{"id":863,"date":"2026-10-04T07:56:16","date_gmt":"2026-10-04T06:56:16","guid":{"rendered":"https:\/\/wpcalc.com\/pt\/fisica\/modulo-de-young\/"},"modified":"2026-10-04T07:56:16","modified_gmt":"2026-10-04T06:56:16","slug":"modulo-de-young","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pt\/fisica\/modulo-de-young\/","title":{"rendered":"Calculadora do M\u00f3dulo de Young"},"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\">Calcular o M\u00f3dulo de Young a partir da tens\u00e3o e da deforma\u00e7\u00e3o<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"863\" data-favorite-title=\"M\u00f3dulo de Young\" data-favorite-url=\"https:\/\/wpcalc.com\/pt\/fisica\/modulo-de-young\/\" data-favorite-excerpt=\"Calcule o m\u00f3dulo de Young (elasticidade) a partir da tens\u00e3o e da deforma\u00e7\u00e3o, ou a partir da for\u00e7a, da \u00e1rea da sec\u00e7\u00e3o transversal e da varia\u00e7\u00e3o de comprimento. Resultados em GPa, MPa, Pa e psi.\" aria-label=\"Adicionar aos favoritos\" data-favorite-icon=\"icon icon-physic\">\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\/pt\/fisica\/modulo-de-young\/\" method=\"POST\" class=\"calculator\" id=\"calculator-863\" data-post=\"863\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Campos de entrada<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">M\u00e9todo de c\u00e1lculo<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"mtodo_de_clculo\"><option value=\"ss\" selected>A partir da tens\u00e3o e da deforma\u00e7\u00e3o<\/option><option value=\"fal\">A partir da for\u00e7a, da \u00e1rea e dos comprimentos<\/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 is-single has-term\" id=\"input-2\"><label for=\"field-2\">Tens\u00e3o (\u03c3)<\/label><div class=\"term\">\u03c3<\/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=\"tenso_\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 200\" min=\"0\" value=\"200\"\/><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"unidade\" aria-label=\"Unidade\"><option value=\"Pa\">Pa<\/option><option value=\"kPa\">kPa<\/option><option value=\"MPa\" selected>MPa<\/option><option value=\"GPa\">GPa<\/option><option value=\"psi\">psi<\/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-2\">For\u00e7a por unidade de \u00e1rea (\u03c3 = F\/A). Deixe em branco se quiser usar o m\u00e9todo da for\u00e7a abaixo.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-3\"><label for=\"field-4\">Deforma\u00e7\u00e3o (\u03b5)<\/label><div class=\"term\">\u03b5<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"deformao_\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 0.001\" min=\"0\" value=\"0.001\"\/><small id=\"calc-info-4\">Deforma\u00e7\u00e3o relativa \u03b5 = \u0394L\/L\u2080 (adimensional). 0,001 = 0,1% de alongamento.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-4\" hidden><label for=\"field-5\">For\u00e7a (F)<\/label><div class=\"term\">F<\/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=\"fora_f\" id=\"field-5\" aria-describedby=\"calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 2000\" min=\"0\" value=\"2000\"\/><div class=\"grid grid-cols-1\"><select id=\"field-6\" name=\"unidade\" aria-label=\"Unidade\"><option value=\"N\" selected>N<\/option><option value=\"kN\">kN<\/option><option value=\"lbf\">lbf<\/option><option value=\"kgf\">kgf<\/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-5\">For\u00e7a de tra\u00e7\u00e3o ou de compress\u00e3o aplicada ao corpo de prova.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-5\" hidden><label for=\"field-7\">\u00c1rea da sec\u00e7\u00e3o transversal (A)<\/label><div class=\"term\">A<\/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=\"rea_da_seco_transversal_a\" id=\"field-7\" aria-describedby=\"calc-info-7\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 100\" min=\"0\" value=\"100\"\/><div class=\"grid grid-cols-1\"><select id=\"field-8\" name=\"unidade\" aria-label=\"Unidade\"><option value=\"mm2\" selected>mm\u00b2<\/option><option value=\"cm2\">cm\u00b2<\/option><option value=\"m2\">m\u00b2<\/option><option value=\"in2\">in\u00b2<\/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-7\">Sec\u00e7\u00e3o transversal original do corpo de prova, perpendicular \u00e0 for\u00e7a. Uma barra de 10 mm \u00d7 10 mm tem A = 100 mm\u00b2.