{"id":100223,"date":"2026-02-15T20:28:12","date_gmt":"2026-02-15T20:28:12","guid":{"rendered":"https:\/\/vibromera.eu\/?post_type=calculator&#038;p=100223"},"modified":"2026-07-15T10:51:21","modified_gmt":"2026-07-15T10:51:21","slug":"shaft-diameter-torsion","status":"publish","type":"calculator","link":"https:\/\/vibromera.eu\/it\/calculators\/shaft-diameter-torsion\/","title":{"rendered":"Documented Circular-Shaft Elastic-Torsion Worksheet"},"content":{"rendered":"\n<script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"WebApplication\",\"name\":\"Documented Circular-Shaft Elastic-Torsion Worksheet\",\"description\":\"Calculate a mathematical outer-diameter lower bound for a uniform circular shaft from externally approved nominal shear-stress and optional twist-rate criteria.\",\"url\":\"https:\/\/vibromera.eu\/calculators\/shaft-diameter-torsion\/\",\"applicationCategory\":\"EngineeringApplication\",\"operatingSystem\":\"Any\",\"isAccessibleForFree\":true,\"inLanguage\":\"en\",\"publisher\":{\"@type\":\"Organization\",\"name\":\"Vibromera\"}}<\/script>\n<style>\n.vbm223{--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}.vbm223 *{box-sizing:border-box}.vbm223 h1,.vbm223 h2,.vbm223 h3{line-height:1.2}.vbm223 h1{font-size:clamp(27px,4vw,42px);margin:8px 0 12px}.vbm223 h2{font-size:22px;margin:0 0 16px}.vbm223 h3{font-size:17px;margin:22px 0 8px}.vbm223 p{margin:8px 0}.vbm223 a{color:var(--blue)}.vbm223-header{text-align:center;padding:32px 18px 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button{border:0;border-radius:9px;padding:11px 17px;font:inherit;font-weight:760;cursor:pointer}.vbm223-primary{background:var(--blue);color:#fff}.vbm223-primary:hover{background:var(--blue2)}.vbm223-secondary{background:#e8eef5;color:var(--ink)}.vbm223-error{display:none;margin-top:14px;padding:11px 13px;border:1px solid #edb5b5;border-radius:8px;background:var(--red-bg);color:var(--red);font-weight:650}.vbm223-error.show{display:block}.vbm223-results{display:none}.vbm223-results.show{display:block}.vbm223-result-grid{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:12px}.vbm223-result{border:1px solid var(--line);border-radius:10px;background:var(--soft);padding:13px}.vbm223-result.primary{grid-column:1\/-1;border-color:#7dacda;background:#eef6ff}.vbm223-result-label{color:var(--muted);font-size:12px;font-weight:700;text-transform:uppercase;letter-spacing:.06em}.vbm223-result-value{font:700 18px\/1.35 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summary{cursor:pointer;font-weight:760}.vbm223-hidden{display:none!important}.vbm223-small{font-size:13px;color:var(--muted)}\n@media(max-width:700px){.vbm223{padding-left:10px;padding-right:10px}.vbm223-card{padding:16px}.vbm223-grid,.vbm223-result-grid{grid-template-columns:1fr}.vbm223-result.primary,.vbm223-field.full{grid-column:auto}.vbm223-combo select{min-width:92px}.vbm223 table{display:block;overflow-x:auto;white-space:normal}}\n<\/style>\n<main class=\"vbm223\">\n  <header class=\"vbm223-header\">\n    <p class=\"vbm223-kicker\">Reference worksheet \u00b7 uniform circular shaft<\/p>\n    <h1>Documented Circular-Shaft Elastic-Torsion Worksheet<\/h1>\n    <p class=\"vbm223-lead\">Calculate a mathematical outer-diameter lower bound from an externally approved nominal elastic surface-shear-stress limit and, optionally, an externally approved elastic twist-rate limit.<\/p>\n    <div class=\"vbm223-badges\"><span class=\"vbm223-badge\">General torsion equations<\/span><span class=\"vbm223-badge\">No material preset<\/span><span class=\"vbm223-badge\">No generated safety factor<\/span><\/div>\n  <\/header>\n\n  <div class=\"vbm223-notice\"><strong>Scope, not shaft certification.