{"id":66710,"date":"2026-10-13T08:41:15","date_gmt":"2026-10-13T05:41:15","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66710"},"modified":"2026-10-13T08:41:15","modified_gmt":"2026-10-13T05:41:15","slug":"shaft-torsion","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/engineering\/shaft-torsion\/","title":{"rendered":"Shaft Torsion"},"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\">Check Shear Stress and Angle of Twist in a Circular Shaft<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66710\" data-favorite-title=\"Shaft Torsion\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/engineering\/shaft-torsion\/\" data-favorite-excerpt=\"Calculate the maximum shear stress and the angle of twist in a solid or hollow circular shaft. Enter the torque, dimensions, length and shear modulus to check a shaft design in seconds.\" 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\/shaft-torsion\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66710\" data-post=\"66710\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"is-single field\" id=\"input-1\"><span class=\"font-semibold\">Loading<\/span><\/div><div class=\"field is-single has-term\" id=\"input-2\"><label for=\"field-2\">Applied torque (T)<\/label><div class=\"term\">T<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"applied_torque_t\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 1000\" min=\"0\" value=\"1000\"\/><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"unit\" aria-label=\"Unit\"><option value=\"nm\" selected>N\u00b7m<\/option><option value=\"knm\">kN\u00b7m<\/option><option value=\"lbft\">lb\u00b7ft<\/option><option value=\"lbin\">lb\u00b7in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-2\">Constant torque transmitted by the shaft.<\/small><\/div><div class=\"is-single field\" id=\"input-3\"><span class=\"font-semibold\">Shaft geometry<\/span><\/div><div class=\"field is-single\" id=\"input-4\"><label for=\"field-5\">Shaft type<\/label><div class=\"grid grid-cols-1\"><select id=\"field-5\" name=\"shaft_type\"><option value=\"solid\" selected>Solid circular shaft<\/option><option value=\"hollow\">Hollow circular shaft (tube)<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field is-single has-term\" id=\"input-5\"><label for=\"field-6\">Outer diameter (D)<\/label><div class=\"term\">D<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"outer_diameter_d\" id=\"field-6\" aria-describedby=\"calc-info-6\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 40\" min=\"0\" value=\"40\"\/><div class=\"grid grid-cols-1\"><select id=\"field-7\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><option value=\"ft\">ft<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-6\">Outside diameter of the shaft cross-section.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-6\" hidden><label for=\"field-8\">Inner diameter (d)<\/label><div class=\"term\">d<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"inner_diameter_d\" id=\"field-8\" aria-describedby=\"calc-info-8\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 20\" min=\"0\" value=\"20\"\/><div class=\"grid grid-cols-1\"><select id=\"field-9\" name=\"unit\" aria-label=\"Unit\"><option value=\"mm\" selected>mm<\/option><option value=\"cm\">cm<\/option><option value=\"m\">m<\/option><option value=\"in\">in<\/option><option value=\"ft\">ft<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-8\">Hollow shaft only: diameter of the bore. Must be smaller than the outer diameter.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-7\"><label for=\"field-10\">Shaft length (L)<\/label><div class=\"term\">L<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"shaft_length_l\" id=\"field-10\" aria-describedby=\"calc-info-10\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 1\" min=\"0\" value=\"1\"\/><div class=\"grid grid-cols-1\"><select id=\"field-11\" name=\"unit\" aria-label=\"Unit\"><option value=\"m\" selected>m<\/option><option value=\"cm\">cm<\/option><option value=\"mm\">mm<\/option><option value=\"ft\">ft<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-10\">Length of the shaft between the point where the torque is applied and the fixed end.