{"id":59947,"date":"2016-12-29T10:25:35","date_gmt":"2016-12-29T08:25:35","guid":{"rendered":"https:\/\/wpcalc.com\/?p=59947"},"modified":"2026-09-30T18:07:34","modified_gmt":"2026-09-30T15:07:34","slug":"cantilever-beam-with-load-at-any-point","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/engineering\/cantilever-beam-with-load-at-any-point\/","title":{"rendered":"Cantilever Beam with Load at Any Point 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\">Analyze Cantilever Deflection with Point Load<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"59947\" data-favorite-title=\"Cantilever Beam with Load at Any Point\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/engineering\/cantilever-beam-with-load-at-any-point\/\" data-favorite-excerpt=\"Calculate the slope and deflection of a cantilever beam with a point load applied at any position along its length. Enter the load, beam properties, load position, and evaluation point to get instant results.\" 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-any-point\/\" method=\"POST\" class=\"calculator\" id=\"calculator-59947\" data-post=\"59947\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">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=\"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\">a<\/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=\"80\"\/><div class=\"unit\" id=\"calc-unit-5\">in<\/div><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">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-6\" aria-describedby=\"calc-unit-6\" step=\"any\" value=\"90\"\/><div class=\"unit\" id=\"calc-unit-6\">in<\/div><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"59947\">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><span class=\"field-value\" id=\"result-1\"><\/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><span class=\"field-value\" id=\"result-2\"><\/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=\"59947\" 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\" 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\">Confused <\/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\">Disappointed <\/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\">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 Load at Any Point 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\\[\n\\theta_{free} = \\frac{P a^2}{2 E I}\n\\]\n\\[\n\\delta(x) =\n\\begin{cases}\n\\dfrac{P x^2 (3a - x)}{6 E I}, &amp; x \\leq a \\\\[6pt]\n\\dfrac{P a^2 (3x - a)}{6 E I}, &amp; x \\geq a\n\\end{cases}\n\\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(P\\) \u2014 applied point load (lbs)<\/li>\n<li>\\(E\\) \u2014 elastic modulus of the beam material (psi)<\/li>\n<li>\\(I\\) \u2014 area moment of inertia of the cross-section (in\u2074)<\/li>\n<li>\\(L\\) \u2014 total length of the beam (in)<\/li>\n<li>\\(a\\) \u2014 distance from the fixed support to the load (in)<\/li>\n<li>\\(x\\) \u2014 distance from the fixed support to the point where deflection is evaluated (in)<\/li>\n<\/ul>\n<p>The slope formula gives the rotation at the free end of the beam. The deflection formula is piecewise: the first expression applies between the support and the load, and the second applies from the load to the free end, where the beam bends in a straight (rigid-body) line.<\/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 Load at Any Point \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a steel cantilever beam is 120 in long with \\(E = 29{,}000{,}000\\) psi and \\(I = 150\\) in\u2074. A 500 lb load acts 80 in from the fixed support. Find the slope at the free end and the deflection 90 in from the support.<\/p>\n<p>\\[ \\theta_{free} = \\frac{P a^2}{2 E I} = \\frac{500 \\times 80^2}{2 \\times 29{,}000{,}000 \\times 150} = \\frac{3{,}200{,}000}{8{,}700{,}000{,}000} = 0.000368 \\text{ rad} \\]<\/p>\n<p>Since the deflection point \\(x = 90\\) in lies beyond the load \\(a = 80\\) in, use the second expression:<\/p>\n<p>\\[ \\delta = \\frac{P a^2 (3x - a)}{6 E I} = \\frac{500 \\times 80^2 \\times (3 \\times 90 - 80)}{6 \\times 29{,}000{,}000 \\times 150} = \\frac{608{,}000{,}000}{26{,}100{,}000{,}000} = 0.0233 \\text{ in} \\]<\/p>\n<p><strong>Result:<\/strong> the slope at the free end is approximately 0.000368 rad (about 0.021\u00b0) and the deflection at 90 in from the support is 0.0233 in downward.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the <strong>Load<\/strong> \\(P\\) in pounds \u2014 the point force applied to the beam.<\/li>\n<li>Enter the <strong>Elastic Modulus<\/strong> \\(E\\) in psi. For steel this is typically 29,000,000 psi; for aluminum about 10,000,000 psi.<\/li>\n<li>Enter the <strong>Moment of Inertia<\/strong> \\(I\\) in in\u2074 for the beam cross-section (available from standard beam tables).