<\/strong> This is a general engineering Saint-Venant circular-shaft worksheet, not an ISO, API, ASME or DIN design\/acceptance calculation. It does not choose a material, shear allowable, load factor, safety factor, fatigue method, keyway factor, standard size or safe machine verdict.<\/div>\n\n  <form id=\"vbm223-form\" class=\"vbm223-card\" autocomplete=\"off\" novalidate>\n    <h2>Documented inputs<\/h2>\n    <div class=\"vbm223-grid\">\n      <div class=\"vbm223-field\"><label for=\"vbm223-T\">Documented design torque magnitude T <span class=\"vbm223-hint\">Enter the already approved governing torque after external load\/duty\/dynamic factors<\/span><\/label><div class=\"vbm223-combo\"><input id=\"vbm223-T\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\" aria-describedby=\"vbm223-number-note\"><select id=\"vbm223-T-unit\" aria-label=\"Torque unit\"><option value=\"Nm\">N\u00b7m<\/option><option value=\"lbfin\">lbf\u00b7in<\/option><option value=\"lbfft\">lbf\u00b7ft<\/option><\/select><\/div><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-k\">Bore ratio k = d\u1d62\/d\u2092 <span class=\"vbm223-hint\">0 for solid; must satisfy 0 \u2264 k &lt; 1<\/span><\/label><input id=\"vbm223-k\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\"><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-tau\">Approved nominal elastic surface-shear-stress limit \u03c4<sub>lim<\/sub> <span class=\"vbm223-hint\">Already includes every externally required design reduction\/factor<\/span><\/label><div class=\"vbm223-combo\"><input id=\"vbm223-tau\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\"><select id=\"vbm223-tau-unit\" aria-label=\"Shear-stress unit\"><option value=\"MPa\">MPa<\/option><option value=\"ksi\">ksi<\/option><\/select><\/div><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-twist-mode\">Twist-rate criterion<\/label><select id=\"vbm223-twist-mode\"><option value=\"off\">Not evaluated<\/option><option value=\"on\">Evaluate approved elastic twist-rate limit<\/option><\/select><\/div>\n      <div class=\"vbm223-field vbm223-hidden\" id=\"vbm223-G-field\"><label for=\"vbm223-G\">Approved shear modulus G <span class=\"vbm223-hint\">For the documented material state and temperature<\/span><\/label><div class=\"vbm223-combo\"><input id=\"vbm223-G\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\" disabled><select id=\"vbm223-G-unit\" aria-label=\"Shear-modulus unit\" disabled><option value=\"GPa\">GPa<\/option><option value=\"Mpsi\">Mpsi<\/option><\/select><\/div><\/div>\n      <div class=\"vbm223-field vbm223-hidden\" id=\"vbm223-phi-field\"><label for=\"vbm223-phi\">Approved twist-rate limit \u03c6\u2032<sub>lim<\/sub><\/label><div class=\"vbm223-combo\"><input id=\"vbm223-phi\" type=\"text\" inputmode=\"decimal\" placeholder=\"blank\" disabled><select id=\"vbm223-phi-unit\" aria-label=\"Twist-rate unit\" disabled><option value=\"degm\">deg\/m<\/option><option value=\"radm\">rad\/m<\/option><option value=\"degft\">deg\/ft<\/option><\/select><\/div><\/div>\n      <p class=\"vbm223-field full vbm223-small\" id=\"vbm223-number-note\">Use a decimal point or decimal comma; signs, thousands separators, units inside number fields and scientific notation are intentionally rejected.