{"id":100222,"date":"2026-02-15T20:28:09","date_gmt":"2026-02-15T20:28:09","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100222"},"modified":"2026-07-15T10:27:59","modified_gmt":"2026-07-15T10:27:59","slug":"shaft-deflection-calculator","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/ar\/calculators\/shaft-deflection-calculator\/","title":{"rendered":"\u0648\u0631\u0642\u0629 \u0639\u0645\u0644 \u0627\u0644\u0627\u0646\u062d\u0646\u0627\u0621 \u062a\u062d\u062a \u0627\u0644\u062d\u0645\u0644 \u0627\u0644\u0646\u0642\u0637\u064a \u0627\u0644\u0645\u0648\u062b\u0642"},"content":{"rendered":"\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Documented Point-Load Shaft-Deflection Worksheet\",\"description\":\"Calculate small-deflection Euler-Bernoulli response for one transverse point load on an ideal simply supported beam or at the free end of an ideal cantilever.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/shaft-deflection-calculator\/\",\"applicationCategory\":\"EngineeringApplication\",\"operatingSystem\":\"Any\",\"isAccessibleForFree\":true,\"inLanguage\":\"en\",\"publisher\":{\"@type\":\"Organization\",\"name\":\"Vibromera\"}}<\/script>\n<style>\n.vbm222{--ink:#172235;--muted:#556276;--line:#d8e0e9;--soft:#f4f7fa;--blue:#135ca8;--blue2:#0d447e;--amber:#8a5700;--amber-bg:#fff7df;--red:#a52828;--red-bg:#fff0f0;--green:#17613a;--green-bg:#edf9f1;max-width:1040px;margin:0 auto;padding:18px 14px 44px;color:var(--ink);font:15px\/1.55 system-ui,-apple-system,\"Segoe UI\",sans-serif}.vbm222 *{box-sizing:border-box}.vbm222 h1,.vbm222 h2,.vbm222 h3{line-height:1.2}.vbm222 h1{font-size:clamp(27px,4vw,42px);margin:8px 0 12px}.vbm222 h2{font-size:22px;margin:0 0 16px}.vbm222 h3{font-size:17px;margin:22px 0 8px}.vbm222 p{margin:8px 0}.vbm222 a{color:var(--blue)}.vbm222-header{text-align:center;padding:32px 18px 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summary{cursor:pointer;font-weight:760}.vbm222-hidden{display:none!important}.vbm222-small{font-size:13px;color:var(--muted)}\n@media(max-width:700px){.vbm222{padding-left:10px;padding-right:10px}.vbm222-card{padding:16px}.vbm222-grid,.vbm222-result-grid{grid-template-columns:1fr}.vbm222-result.primary,.vbm222-field.full{grid-column:auto}.vbm222-combo select{min-width:82px}.vbm222 table{display:block;overflow-x:auto;white-space:normal}}\n<\/style>\n<main class=\"vbm222\">\n  <header class=\"vbm222-header\">\n    <p class=\"vbm222-kicker\">Reference worksheet \u00b7 one static point load<\/p>\n    <h1>Documented Point-Load Shaft-Deflection Worksheet<\/h1>\n    <p class=\"vbm222-lead\">Calculate the small-deflection Euler\u2013Bernoulli response of a constant-EI member for one transverse point load: at any position between ideal simple supports, or at the free end of an ideal cantilever.<\/p>\n    <div class=\"vbm222-badges\"><span class=\"vbm222-badge\">General beam equations<\/span><span class=\"vbm222-badge\">No generated allowance<\/span><span class=\"vbm222-badge\">Strict input validation<\/span><\/div>\n  <\/header>\n\n  <div class=\"vbm222-notice\"><strong>Scope, not acceptance.<\/strong> This is a general engineering beam worksheet, not an ISO, API, ANSI\/ASME or DIN acceptance calculation. It does not generate a permissible shaft deflection, bearing\/seal clearance margin, critical speed, runout, fatigue life or safe operating verdict.