1 <span class=\"vc-label-hint\">(\u00d710\u207b\u2076 \/\u00b0C)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-alpha\" inputmode=\"decimal\" step=\"any\" min=\"0\" placeholder=\"e.g. 12\" value=\"12\">\n            <\/div>\n            <div class=\"vc-field\">\n                <label class=\"vc-label\" for=\"vc-dT\">Temperature Change \u0394T <span class=\"vc-label-hint\">(\u00b0C)<\/span><\/label>\n                <input class=\"vc-input\" type=\"number\" id=\"vc-dT\" inputmode=\"decimal\" step=\"any\" placeholder=\"e.g. 100\" value=\"100\">\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-E=\"200\" data-a=\"12\" data-dT=\"100\">Steel \u0394T=100\u00b0C<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-E=\"193\" data-a=\"17.3\" data-dT=\"150\">SS304 \u0394T=150\u00b0C<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-E=\"69\" data-a=\"23.6\" data-dT=\"80\">Al 6061 \u0394T=80\u00b0C<\/button>\n            <button type=\"button\" class=\"vc-preset-btn\" data-E=\"117\" data-a=\"16.6\" data-dT=\"200\">Copper \u0394T=200\u00b0C<\/button>\n        <\/div>\n    <\/form>\n    <div class=\"vc-results\" id=\"vc-results\">\n        <div class=\"vc-results-head\"><h2 class=\"vc-results-title\">Results<\/h2><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><\/div>\n        <div class=\"vc-result-grid\">\n            <div class=\"vc-rcard vc-rcard-primary\"><div class=\"vc-rcard-label\">Thermal Stress \u03c3<\/div><div class=\"vc-rcard-value\" id=\"vc-r-stress\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Thermal Strain \u03b5<\/div><div class=\"vc-rcard-value\" id=\"vc-r-strain\">\u2014<\/div><\/div>\n            <div class=\"vc-rcard\"><div class=\"vc-rcard-label\">Stress Type<\/div><div class=\"vc-rcard-value\" id=\"vc-r-type\">\u2014<\/div><\/div>\n        <\/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>Thermal Stress in Constrained Members<\/h3>\n        <div class=\"vc-formula-box\" id=\"vc-katex-1\"><\/div>\n        <div class=\"vc-formula-box\" id=\"vc-katex-2\"><\/div>\n        <h3>Common CTE Values<\/h3>\n        <table class=\"vc-table\">\n            <thead><tr><th>Material<\/th><th>E (GPa)<\/th><th>\u03b1 (\u00d710\u207b\u2076\/\u00b0C)<\/th><\/tr><\/thead>\n            <tbody>\n                <tr><td>Carbon Steel<\/td><td>200<\/td><td>12<\/td><\/tr>\n                <tr><td>Stainless Steel 304<\/td><td>193<\/td><td>17.3<\/td><\/tr>\n                <tr><td>Aluminum 6061<\/td><td>69<\/td><td>23.6<\/td><\/tr>\n                <tr><td>Copper<\/td><td>117<\/td><td>16.6<\/td><\/tr>\n                <tr><td>Titanium<\/td><td>114<\/td><td>8.6<\/td><\/tr>\n                <tr><td>Brass<\/td><td>100<\/td><td>19<\/td><\/tr>\n            <\/tbody>\n        <\/table>\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\/thermal-expansion-calculator\/\">Thermal Expansion<\/a>\n        <a class=\"vc-related-link\" href=\"\/calculators\/stress-calculator\/\">Stress Calculator<\/a>\n        <a class=\"vc-related-link\" href=\"\/calculators\/safety-factor-calculator\/\">Safety Factor<\/a>\n        <a class=\"vc-related-link\" href=\"\/calculators\/pressure-vessel-calculator\/\">Pressure Vessel<\/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;\">Thermal stress formula. 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='thermal-stress-calculator';\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\nvar mats={steel:{E:200,a:12},ss:{E:193,a:17.3},al:{E:69,a:23.6},cu:{E:117,a:16.6},ti:{E:114,a:8.6},brass:{E:100,a:19}};\n\n$('vc-mat').addEventListener('change',function(){\n    var m=mats[this.value];\n    if(m){$('vc-E').value=m.E;$('vc-alpha').value=m.a}\n    calculate();\n});\n\nfunction calculate(){\n    var E=parseFloat($('vc-E').value),alpha=parseFloat($('vc-alpha').value),dT=parseFloat($('vc-dT').value);\n    if(!E||E<=0||!alpha||alpha<=0||isNaN(dT)||dT===0){$('vc-results').classList.remove('vc-visible');return}\n\n    var strain=alpha*1e-6*Math.abs(dT);\n    var stress=E*1e3*alpha*1e-6*Math.abs(dT);\/\/ MPa\n    var type=dT>0?'Compressive (heating)':'Tensile (cooling)';\n\n    $('vc-r-stress').innerHTML=fmt(stress,2)+' <span class=\"vc-rcard-unit\">MPa<\/span>';\n    $('vc-r-strain').innerHTML=fmt(strain*1e6,1)+' <span class=\"vc-rcard-unit\">\u03bc\u03b5<\/span> <span class=\"vc-rcard-unit\">('+fmt(strain*100,4)+'%)<\/span>';\n    $('vc-r-type').innerHTML='<span style=\"font-size:14px\">'+type+'<\/span>';\n    $('vc-results').classList.add('vc-visible');\n\n    saveHistory({E:E,alpha:alpha,dT:dT,stress:stress,_summary:'\u03c3='+fmt(stress,1)+' MPa, \u0394T='+dT+'\u00b0C'});\n}\n\ndocument.querySelectorAll('.vc-preset-btn').forEach(function(btn){btn.addEventListener('click',function(e){e.preventDefault();\n    $('vc-E').value=this.getAttribute('data-E');$('vc-alpha').value=this.getAttribute('data-a');$('vc-dT').value=this.getAttribute('data-dT');$('vc-mat').value='custom';calculate();\n})});\n\n$('vc-form').addEventListener('input',calculate);$('vc-form').addEventListener('change',calculate);\n\n$('vc-copy-btn').addEventListener('click',function(){\n    var text='Thermal Stress\\n\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\u2501\\nStress: '+$('vc-r-stress').textContent+'\\nStrain: '+$('vc-r-strain').textContent+'\\nType: '+$('vc-r-type').