<\/p>\n\n      <div class=\"vbm223-field\"><label for=\"vbm223-project\">Project \/ equipment<\/label><input id=\"vbm223-project\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-shaft\">Shaft \/ section identifier<\/label><input id=\"vbm223-shaft\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-torque-source\">Torque source, operating state and factors<\/label><input id=\"vbm223-torque-source\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-limit-source\">Shear-limit source and exact location<\/label><input id=\"vbm223-limit-source\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-twist-source\">G \/ twist-limit source <span class=\"vbm223-hint\">Write not used when criterion is off<\/span><\/label><input id=\"vbm223-twist-source\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-geometry-source\">Circular-section and bore-ratio basis<\/label><input id=\"vbm223-geometry-source\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-exclusions\">Excluded loads\/features and companion checks<\/label><input id=\"vbm223-exclusions\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-prepared\">Prepared \/ reviewed by<\/label><input id=\"vbm223-prepared\" type=\"text\" placeholder=\"required\"><\/div>\n      <div class=\"vbm223-field\"><label for=\"vbm223-date\">Record date<\/label><input id=\"vbm223-date\" type=\"text\" placeholder=\"YYYY-MM-DD\"><\/div>\n      <div class=\"vbm223-field full\"><label for=\"vbm223-notes\">Notes <span class=\"vbm223-hint\">Optional: torque direction, temperature, manufacturing allowance, size series<\/span><\/label><textarea id=\"vbm223-notes\"><\/textarea><\/div>\n    <\/div>\n\n    <div class=\"vbm223-checks\">\n      <label class=\"vbm223-check\"><input type=\"checkbox\" id=\"vbm223-c1\"><span>The analysed segment is straight, homogeneous, prismatic and circular, with constant d\u2092, d\u1d62, G and internal torque over the stated region.<\/span><\/label>\n      <label class=\"vbm223-check\"><input type=\"checkbox\" id=\"vbm223-c2\"><span>Small-strain linear-elastic Saint-Venant torsion is appropriate; warping restraint, plasticity and noncircular\/open-section behavior are outside this calculation.<\/span><\/label>\n      <label class=\"vbm223-check\"><input type=\"checkbox\" id=\"vbm223-c3\"><span>The entered torque and limits are externally approved design inputs; no material preset or extra safety\/load factor is to be invented by this worksheet.<\/span><\/label>\n      <label class=\"vbm223-check\"><input type=\"checkbox\" id=\"vbm223-c4\"><span>I understand that the result is only an ideal mathematical lower bound; keyways, shoulders, holes, fits, combined loads, fatigue, dynamics and manufacturing\/standard-size selection require separate controlled checks.<\/span><\/label>\n    <\/div>\n    <div class=\"vbm223-actions\"><button type=\"submit\" class=\"vbm223-primary\">Calculate documented criteria<\/button><button type=\"button\" class=\"vbm223-secondary\" id=\"vbm223-clear\">Clear<\/button><\/div>\n    <div id=\"vbm223-error\" class=\"vbm223-error\" role=\"alert\"><\/div>\n  <\/form>\n\n  <section id=\"vbm223-results\" class=\"vbm223-card vbm223-results\" aria-live=\"polite\">\n    <h2>Reference results<\/h2>\n    <div class=\"vbm223-result-grid\">\n      <div class=\"vbm223-result primary\"><div class=\"vbm223-result-label\">Mathematical outer-diameter lower bound<\/div><div class=\"vbm223-result-value\" id=\"vbm223-r-d\">\u2014<\/div><\/div>\n      <div class=\"vbm223-result\"><div class=\"vbm223-result-label\">Stress-criterion diameter d<sub>\u03c4<\/sub><\/div><div class=\"vbm223-result-value\" id=\"vbm223-r-dtau\">\u2014<\/div><\/div>\n      <div class=\"vbm223-result\"><div class=\"vbm223-result-label\">Twist-criterion diameter d<sub>\u03c6<\/sub><\/div><div class=\"vbm223-result-value\" id=\"vbm223-r-dphi\">\u2014<\/div><\/div>\n      <div class=\"vbm223-result\"><div class=\"vbm223-result-label\">Governing enabled criterion<\/div><div class=\"vbm223-result-value\" id=\"vbm223-r-governing\">\u2014<\/div><\/div>\n      <div class=\"vbm223-result\"><div class=\"vbm223-result-label\">Corresponding inner diameter<\/div><div