{"id":2669,"date":"2026-10-15T02:28:28","date_gmt":"2026-10-15T01:28:28","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/inzynieria\/belka-wolnopodparta-obciazenie-w-srodku\/"},"modified":"2026-10-15T02:28:28","modified_gmt":"2026-10-15T01:28:28","slug":"belka-wolnopodparta-obciazenie-w-srodku","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/inzynieria\/belka-wolnopodparta-obciazenie-w-srodku\/","title":{"rendered":"Belka swobodnie podparta z si\u0142\u0105 skupion\u0105 w \u015brodku"},"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\">Reakcje podp\u00f3r, moment zginaj\u0105cy i ugi\u0119cie belki<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2669\" data-favorite-title=\"Belka swobodnie podparta z obci\u0105\u017ceniem w \u015brodku\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/inzynieria\/belka-wolnopodparta-obciazenie-w-srodku\/\" data-favorite-excerpt=\"Oblicz reakcje podp\u00f3r, maksymalny moment zginaj\u0105cy i ugi\u0119cie w \u015brodku belki swobodnie podpartej, obci\u0105\u017conej si\u0142\u0105 skupion\u0105 w \u015brodku rozpi\u0119to\u015bci. Wpisz rozpi\u0119to\u015b\u0107, obci\u0105\u017cenie, modu\u0142 Younga i moment bezw\u0142adno\u015bci.\" aria-label=\"Dodaj do ulubionych\" data-favorite-icon=\"icon icon-civil\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/pl\/inzynieria\/belka-wolnopodparta-obciazenie-w-srodku\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2669\" data-post=\"2669\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Rozpi\u0119to\u015b\u0107 (L)<\/label><div class=\"term\">L<\/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=\"rozpito_l\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0.01\" value=\"4\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"m\" selected>m<\/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-1\">Odleg\u0142o\u015b\u0107 mi\u0119dzy dwiema podporami<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-3\">Si\u0142a skupiona (P)<\/label><div class=\"term\">P<\/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=\"sia_skupiona_p\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"1.0E-6\" value=\"10\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"kN\" selected>kN<\/option><option value=\"N\">N<\/option><option value=\"lbf\">lbf<\/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-3\">Przy\u0142o\u017cona w \u015brodku rozpi\u0119to\u015bci<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-5\">Modu\u0142 Younga (E)<\/label><div class=\"term\">E<\/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=\"modu_younga_e\" id=\"field-5\" aria-describedby=\"calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"1.0E-6\" value=\"200\"\/><div class=\"grid grid-cols-1\"><select id=\"field-6\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"GPa\" selected>GPa<\/option><option value=\"MPa\">MPa<\/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-5\">Stal \u2248 200 GPa (29 000 ksi), aluminium \u2248 69 GPa, drewno \u2248 11 GPa<\/small><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-7\">Moment bezw\u0142adno\u015bci przekroju (I)<\/label><div class=\"term\">I<\/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=\"moment_bezwadnoci_przekroju_i\" id=\"field-7\" aria-describedby=\"calc-info-7\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"1.0E-6\" value=\"1000\"\/><div class=\"grid grid-cols-1\"><select