{"id":2625,"date":"2024-08-07T19:17:06","date_gmt":"2024-08-07T13:47:06","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=2625"},"modified":"2025-07-16T15:25:35","modified_gmt":"2025-07-16T09:55:35","slug":"shear-stress-distribution-in-circular-section","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section","title":{"rendered":"Shear Stress Distribution in Circular Section"},"content":{"rendered":"<h2 style=\"text-align: justify;\">Solid circular shaft<\/h2>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2626 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/solid-circular-shaft.jpg\" alt=\"Solid Circular Shaft\" width=\"517\" height=\"224\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/solid-circular-shaft.jpg 517w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/solid-circular-shaft-300x130.jpg 300w\" sizes=\"auto, (max-width: 517px) 100vw, 517px\" \/><\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_1 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#Hollow-circular-shaft\" >Hollow circular shaft<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#Thin-circular-tube-with-mean-radius-R\" >Thin circular tube with mean radius R<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#Composite-circular-shaft\" >Composite circular shaft<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#DESIGN-OF-SHAFT\" >DESIGN OF SHAFT<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#Strength-criteria\" >Strength criteria<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#SERIES-COMBINATION-OF-SHAFT\" >SERIES COMBINATION OF SHAFT<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#PARALLEL-COMBINATION-OF-SHAFT\" >PARALLEL COMBINATION OF SHAFT<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#STRAIN-ENERGY-IN-TORSION\" >STRAIN ENERGY IN TORSION<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#SHAFT-SUBJECTED-TO-COMBINED-BENDING-MOMENT-AND-TWISTING-MOMENT\" >SHAFT SUBJECTED TO COMBINED BENDING MOMENT AND TWISTING MOMENT<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#Equivalent-Bending-Moment\" >Equivalent Bending Moment<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#Equivalent-Torque\" >Equivalent Torque<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#SHAFT-SUBJECTED-TO-COMBINED-AXIAL-FORCE-AND-TORSIONAL-MOMENT\" >SHAFT SUBJECTED TO COMBINED AXIAL FORCE AND TORSIONAL MOMENT<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#Effect-of-axial-force\" >Effect of axial force<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/shear-stress-distribution-in-circular-section\/#Effect-of-torsion\" >Effect of torsion<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Hollow-circular-shaft\"><\/span>Hollow circular shaft <img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-2627 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/hollow-circular-shaft.jpg\" alt=\"Hollow circular shaft\" width=\"337\" height=\"178\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/hollow-circular-shaft.jpg 337w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/hollow-circular-shaft-300x158.jpg 300w\" sizes=\"auto, (max-width: 337px) 100vw, 337px\" \/><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">D<sub>i<\/sub> = internal diameter<br \/>\nD<sub>o<\/sub> = outer diameter<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2628 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/outer-diameter.jpg\" alt=\"Outer Diameter\" width=\"168\" height=\"117\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/outer-diameter.jpg 168w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/outer-diameter-130x90.jpg 130w\" sizes=\"auto, (max-width: 168px) 100vw, 168px\" \/><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Thin-circular-tube-with-mean-radius-R\"><\/span>Thin circular tube with mean radius R <img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-2630 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/shear-stress-2.jpg\" alt=\"Shear Stress\" width=\"330\" height=\"181\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/shear-stress-2.jpg 330w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/shear-stress-2-300x165.jpg 300w\" sizes=\"auto, (max-width: 330px) 100vw, 330px\" \/><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Let t is thickness of tube<\/p>\n<p style=\"text-align: justify;\">I<sub>P<\/sub> = Ar<sup>2<br \/>\n<\/sup>A = (2\u03c0R) \u00d7 t<\/p>\n<p style=\"text-align: justify;\">I<sub>P<\/sub> = (2\u03c0Rt)R<sup>2<\/sup> = 2\u03c0R<sup>3<\/sup>t<\/p>\n<p style=\"text-align: justify;\">Z<sub>P<\/sub> = I<sub>P<\/sub>\/R<sub>max<\/sub> = 2\u03c0R<sup>3<\/sup>t\/ R = 2\u03c0R<sup>2<\/sup>t<\/p>\n<p style=\"text-align: justify;\">The shear stress distribution is assumed uniform across the thickness and is given by<\/p>\n<p style=\"text-align: justify;\">\u03c4 = T\/Z<sub>P<\/sub><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Composite-circular-shaft\"><\/span>Composite circular shaft<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Consider a composite circular shaft made from two materials whose modulus of rigidity are G1 and G2 In composite shaft, total torque T is shared by both shaft.