<\/small><\/div><div class=\"is-single field\" id=\"input-8\"><span class=\"font-semibold\">Material<\/span><\/div><div class=\"field is-single has-term\" id=\"input-9\"><label for=\"field-13\">Shear modulus (G)<\/label><div class=\"term\">G<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"shear_modulus_g\" id=\"field-13\" aria-describedby=\"calc-info-13\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 80\" min=\"0\" value=\"80\"\/><div class=\"grid grid-cols-1\"><select id=\"field-14\" name=\"unit\" aria-label=\"Unit\"><option value=\"gpa\" selected>GPa<\/option><option value=\"mpa\">MPa<\/option><option value=\"psi\">psi<\/option><option value=\"ksi\">ksi<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-13\">Modulus of rigidity of the shaft material. Typical values: steel \u2248 80 GPa (11.6\u00b710\u2076 psi), aluminum \u2248 27 GPa, copper \u2248 45 GPa.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66710\">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> Torsion Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Maximum shear stress (\u03c4max)<\/span><span class=\"result-term\">\u03c4max<\/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 is-column\" id=\"output-2\"><span class=\"field-title\"><span>Angle of twist (\u03c6)<\/span><span class=\"result-term\">\u03c6<\/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=\"66710\" 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>Shaft Torsion 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\\[ J = \\frac{\\pi D^4}{32} \\quad (\\text{solid}) \\qquad J = \\frac{\\pi (D^4 - d^4)}{32} \\quad (\\text{hollow}) \\]\n\\[ \\tau_{max} = \\frac{T \\cdot c}{J}, \\quad c = \\frac{D}{2} \\qquad \\varphi = \\frac{T \\cdot L}{J \\cdot G} \\]<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(T\\) \u2014 applied torque (N\u00b7m)<\/li>\n<li>\\(D\\) \u2014 outer diameter of the shaft (m)<\/li>\n<li>\\(d\\) \u2014 inner (bore) diameter for a hollow shaft (m)<\/li>\n<li>\\(c\\) \u2014 distance from the centre to the outer surface (m)<\/li>\n<li>\\(J\\) \u2014 polar moment of inertia of the cross-section (m\u2074)<\/li>\n<li>\\(G\\) \u2014 shear modulus (modulus of rigidity) of the material (Pa)<\/li>\n<li>\\(L\\) \u2014 length of the shaft between the applied torque and the fixed end (m)<\/li>\n<li>\\(\\tau_{max}\\) \u2014 maximum shear stress at the outer surface (Pa)<\/li>\n<li>\\(\\varphi\\) \u2014 angle of twist over the length \\(L\\) (rad)<\/li>\n<\/ul>\n<p>The first two expressions describe the torsion of a circular shaft: the shear stress grows linearly from zero at the centre to a maximum at the outer surface, and the shaft twists by an angle that is proportional to the torque and length and inversely proportional to the torsional stiffness \\(JG\\).<\/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>Shaft Torsion \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a solid steel shaft of diameter \\(D = 40\\) mm and length \\(L = 1\\) m transmits a constant torque of \\(T = 1000\\) N\u00b7m. The shear modulus of steel is \\(G = 80\\) GPa.<\/p>\n<p>\\[ J = \\frac{\\pi (0.04)^4}{32} = 2.513 \\times 10^{-7}\\ \\text{m}^4 \\]<\/p>\n<p>\\[ \\tau_{max} = \\frac{1000 \\times 0.02}{2.513 \\times 10^{-7}} = 79.6 \\times 10^6\\ \\text{Pa} \\approx 79.6\\ \\text{MPa} \\]<\/p>\n<p>\\[ \\varphi = \\frac{1000 \\times 1}{2.513 \\times 10^{-7} \\times 80 \\times 10^9} = 0.04974\\ \\text{rad} \\approx 2.85^\\circ \\]<\/p>\n<p><strong>Result:<\/strong> the maximum shear stress is about 79.6 MPa (\u2248 11.5 ksi) at the outer surface, and the shaft twists through approximately 2.85\u00b0 (0.0497 rad) over its 1 m length \u2014 about 2.85\u00b0 per metre.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Shaft Torsion Calculator<\/h2>\n<ol>\n<li>Enter the <strong>Applied torque (T)<\/strong> and select its unit \u2014 N\u00b7m, kN\u00b7m, lb\u00b7ft or lb\u00b7in.<\/li>\n<li>Choose the <strong>Shaft type<\/strong>: solid circular shaft or hollow circular shaft (tube). For a hollow shaft, also enter the <strong>Inner diameter (d)<\/strong>, which must be smaller than the outer diameter.<\/li>\n<li>Enter the <strong>Outer diameter (D)<\/strong> and the <strong>Shaft length (L)<\/strong> with their units (mm, cm, m, in or ft).<\/li>\n<li>Enter the <strong>Shear modulus (G)<\/strong> of the material \u2014 steel \u2248 80 GPa, aluminum \u2248 27 GPa, copper \u2248 45 GPa \u2014 and choose the unit (GPa, MPa, psi or ksi).<\/li>\n<li>Read the results: the <strong>maximum shear stress (\u03c4max)<\/strong> in MPa, ksi and psi, and the <strong>angle of twist (\u03c6)<\/strong> in degrees, radians, degrees per metre and degrees per foot.<\/li>\n<\/ol>\n<h2>What Is Shaft Torsion?