id=\"field-8\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"cm4\" selected>cm\u2074<\/option><option value=\"mm4\">mm\u2074<\/option><option value=\"in4\">in\u2074<\/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\">Mom\u0119t bezw\u0142adno\u015bci pola przekroju belki wzgl\u0119dem osi zginania<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"2669\">Oblicz<\/button><button type=\"reset\">Wyczy\u015b\u0107<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Obliczaj automatycznie<\/label><small>Po w\u0142\u0105czeniu wynik aktualizuje si\u0119 automatycznie po zmianie dowolnej warto\u015bci.<\/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> Wyniki dla belki<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Reakcje podp\u00f3r (ka\u017cda)<\/span><span class=\"result-term\">R<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Maksymalny moment zginaj\u0105cy<\/span><span class=\"result-term\">M<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>Maksymalne ugi\u0119cie (w \u015brodku rozpi\u0119to\u015bci)<\/span><span class=\"result-term\">\u03b4<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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\">Kopiuj link do tego obliczenia<\/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>Zostaw komentarz<\/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=\"2669\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Jak oceniasz ten wynik?<\/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\">Lubi\u0119 to <\/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\">Pomocne <\/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\">Niejasne <\/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\">Rozczarowanie <\/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\">Niedok\u0142adne <\/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>Wzory dla belki swobodnie podpartej z obci\u0105\u017ceniem w \u015brodku<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Wz\u00f3r<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ R = \\frac{P}{2} \\qquad M_{max} = \\frac{P \\, L}{4} \\qquad \\delta_{max} = \\frac{P \\, L^3}{48 \\, E \\, I} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Oznaczenia:<\/strong><\/p>\n<ul>\n<li>\\(R\\) \u2014 reakcja na ka\u017cdej z podp\u00f3r (N)<\/li>\n<li>\\(P\\) \u2014 si\u0142a skupiona przy\u0142o\u017cona w \u015brodku rozpi\u0119to\u015bci (N)<\/li>\n<li>\\(L\\) \u2014 rozpi\u0119to\u015b\u0107 mi\u0119dzy podporami (m)<\/li>\n<li>\\(M_{max}\\) \u2014 maksymalny moment zginaj\u0105cy, wyst\u0119puj\u0105cy pod si\u0142\u0105 (N\u00b7m)<\/li>\n<li>\\(\\delta_{max}\\) \u2014 maksymalne ugi\u0119cie w \u015brodku rozpi\u0119to\u015bci (m)<\/li>\n<li>\\(E\\) \u2014 modu\u0142 Younga materia\u0142u belki (Pa)<\/li>\n<li>\\(I\\) \u2014 moment bezw\u0142adno\u015bci przekroju (m\u2074)<\/li>\n<\/ul>\n<p>&lt;Poniewa\u017c si\u0142a dzia\u0142a dok\u0142adnie w po\u0142owie odleg\u0142o\u015bci mi\u0119dzy dwiema identycznymi podporami, symetria rozdziela j\u0105 r\u00f3wno pomi\u0119dzy nie. Moment zginaj\u0105cy i ugi\u0119cie osi\u0105gaj\u0105 maksimum w \u015brodku rozpi\u0119to\u015bci, tam gdzie przy\u0142o\u017cona jest si\u0142a.