<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2631 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/shear-strain-1-1.jpg\" alt=\"Shear Strain\" width=\"774\" height=\"206\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/shear-strain-1-1.jpg 774w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/shear-strain-1-1-300x80.jpg 300w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/shear-strain-1-1-768x204.jpg 768w\" sizes=\"auto, (max-width: 774px) 100vw, 774px\" \/><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"DESIGN-OF-SHAFT\"><\/span>DESIGN OF SHAFT<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Shaft is designed on the basis of following two criteria,<\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Strength-criteria\"><\/span>Strength criteria<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\">\u03c4\u2264\u03c4<sub>P<\/sub><\/p>\n<p style=\"text-align: justify;\">where, \u03c4<sub>P<\/sub> is permissible shear stress, \u03c4 is given as,<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2632 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/strength.jpg\" alt=\"Strength\" width=\"152\" height=\"57\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/strength.jpg 152w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/strength-150x57.jpg 150w\" sizes=\"auto, (max-width: 152px) 100vw, 152px\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Stiffness criteria<\/strong><\/p>\n<p style=\"text-align: justify;\">\u03b8\u2264\u03b8<sub>P<\/sub><\/p>\n<p style=\"text-align: justify;\">where, \u03b8<sub>P<\/sub> is permissible angle of twist.<\/p>\n<p style=\"text-align: justify;\">Where, \u03b8 is given by,<\/p>\n<p style=\"text-align: justify;\">T\/I<sub>P<\/sub> = G\u03b8\/L<\/p>\n<p style=\"text-align: justify;\">\u03b8 = TL\/GI<sub>P<\/sub><\/p>\n<p style=\"text-align: justify;\">The diameter of shaft will be greater value that is calculated by strength criteria or stiffness criteria.<\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"SERIES-COMBINATION-OF-SHAFT\"><\/span>SERIES COMBINATION OF SHAFT <img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-2650 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/series-combination.jpg\" alt=\"Series Combination\" width=\"400\" height=\"192\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/series-combination.jpg 400w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/series-combination-300x144.jpg 300w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Series combination of shafts is done to transmit torque from one shaft to another shaft. For series connection couplings are used. Torque in all shaft connected in series will be equal.<\/p>\n<p style=\"text-align: justify;\">T<sub>1<\/sub> = T<sub>2<\/sub> = T<\/p>\n<p style=\"text-align: justify;\">Total angle of twist from A to C<\/p>\n<p style=\"text-align: justify;\">\u03b8<sub>AC<\/sub> = \u03b8<sub>AB<\/sub> + \u03b8<sub>BC<\/sub><\/p>\n<p style=\"text-align: justify;\">\u03b8<sub>AC<\/sub> = TL<sub>1<\/sub>\/G<sub>1<\/sub>I<sub>P1<\/sub> + TL2\/G<sub>2<\/sub>I<sub>P2<\/sub><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"PARALLEL-COMBINATION-OF-SHAFT\"><\/span>PARALLEL COMBINATION OF SHAFT <img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-2651 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/parallel-combination.jpg\" alt=\"Parallel Combination\" width=\"311\" height=\"222\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/parallel-combination.jpg 311w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/parallel-combination-300x214.jpg 300w\" sizes=\"auto, (max-width: 311px) 100vw, 311px\" \/><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">When one shaft is inside the other shaft, then two shafts are said to be connected in parallel. Parallel combination is done by providing keys between the shafts.<\/p>\n<p style=\"text-align: justify;\">Let I<sub>P1<\/sub> and G<sub>1<\/sub> are properties of outer shaft and I<sub>P2<\/sub> and G<sub>2<\/sub> are properties of inner shaft.<\/p>\n<p style=\"text-align: justify;\">Since there is no relative motion between them. Then angle of twist in both shaft will be equal.<\/p>\n<p style=\"text-align: justify;\">\u03b8<sub>1<\/sub> = \u03b8<sub>2<\/sub><\/p>\n<p style=\"text-align: justify;\">T<sub>1<\/sub>L\/G<sub>1<\/sub>I<sub>P1<\/sub> = T<sub>2<\/sub>L\/G<sub>2<\/sub>I<sub>P2<\/sub><\/p>\n<p style=\"text-align: justify;\">The total torque is shared by both shafts. Hence<\/p>\n<p style=\"text-align: justify;\">T<sub>1<\/sub> + T<sub>2<\/sub> = T<\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"STRAIN-ENERGY-IN-TORSION\"><\/span>STRAIN ENERGY IN TORSION<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">The torsional strain energy of a shaft is equal to the work done in twisting<\/p>\n<p style=\"text-align: justify;\">U = 1\/2T\u03b8<\/p>\n<p style=\"text-align: justify;\">where, T = Applied torque, \u03b8 = Angle of twisting<\/p>\n<p style=\"text-align: justify;\">Due to torque, shear stresses are developed in metal. Hence strain energy due to torque can be represented in terms of shear stress.