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=$('vc-copy-btn');b.classList.add('vc-copied');b.innerHTML='<svg width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><polyline points=\"20 6 9 17 4 12\"\/><\/svg> Copied!';setTimeout(function(){b.classList.remove('vc-copied');b.innerHTML='<svg width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><rect x=\"9\" y=\"9\" width=\"13\" height=\"13\" rx=\"2\"\/><path d=\"M5 15H4a2 2 0 01-2-2V4a2 2 0 012-2h9a2 2 0 012 2v1\"\/><\/svg> Copy';},2000)})}\n});\n\ndocument.querySelectorAll('.vc-section-toggle').forEach(function(btn){btn.addEventListener('click',function(){var s=this.closest('.vc-section');s.classList.toggle('vc-open');this.setAttribute('aria-expanded',s.classList.contains('vc-open'))})});\n\nvar faqData=[\n    {q:'What is thermal stress?',a:'<p>Thermal stress develops when a material is heated or cooled but cannot freely expand\/contract. For full constraint: \u03c3 = E \u00d7 \u03b1 \u00d7 \u0394T.<\/p>'},\n    {q:'What is the coefficient of thermal expansion?',a:'<p>CTE (\u03b1) is the fractional change in length per degree. Steel: ~12\u00d710\u207b\u2076\/\u00b0C, Aluminum: ~23.6\u00d710\u207b\u2076\/\u00b0C, Copper: ~16.6\u00d710\u207b\u2076\/\u00b0C.<\/p>'},\n    {q:'When does thermal stress occur?',a:'<p>Pipes fixed at both ends, welded rails, bonded dissimilar materials, thick sections with temperature gradients, or any restrained structure.<\/p>'},\n    {q:'How can thermal stress be reduced?',a:'<p>Expansion joints, sliding supports, lower CTE materials, insulation to reduce \u0394T, flexible connections, or matching CTE in bimetallic systems.<\/p>'},\n    {q:'Is thermal stress compressive when heating?',a:'<p>Yes, for a fully constrained member. Heating \u2192 material wants to expand \u2192 constraint prevents it \u2192 compressive stress. Cooling causes tensile stress.<\/p>'}\n];\nvar fl=$('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('\\\\sigma = E \\\\cdot \\\\alpha \\\\cdot \\\\Delta T \\\\quad [\\\\text{MPa}]',$('vc-katex-1'),{displayMode:true,throwOnError:false});\n    katex.render('\\\\varepsilon = \\\\alpha \\\\cdot \\\\Delta T',$('vc-katex-2'),{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>\u09b8\u0982\u0995\u09c0\u09b0\u09cd\u09a3 \u09b8\u09a6\u09b8\u09cd\u09af\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09ac\u09bf\u09a8\u09be\u09ae\u09c2\u09b2\u09cd\u09af\u09c7 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09a4\u09be\u09aa\u09c0\u09af\u09bc \u099a\u09be\u09aa \u0995\u09cd\u09af\u09be\u09b2\u0995\u09c1\u09b2\u09c7\u099f\u09b0\u0964 \u03c3 = E \u00d7 \u03b1 \u00d7 \u0394T\u0964 \u09b8\u09cd\u09a5\u09bf\u09a4\u09bf\u09b8\u09cd\u09a5\u09be\u09aa\u0995 \u09ae\u09a1\u09c1\u09b2\u09be\u09b8, \u09a4\u09be\u09aa\u09c0\u09af\u09bc \u09aa\u09cd\u09b0\u09b8\u09be\u09b0\u09a3\u09c7\u09b0 \u0997\u09c1\u09a3\u09be\u0999\u09cd\u0995 \u098f\u09ac\u0982 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u09b2\u09bf\u0996\u09c1\u09a8\u0964<\/p>","protected":false},"featured_media":0,"template":"","meta":{"ai_generated_summary":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-100257","calculator","type-calculator","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/calculator\/100257","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/calculator"}],"about":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/types\/calculator"}],"version-history":[{"count":1,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/calculator\/100257\/revisions"}],"predecessor-version":[{"id":100570,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/calculator\/100257\/revisions\/100570"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/media?parent=100257"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/categories?post=100257"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/tags?post=100257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}