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-6\" hidden><label for=\"field-9\">Comprimento inicial (L\u2080)<\/label><div class=\"term\">L\u2080<\/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=\"comprimento_inicial_l\" id=\"field-9\" aria-describedby=\"calc-info-9\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 1000\" min=\"0\" value=\"1000\"\/><div class=\"grid grid-cols-1\"><select id=\"field-10\" name=\"unidade\" aria-label=\"Unidade\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><option value=\"ft\">ft<\/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-9\">Comprimento do corpo de prova antes da aplica\u00e7\u00e3o da for\u00e7a (comprimento de refer\u00eancia).<\/small><\/div><div class=\"field is-single has-term\" id=\"input-7\" hidden><label for=\"field-11\">Varia\u00e7\u00e3o de comprimento (\u0394L)<\/label><div class=\"term\">\u0394L<\/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=\"variao_de_comprimento_l\" id=\"field-11\" aria-describedby=\"calc-info-11\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 0.1\" min=\"0\" value=\"0.1\"\/><div class=\"grid grid-cols-1\"><select id=\"field-12\" name=\"unidade\" aria-label=\"Unidade\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><option value=\"ft\">ft<\/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-11\">Alongamento (positivo) ou compress\u00e3o do corpo de prova sob a a\u00e7\u00e3o da for\u00e7a. N\u00e3o pode ser zero.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalcular\" id=\"calculate-button\" data-post=\"863\">Calcular<\/button><button type=\"reset\">Redefinir<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calcular automaticamente<\/label><small>Se ativado, o resultado \u00e9 atualizado automaticamente quando voc\u00ea altera qualquer valor.<\/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> Resultados do M\u00f3dulo de Young<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>M\u00f3dulo de elasticidade de Young<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-2\"><span class=\"field-title\"><span>F\u00f3rmula, resumo tens\u00e3o\u2013deforma\u00e7\u00e3o e notas<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><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\">Copiar link para este c\u00e1lculo<\/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>Deixar um coment\u00e1rio<\/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=\"863\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">O que voc\u00ea achou deste resultado?<\/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\">Curti <\/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\">\u00datil <\/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\">Confuso <\/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\">Decepcionado <\/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\">Impreciso <\/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>F\u00f3rmula do M\u00f3dulo de Young<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">F\u00f3rmula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ E = \\frac{\\sigma}{\\varepsilon} = \\frac{F \/ A}{\\Delta L \/ L_0} = \\frac{F \\, L_0}{A \\, \\Delta L} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Onde:<\/strong><\/p>\n<ul>\n<li>\\(E\\) \u2014 m\u00f3dulo de Young, ou m\u00f3dulo de elasticidade (Pa)<\/li>\n<li>\\(\\sigma\\) \u2014 tens\u00e3o, a for\u00e7a por unidade de \u00e1rea \\(\\sigma = F\/A\\) (Pa)<\/li>\n<li>\\(\\varepsilon\\) \u2014 deforma\u00e7\u00e3o espec\u00edfica, a deforma\u00e7\u00e3o relativa \\(\\varepsilon = \\Delta L \/ L_0\\) (adimensional)<\/li>\n<li>\\(F\\) \u2014 for\u00e7a de tra\u00e7\u00e3o ou compress\u00e3o aplicada (N)<\/li>\n<li>\\(A\\) \u2014 \u00e1rea da sec\u00e7\u00e3o transversal original do corpo de prova (m\u00b2)<\/li>\n<li>\\(L_0\\) \u2014 comprimento original do corpo de prova (m)<\/li>\n<li>\\(\\Delta L\\) \u2014 varia\u00e7\u00e3o de comprimento sob a carga (m)<\/li>\n<\/ul>\n<p>A f\u00f3rmula exprime a lei de Hooke na sua forma de raz\u00e3o: o m\u00f3dulo de Young \u00e9 a raz\u00e3o constante entre a tens\u00e3o e a deforma\u00e7\u00e3o na regi\u00e3o el\u00e1stica, ou seja, o declive da parte retil\u00ednea do diagrama tens\u00e3o\u2013deforma\u00e7\u00e3o.