<\/p>\n<p style=\"text-align: justify;\">Strain energy per unit volume = \u03c4<sup>2<\/sup>\/2G<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2652 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/shear-resilience.jpg\" alt=\"Shear Resilience\" width=\"602\" height=\"242\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/shear-resilience.jpg 602w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/shear-resilience-300x121.jpg 300w\" sizes=\"auto, (max-width: 602px) 100vw, 602px\" \/><\/p>\n<p style=\"text-align: justify;\">Consider a elemental tube of radius r and thickness dr and length L<\/p>\n<p style=\"text-align: justify;\">Volume dV = (2\u03c0r)dr.L<\/p>\n<p style=\"text-align: justify;\">\u2234 Strain energy stored in elemental volume dV is given by,<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2653 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/strain-energy-2.jpg\" alt=\"Strain Energy\" width=\"526\" height=\"128\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/strain-energy-2.jpg 526w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/strain-energy-2-300x73.jpg 300w\" sizes=\"auto, (max-width: 526px) 100vw, 526px\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Case I: Strain energy in terms of torque<\/strong><\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2654 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/torque.jpg\" alt=\"Torque\" width=\"836\" height=\"347\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/torque.jpg 836w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/torque-300x125.jpg 300w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/torque-768x319.jpg 768w\" sizes=\"auto, (max-width: 836px) 100vw, 836px\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Case II: Strain energy in terms of \u03c4<sub>max<\/sub><\/strong><\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2655 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/strain-energy-3.jpg\" alt=\"Strain energy\" width=\"823\" height=\"365\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/strain-energy-3.jpg 823w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/strain-energy-3-300x133.jpg 300w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/strain-energy-3-768x341.jpg 768w\" sizes=\"auto, (max-width: 823px) 100vw, 823px\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Case III: Strain energy in hollow shaft in terms of \u03c4<sub>max<\/sub><\/strong><\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2656 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/hollow-shaft-.jpg\" alt=\"Hollow Shaft \" width=\"815\" height=\"283\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/hollow-shaft-.jpg 815w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/hollow-shaft--300x104.jpg 300w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/hollow-shaft--768x267.jpg 768w\" sizes=\"auto, (max-width: 815px) 100vw, 815px\" \/><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"SHAFT-SUBJECTED-TO-COMBINED-BENDING-MOMENT-AND-TWISTING-MOMENT\"><\/span>SHAFT SUBJECTED TO COMBINED BENDING MOMENT AND TWISTING MOMENT<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">When the shaft is subjected to combined bending moment and twisting moment, the maximum stresses induced in shaft are due to the combined effect of shear stress (\u03c4) and bending stress (\u03c3<sub>b<\/sub>)<\/p>\n<p style=\"text-align: justify;\"><strong>(i) Effect of pure bending<\/strong><\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2657 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/pure-bending.jpg\" alt=\"Pure Bending\" width=\"523\" height=\"256\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/pure-bending.jpg 523w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/pure-bending-300x147.jpg 300w\" sizes=\"auto, (max-width: 523px) 100vw, 523px\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) Effect of pure torsion<\/strong><\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2658 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/Pure-Torsion.jpg\" alt=\"Pure Torsion\" width=\"516\" height=\"290\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/Pure-Torsion.jpg 516w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/Pure-Torsion-300x169.jpg 300w\" sizes=\"auto, (max-width: 516px) 100vw, 516px\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) Combined effect of bending and twisting:\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\">Under the effect of bending moment (M) and torsional moment (T), the stresses at extreme bottom or top fibre construct a stress element as shown in figure.<\/p>\n<p style=\"text-align: justify;\"><strong><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2659 size-full aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/twisting.jpg\" alt=\"Twisting\" width=\"276\" height=\"164\" \/><\/strong><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2660 size-full aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/principal-stresses-1.jpg\" alt=\"Principal Stresses\" width=\"597\" height=\"455\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/principal-stresses-1.jpg 597w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/principal-stresses-1-300x229.jpg 300w\" sizes=\"auto, (max-width: 597px) 100vw, 597px\" \/><\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Equivalent-Bending-Moment\"><\/span>Equivalent Bending Moment<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\">It is that bending moment which produces same maximum normal stress as produced by combined effect of bending and twisting.