{"id":59952,"date":"2016-12-29T10:41:23","date_gmt":"2016-12-29T08:41:23","guid":{"rendered":"https:\/\/wpcalc.com\/?p=59952"},"modified":"2026-09-30T18:41:13","modified_gmt":"2026-09-30T15:41:13","slug":"cantilever-beam-with-uniform-load","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/engineering\/cantilever-beam-with-uniform-load\/","title":{"rendered":"Uniform Load on Cantilever Beam 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\">Deflection of Beam with Uniform Load<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"59952\" data-favorite-title=\"Cantilever Beam with Uniform Load\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/engineering\/cantilever-beam-with-uniform-load\/\" data-favorite-excerpt=\"Calculates the slope at the free end and the deflection at any point along a cantilever beam carrying a uniform load. Enter the applied load, elastic modulus, moment of inertia, beam length and position.\" 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-uniform-load\/\" method=\"POST\" class=\"calculator\" id=\"calculator-59952\" data-post=\"59952\"><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\">Load Position<\/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=\"load_position\" 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=\"59952\">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=\"59952\" 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 with Uniform Load 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\\[ P_0 = \\frac{P \\cdot L}{L - x} \\\\ \\theta = \\frac{P_0 \\cdot L^3}{6 \\cdot E \\cdot I} \\\\ \\delta = \\frac{P_0 \\cdot x^2 \\cdot (x^3 + 6L^2 - 4Lx)}{24 \\cdot E \\cdot I} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(P\\) \u2014 total applied load (lbs)<\/li>\n<li>\\(E\\) \u2014 elastic modulus of the beam material (psi)<\/li>\n<li>\\(I\\) \u2014 moment of inertia of the cross-section (in\u2074)<\/li>\n<li>\\(L\\) \u2014 length of the cantilever, from the fixed end to the free end (in)<\/li>\n<li>\\(x\\) \u2014 position along the beam, measured from the fixed end, where the deflection is evaluated (in)<\/li>\n<li>\\(P_0\\) \u2014 equivalent uniform load derived from \\(P\\), \\(L\\) and \\(x\\)<\/li>\n<\/ul>\n<p>The calculator first converts the applied load into an equivalent uniformly distributed load \\(P_0\\), then applies the standard cantilever expressions to find the slope at the free end \\(\\theta\\) and the deflection \\(\\delta\\) at the chosen position \\(x\\).<\/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 with Uniform Load \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a steel cantilever beam with \\(E = 29{,}000{,}000\\) psi and \\(I = 150\\) in\u2074 is 120 in long. A total load of 500 lbs is applied, and the deflection is evaluated at \\(x = 100\\) in from the fixed end.<\/p>\n<p>\\[ P_0 = \\frac{500 \\cdot 120}{120 - 100} = \\frac{60{,}000}{20} = 3000 \\text{ lbs} \\]<\/p>\n<p>\\[ \\theta = \\frac{3000 \\cdot 120^3}{6 \\cdot 29{,}000{,}000 \\cdot 150} = \\frac{5.184 \\times 10^9}{2.61 \\times 10^{10}} \\approx 0.1986 \\text{ rad} \\]<\/p>\n<p>\\[ \\delta = \\frac{3000 \\cdot 100^2 \\cdot (100^3 + 6 \\cdot 120^2 - 4 \\cdot 120 \\cdot 100)}{24 \\cdot 29{,}000{,}000 \\cdot 150} \\approx 298.39 \\text{ in} \\]<\/p>\n<p><strong>Result:<\/strong> the slope at the free end is approximately 0.1986 rad and the deflection at 100 in from the fixed end is approximately 298.39 in. A deflection of this magnitude is far beyond typical serviceability limits (such as \\(L\/240\\) or \\(L\/360\\)), which shows immediately that the beam is much too flexible for this loading.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the total <strong>Applied Load<\/strong> (\\(P\\)) in pounds acting on the beam.<\/li>\n<li>Enter the <strong>Elastic Modulus<\/strong> (\\(E\\)) in psi. For structural steel, use 29,000,000 psi; see the reference table below for other materials.<\/li>\n<li>Enter the <strong>Moment of Inertia<\/strong> (\\(I\\)) of the beam cross-section in in\u2074. This value comes from the shape's properties (for a rectangle, \\(I = bh^3\/12\\)).<\/li>\n<li>Enter the <strong>Beam Length<\/strong> (\\(L\\)) in inches, measured from the fixed end to the free end.<\/li>\n<li>Enter the <strong>Load Position<\/strong> (\\(x\\)) in inches \u2014 the distance from the fixed end at which you want the deflection. This value must be smaller than the beam length.<\/li>\n<li>Read the results: the <strong>Slope at Free End<\/strong> in radians and the <strong>Deflection<\/strong> in inches at the selected position.