class=\"vbm223-result-value\" id=\"vbm223-r-di\">\u2014<\/div><\/div>\n      <div class=\"vbm223-result\"><div class=\"vbm223-result-label\">Polar second moment J at bound<\/div><div class=\"vbm223-result-value\" id=\"vbm223-r-J\">\u2014<\/div><\/div>\n      <div class=\"vbm223-result\"><div class=\"vbm223-result-label\">Nominal surface shear stress at bound<\/div><div class=\"vbm223-result-value\" id=\"vbm223-r-tau\">\u2014<\/div><\/div>\n      <div class=\"vbm223-result\"><div class=\"vbm223-result-label\">Elastic twist rate at bound<\/div><div class=\"vbm223-result-value\" id=\"vbm223-r-phi\">\u2014<\/div><\/div>\n    <\/div>\n    <div class=\"vbm223-status\" id=\"vbm223-r-status\"><\/div>\n  <\/section>\n\n  <section class=\"vbm223-card\">\n    <h2>Equations and definitions<\/h2>\n    <p>Let d\u2092 be outer diameter, d\u1d62 inner diameter, k = d\u1d62\/d\u2092, T the constant internal torque magnitude, \u03c4<sub>lim<\/sub> the externally approved nominal elastic surface-shear-stress limit, G the approved shear modulus, and \u03c6\u2032<sub>lim<\/sub> the approved twist angle per unit length in rad\/m.<\/p>\n    <div class=\"vbm223-formula\">J = \u03c0(d\u2092\u2074 \u2212 d\u1d62\u2074)\/32 = \u03c0d\u2092\u2074(1 \u2212 k\u2074)\/32<br>\u03c4max = T(d\u2092\/2)\/J = 16T\/[\u03c0d\u2092\u00b3(1 \u2212 k\u2074)]<br>\u03c6\u2032 = d\u03c6\/dx = T\/(GJ)<\/div>\n    <p>Solving the first criterion for d\u2092 gives:<\/p>\n    <div class=\"vbm223-formula\">d\u03c4 = {16T\/[\u03c0\u03c4lim(1 \u2212 k\u2074)]}^(1\/3)<\/div>\n    <p>When the optional twist-rate criterion is enabled:<\/p>\n    <div class=\"vbm223-formula\">d\u03c6 = {32T\/[\u03c0G\u03c6\u2032lim(1 \u2212 k\u2074)]}^(1\/4)<br>d\u2092,lower = max(d\u03c4, d\u03c6)<\/div>\n    <p>With only the stress criterion enabled, d\u2092,lower = d\u03c4. Diameter is not automatically rounded: selecting a production size, tolerance and manufacturing allowance is an external design action.<\/p>\n    <h3>Dimensional check<\/h3>\n    <p>T\/\u03c4 has dimension length\u00b3, so d\u03c4 has dimension length. T\/(G\u03c6\u2032) has dimension length\u2074 because \u03c6\u2032 has dimension 1\/length, so d\u03c6 has dimension length. These are general elastic torsion equations, not formulas issued by ISO.<\/p>\n  <\/section>\n\n  <section class=\"vbm223-card\">\n    <h2>Model boundary<\/h2>\n    <ul>\n      <li>Use the governing internal design torque for the analysed segment\u2014not automatically motor nameplate torque, average torque or power\/RPM torque. Startup, reversal, braking, shock, transient and duty-cycle effects must already be handled by the controlled load basis.<\/li>\n      <li>The worksheet does not derive \u03c4<sub>lim<\/sub> from tensile yield or ultimate strength. The University of Maryland example explicitly warns that tensile-test strengths cannot be directly compared with shear stress.<\/li>\n      <li>No material\/shear-modulus table, 0.577 multiplier, 50\u201360% rule, 1.5\u20133.0 safety factor or keyway K<sub>t<\/sub> range is generated. Those depend on material data, failure theory, loading, fatigue method, geometry and controlled design rules.<\/li>\n      <li>Combined bending and torsion are not replaced by one universal \u201cequivalent torque\u201d here. A compatible combined-stress\/fatigue method must keep its stress normalization, load factors and allowables consistent.<\/li>\n      <li>Stress concentrations at keyways, splines, shoulders, grooves, holes and fits; contact\/fretting; fatigue; yielding; buckling; critical speed; lateral vibration; thermal effects and bearing\/coupling loads are outside the model.