<br \/>\nIf M<sub>e<\/sub> the equivalent bending moment. Then maximum normal stress produced will be<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2661 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/equivalent-bending.jpg\" alt=\"Equivalent Bending\" width=\"380\" height=\"161\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/equivalent-bending.jpg 380w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/equivalent-bending-300x127.jpg 300w\" sizes=\"auto, (max-width: 380px) 100vw, 380px\" \/><\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Equivalent-Torque\"><\/span>Equivalent Torque<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\">It is that torque which produces same maximum shear stress as produced by combined effect of bending and twisting.<br \/>\nIf T<sub>e<\/sub> be the equivalent torque then maximum shear stress produced will be<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2662 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/equivalent-torque.jpg\" alt=\"Equivalent Torque\" width=\"205\" height=\"187\" \/><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"SHAFT-SUBJECTED-TO-COMBINED-AXIAL-FORCE-AND-TORSIONAL-MOMENT\"><\/span>SHAFT SUBJECTED TO COMBINED AXIAL FORCE AND TORSIONAL MOMENT<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2663 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/axial-force.jpg\" alt=\"Axial Force\" width=\"339\" height=\"182\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/axial-force.jpg 339w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/axial-force-300x161.jpg 300w\" sizes=\"auto, (max-width: 339px) 100vw, 339px\" \/><\/p>\n<p style=\"text-align: justify;\">Consider a shaft subjected to an axial force and torsional moment as shown in figure.<\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Effect-of-axial-force\"><\/span>Effect of axial force<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2665 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/axial-force-1.jpg\" alt=\"Axial Force\" width=\"497\" height=\"94\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/axial-force-1.jpg 497w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/axial-force-1-300x57.jpg 300w\" sizes=\"auto, (max-width: 497px) 100vw, 497px\" \/><\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Effect-of-torsion\"><\/span>Effect of torsion<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2666 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/torsion.jpg\" alt=\"Torsion\" width=\"242\" height=\"101\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Combined effect of direct force and torsion<\/strong><\/p>\n<p style=\"text-align: justify;\">Consider any point on surface of shaft which will be in a state as shown below:<br \/>\nTherefore, principal stresses can be given as,<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2667 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/torsion-1.jpg\" alt=\"Torsion\" width=\"263\" height=\"293\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Elastic Instability and Critical Load<\/strong><\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2668 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/critical-load.jpg\" alt=\"Critical Load\" width=\"898\" height=\"287\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/critical-load.jpg 898w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/critical-load-300x96.jpg 300w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/critical-load-768x245.jpg 768w\" sizes=\"auto, (max-width: 898px) 100vw, 898px\" \/><\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2669 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/unstable-equilibrium.jpg\" alt=\"Unstable Equilibrium\" width=\"368\" height=\"211\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/unstable-equilibrium.jpg 368w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/unstable-equilibrium-300x172.jpg 300w\" sizes=\"auto, (max-width: 368px) 100vw, 368px\" \/><\/p>\n<p style=\"text-align: center;\"><a class=\"btn btn-danger\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/ce\/strength-of-material\/transformation-equations\/\" target=\"_blank\" rel=\"noopener\">&lt;&lt; Previous<\/a> | <a class=\"btn btn-success\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/ce\/strength-of-material\/euler-theory\/\" target=\"_blank\" rel=\"noopener\"> Next &gt;&gt;<\/a><br \/>\n<strong> Must Read: <\/strong> <a href=\"https:\/\/study.madeeasy.in\/subjects\/what-is-strength-of-material\/\" target=\"_blank\" rel=\"noopener\"><strong>What is Strength of Material?<\/strong><\/a><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solid circular shaft Hollow circular shaft Di = internal diameter Do = outer diameter Thin circular tube with mean radius<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[685,2],"tags":[722,723,721,724],"class_list":["post-2625","post","type-post","status-publish","format-standard","hentry","category-strength-of-material","category-ce","tag-axial-force","tag-critical-load","tag-shear-resilience","tag-unstable-equilibrium"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/2625","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=2625"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/2625\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=2625"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=2625"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=2625"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}