<\/h2>\n<p>When a torque is applied along the axis of a shaft \u2014 as happens in every transmission shaft, drive line, drill string or propeller shaft \u2014 the material is twisted. Each cross-section rotates relative to the neighbouring one, and this rotation is resisted by shear stresses distributed over the cross-section. For circular shafts (solid or with a concentric bore) the analysis is exact and simple: the shear stress varies linearly from zero at the centre to a maximum at the outer surface, which is why shafts and tubes fail or yield first on their outside.<\/p>\n<p>Two quantities matter in a torsion check. The first is the <strong>maximum shear stress<\/strong>, \\(\\tau_{max} = Tc\/J\\): it must stay below the allowable shear stress of the material. For ductile steels, the yield strength in shear is roughly 0.5\u20130.6 times the tensile yield strength, and a suitable safety factor is applied on top. The second is the <strong>angle of twist<\/strong>, \\(\\varphi = TL\/JG\\): even a shaft that is strong enough may twist too much, causing misalignment, vibration, wind-up in control systems or positional error in precision drives.<\/p>\n<p>The geometry term \\(J\\), the polar moment of inertia, scales with the fourth power of the diameter. Doubling the diameter of a solid shaft reduces the stress and the twist by a factor of 16. This is why hollow shafts are so effective: removing the lightly stressed material near the centre costs very little stiffness but saves a great deal of weight, which is why drive shafts, roll shafts and many aerospace parts are tubes.<\/p>\n<p>When interpreting the result, compare \\(\\tau_{max}\\) with the material's allowable shear stress, and compare \\(\\varphi\\) with your twist limit. As a traditional rule of thumb, machine shafts are limited to about 1\u00b0 of twist over a length of 20 diameters; precision and long-line shafting often needs much tighter limits.<\/p>\n<h2>Assumptions, Limitations and Practical Tips<\/h2>\n<ul>\n<li><strong>Circular cross-sections only.<\/strong> The formulas are exact for solid or hollow circular shafts with a concentric bore. Non-circular sections (rectangular bars, keyways, splines) twist very differently and need separate methods.<\/li>\n<li><strong>Uniform shaft, constant torque.<\/strong> The calculation assumes a constant cross-section, a single material and a torque that does not change along the length. For stepped shafts or several torques, compute each segment separately and add the twist angles.<\/li>\n<li><strong>Linear elastic behaviour.<\/strong> Stresses must remain below the elastic limit; the result does not predict yielding or plastic collapse.<\/li>\n<li><strong>Keyways, holes and shoulders<\/strong> concentrate stress. A real shaft with a keyseat or a shoulder may see local stresses 1.5\u20133 times the nominal value, so allow for stress concentration factors in critical designs.<\/li>\n<li><strong>Check both criteria.<\/strong> A short, stout shaft is usually governed by stress; a long, slender shaft is often governed by twist. Also remember that a long shaft can fail by buckling or whirling \u2014 a related check is our <a href=\"https:\/\/wpcalc.com\/en\/engineering\/euler-buckling-load\/\">Euler buckling load calculator<\/a>.<\/li>\n<li>For shafts that also carry bending loads (gears, belts, pulleys), superimpose the bending stresses from a beam analysis \u2014 see the <a href=\"https:\/\/wpcalc.com\/en\/engineering\/simply-supported-beam-with-center-point-load\/\">simply supported beam with center point load<\/a> and <a href=\"https:\/\/wpcalc.com\/en\/engineering\/simply-supported-beam-with-uniform-load\/\">uniformly loaded beam<\/a> calculators.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the maximum shear stress and the angle of twist in a solid or hollow circular shaft. Enter the torque, dimensions, length and shear modulus to check a shaft design in seconds.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[4],"tags":[97],"class_list":["post-66710","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\/66710","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\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=66710"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66710\/revisions"}],"predecessor-version":[{"id":66711,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66710\/revisions\/66711"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66710"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66710"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66710"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}