<\/li>\n<\/ol>\n<h2>About Cantilever Beams with Uniform Loads<\/h2>\n<p>A cantilever beam is fixed at one end and completely free at the other, so all support reactions \u2014 moment, shear and deflection resistance \u2014 must come from the single fixed end. Balconies, canopies, overhanging floor joists, brackets and pergola arms are all typical cantilever structures.<\/p>\n<p>A <em>uniform load<\/em> is spread continuously along the beam rather than concentrated at one point. Common examples include the self-weight of the beam itself, a masonry wall built on top of an overhang, accumulated snow on a canopy, or soil pressure behind a retaining bracket. Because the load is distributed, the internal shear and moment build up gradually from the free end toward the fixed end, where both reach their maximum values.<\/p>\n<p>The two results tell you how the beam deforms. The <strong>slope<\/strong> \\(\\theta\\) is the rotation of the beam at the free end in radians; small values (a few thousandths of a radian) are normal in service. The <strong>deflection<\/strong> \\(\\delta\\) is the vertical displacement at the position you selected. Deflection is extremely sensitive to length \u2014 it grows roughly with the fourth power of the span \u2014 and to the stiffness \\(EI\\). Increasing the beam depth is the most effective fix, since \\(I\\) grows with the cube of the height: doubling the depth of a rectangular beam multiplies its stiffness by eight.<\/p>\n<p>To judge whether a result is acceptable, compare the deflection with common code limits such as \\(L\/240\\) for total load or \\(L\/360\\) for live load alone. If you are sizing a whole floor or roof system rather than a single member, the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/maximum-floor-live-load-capacity\/\">Maximum Floor Live Load Capacity<\/a> calculator and the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/rafter-length\/\">Rafter Length<\/a> calculator may also be useful.<\/p>\n<h2>Assumptions, Limitations and Practical Tips<\/h2>\n<ul>\n<li>The beam is made of a linear-elastic material, has a constant cross-section, and deflects only slightly (small-deflection theory).<\/li>\n<li>The fixed end is perfectly rigid; in reality, connection flexibility and support rotation can noticeably increase real deflections.<\/li>\n<li>Load Position \\(x\\) must be less than Beam Length \\(L\\). As \\(x\\) approaches \\(L\\), the equivalent load \\(P_0\\) grows very rapidly and results lose practical meaning.<\/li>\n<li>Units must be consistent: load in lbs, dimensions in inches, modulus in psi. If you work in metric units (N, mm, MPa), the same formulas apply but every input must be converted.<\/li>\n<li>This tool checks <em>serviceability<\/em> (deflection and slope) only. Bending and shear stresses must be verified separately against the material's allowable values before the design is considered safe.<\/li>\n<li>Only vertical loads are considered; lateral loads, torsion, dynamics and vibration are outside the scope of this calculation.<\/li>\n<\/ul>\n<p>Typical elastic modulus values for common materials:<\/p>\n<table>\n<thead>\n<tr>\n<th>Material<\/th>\n<th>Elastic Modulus \\(E\\) (psi)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Structural steel<\/td>\n<td>29,000,000<\/td>\n<\/tr>\n<tr>\n<td>Aluminum<\/td>\n<td>10,000,000<\/td>\n<\/tr>\n<tr>\n<td>Reinforced concrete<\/td>\n<td>3,500,000<\/td>\n<\/tr>\n<tr>\n<td>Wood (Douglas fir)<\/td>\n<td>1,600,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When a cantilever supports a wall or slab, remember that the supporting structure below must also carry the load \u2014 the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/concrete-slab-maximum-wall-load\/\">Concrete Slab Maximum Wall Load<\/a> calculator can help check that side of the problem.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculates the slope at the free end and the deflection at any point along a cantilever beam carrying a uniform load. Enter the applied load, elastic modulus, moment of inertia, beam length and position.<\/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-59952","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\/59952","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=59952"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/59952\/revisions"}],"predecessor-version":[{"id":66025,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/59952\/revisions\/66025"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=59952"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=59952"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=59952"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}