<\/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>Belka swobodnie podparta z obci\u0105\u017ceniem w \u015brodku \u2013 przyk\u0142ad obliczeniowy<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> stalowa belka o rozpi\u0119to\u015bci 4 m przenosi si\u0142\u0119 skupion\u0105 10 kN przy\u0142o\u017con\u0105 w \u015brodku. Belka ma modu\u0142 Younga \\(E = 200\\) GPa i moment bezw\u0142adno\u015bci \\(I = 1000\\) cm\u2074 = \\(1 \\times 10^{-5}\\) m\u2074.<\/p>\n<p>\\[ R = \\frac{P}{2} = \\frac{10\\,000\\ \\text{N}}{2} = 5\\ \\text{kN} \\]<\/p>\n<p>\\[ M_{max} = \\frac{P L}{4} = \\frac{10\\,000 \\times 4}{4} = 10\\ \\text{kN\u00b7m} \\]<\/p>\n<p>\\[ \\delta_{max} = \\frac{P L^3}{48 E I} = \\frac{10\\,000 \\times 4^3}{48 \\times 200 \\times 10^9 \\times 1 \\times 10^{-5}} \\approx 6.67 \\times 10^{-3}\\ \\text{m} \\]<\/p>\n<p><strong>Wynik:<\/strong> ka\u017cda podpora przenosi 5 kN, maksymalny moment zginaj\u0105cy wynosi 10 kN\u00b7m w \u015brodku rozpi\u0119to\u015bci, a belka ugina si\u0119 o oko\u0142o 6,67 mm (ok. 0,26 cala) w miejscu przy\u0142o\u017cenia si\u0142y.<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wpisz <strong>Rozpi\u0119to\u015b\u0107 (L)<\/strong> \u2014 odleg\u0142o\u015b\u0107 mi\u0119dzy dwiema podporami \u2014 i wybierz jednostk\u0119 (m lub ft).<\/li>\n<li>Wpisz <strong>Si\u0142\u0119 skupion\u0105 (P)<\/strong> przy\u0142o\u017con\u0105 w \u015brodku rozpi\u0119to\u015bci i wybierz jej jednostk\u0119 (kN, N lub lbf).<\/li>\n<li>Wpisz <strong>Modu\u0142 Younga (E)<\/strong> materia\u0142u. Stal ma oko\u0142o 200 GPa, aluminium oko\u0142o 69 GPa, drewno oko\u0142o 11 GPa.<\/li>\n<li>Wpisz <strong>Moment bezw\u0142adno\u015bci (I)<\/strong> przekroju belki wzgl\u0119dem osi zginania, w cm\u2074, mm\u2074 lub in\u2074.<\/li>\n<li>Odczytaj wyniki: reakcje podp\u00f3r, maksymalny moment zginaj\u0105cy oraz maksymalne ugi\u0119cie w \u015brodku rozpi\u0119to\u015bci.<\/li>\n<\/ol>\n<h2>Czym jest belka swobodnie podparta z obci\u0105\u017ceniem w \u015brodku?<\/h2>\n<p>Belka swobodnie podparta opiera si\u0119 na dw\u00f3ch podporach \u2014 zazwyczaj przegubowej na jednym ko\u0144cu i ruchomej na drugim. \u017badna z podp\u00f3r nie utrudnia obrotu, wi\u0119c belka mo\u017ce swobodnie si\u0119 zgina\u0107, a jej ko\u0144ce mog\u0105 si\u0119 obraca\u0107. To jedno z najcz\u0119stszych uproszcze\u0144 w in\u017cynierii budowlanej i mechanicznej: dobrze odzwierciedla legary pod\u0142ogowe oparte na dw\u00f3ch \u015bcianach, wa\u0142y maszyn le\u017c\u0105ce na dw\u00f3ch \u0142o\u017cyskach oraz wiele element\u00f3w maszyn i ram.<\/p>\n<p>Gdy pojedyncza si\u0142a skupiona \\(P\\) dzia\u0142a dok\u0142adnie w \u015brodku rozpi\u0119to\u015bci, zadanie staje si\u0119 symetryczne. Ka\u017cda podpora przenosi dok\u0142adnie po\u0142ow\u0119 obci\u0105\u017cenia, wi\u0119c \\(R = P\/2\\). Moment zginaj\u0105cy ro\u015bnie liniowo od zera na ka\u017cdej podporze do maksimum \\(M_{max} = PL\/4\\) bezpo\u015brednio pod si\u0142\u0105 \u2014 to miejsce najwi\u0119kszych napr\u0119\u017ce\u0144 w belce, w kt\u00f3rym zacz\u0119\u0142oby si\u0119 zniszczenie lub uplastycznienie materia\u0142u.<\/p>\n<p>Ugi\u0119cie zale\u017cy od trzeciej pot\u0119gi rozpi\u0119to\u015bci: \\(\\delta_{max} = PL^3\/(48EI)\\). Podwojenie rozpi\u0119to\u015bci zwi\u0119ksza ugi\u0119cie o\u015bmiokrotnie \u2014 dlatego tak trudno zachowa\u0107 sztywno\u015b\u0107 przy du\u017cych rozpi\u0119to\u015bciach. Iloczyn \\(EI\\), zwany <em>sztywno\u015bci\u0105 na zginanie<\/em>, opisuje op\u00f3r belki przeciwko zginaniu: sztywniejszy materia\u0142 (wy\u017csze \\(E\\)) lub wy\u017cszy przekr\u00f3j (wi\u0119ksze \\(I\\)) zmniejszaj\u0105 ugi\u0119cie. Je\u015bli chcesz przeanalizowa\u0107 sztywno\u015b\u0107 materia\u0142u osobno, <a href=\"https:\/\/wpcalc.com\/pl\/inzynieria\/czestotliwosc-rezonansowa-sprezyny-modul\/\">kalkulator modu\u0142u Younga<\/a> obejmuje pokrewne obliczenia w\u0142a\u015bciwo\u015bci materia\u0142owych.