<\/div>\n\n  <form id=\"vbm222-form\" class=\"vbm222-card\" autocomplete=\"off\" novalidate>\n    <h2>Documented inputs<\/h2>\n    <div class=\"vbm222-grid\">\n      <div class=\"vbm222-field\"><label for=\"vbm222-support\">Ideal support\/load case<\/label><select id=\"vbm222-support\"><option value=\"simple\">Pin + roller; point load at position a<\/option><option value=\"cantilever\">Fixed cantilever; point load at free end<\/option><\/select><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-section\">Section input<\/label><select id=\"vbm222-section\"><option value=\"I\">Documented second moment I<\/option><option value=\"solid\">Uniform solid circular diameter d<\/option><\/select><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-L\">Span\/length L<\/label><div class=\"vbm222-combo\"><input id=\"vbm222-L\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\" aria-describedby=\"vbm222-number-note\"><select id=\"vbm222-L-unit\" aria-label=\"Span unit\"><option value=\"mm\">mm<\/option><option value=\"in\">in<\/option><\/select><\/div><\/div>\n      <div class=\"vbm222-field\" id=\"vbm222-a-field\"><label for=\"vbm222-a\">Load position a from left support <span class=\"vbm222-hint\">Must satisfy 0 &lt; a &lt; L<\/span><\/label><div class=\"vbm222-combo\"><input id=\"vbm222-a\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\"><select id=\"vbm222-a-unit\" aria-label=\"Load-position unit\"><option value=\"mm\">mm<\/option><option value=\"in\">in<\/option><\/select><\/div><\/div>\n      <div class=\"vbm222-field\" id=\"vbm222-I-field\"><label for=\"vbm222-I\">Second moment of area I <span class=\"vbm222-hint\">About the bending axis; not polar J<\/span><\/label><div class=\"vbm222-combo\"><input id=\"vbm222-I\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\"><select id=\"vbm222-I-unit\" aria-label=\"Second-moment unit\"><option value=\"mm4\">mm\u2074<\/option><option value=\"in4\">in\u2074<\/option><\/select><\/div><\/div>\n      <div class=\"vbm222-field vbm222-hidden\" id=\"vbm222-d-field\"><label for=\"vbm222-d\">Uniform solid diameter d<\/label><div class=\"vbm222-combo\"><input id=\"vbm222-d\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\" disabled><select id=\"vbm222-d-unit\" aria-label=\"Diameter unit\" disabled><option value=\"mm\">mm<\/option><option value=\"in\">in<\/option><\/select><\/div><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-E\">Young\u2019s modulus E<\/label><div class=\"vbm222-combo\"><input id=\"vbm222-E\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\"><select id=\"vbm222-E-unit\" aria-label=\"Young modulus unit\"><option value=\"GPa\">GPa<\/option><option value=\"Mpsi\">Mpsi<\/option><\/select><\/div><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-P\">Transverse point-load magnitude P <span class=\"vbm222-hint\">Zero is allowed as a baseline<\/span><\/label><div class=\"vbm222-combo\"><input id=\"vbm222-P\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\"><select id=\"vbm222-P-unit\" aria-label=\"Point-load unit\"><option value=\"N\">N<\/option><option value=\"lbf\">lbf<\/option><\/select><\/div><\/div>\n      <p class=\"vbm222-field full vbm222-small\" id=\"vbm222-number-note\">Use a decimal point or decimal comma; thousands separators and scientific notation are intentionally rejected.<\/p>\n\n      <div class=\"vbm222-field\"><label for=\"vbm222-project\">Project \/ equipment<\/label><input id=\"vbm222-project\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-shaft\">Member \/ shaft identifier<\/label><input id=\"vbm222-shaft\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-geometry-source\">Geometry or I source<\/label><input id=\"vbm222-geometry-source\" type=\"text\" placeholder=\"drawing, model or measurement record\"><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-load-source\">Load source and operating state<\/label><input id=\"vbm222-load-source\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-modulus-source\">E source and temperature\/state<\/label><input id=\"vbm222-modulus-source\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-support-source\">Support-model basis<\/label><input id=\"vbm222-support-source\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-criteria\">Controlled criterion \/ analysis