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.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>M\u00f3dulo de Young \u2013 Exemplo de C\u00e1lculo<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemplo:<\/strong> uma barra de a\u00e7o \u00e9 carregada de modo que a tens\u00e3o seja de 200 MPa e ela se alongue 0,1% do seu comprimento original.<\/p>\n<p>\\[ \\varepsilon = 0.1\\% = 0.001 \\]<\/p>\n<p>\\[ E = \\frac{\\sigma}{\\varepsilon} = \\frac{200\\,000\\,000\\ \\text{Pa}}{0.001} = 200\\,000\\,000\\,000\\ \\text{Pa} \\]<\/p>\n<p><strong>Resultado:<\/strong> \\(E = 200\\ \\text{GPa}\\) \u2014 um valor t\u00edpico para o a\u00e7o estrutural.<\/p>\n<p>\n\r\n<\/section>\n<h2>Como Utilizar a Calculadora<\/h2>\n<ol>\n<li>Abra o menu <strong>M\u00e9todo de c\u00e1lculo<\/strong> e escolha <em>A partir da tens\u00e3o e da deforma\u00e7\u00e3o<\/em> se j\u00e1 conhece estes dois valores, ou <em>A partir da for\u00e7a, \u00e1rea e comprimentos<\/em> se quiser que a calculadora determine a tens\u00e3o e a deforma\u00e7\u00e3o por si.<\/li>\n<li>No primeiro m\u00e9todo, introduza a <strong>Tens\u00e3o (\u03c3)<\/strong> com a sua unidade (Pa, kPa, MPa, GPa ou psi) e a <strong>Deforma\u00e7\u00e3o (\u03b5)<\/strong> como fra\u00e7\u00e3o decimal \u2014 0,001 significa um alongamento de 0,1%.<\/li>\n<li>No segundo m\u00e9todo, introduza a <strong>For\u00e7a (F)<\/strong> (N, kN, lbf ou kgf), a <strong>\u00c1rea da sec\u00e7\u00e3o transversal (A)<\/strong> (mm\u00b2, cm\u00b2, m\u00b2 ou in\u00b2), o <strong>Comprimento original (L\u2080)<\/strong> e a <strong>Varia\u00e7\u00e3o de comprimento (\u0394L)<\/strong> em unidades de comprimento coerentes.<\/li>\n<li>Leia o resultado: o m\u00f3dulo de Young \u00e9 apresentado em GPa, MPa, Pa, psi, ksi e psf, juntamente com um resumo da tens\u00e3o e da deforma\u00e7\u00e3o utilizadas no c\u00e1lculo.<\/li>\n<li>Consulte a nota de interpreta\u00e7\u00e3o, que compara o seu resultado com valores t\u00edpicos de materiais comuns como o a\u00e7o, o alum\u00ednio, o bet\u00e3o e a borracha.<\/li>\n<\/ol>\n<h2>O Que \u00c9 o M\u00f3dulo de Young?<\/h2>\n<p>O m\u00f3dulo de Young (tamb\u00e9m chamado m\u00f3dulo de elasticidade) \u00e9 uma propriedade do material que descreve a sua rigidez \u2014 a sua resist\u00eancia a ser esticado ou comprimido. Define-se como a raz\u00e3o entre a tens\u00e3o e a deforma\u00e7\u00e3o na regi\u00e3o el\u00e1stica da deforma\u00e7\u00e3o, em que o material regressa \u00e0 sua forma original assim que a carga \u00e9 removida. Quanto maior o m\u00f3dulo, maior a for\u00e7a necess\u00e1ria para produzir a mesma deforma\u00e7\u00e3o.<\/p>\n<p>A tens\u00e3o \\(\\sigma = F\/A\\) mede a intensidade da carga \u2014 a for\u00e7a distribu\u00edda pela \u00e1rea da sec\u00e7\u00e3o transversal \u2014 enquanto a deforma\u00e7\u00e3o \\(\\varepsilon = \\Delta L \/ L_0\\) mede a resposta \u2014 o alongamento ou encurtamento em rela\u00e7\u00e3o ao comprimento original. Como a deforma\u00e7\u00e3o \u00e9 adimensional, o m\u00f3dulo de Young tem as mesmas unidades da tens\u00e3o, habitualmente gigapascal (GPa) na pr\u00e1tica m\u00e9trica ou libras por polegada quadrada (psi) no sistema imperial.<\/p>\n<p>O resultado indica o declive da parte linear do diagrama tens\u00e3o\u2013deforma\u00e7\u00e3o. Um valor de 200 GPa significa que uma barra de 1 metro do material se alonga apenas 5 micr\u00f3metros por cada megapascal de tens\u00e3o aplicada. Os valores t\u00edpicos variam entre cerca de 0,01 GPa para a borracha macia e cerca de 1100 GPa para o diamante; o a\u00e7o estrutural situa-se perto de 200 GPa, as ligas de alum\u00ednio em torno de 70 GPa e o bet\u00e3o em torno de 30 GPa. Conhecer o m\u00f3dulo de um material \u00e9 essencial na engenharia de estruturas para prever deforma\u00e7\u00f5es (flechas), na ci\u00eancia dos materiais para caracterizar novos materiais e em qualquer situa\u00e7\u00e3o em que a deforma\u00e7\u00e3o el\u00e1stica importa \u2014 de pontes a molas.<\/p>\n<h2>Hip\u00f3teses, Limita\u00e7\u00f5es e Dicas Pr\u00e1ticas<\/h2>\n<ul>\n<li><strong>Apenas na regi\u00e3o el\u00e1stica linear.