{"id":59950,"date":"2016-12-29T10:34:32","date_gmt":"2016-12-29T08:34:32","guid":{"rendered":"https:\/\/wpcalc.com\/?p=59950"},"modified":"2026-09-30T18:05:47","modified_gmt":"2026-09-30T15:05:47","slug":"cantilever-beam-with-load-at-free-end","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/engineering\/cantilever-beam-with-load-at-free-end\/","title":{"rendered":"Cantilever Beam Load at Free End Calculator"},"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\">Free End Load Cantilever Analysis<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"59950\" data-favorite-title=\"Cantilever Beam with Load at Free End\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/engineering\/cantilever-beam-with-load-at-free-end\/\" data-favorite-excerpt=\"Calculate the slope at the free end and the deflection at any point along a cantilever beam with a point load applied at its tip. Enter the load, elastic modulus, moment of inertia, beam length, and the position of interest.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-civil\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/engineering\/cantilever-beam-with-load-at-free-end\/\" method=\"POST\" class=\"calculator\" id=\"calculator-59950\" data-post=\"59950\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Applied Load<\/label><div class=\"term\">P<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"applied_load\" id=\"field-1\" aria-describedby=\"calc-unit-1\" step=\"any\" value=\"500\"\/><div class=\"unit\" id=\"calc-unit-1\">lbs<\/div><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Elastic Modulus<\/label><div class=\"term\">E<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"elastic_modulus\" id=\"field-2\" aria-describedby=\"calc-unit-2\" step=\"any\" value=\"29000000\"\/><div class=\"unit\" id=\"calc-unit-2\">psi<\/div><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Moment of Inertia<\/label><div class=\"term\">I<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"moment_of_inertia\" id=\"field-3\" aria-describedby=\"calc-unit-3\" step=\"any\" value=\"150\"\/><div class=\"unit\" id=\"calc-unit-3\">in\u2074<\/div><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">Beam Length<\/label><div class=\"term\">L<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"beam_length\" id=\"field-4\" aria-describedby=\"calc-unit-4\" step=\"any\" value=\"120\"\/><div class=\"unit\" id=\"calc-unit-4\">in<\/div><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">Deflection Point<\/label><div class=\"term\">x<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"deflection_point\" id=\"field-5\" aria-describedby=\"calc-unit-5\" step=\"any\" value=\"100\"\/><div class=\"unit\" id=\"calc-unit-5\">in<\/div><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"59950\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Slope at Free End<\/span><span class=\"result-term\">\u03b8<\/span><\/span><span class=\"field-value\" id=\"result-1\" data-unit=\"rad\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Deflection<\/span><span class=\"result-term\">\u03b4<\/span><\/span><span class=\"field-value\" id=\"result-2\" data-unit=\"in\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"59950\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" 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data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Cantilever Beam Load at Free End Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\theta = \\frac{P \\, L^2}{2 E I} \\qquad \\delta_x = \\frac{P \\, x^2 (3L - x)}{6 E I} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(P\\) \u2014 applied load at the free end (lbs or N)<\/li>\n<li>\\(L\\) \u2014 length of the beam from the fixed support to the free end (in or m)<\/li>\n<li>\\(E\\) \u2014 elastic modulus of the material (psi or Pa)<\/li>\n<li>\\(I\\) \u2014 moment of inertia of the beam cross-section (in\u2074 or m\u2074)<\/li>\n<li>\\(x\\) \u2014 distance from the fixed end to the point where deflection is evaluated<\/li>\n<li>\\(\\theta\\) \u2014 slope (rotation) at the free end, in radians<\/li>\n<li>\\(\\delta_x\\) \u2014 vertical deflection at position \\(x\\)<\/li>\n<\/ul>\n<p>The first formula gives the rotation of the beam at its tip; the second gives the deflection at any distance \\(x\\) from the fixed support. Setting \\(x = L\\) in the second formula yields the familiar maximum tip deflection \\(\\delta_{max} = \\frac{P L^3}{3 E I}\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Cantilever Beam Load at Free End \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a steel cantilever beam, 120 in long, carries a 500 lb load at its free end. The steel has an elastic modulus of 29,000,000 psi and the cross-section has a moment of inertia of 150 in\u2074. Find the slope at the free end and the deflection 100 in from the fixed support.<\/p>\n<p>\\[ \\theta = \\frac{P L^2}{2 E I} = \\frac{500 \\times 120^2}{2 \\times 29{,}000{,}000 \\times 150} = \\frac{7{,}200{,}000}{8{,}700{,}000{,}000} = 0.008276 \\text{ rad} \\]<\/p>\n<p>\\[ \\delta = \\frac{P x^2 (3L - x)}{6 E I} = \\frac{500 \\times 100^2 \\times (3 \\times 120 - 100)}{6 \\times 29{,}000{,}000 \\times 150} = \\frac{1{,}300{,}000{,}000}{26{,}100{,}000{,}000} = 0.0498 \\text{ in} \\]<\/p>\n<p><strong>Result:<\/strong> the slope at the free end is 0.0083 rad (about 0.47\u00b0), and the beam deflects 0.0498 in at 100 in from the wall. At the very tip (\\(x = L\\)) the deflection reaches its maximum of 0.0662 in.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the <strong>Applied Load<\/strong> \\(P\\) in lbs \u2014 the force acting at the free end of the beam.