reference<\/label><input id=\"vbm222-criteria\" type=\"text\" placeholder=\"required; write none if exploratory\"><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-prepared\">Prepared by<\/label><input id=\"vbm222-prepared\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm222-field\"><label for=\"vbm222-date\">Record date<\/label><input id=\"vbm222-date\" type=\"text\" placeholder=\"YYYY-MM-DD\"><\/div>\n      <div class=\"vbm222-field full\"><label for=\"vbm222-notes\">Notes <span class=\"vbm222-hint\">Optional: load direction, temperature, support details, exclusions<\/span><\/label><textarea id=\"vbm222-notes\"><\/textarea><\/div>\n    <\/div>\n\n    <div class=\"vbm222-checks\">\n      <label class=\"vbm222-check\"><input type=\"checkbox\" id=\"vbm222-c1\"><span>The member is straight, homogeneous and prismatic over L; E and I are constant for the stated bending plane.<\/span><\/label>\n      <label class=\"vbm222-check\"><input type=\"checkbox\" id=\"vbm222-c2\"><span>Small-deflection, linearly elastic Euler\u2013Bernoulli bending is appropriate; shear deformation, support compliance and geometric nonlinearity are negligible.<\/span><\/label>\n      <label class=\"vbm222-check\"><input type=\"checkbox\" id=\"vbm222-c3\"><span>The selected ideal support and single point-load case represent the documented physical load path; other loads are excluded from this calculation.<\/span><\/label>\n      <label class=\"vbm222-check\"><input type=\"checkbox\" id=\"vbm222-c4\"><span>I understand that these results are reference values, not an allowable-deflection, strength, fatigue, critical-speed or machine-acceptance decision.<\/span><\/label>\n    <\/div>\n    <div class=\"vbm222-actions\"><button type=\"submit\" class=\"vbm222-primary\">Calculate documented case<\/button><button type=\"button\" class=\"vbm222-secondary\" id=\"vbm222-clear\">Clear<\/button><\/div>\n    <div id=\"vbm222-error\" class=\"vbm222-error\" role=\"alert\"><\/div>\n  <\/form>\n\n  <section id=\"vbm222-results\" class=\"vbm222-card vbm222-results\" aria-live=\"polite\">\n    <h2>Reference results<\/h2>\n    <div class=\"vbm222-result-grid\">\n      <div class=\"vbm222-result primary\"><div class=\"vbm222-result-label\">Maximum downward deflection<\/div><div class=\"vbm222-result-value\" id=\"vbm222-r-max\">\u2014<\/div><\/div>\n      <div class=\"vbm222-result\"><div class=\"vbm222-result-label\">Location of maximum from left\/fixed end<\/div><div class=\"vbm222-result-value\" id=\"vbm222-r-x\">\u2014<\/div><\/div>\n      <div class=\"vbm222-result\"><div class=\"vbm222-result-label\">Deflection at the load point<\/div><div class=\"vbm222-result-value\" id=\"vbm222-r-load\">\u2014<\/div><\/div>\n      <div class=\"vbm222-result\"><div class=\"vbm222-result-label\">Largest support\/end rotation magnitude<\/div><div class=\"vbm222-result-value\" id=\"vbm222-r-theta\">\u2014<\/div><\/div>\n      <div class=\"vbm222-result\"><div class=\"vbm222-result-label\">Maximum bending-moment magnitude<\/div><div class=\"vbm222-result-value\" id=\"vbm222-r-moment\">\u2014<\/div><\/div>\n      <div class=\"vbm222-result\"><div class=\"vbm222-result-label\">Maximum elastic surface bending stress<\/div><div class=\"vbm222-result-value\" id=\"vbm222-r-stress\">\u2014<\/div><\/div>\n      <div class=\"vbm222-result\"><div class=\"vbm222-result-label\">Second moment used<\/div><div class=\"vbm222-result-value\" id=\"vbm222-r-I\">\u2014<\/div><\/div>\n      <div class=\"vbm222-result\"><div class=\"vbm222-result-label\">Vertical reactions<\/div><div class=\"vbm222-result-value\" id=\"vbm222-r-reactions\">\u2014<\/div><\/div>\n    <\/div>\n    <div class=\"vbm222-status\" id=\"vbm222-r-status\"><\/div>\n  <\/section>\n\n  <section class=\"vbm222-card\">\n    <h2>Equations and definitions<\/h2>\n    <p>Let L be span, a the distance from the left support to P, b = L \u2212 a, E Young\u2019s modulus, I the centroidal area second moment about the bending axis, and x the distance from the left support. Magnitudes are reported for one transverse load.