<\/li>\n      <li>The hollow\/solid geometry ratios for the same d\u2092 are (1 \u2212 k\u2074) for J and nominal torsional section modulus, and (1 \u2212 k\u00b2) for cross-sectional area. They are geometric comparisons, not a claim that two shafts have equal fatigue strength or mass after redesign.<\/li>\n    <\/ul>\n  <\/section>\n\n  <section class=\"vbm223-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>T = GJ d\u03c6\/dx; \u03c4 = Tr\/J; circular\/cylindrical-tube scope<\/td><td>General linear-elastic circular-shaft torsion; not an ISO formula<\/td><td>MIT OCW 16.01, M9 Shafts: Torsion of Circular Shafts, PDF pages 5\u20136<\/td><\/tr>\n      <tr><td>Solid circle \u03c4max = 16T\/(\u03c0d\u00b3); published T = 5400 lbf\u00b7in gives 27.50 ksi at d = 1.000 in<\/td><td>Published formula and numerical example<\/td><td>University of Maryland, Mechanics of Materials\u2014101, Example 4.3<\/td><\/tr>\n      <tr><td>Published elastic benchmark: d = 40 mm, \u03c4 = 130 MPa, T = 1633.63 N\u00b7m, G inferred from \u03c4\/\u03b3 = 80 GPa, twist \u2248 4.66\u00b0 over 1 m<\/td><td>Published numerical example<\/td><td>Mississippi State University, Pure Torsion Sample Test, problem 1(a)<\/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:\/\/ocw.mit.edu\/courses\/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006\/b86998d8512c92fe3d0ce0f112a7d3d1_gm9_10notes.pdf\" rel=\"noopener\" target=\"_blank\">MIT OCW 16.01, M9 Shafts: Torsion of Circular Shafts<\/a>.<\/li>\n      <li><a href=\"https:\/\/huginn.eng.umd.edu\/ebooks\/ENES220_2018\/resources\/_pdfs_\/Chapter_4_rev_1_5th_ed_9_10_2016_6.pdf\" rel=\"noopener\" target=\"_blank\">University of Maryland, Mechanics of Materials\u2014101, Example 4.3<\/a>.<\/li>\n      <li><a href=\"https:\/\/www.ae.msstate.edu\/tupas\/SA2\/test1_text.html\" rel=\"noopener\" target=\"_blank\">Mississippi State University, Pure Torsion Sample Test<\/a>.<\/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>.<\/li>\n    <\/ul>\n    <p><strong>Accessed:<\/strong> 15 July 2026. Every actual torque factor, material property, allowable, twist limit, stress-concentration\/fatigue factor and acceptance decision remains <strong>NEEDS_LICENSED_SOURCE<\/strong> until the controlled project\/OEM\/code basis is supplied.<\/p>\n    <h3>Published benchmark entry<\/h3>\n    <p>Enter T = 1633.63 N\u00b7m, k = 0, \u03c4<sub>lim<\/sub> = 130 MPa, enable twist, G = 80 GPa and \u03c6\u2032<sub>lim<\/sub> = 4.66 deg\/m. The stress criterion gives 40.0000149 mm and governs by the rounding of the published inputs; calculated twist at that diameter is 4.65528 deg\/m, consistent with the source\u2019s 40 mm and 4.66\u00b0 over 1 m.<\/p>\n  <\/section>\n\n  <section class=\"vbm223-card\">\n    <h2>Questions<\/h2>\n    <details><summary>Why does the worksheet not offer steel grades and safety factors?<\/summary><p>A grade name alone does not establish heat treatment, product form, temperature, statistical basis, shear allowable, fatigue condition or design code. The old presets mixed nominal tensile properties, guessed shear limits and a second safety-factor division. This version accepts one controlled limit only.<\/p><\/details>\n    <details><summary>Is the larger lower-bound diameter a final drawing size?<\/summary><p>No. It is the exact ideal-model lower bound for the enabled input criteria. A competent design must select a manufactured size\/tolerance and verify every omitted feature, load case, strength\/fatigue criterion, fit and dynamic requirement.<\/p><\/details>\n    <details><summary>Can I use the old combined-bending \u201cequivalent torque\u201d formula?<\/summary><p>Not without a controlled method. Expressions normalized to bending stress and expressions normalized to torsional shear stress are not interchangeable. Substituting one into the other can change a factor of two and under-size a shaft.