<\/strong> A f\u00f3rmula \\(E = \\sigma\/\\varepsilon\\) s\u00f3 \u00e9 v\u00e1lida abaixo do limite de proporcionalidade, onde tens\u00e3o e deforma\u00e7\u00e3o s\u00e3o proporcionais. Al\u00e9m dele, o material escoa ou deforma-se de forma n\u00e3o linear e um \u00fanico m\u00f3dulo j\u00e1 n\u00e3o descreve o seu comportamento.<\/li>\n<li><strong>Material homog\u00e9neo e isotr\u00f3pico.<\/strong> O c\u00e1lculo assume que a rigidez \u00e9 a mesma em todos os pontos e em todas as dire\u00e7\u00f5es. A madeira, os comp\u00f3sitos e os materiais refor\u00e7ados com fibras s\u00e3o anisotr\u00f3picos \u2014 o seu m\u00f3dulo no sentido das fibras pode ser v\u00e1rias vezes superior ao do sentido perpendicular.<\/li>\n<li><strong>Deforma\u00e7\u00f5es pequenas.<\/strong> Utilize a \u00e1rea da sec\u00e7\u00e3o transversal original \\(A\\) e o comprimento original \\(L_0\\), e n\u00e3o os valores medidos sob carga. Para grandes deforma\u00e7\u00f5es, a defini\u00e7\u00e3o de tens\u00e3o e deforma\u00e7\u00e3o convencionais torna-se imprecisa.<\/li>\n<li><strong>Valores positivos.<\/strong> Introduza o m\u00f3dulo (valor absoluto) do alongamento ou do encurtamento; a calculadora rejeita valores nulos ou negativos. Na pr\u00e1tica, os m\u00f3dulos de tra\u00e7\u00e3o e de compress\u00e3o do mesmo material costumam ser pr\u00f3ximos, mas alguns materiais (p. ex., o bet\u00e3o) comportam-se de forma muito diferente \u00e0 compress\u00e3o e \u00e0 tra\u00e7\u00e3o.<\/li>\n<li><strong>Efeitos de temperatura e de velocidade de carregamento.<\/strong> O m\u00f3dulo de Young diminui com a temperatura e pode depender da rapidez de aplica\u00e7\u00e3o da carga, especialmente nos pol\u00edmeros. Os valores das tabelas de refer\u00eancia s\u00e3o normalmente medidos \u00e0 temperatura ambiente sob carregamento lento.<\/li>\n<\/ul>\n<h2>Valores T\u00edpicos do M\u00f3dulo de Young<\/h2>\n<table>\n<thead>\n<tr>\n<th>Material<\/th>\n<th>M\u00f3dulo de Young (GPa)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Borracha (natural)<\/td>\n<td>0,01 \u2013 0,1<\/td>\n<\/tr>\n<tr>\n<td>Nylon<\/td>\n<td>2 \u2013 4<\/td>\n<\/tr>\n<tr>\n<td>Madeira (no sentido das fibras)<\/td>\n<td>9 \u2013 16<\/td>\n<\/tr>\n<tr>\n<td>Bet\u00e3o<\/td>\n<td>30 \u2013 40<\/td>\n<\/tr>\n<tr>\n<td>Liga de alum\u00ednio<\/td>\n<td>69 \u2013 79<\/td>\n<\/tr>\n<tr>\n<td>Laton<\/td>\n<td>100 \u2013 110<\/td>\n<\/tr>\n<tr>\n<td>Cobre<\/td>\n<td>110 \u2013 130<\/td>\n<\/tr>\n<tr>\n<td>A\u00e7o (estrutural)<\/td>\n<td>190 \u2013 210<\/td>\n<\/tr>\n<tr>\n<td>Tungst\u00e9nio<\/td>\n<td>400 \u2013 411<\/td>\n<\/tr>\n<tr>\n<td>Diamante<\/td>\n<td>1050 \u2013 1200<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Se estiver a analisar as for\u00e7as numa estrutura carregada, tamb\u00e9m poder\u00e1 achar \u00fateis a nossa <a href=\"https:\/\/wpcalc.com\/pt\/fisica\/calculadora-tencao-em-cordas\/\">calculadora de tens\u00e3o em cabos<\/a> e a <a href=\"https:\/\/wpcalc.com\/pt\/fisica\/calculadora-forca-atrito\/\">calculadora de for\u00e7a de atrito<\/a>. Para a energia armazenada num corpo el\u00e1stico deformado, consulte a <a href=\"https:\/\/wpcalc.com\/pt\/fisica\/calculadora-energia-potencial-elastica\/\">calculadora de energia potencial el\u00e1stica<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calcule o m\u00f3dulo de Young (elasticidade) a partir da tens\u00e3o e da deforma\u00e7\u00e3o, ou a partir da for\u00e7a, da \u00e1rea da sec\u00e7\u00e3o transversal e da varia\u00e7\u00e3o de comprimento. Resultados em GPa, MPa, Pa e psi.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[10],"tags":[55],"class_list":["post-863","post","type-post","status-publish","format-standard","hentry","category-fisica","tag-mecanica"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/863","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/comments?post=863"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/863\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/media?parent=863"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/categories?post=863"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/tags?post=863"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}