<\/p>\n    <h3>Simply supported member; point load at a<\/h3>\n    <div class=\"vbm222-formula\">R\u2090 = Pb\/L; R\u1d66 = Pa\/L; Mmax = Pab\/L<br>\u03b4(a) = Pa\u00b2b\u00b2\/(3EIL)<br>0 \u2264 x \u2264 a: \u03b4(x) = Pbx(L\u00b2 \u2212 b\u00b2 \u2212 x\u00b2)\/(6LEI)<br>a \u2264 x \u2264 L: \u03b4(x) = Pa(L \u2212 x)[L\u00b2 \u2212 a\u00b2 \u2212 (L \u2212 x)\u00b2]\/(6LEI)<\/div>\n    <p>The maximum is found from d\u03b4\/dx = 0 in the valid side of the piecewise field. If a \u2264 L\/2, x<sub>max<\/sub> = L \u2212 \u221a[(L\u00b2 \u2212 a\u00b2)\/3]; if a \u2265 L\/2, x<sub>max<\/sub> = \u221a[(L\u00b2 \u2212 b\u00b2)\/3]. Therefore \u03b4(a) is generally <em>not<\/em> the maximum when a \u2260 L\/2.<\/p>\n    <div class=\"vbm222-formula\">|\u03b8\u2090| = Pab(L + b)\/(6LEI)<br>|\u03b8\u1d66| = Pab(L + a)\/(6LEI)<br>|\u03b8|max = max(|\u03b8\u2090|, |\u03b8\u1d66|)<\/div>\n    <h3>Fixed cantilever; point load at free end<\/h3>\n    <div class=\"vbm222-formula\">\u03b4(x) = Px\u00b2(3L \u2212 x)\/(6EI)<br>\u03b4max = \u03b4(L) = PL\u00b3\/(3EI); |\u03b8(L)| = PL\u00b2\/(2EI); Mmax = PL<\/div>\n    <h3>Section quantities<\/h3>\n    <div class=\"vbm222-formula\">Solid circular section: I = \u03c0d\u2074\/64; \u03c3max = Mmax(d\/2)\/I = 32Mmax\/(\u03c0d\u00b3)<\/div>\n    <p>In direct-I mode the worksheet does not calculate surface stress because I alone does not define the extreme-fibre distance or section modulus. The displayed stress is an ideal elastic bending stress only; it omits stress concentrations, keyways, shoulders, residual stress, axial\/torsional\/shear stress and combined-stress criteria.<\/p>\n    <h3>Dimensional check<\/h3>\n    <p>P has dimension force, E force\/area and I length\u2074, so PL\u00b3\/(EI) has dimension length. M has force\u00b7length and Mc\/I has dimension force\/area. These equations are not formulas issued by ISO.<\/p>\n  <\/section>\n\n  <section class=\"vbm222-card\">\n    <h2>Model boundary<\/h2>\n    <ul>\n      <li>No distributed load or shaft self-weight is included. For a verified full-span uniform-load\/self-weight worksheet, use the <a href=\"\/calculators\/rotor-deflection\/\">related uniform-load deflection calculator<\/a>.<\/li>\n      <li>Stepped\/hollow shafts, overhangs, disks, multiple loads, bearing stiffness, shear deformation, contact, large displacement and rotor-dynamic effects require an appropriate beam, finite-element or rotordynamic model.<\/li>\n      <li>The former universal L\/3000, L\/5000 and L\/10000 labels are not used: an allowable value must come from the controlled design basis for the actual bearings, seals, gears, couplings, process and operating state.<\/li>\n      <li>No critical speed is inferred from an arbitrary static load. A gravity-deflection relation is valid only under its own mass\/load and modal assumptions; this worksheet does not establish them.<\/li>\n      <li>For a solid circle, changing d while P, E and L stay fixed makes deflection scale as 1\/d\u2074. That statement does not apply unchanged to self-weight because the load then also changes with area.