<\/p><\/details>\n  <\/section>\n<\/main>\n<script>\n(function(){\n'use strict';\nvar form=document.getElementById('vbm223-form'),error=document.getElementById('vbm223-error'),results=document.getElementById('vbm223-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 outside the finite numeric range.')}\nfunction calculate(v){\n  var IN=0.0254,FT=0.3048,LBF=4.4482216152605,PSI=LBF\/(IN*IN);\n  if(!['Nm','lbfin','lbfft'].includes(v.TUnit)||!['MPa','ksi'].includes(v.tauUnit)||!['off','on'].includes(v.twistMode)||!['GPa','Mpsi'].includes(v.GUnit)||!['degm','radm','degft'].includes(v.phiUnit))throw new Error('Select valid engineering units and criteria.');\n  var torqueFactor=v.TUnit==='Nm'?1:v.TUnit==='lbfin'?LBF*IN:LBF*FT,T=strictDecimal(v.T,'Design torque T','positive')*torqueFactor,k=strictDecimal(v.k,'Bore ratio k','nonnegative');if(k>=1)throw new Error('Bore ratio k must be less than 1.');\n  var tauLim=strictDecimal(v.tau,'Shear-stress limit','positive')*(v.tauUnit==='MPa'?1e6:1000*PSI),shape=1-Math.pow(k,4),dTau=Math.pow(16*T\/(Math.PI*tauLim*shape),1\/3),G=null,phiLim=null,dPhi=null;\n  if(v.twistMode==='on'){G=strictDecimal(v.G,'Shear modulus G','positive')*(v.GUnit==='GPa'?1e9:1e6*PSI);var rawPhi=strictDecimal(v.phi,'Twist-rate limit','positive');phiLim=v.phiUnit==='radm'?rawPhi:v.phiUnit==='degm'?rawPhi*Math.PI\/180:rawPhi*Math.PI\/180\/FT;dPhi=Math.pow(32*T\/(Math.PI*G*phiLim*shape),1\/4)}\n  var d=dPhi===null?dTau:Math.max(dTau,dPhi),di=k*d,J=Math.PI*Math.pow(d,4)*shape\/32,tauActual=16*T\/(Math.PI*Math.pow(d,3)*shape),phiActual=dPhi===null?null:T\/(G*J),governing;\n  if(dPhi===null)governing='stress (only enabled criterion)';else{var tol=Math.max(dTau,dPhi)*Number.EPSILON*64;governing=Math.abs(dTau-dPhi)<=tol?'equal within numeric precision':dTau>dPhi?'stress':'twist rate'}\n  finite([T,k,tauLim,shape,dTau,d,di,J,tauActual].concat(G===null?[]:[G,phiLim,dPhi,phiActual]));\n  return{TUnit:v.TUnit,tauUnit:v.tauUnit,twistMode:v.twistMode,GUnit:v.GUnit,phiUnit:v.phiUnit,T:T,k:k,tauLim:tauLim,shape:shape,dTau:dTau,G:G,phiLim:phiLim,dPhi:dPhi,d:d,di:di,J:J,tauActual:tauActual,phiActual:phiActual,governing:governing};\n}\nfunction requiredText(id,label){var value=document.getElementById(id).value.trim();if(!value)throw new Error(label+' is required.');return value}\nfunction values(){return{T:document.getElementById('vbm223-T').value,TUnit:document.getElementById('vbm223-T-unit').value,k:document.getElementById('vbm223-k').value,tau:document.getElementById('vbm223-tau').value,tauUnit:document.getElementById('vbm223-tau-unit').value,twistMode:document.getElementById('vbm223-twist-mode').value,G:document.getElementById('vbm223-G').value,GUnit:document.getElementById('vbm223-G-unit').value,phi:document.getElementById('vbm223-phi').value,phiUnit:document.getElementById('vbm223-phi-unit').value}}\nfunction fmt(n){if(!Number.isFinite(n))return'\u2014';if(n===0)return'0';var a=Math.abs(n);return(a>=1e8||a<1e-6?n.toExponential(7):n.toPrecision(8)).replace(\/(\\.\\d*?[1-9])0+(e|$)\/,'$1$2').replace(\/\\.0+(e|$)\/,'$1')}\nfunction setText(id,text){document.getElementById(id).textContent=text}\nfunction diameterText(d){return fmt(d*1000)+' mm \u00b7 '+fmt(d\/0.0254)+' in'}\nfunction render(r){\n  setText('vbm223-r-d',diameterText(r.d));setText('vbm223-r-dtau',diameterText(r.dTau));setText('vbm223-r-dphi',r.dPhi===null?'not evaluated':diameterText(r.dPhi));setText('vbm223-r-governing',r.governing);setText('vbm223-r-di',diameterText(r.di));\n  setText('vbm223-r-J',fmt(r.J*1e12)+' mm\u2074 \u00b7 '+fmt(r.J\/Math.pow(0.0254,4))+' in\u2074');setText('vbm223-r-tau',fmt(r.tauActual\/1e6)+' MPa \u00b7 '+fmt(r.tauActual\/6894757.293168361)+' ksi');setText('vbm223-r-phi',r.phiActual===null?'not evaluated':fmt(r.phiActual)+' rad\/m \u00b7 '+fmt(r.phiActual*180\/Math.PI)+' deg\/m');\n  setText('vbm223-r-status','No rounding, material choice, safety factor or acceptance verdict has been added. Project: '+requiredText('vbm223-project','Project \/ equipment')+'; shaft: '+requiredText('vbm223-shaft','Shaft \/ section identifier')+'.');results.classList.add('show');\n}\nfunction setError(message){error.textContent=message;error.classList.add('show');results.classList.remove('show')}\nfunction clearError(){error.textContent='';error.classList.remove('show')}\nfunction updateFields(){var on=document.getElementById('vbm223-twist-mode').value==='on';['G','phi'].forEach(function(n){document.getElementById('vbm223-'+n+'-field').classList.toggle('vbm223-hidden',!on);document.getElementById('vbm223-'+n).disabled=!on;document.getElementById('vbm223-'+n+'-unit').disabled=!on});results.classList.remove('show');clearError()}\nform.addEventListener('submit',function(e){e.preventDefault();clearError();try{['vbm223-project','vbm223-shaft','vbm223-torque-source','vbm223-limit-source','vbm223-twist-source','vbm223-geometry-source','vbm223-exclusions','vbm223-prepared','vbm223-date'].forEach(function(id){requiredText(id,document.querySelector('label[for=\"'+id+'\"]').textContent.trim())});['vbm223-c1','vbm223-c2','vbm223-c3','vbm223-c4'].forEach(function(id){if(!document.getElementById(id).checked)throw new Error('Confirm all four model and use statements.');});render(calculate(values()))}catch(ex){setError(ex.message||'Unable to calculate.')}});\nform.addEventListener('keydown',function(e){if(e.key==='Enter'&&e.target.tagName!=='TEXTAREA'&&e.target.tagName!=='BUTTON'){e.preventDefault();form.requestSubmit()}});\ndocument.getElementById('vbm223-twist-mode').addEventListener('change',updateFields);document.getElementById('vbm223-clear').addEventListener('click',function(){form.reset();form.querySelectorAll('input[type=text],textarea').forEach(function(el){el.value=''});results.classList.remove('show');clearError();updateFields();document.getElementById('vbm223-T').focus()});form.querySelectorAll('input,select,textarea').forEach(function(el){if(el.id!=='vbm223-twist-mode')el.addEventListener('input',function(){results.classList.remove('show');clearError()})});updateFields();\n})();\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>Calculate an ideal circular-shaft diameter lower bound from externally approved nominal shear-stress and optional twist-rate criteria, without material or safety-factor presets.<\/p>","protected":false},"featured_media":0,"template":"","meta":{"ai_generated_summary":"","footnotes":""},"categories":[],"tags":[],"class_list":["post-100223","calculator","type-calculator","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/it\/wp-json\/wp\/v2\/calculator\/100223","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/it\/wp-json\/wp\/v2\/calculator"}],"about":[{"href":"https:\/\/vibromera.eu\/it\/wp-json\/wp\/v2\/types\/calculator"}],"version-history":[{"count":3,"href":"https:\/\/vibromera.eu\/it\/wp-json\/wp\/v2\/calculator\/100223\/revisions"}],"predecessor-version":[{"id":102568,"href":"https:\/\/vibromera.eu\/it\/wp-json\/wp\/v2\/calculator\/100223\/revisions\/102568"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/it\/wp-json\/wp\/v2\/media?parent=100223"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/it\/wp-json\/wp\/v2\/categories?post=100223"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/it\/wp-json\/wp\/v2\/tags?post=100223"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}