<\/li>\n<li>Enter the <strong>Beam Length<\/strong> \\(L\\) in inches, measured from the fixed support to the free end.<\/li>\n<li>Enter the <strong>Load Position<\/strong> \\(a\\) in inches \u2014 the distance from the fixed support to where the load is applied. It must be between 0 and \\(L\\).<\/li>\n<li>Enter the <strong>Deflection Point<\/strong> \\(x\\) in inches \u2014 the distance from the fixed support where you want the deflection.<\/li>\n<li>Read the <strong>Slope at Free End<\/strong> (in radians) and the <strong>Deflection<\/strong> (in inches) below the inputs.<\/li>\n<\/ol>\n<h2>What Is a Cantilever Beam with a Load at Any Point?<\/h2>\n<p>A cantilever beam is fixed rigidly at one end and free at the other. In many practical situations the load does not act at the free tip \u2014 for example, a pipe or duct supported partway along a bracket, a sign mounted mid-span on a pole arm, or a machine bolted to a platform extension. In these cases the load sits at a distance \\(a\\) from the fixed support, and the beam beyond the load carries no bending moment; it simply rotates as a rigid body.<\/p>\n<p>This changes the deflected shape compared to a tip-loaded cantilever. Between the support and the load, the beam follows a curved elastic shape described by the first formula. Beyond the load, the beam remains straight but sloped, so any point in that region deflects according to the second formula. The maximum deflection always occurs at the free end, and its value is found by substituting \\(x = L\\) into the second expression: \\(\\delta_{max} = \\frac{P a^2 (3L - a)}{6EI}\\).<\/p>\n<p>The slope at the free end depends only on the load magnitude and its distance from the support, not on where along the beam you measure it afterward \u2014 the portion beyond the load rotates uniformly. A small slope in radians means a nearly horizontal beam; multiply by \\(57.3\\) to convert to degrees if needed.<\/p>\n<p>Interpreting the result is usually a matter of comparing deflection against a limit. Many design codes restrict live-load deflection of beams to \\(L\/360\\) or \\(L\/240\\). For the example beam above, \\(L\/360 = 120\/360 = 0.333\\) in, so a 0.0233 in deflection is comfortably within that limit. When checking a floor or platform more broadly, you may also want to verify the overall load capacity with the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/maximum-floor-live-load-capacity\/\">Maximum Floor Live Load Capacity<\/a> calculator.<\/p>\n<h2>Assumptions and Limitations<\/h2>\n<ul>\n<li><strong>Small deflection theory:<\/strong> the formulas assume deflections are small relative to the beam length and that the material remains linearly elastic (below yield).<\/li>\n<li><strong>Rigid, perfectly fixed support:<\/strong> any rotation or flexibility at the anchor increases real deflections beyond the calculated values.<\/li>\n<li><strong>Point load only:<\/strong> the beam's own weight and any distributed loads are ignored. For heavy sections, add the self-weight as a separate check.<\/li>\n<li><strong>Constant cross-section:<\/strong> \\(E\\) and \\(I\\) are uniform along the entire length. A tapered or stepped beam requires a more detailed analysis.<\/li>\n<li><strong>Units must be consistent:<\/strong> this calculator works in pounds, psi, inches and in\u2074. If your data is in metric units, convert first or interpret the outputs accordingly (the formulas themselves are unit-agnostic as long as inputs are consistent).<\/li>\n<li><strong>Sign convention:<\/strong> the deflection result is a magnitude; downward deflection under a downward load is implied.<\/li>\n<\/ul>\n<p>If your design involves loads applied directly to a slab rather than a cantilever member, see the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/concrete-slab-maximum-wall-load\/\">Concrete Slab Maximum Wall Load<\/a> calculator, or check the required quantities for the supporting structure with the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/concrete-yardage\/\">Concrete Yardage<\/a> tool.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the slope and deflection of a cantilever beam with a point load applied at any position along its length. Enter the load, beam properties, load position, and evaluation point to get instant results.<\/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-59947","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\/59947","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=59947"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/59947\/revisions"}],"predecessor-version":[{"id":65950,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/59947\/revisions\/65950"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=59947"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=59947"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=59947"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}