<\/li>\n    <\/ul>\n  <\/section>\n\n  <section class=\"vbm222-card\">\n    <h2>Source traceability<\/h2>\n    <table><thead><tr><th>Claim<\/th><th>Classification<\/th><th>Evidence<\/th><\/tr><\/thead><tbody>\n      <tr><td>Piecewise deflection and true maximum for a point load between simple supports; end-loaded cantilever field and maximum<\/td><td>General Euler\u2013Bernoulli beam solutions; not ISO formulas<\/td><td>MIT, Engineering Mechanics for Structures, \u00a710.1, pp. 268\u2013270<\/td><\/tr>\n      <tr><td>Published center-load benchmark: P = 10 kip, L = 60 in, E = 30,000 ksi, I = 416.7 in\u2074 gives 0.0036 in by PL\u00b3\/(48EI)<\/td><td>Published numerical example<\/td><td>NASA NTRS document 19700016059, report page 75 (PDF page 81), equation (53)<\/td><\/tr>\n      <tr><td>1 in = 0.0254 m; 1 lbf = 4.4482216152605 N under standard gravity<\/td><td>Exact SI conversion<\/td><td>NIST SP 811, Appendix B.8 and footnotes<\/td><\/tr>\n    <\/tbody><\/table>\n    <ul>\n      <li><a href=\"https:\/\/web.mit.edu\/1.105\/emech10_04.pdf\" rel=\"noopener\" target=\"_blank\">MIT, Engineering Mechanics for Structures, Chapter 10: Deflections due to Bending<\/a> \u2014 official public university material; the point-load cases appear on book pages 268\u2013270.<\/li>\n      <li><a href=\"https:\/\/ntrs.nasa.gov\/api\/citations\/19700016059\/downloads\/19700016059.pdf\" rel=\"noopener\" target=\"_blank\">NASA Technical Reports Server document 19700016059<\/a> \u2014 published classical-method benchmark on report page 75.<\/li>\n      <li><a href=\"https:\/\/www.nist.gov\/pml\/special-publication-811\/nist-guide-si-appendix-b-conversion-factors\/nist-guide-si-appendix-b8\" rel=\"noopener\" target=\"_blank\">NIST SP 811, Appendix B.8<\/a> \u2014 conversion factors.<\/li>\n    <\/ul>\n    <p><strong>Accessed:<\/strong> 15 July 2026. No standard-based permissible-deflection table was verified for this generic model. Any claimed universal allowance remains <strong>NEEDS_LICENSED_SOURCE<\/strong> until the controlled equipment\/project criterion is supplied.<\/p>\n    <h3>Published benchmark entry<\/h3>\n    <p>Select simple supports and direct I. Enter L = 60 in, a = 30 in, I = 416.7 in\u2074, E = 30 Mpsi and P = 10000 lbf. The worksheet should return approximately 0.003599712 in, which rounds to the report\u2019s 0.0036 in.<\/p>\n  <\/section>\n\n  <section class=\"vbm222-card\">\n    <h2>Questions<\/h2>\n    <details><summary>Why is the offset-load maximum not at the load?<\/summary><p>The load point is where bending moment is largest, but maximum deflection occurs where slope is zero. Except at midspan, those locations differ; the worksheet evaluates the correct side of the piecewise elastic curve.<\/p><\/details>\n    <details><summary>Can this determine a shaft critical speed?<\/summary><p>No. This is a static one-load beam calculation. Rotor critical speed depends on the distributed mass, disks, bearings\/support coefficients, gyroscopic effects and mode shape. Applying a gravity-deflection shortcut to an unrelated load case is not justified.<\/p><\/details>\n    <details><summary>Does the calculated stress prove the shaft is safe?<\/summary><p>No. It is only the nominal elastic surface stress for a uniform solid circle in the stated bending plane. Strength and fatigue checks need material allowables, combined loads, stress concentrations, mean\/alternating components, surface\/size\/reliability effects and the controlled design method.<\/p><\/details>\n  <\/section>\n<\/main>\n<script>\n(function(){\n'use strict';\nvar form=document.getElementById('vbm222-form'),error=document.getElementById('vbm222-error'),results=document.getElementById('vbm222-results');\nfunction strictDecimal(value,label,kind){var s=String(value==null?'':value).trim();if(!\/^(?:\\d+(?:[.,]\\d*)?|[.,]\\d+)$\/.test(s))throw new Error(label+' must be one plain decimal number.');var n=Number(s.replace(',','.'));if(!Number.isFinite(n))throw new Error(label+' is outside the finite numeric range.');if(kind==='positive'&&n<=0)throw new Error(label+' must be greater than zero.');if(kind==='nonnegative'&&n<0)throw new Error(label+' must be zero or greater.');return n}\nfunction finite(values){if(!values.every(Number.isFinite))throw new Error('The calculation is 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