{"id":100211,"date":"2026-02-15T20:27:37","date_gmt":"2026-02-15T20:27:37","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100211"},"modified":"2026-07-14T22:03:11","modified_gmt":"2026-07-14T22:03:11","slug":"rotor-deflection","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/id\/calculators\/rotor-deflection\/","title":{"rendered":"Simply Supported Shaft Deflection Worksheet"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"WebApplication\",\n  \"name\": \"Simply Supported Beam and Solid-Shaft Deflection Worksheet\",\n  \"description\": \"Calculate the small-deflection Euler-Bernoulli midspan displacement of a uniform simply supported beam under a full-span uniform load and a center point load, with an optional uniform solid-shaft self-weight helper.\",\n  \"url\": \"https:\/\/vibromera.eu\/calculators\/rotor-deflection\/\",\n  \"applicationCategory\": \"EngineeringApplication\",\n  \"operatingSystem\": \"Any\",\n  \"isAccessibleForFree\": true,\n  \"inLanguage\": \"en\",\n  \"creator\": {\"@type\": \"Organization\", \"name\": \"Vibromera\", \"url\": \"https:\/\/vibromera.eu\/\"},\n  \"featureList\": [\n    \"Direct E-I uniform-load and center-load model\",\n    \"Optional uniform solid circular shaft self-weight helper\",\n    \"Exact SI and U.S. customary unit normalization\",\n    \"Strict input validation and documented model boundary\"\n  ]\n}\n<\/script>\n\n<style>\n.vbm211{--ink:#17212b;--muted:#55616d;--line:#d7dee5;--soft:#f4f7f9;--paper:#fff;--blue:#165d89;--blue2:#0b466b;--warn:#8a5400;--warnbg:#fff6df;--ok:#16613b;--okbg:#edf8f2;max-width:1000px;margin:0 auto;padding:20px 16px 44px;color:var(--ink);font:15px\/1.58 system-ui,-apple-system,\"Segoe UI\",sans-serif}\n.vbm211 *{box-sizing:border-box}.vbm211 h1,.vbm211 h2,.vbm211 h3{line-height:1.2}.vbm211 a{color:var(--blue)}\n.vbm211-hero{text-align:center;padding:34px 18px 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6px;font-weight:800}.vbm211-check{display:flex;align-items:flex-start;gap:9px;margin:8px 0;color:var(--muted)}.vbm211-check input{width:auto;margin-top:4px;flex:0 0 auto}.vbm211-model-note{grid-column:1\/-1;padding:12px 14px;border:1px solid #bdd4e3;border-radius:7px;background:#eef7fc;color:#24485f}\n.vbm211-actions{display:flex;gap:10px;flex-wrap:wrap;margin-top:18px}.vbm211-btn{width:auto!important;padding:11px 18px!important;border:1px solid var(--blue)!important;border-radius:7px!important;background:var(--blue)!important;color:#fff!important;font-weight:800!important;cursor:pointer}.vbm211-btn:hover{background:var(--blue2)!important}.vbm211-btn.secondary{border-color:#8795a1!important;background:#fff!important;color:var(--ink)!important}.vbm211-error{display:none;margin-top:15px;padding:12px 14px;border-left:4px solid #a52222;border-radius:5px;background:#fff0f0;color:#7e1818}.vbm211-error.show{display:block}\n.vbm211-results{display:none}.vbm211-results.show{display:block}.vbm211-result-grid{display:grid;grid-template-columns:repeat(3,minmax(0,1fr));gap:12px}.vbm211-result{padding:15px;border:1px solid var(--line);border-radius:8px;background:var(--soft)}.vbm211-result.primary{grid-column:1\/-1;border:2px solid var(--blue);background:#edf7fc}.vbm211-result-label{color:var(--muted);font-size:12px;font-weight:800;text-transform:uppercase;letter-spacing:.05em}.vbm211-result-value{margin-top:4px;font:750 21px\/1.25 ui-monospace,monospace;overflow-wrap:anywhere}.vbm211-result.primary .vbm211-result-value{color:var(--blue2);font-size:29px}.vbm211-result-note{margin-top:4px;color:var(--muted);font-size:12px}.vbm211-pass{margin-top:15px;padding:12px 14px;border-left:4px solid var(--ok);border-radius:5px;background:var(--okbg);color:#194b33}\n.vbm211-section{margin-top:25px;padding:21px;border:1px solid var(--line);border-radius:10px;background:#fff}.vbm211-section h2{margin:0 0 13px;font-size:22px}.vbm211-section h3{margin:20px 0 8px;font-size:17px}.vbm211-formula{margin:10px 0;padding:13px 15px;border:1px solid var(--line);border-radius:7px;background:var(--soft);font:650 16px\/1.55 ui-monospace,monospace;text-align:center;overflow-x:auto}.vbm211-tablewrap{overflow-x:auto}.vbm211 table{width:100%;border-collapse:collapse;font-size:13px}.vbm211 th,.vbm211 td{padding:10px;border:1px solid var(--line);text-align:left;vertical-align:top}.vbm211 th{background:var(--soft)}.vbm211-source-list li{margin:8px 0}.vbm211 details{margin:9px 0;border:1px solid var(--line);border-radius:7px;background:#fff}.vbm211 summary{padding:12px 14px;font-weight:750;cursor:pointer}.vbm211 details p{padding:0 14px 13px;margin:0;color:var(--muted)}.vbm211-footer{margin-top:25px;padding-top:18px;border-top:1px solid var(--line);color:var(--muted);font-size:13px}\n@media(max-width:720px){.vbm211-grid,.vbm211-result-grid{grid-template-columns:1fr}.vbm211-result.primary,.vbm211-field.full,.vbm211-fieldset,.vbm211-model-note{grid-column:1}.vbm211-body,.vbm211-section{padding:16px}.vbm211-combo{grid-template-columns:minmax(0,1fr) 105px}}\n@media print{.vbm211-actions{display:none}.vbm211-card,.vbm211-section{box-shadow:none;break-inside:avoid}.vbm211-results.show{display:block}}\n<\/style>\n\n<main class=\"vbm211\" id=\"vbm-c100211\">\n  <header class=\"vbm211-hero\">\n    <p class=\"vbm211-kicker\">Reference worksheet \u00b7 static beam model<\/p>\n    <h1>Simply Supported Beam \/ Solid-Shaft Deflection Worksheet<\/h1>\n    <p class=\"vbm211-lead\">Calculate the small-deflection midspan displacement of a uniform, prismatic, simply supported Euler\u2013Bernoulli beam under a full-span uniform load and a point load located exactly at midspan.<\/p>\n    <div class=\"vbm211-badges\"><span class=\"vbm211-badge\">qL\u2074 \/ EI<\/span><span class=\"vbm211-badge\">FL\u00b3 \/ EI<\/span><span class=\"vbm211-badge\">strict units<\/span><span class=\"vbm211-badge\">no acceptance limit<\/span><\/div>\n  <\/header>\n\n  <div class=\"vbm211-notice\"><strong>Scope, not certification.<\/strong> These are general engineering beam equations, not an ISO or API acceptance calculation. The worksheet does not determine an allowable sag, bearing-clearance margin, runout, rotor critical speed, stress, fatigue, rubbing risk, or safe operating condition.<\/div>\n\n  <section class=\"vbm211-card\" aria-labelledby=\"vbm211-input-title\">\n    <div class=\"vbm211-card-h\"><h2 id=\"vbm211-input-title\">Controlled inputs<\/h2><p>Use the direct model when E, I and loading are already documented. Use the helper only for a uniform solid circular shaft whose self-weight is represented by a uniform line load. Changing a unit clears its paired numeric field and the previous result, so an existing number is never silently reinterpreted.