[{"data":1,"prerenderedAt":24},["ShallowReactive",2],{"formula-en-bending-torsion-equivalent-stress":3},{"formula":4,"related_topics":12,"alternates":21},{"id":5,"code":6,"slug":6,"title":7,"formula":8,"locale":9,"seo_title":10,"seo_description":10,"content_html":11},98,"bending-torsion-equivalent-stress","Equivalent Stress under Combined Bending and Torsion","σeq = √(σb² + 3τt²)","en",null,"\u003Cdiv class=\"formula-chunk\">\u003Cp>At a critical point of a circular shaft subjected to bending and torsion, the bending moment produces normal stress $\\sigma_b$ and the torque produces shear stress $\\tau_t$:\u003C\u002Fp>\u003Cp>$$\\sigma_b=\\frac{M_b}{W},\\qquad \\tau_t=\\frac{T}{W_p}.$$\u003C\u002Fp>\u003Cp>For this plane stress state, the von Mises equivalent stress is:\u003C\u002Fp>\u003Cp>$$\\sigma_{\\mathrm{eq,VM}}=\\sqrt{\\sigma_b^2+3\\tau_t^2}.$$\u003C\u002Fp>\u003Cp>According to the Tresca criterion:\u003C\u002Fp>\u003Cp>$$\\sigma_{\\mathrm{eq,T}}=\\sqrt{\\sigma_b^2+4\\tau_t^2}.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$M_b$\u003C\u002Fstrong> — resultant bending moment;\u003C\u002Fli>\u003Cli>\u003Cstrong>$T$\u003C\u002Fstrong> — torque;\u003C\u002Fli>\u003Cli>\u003Cstrong>$W$\u003C\u002Fstrong> — bending section modulus;\u003C\u002Fli>\u003Cli>\u003Cstrong>$W_p$\u003C\u002Fstrong> — polar section modulus.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>The resulting equivalent stress is compared with an allowable value appropriate to the material and the adopted design method.\u003C\u002Fp>\u003Cp>The calculator uses the von Mises criterion for the bending-plus-torsion plane stress state.\u003C\u002Fp>\u003Cdiv data-formula-calculator-config=\"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\">\u003C\u002Fdiv>\u003C\u002Fdiv>",[13,17],{"id":14,"name":15,"path":16},68,"Combined Bending and Torsion","strength-of-materials\u002Fcombined-loading\u002Fcombined-bending-and-torsion",{"id":18,"name":19,"path":20},65,"Combined Loading","strength-of-materials\u002Fcombined-loading",{"en":22,"uk":23},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Fformulas\u002Fbending-torsion-equivalent-stress","https:\u002F\u002Fmechclassroom.com\u002Fformulas\u002Fbending-torsion-equivalent-stress",1787712543823]