<\/li>\n<li>Enter the <strong>Elastic Modulus<\/strong> \\(E\\) in psi. For steel use about 29,000,000 psi; for aluminum about 10,000,000 psi; for wood roughly 1,000,000\u20131,600,000 psi depending on the species.<\/li>\n<li>Enter the <strong>Moment of Inertia<\/strong> \\(I\\) in in\u2074 for the beam cross-section. For a rectangular section, \\(I = \\frac{b h^3}{12}\\), where \\(h\\) is the dimension in the direction of bending.<\/li>\n<li>Enter the <strong>Beam Length<\/strong> \\(L\\) in inches, measured from the fixed support to the point where the load is applied.<\/li>\n<li>Enter the <strong>Deflection Point<\/strong> \\(x\\) in inches \u2014 the distance from the fixed end where you want to know the deflection. Use \\(x = L\\) for the maximum deflection at the tip.<\/li>\n<li>Read the <strong>Slope at Free End<\/strong> (in radians) and the <strong>Deflection<\/strong> at your chosen point (in inches).<\/li>\n<\/ol>\n<h2>What Is a Cantilever Beam with a Tip Load?<\/h2>\n<p>A cantilever beam is fixed rigidly at one end \u2014 built into a wall, column, or foundation \u2014 and completely free at the other. When a single concentrated load \\(P\\) acts at the free end, the beam bends downward, with zero deflection and zero slope at the fixed end and the largest movement at the tip. This is one of the most fundamental cases in structural mechanics, and it appears everywhere in practice: balcony slabs, aircraft wings, diving boards, shelf brackets, sign posts, and robotic arms all behave essentially as cantilevers.<\/p>\n<p>The calculator is based on the standard Euler\u2013Bernoulli beam theory solution for this loading case. The deflected shape is a smooth cubic curve: nothing moves at the wall, and the deflection grows rapidly toward the free end. Because the deflection formula contains \\(x^2\\) and \\(x^3\\)-type terms, doubling the load or the length has a dramatic effect \u2014 deflection scales linearly with \\(P\\) but with the <em>cube<\/em> of the length. A beam twice as long deflects eight times as much under the same tip load.<\/p>\n<p>The <strong>slope<\/strong> result tells you how much the beam rotates at the tip, in radians. Small values are normal in stiff structures; if you need degrees, multiply the radian value by \\(\\frac{180}{\\pi} \\approx 57.3\\). The <strong>deflection<\/strong> result tells you how far the beam sags at your chosen point. Most design codes limit deflection to a fraction of the span \u2014 for example \\(L\/240\\) or \\(L\/360\\) for live load in building design. For the example beam above, \\(L\/240 = 0.5\\) in, so a tip deflection of 0.066 in would easily pass such a limit.<\/p>\n<h2>Assumptions and Limitations<\/h2>\n<ul>\n<li><strong>Small deflections:<\/strong> the formulas assume the slope is small (typically well under 0.1 rad) and that the load is applied vertically. For large rotations the results become inaccurate.<\/li>\n<li><strong>Linear elastic material:<\/strong> the material must stay below its yield stress, and \\(E\\) must be constant along the beam.<\/li>\n<li><strong>Constant cross-section:<\/strong> \\(I\\) is assumed uniform over the full length. A tapered or stepped beam needs a more detailed analysis.<\/li>\n<li><strong>Rigid fixed support:<\/strong> the wall or support is assumed perfectly rigid. In reality, connection flexibility or rotation at the base increases the actual deflection.<\/li>\n<li><strong>Load at the very tip only:<\/strong> if the load is distributed, applied at an intermediate point, or combined with other loads, use superposition carefully or a different loading case. Self-weight of the beam is not included unless you add it as an equivalent load.<\/li>\n<li><strong>Deflection only:<\/strong> this tool does not compute bending moment or shear. For a tip load the maximum bending moment is \\(M = P L\\) at the fixed end \u2014 always check stresses separately before finalizing a design.<\/li>\n<\/ul>\n<p>When sizing real members, deflection is only one check among several. You may also want to verify load capacity limits, for example with the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/maximum-floor-live-load-capacity\/\">Maximum Floor Live Load Capacity Calculator<\/a>, or check a supported slab with the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/concrete-slab-maximum-wall-load\/\">Concrete Slab Maximum Wall Load Calculator<\/a>. If your cantilever is part of a roof structure, the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/rafter-length\/\">Rafter Length Calculator<\/a> can help you lay out the supporting members.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the slope at the free end and the deflection at any point along a cantilever beam with a point load applied at its tip. Enter the load, elastic modulus, moment of inertia, beam length, and the position of interest.<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[4],"tags":[97],"class_list":["post-59950","post","type-post","status-publish","format-standard","hentry","category-engineering","tag-beams"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/59950","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=59950"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/59950\/revisions"}],"predecessor-version":[{"id":65946,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/59950\/revisions\/65946"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=59950"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=59950"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=59950"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}