<\/p><\/div>\n    <div class=\"vbm211-body\">\n      <form id=\"vbm211-form\" novalidate>\n        <div class=\"vbm211-grid\">\n          <div class=\"vbm211-field full\"><label for=\"vbm211-model\">Model<\/label><select id=\"vbm211-model\"><option value=\"direct\">Direct documented E\u2013I beam<\/option><option value=\"solid\">Uniform solid circular shaft self-weight helper<\/option><\/select><\/div>\n          <div class=\"vbm211-model-note\" id=\"vbm211-model-note\">Direct inputs: span L, Young\u2019s modulus E, second moment of area I, full-span uniform line load q, and an optional downward point load F at midspan.<\/div>\n\n          <div class=\"vbm211-field\"><label for=\"vbm211-L\">Support span L<\/label><div class=\"vbm211-combo\"><input id=\"vbm211-L\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><select id=\"vbm211-L-unit\" aria-label=\"Span unit\"><option value=\"mm\">mm<\/option><option value=\"in\">in<\/option><\/select><\/div><\/div>\n          <div class=\"vbm211-field\"><label for=\"vbm211-E\">Documented Young\u2019s modulus E<\/label><div class=\"vbm211-combo\"><input id=\"vbm211-E\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><select id=\"vbm211-E-unit\" aria-label=\"Young modulus unit\"><option value=\"GPa\">GPa<\/option><option value=\"ksi\">ksi<\/option><\/select><\/div><\/div>\n\n          <div class=\"vbm211-field\" data-direct><label for=\"vbm211-I\">Documented second moment of area I<\/label><div class=\"vbm211-combo\"><input id=\"vbm211-I\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><select id=\"vbm211-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=\"vbm211-field\" data-direct><label for=\"vbm211-q\">Uniform line load q <span class=\"vbm211-hint\">(zero allowed)<\/span><\/label><div class=\"vbm211-combo\"><input id=\"vbm211-q\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><select id=\"vbm211-q-unit\" aria-label=\"Uniform load unit\"><option value=\"N\/m\">N\/m<\/option><option value=\"N\/mm\">N\/mm<\/option><option value=\"lbf\/in\">lbf\/in<\/option><\/select><\/div><\/div>\n\n          <div class=\"vbm211-field\" data-solid hidden><label for=\"vbm211-d\">Uniform solid shaft diameter d<\/label><div class=\"vbm211-combo\"><input id=\"vbm211-d\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><select id=\"vbm211-d-unit\" aria-label=\"Diameter unit\"><option value=\"mm\">mm<\/option><option value=\"in\">in<\/option><\/select><\/div><\/div>\n          <div class=\"vbm211-field\" data-solid hidden><label for=\"vbm211-rho\">Documented mass density \u03c1<\/label><div class=\"vbm211-combo\"><input id=\"vbm211-rho\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><select id=\"vbm211-rho-unit\" aria-label=\"Density unit\"><option value=\"kg\/m3\">kg\/m\u00b3<\/option><option value=\"lbm\/in3\">lbm\/in\u00b3<\/option><\/select><\/div><\/div>\n\n          <div class=\"vbm211-field\"><label for=\"vbm211-F\">Downward point load F at exactly L\/2 <span class=\"vbm211-hint\">(zero allowed)<\/span><\/label><div class=\"vbm211-combo\"><input id=\"vbm211-F\" inputmode=\"decimal\" autocomplete=\"off\" placeholder=\"blank\"><select id=\"vbm211-F-unit\" aria-label=\"Point load unit\"><option value=\"N\">N<\/option><option value=\"lbf\">lbf<\/option><\/select><\/div><\/div>\n          <div class=\"vbm211-field\"><label for=\"vbm211-id\">Machine \/ shaft \/ calculation ID<\/label><input id=\"vbm211-id\" autocomplete=\"off\" placeholder=\"required record\"><\/div>\n          <div class=\"vbm211-field\"><label for=\"vbm211-geometry-source\">Geometry \/ I source<\/label><input id=\"vbm211-geometry-source\" autocomplete=\"off\" placeholder=\"drawing, revision, measurement or model\"><\/div>\n          <div class=\"vbm211-field\"><label for=\"vbm211-material-source\">E source