<\/p>\n<p>Interpretuj\u0105c wynik, por\u00f3wnaj ugi\u0119cie z dopuszczalnym limitem dla twojej konstrukcji. Wiele norm budowlanych ogranicza ugi\u0119cie od obci\u0105\u017ce\u0144 u\u017cytkowych do \\(L\/360\\) lub \\(L\/240\\) rozpi\u0119to\u015bci; dla rozpi\u0119to\u015bci 4 m z powy\u017cszego przyk\u0142adu \\(L\/360 \\approx 11.1\\) mm, wi\u0119c ugi\u0119cie 6,67 mm spe\u0142nia ten warunek.<\/p>\n<h2>Za\u0142o\u017cenia i ograniczenia<\/h2>\n<ul>\n<li>Si\u0142a musi dzia\u0142a\u0107 <strong>dok\u0142adnie w \u015brodku rozpi\u0119to\u015bci<\/strong>. Dla si\u0142y przy\u0142o\u017conej w dowolnym innym miejscu wzory si\u0119 zmieniaj\u0105 \u2014 reakcje wynosz\u0105 wtedy \\(R_1 = Pb\/L\\), \\(R_2 = Pa\/L\\).<\/li>\n<li>Belka jest pryzmatyczna (o sta\u0142ym przekroju), jednorodna i liniowo spr\u0119\u017cysta \u2014 ugi\u0119cia s\u0105 ma\u0142e w por\u00f3wnaniu z rozpi\u0119to\u015bci\u0105.<\/li>\n<li>Ci\u0119\u017car w\u0142asny belki jest pomijany. Je\u015bli jest on istotny wobec przy\u0142o\u017conego obci\u0105\u017cenia, dodaj go jako obci\u0105\u017cenie r\u00f3wnomiernie roz\u0142o\u017cone w osobnej analizie.<\/li>\n<li>Podpory s\u0105 idealne: przenosz\u0105 tylko obci\u0105\u017cenie pionowe i nie stawiaj\u0105 \u017cadnego oporu obrotowi ani nie stanowi\u0105 utwierdzenia.<\/li>\n<li>Uwzgl\u0119dniane s\u0105 wy\u0142\u0105cznie obci\u0105\u017cenia pionowe; si\u0142y poziome, skr\u0119canie i efekty dynamiczne wykraczaj\u0105 poza zakres tych wzor\u00f3w.<\/li>\n<\/ul>\n<p>W sytuacjach dynamicznych \u2014 na przyk\u0142ad gdy masa oparta na belce pe\u0142ni rol\u0119 spr\u0119\u017cyny \u2014 statyczny wynik z tego kalkulatora jest tylko punktem wyj\u015bcia. Do analizy drga\u0144 zajrzyj do <a href=\"https:\/\/wpcalc.com\/pl\/inzynieria\/masa-sprezyny\/\">kalkulatora uk\u0142adu masa\u2013spr\u0119\u017cyna<\/a> oraz <a href=\"https:\/\/wpcalc.com\/pl\/inzynieria\/czestotliwosc-rezonansowa-sprezyny\/\">kalkulatora cz\u0119stotliwo\u015bci rezonansowej spr\u0119\u017cyny<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Oblicz reakcje podp\u00f3r, maksymalny moment zginaj\u0105cy i ugi\u0119cie w \u015brodku belki swobodnie podpartej, obci\u0105\u017conej si\u0142\u0105 skupion\u0105 w \u015brodku rozpi\u0119to\u015bci. Wpisz rozpi\u0119to\u015b\u0107, obci\u0105\u017cenie, modu\u0142 Younga i moment bezw\u0142adno\u015bci.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[84],"class_list":["post-2669","post","type-post","status-publish","format-standard","hentry","category-inzynieria","tag-belki"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2669","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/comments?post=2669"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2669\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=2669"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=2669"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=2669"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}