and material condition<\/label><input id=\"vbm211-material-source\" autocomplete=\"off\" placeholder=\"specification, report, temperature\/condition\"><\/div>\n          <div class=\"vbm211-field full\"><label for=\"vbm211-load-source\">Load source <span id=\"vbm211-load-source-hint\" class=\"vbm211-hint\">(q and center F)<\/span><\/label><input id=\"vbm211-load-source\" autocomplete=\"off\" placeholder=\"load case \/ density source \/ revision\"><\/div>\n\n          <fieldset class=\"vbm211-fieldset\"><legend>Required model confirmation<\/legend>\n            <label class=\"vbm211-check\"><input type=\"checkbox\" id=\"vbm211-c1\"> <span>The member is straight, prismatic and uniform between ideal simple supports; E and I are constant about the bending axis.<\/span><\/label>\n            <label class=\"vbm211-check\"><input type=\"checkbox\" id=\"vbm211-c2\"> <span>Small-deflection, linearly elastic Euler\u2013Bernoulli bending and load superposition are appropriate; shear deformation, support\/bearing compliance and geometric nonlinearity are negligible.<\/span><\/label>\n            <label class=\"vbm211-check\"><input type=\"checkbox\" id=\"vbm211-c3\"> <span>q is uniform over the full span and downward; F is downward and located exactly at midspan. The solid helper additionally assumes a uniform solid circle and conventional standard gravity g\u2080 = 9.80665 m\/s\u00b2.<\/span><\/label>\n          <\/fieldset>\n        <\/div>\n        <div class=\"vbm211-actions\"><button class=\"vbm211-btn\" type=\"submit\">Calculate documented case<\/button><button class=\"vbm211-btn secondary\" type=\"button\" id=\"vbm211-reset\">Clear<\/button><\/div>\n        <div class=\"vbm211-error\" id=\"vbm211-error\" role=\"alert\" aria-live=\"assertive\"><\/div>\n      <\/form>\n    <\/div>\n  <\/section>\n\n  <section class=\"vbm211-card vbm211-results\" id=\"vbm211-results\" aria-live=\"polite\">\n    <div class=\"vbm211-card-h\"><h2>Reference result<\/h2><p id=\"vbm211-result-context\"><\/p><\/div>\n    <div class=\"vbm211-body\">\n      <div class=\"vbm211-result-grid\">\n        <div class=\"vbm211-result primary\"><div class=\"vbm211-result-label\">Total downward midspan displacement<\/div><div class=\"vbm211-result-value\" id=\"vbm211-r-total\">\u2014<\/div><div class=\"vbm211-result-note\" id=\"vbm211-r-total-alt\">\u2014<\/div><\/div>\n        <div class=\"vbm211-result\"><div class=\"vbm211-result-label\">Uniform-load contribution<\/div><div class=\"vbm211-result-value\" id=\"vbm211-r-q\">\u2014<\/div><\/div>\n        <div class=\"vbm211-result\"><div class=\"vbm211-result-label\">Center-load contribution<\/div><div class=\"vbm211-result-value\" id=\"vbm211-r-f\">\u2014<\/div><\/div>\n        <div class=\"vbm211-result\"><div class=\"vbm211-result-label\">Normalized q<\/div><div class=\"vbm211-result-value\" id=\"vbm211-r-line\">\u2014<\/div><\/div>\n        <div class=\"vbm211-result\"><div class=\"vbm211-result-label\">Normalized I<\/div><div class=\"vbm211-result-value\" id=\"vbm211-r-i\">\u2014<\/div><\/div>\n        <div class=\"vbm211-result\"><div class=\"vbm211-result-label\">Span ratio \u03b4\/L<\/div><div class=\"vbm211-result-value\" id=\"vbm211-r-ratio\">\u2014<\/div><\/div>\n        <div class=\"vbm211-result\" data-solid-result hidden><div class=\"vbm211-result-label\">Uniform shaft mass<\/div><div class=\"vbm211-result-value\" id=\"vbm211-r-mass\">\u2014<\/div><\/div>\n      <\/div>\n      <div class=\"vbm211-pass\">No pass\/fail limit is applied. Compare the result only with the controlled drawing, equipment specification, bearing\/seal geometry and the applicable licensed project or machine criteria.<\/div>\n    <\/div>\n  <\/section>\n\n  <section class=\"vbm211-section\">\n    <h2>Equations and dimensional boundary<\/h2>\n    <p>For a constant-EI simply supported beam, a downward uniform line load q over the whole span and a downward center point load F are superposed:<\/p>\n    <div class=\"vbm211-formula\">\u03b4q = 5 q L\u2074 \/ (384 E I) &nbsp;&nbsp;\u00b7&nbsp;&nbsp; \u03b4F = F L\u00b3 \/ (48 E I) &nbsp;&nbsp;\u00b7&nbsp;&nbsp; \u03b4total = \u03b4q + \u03b4F<\/div>\n    <p>q has dimension force\/length, E has force\/area, and I has length\u2074. Each quotient therefore has dimension length. For nonnegative symmetric loads, the reported location is midspan and is the maximum downward deflection within this ideal model.<\/p>\n    <h3>Optional uniform solid-shaft helper<\/h3>\n    <div class=\"vbm211-formula\">A = \u03c0d\u00b2\/4 &nbsp;&nbsp;\u00b7&nbsp;&nbsp; I = \u03c0d\u2074\/64 &nbsp;&nbsp;\u00b7&nbsp;&nbsp; qself = \u03c1 A g\u2080, where g\u2080 = 9.80665 m\/s\u00b2<\/div>\n    <p>The helper uses mass density, not weight density. For its self-weight term, substituting A and I gives \u03b4self = 5\u03c1g\u2080L\u2074\/(24Ed\u00b2). Thus self-weight deflection scales as 1\/d\u00b2, while the center-force contribution scales as 1\/d\u2074. Increasing d from 50 mm to 60 mm reduces those terms by 30.56% and 51.77%, respectively\u2014not one universal percentage.<\/p>\n    <h3>What is excluded<\/h3>\n    <p>Stepped or hollow shafts, overhangs, disks at arbitrary locations, multiple loads, nonuniform E\/I, real bearing stiffness, shear deformation, local gravity, thermal bow, residual stress, assembly preload, contact, plasticity, large displacement, spin softening\/stiffening, gyroscopic effects and rotor-dynamic response are outside this worksheet. Use a beam\/FE\/rotordynamic model that represents those features.<\/p>\n  <\/section>\n\n  <section class=\"vbm211-section\">\n    <h2>Independent published check<\/h2>\n    <p>A Missouri University of Science and Technology instructional example reports a simply supported uniform-load case with q = 3 kip\/ft, L = 10 ft, E = 29,000.02 ksi and I = 984 in\u2074. Enter the equivalent direct inputs q = 250 lbf\/in, L = 120 in, E = 29,000.02 ksi, I = 984 in\u2074 and F = 0. The worksheet should reproduce approximately 0.02365 in (23.65 mil). This is a formula check, not an allowable-deflection statement.<\/p>\n  <\/section>\n\n  <section class=\"vbm211-section\">\n    <h2>Source classification<\/h2>\n    <div class=\"vbm211-tablewrap\"><table><thead><tr><th>Claim<\/th><th>Classification<\/th><th>Source<\/th><\/tr><\/thead><tbody>\n      <tr><td>5qL\u2074\/(384EI), simply supported full-span uniform load<\/td><td>General Euler\u2013Bernoulli beam solution; not an ISO formula<\/td><td>MIT OCW 2.082 notes; MIT OCW 1.050 formula sheet<\/td><\/tr>\n      <tr><td>FL\u00b3\/(48EI), center point load<\/td><td>General Euler\u2013Bernoulli beam solution; not an ISO formula<\/td><td>MIT OCW 2.72 lecture 3 \/ NPTEL vibration course<\/td><\/tr>\n      <tr><td>I = \u03c0(d\u2092\u2074\u2212d\u1d62\u2074)\/64<\/td><td>Geometric second moment about a centroidal diameter<\/td><td>MIT OCW 2.72 lecture 3<\/td><\/tr>\n      <tr><td>Linear superposition<\/td><td>Conditional model operation<\/td><td>MIT OCW 2.72 lecture 3 states linearity, load independence and small geometry change assumptions<\/td><\/tr>\n      <tr><td>g\u2080 and unit factors<\/td><td>Conventional metrology \/ exact definitions<\/td><td>BIPM CGPM Resolution 3-2; NIST SP 811 Appendix B<\/td><\/tr>\n    <\/tbody><\/table><\/div>\n    <ul class=\"vbm211-source-list\">\n      <li><a href=\"https:\/\/ocw.mit.edu\/courses\/2-082-ship-structural-analysis-design-13-122-spring-2003\/737b352ff63d7f434bc26dbde30f8775_notes_27_buckling.pdf\" rel=\"noopener\" target=\"_blank\">MIT OCW 2.082, Ship Structural Analysis &amp; Design, buckling notes<\/a> \u2014 page 1 shows the simply supported uniform-load maximum-deflection expression.<\/li>\n      <li><a href=\"https:\/\/ocw.mit.edu\/courses\/2-72-elements-of-mechanical-design-spring-2009\/36eed65f96add829d317d8d83b80bf30_MIT2_72s09_lec03.pdf\" rel=\"noopener\" target=\"_blank\">MIT OCW 2.72, Elements of Mechanical Design, lecture 3<\/a> \u2014 center-load displacement, circular-section I and superposition assumptions.<\/li>\n      <li><a href=\"https:\/\/ocw.mit.edu\/courses\/1-050-solid-mechanics-fall-2004\/fd4eff39aec922b8c07660006f40686e_pset04_11.pdf\" rel=\"noopener\" target=\"_blank\">MIT OCW 1.050, Solid Mechanics, Problem Set 11<\/a> \u2014 beam displacement formula sheet.<\/li>\n      <li><a href=\"https:\/\/web.mst.edu\/jthomas\/classes\/3201\/robot\/inertia\/index.html\" rel=\"noopener\" target=\"_blank\">Missouri S&amp;T, Computer Modeling lab<\/a> \u2014 published numerical uniform-load check.<\/li>\n      <li><a href=\"https:\/\/www.bipm.org\/en\/committees\/cg\/cgpm\/3-1901\/resolution-2\" rel=\"noopener\" target=\"_blank\">BIPM, 3rd CGPM Resolution 2<\/a> \u2014 conventional standard gravity 9.80665 m\/s\u00b2 and mass\/weight distinction.<\/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 inch, pound and pound-force conversion basis.<\/li>\n    <\/ul>\n    <p><strong>Accessed:<\/strong> 14 July 2026. The MIT PDF pages used above were rendered and visually checked during the audit.<\/p>\n  <\/section>\n\n  <section class=\"vbm211-section\">\n    <h2>Questions this worksheet does not answer<\/h2>\n    <details><summary>What shaft deflection is acceptable?<\/summary><p>There is no universal 25\u201350 \u03bcm, 0.1 mm or \u201c10% of bearing clearance\u201d limit that this page can safely apply. Acceptance depends on the actual bearing, seal, coupling, rotor geometry, load case, operating state, measurement definition and controlled equipment\/project requirements.<\/p><\/details>\n    <details><summary>Is static sag the same as runout or 1\u00d7 vibration?<\/summary><p>No. Static elastic deflection, indicated runout and synchronous vibration are different quantities. A universal \u201cTIR = 2 \u00d7 sag\u201d or \u201csag produces 1\u00d7 vibration\u201d rule is not used here; the measurement setup and rotating-system model must be defined.<\/p><\/details>\n    <details><summary>Does increasing diameter always give a fourth-power reduction?<\/summary><p>No. With fixed external q or F, I drives a fourth-power dependence for a solid circle. When q itself is the shaft\u2019s own weight, q grows with d\u00b2, so the resulting self-weight sag varies as 1\/d\u00b2. Mixed loads have no single diameter exponent.<\/p><\/details>\n  <\/section>\n\n  <footer class=\"vbm211-footer\">Revision: scientific audit 2026-07-14 \u00b7 English source page \u00b7 Reference calculation only. 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