[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"formula-en-beam-curvature-moment":3},{"formula":4,"related_topics":12,"alternates":29},{"id":5,"code":6,"slug":6,"title":7,"formula":8,"locale":9,"seo_title":10,"seo_description":10,"content_html":11},86,"beam-curvature-moment","Beam Curvature, Bending Moment, and Flexural Rigidity","M = EI \u002F ρ","en",null,"\u003Cdiv class=\"formula-chunk\">\u003Cp>In classical linear-elastic bending theory, curvature of the deflected beam axis is related to bending moment by:\u003C\u002Fp>\u003Cp>$$\\kappa=\\frac{1}{\\rho}=\\frac{M}{EI},$$\u003C\u002Fp>\u003Cp>where the sign of the right-hand side depends on the adopted conventions for bending moment and deflection.\u003C\u002Fp>\u003Cp>For small rotations, curvature is approximately:\u003C\u002Fp>\u003Cp>$$\\kappa\\approx w''(x).$$\u003C\u002Fp>\u003Cp>Therefore, the differential equation of the elastic curve is often written as:\u003C\u002Fp>\u003Cp>$$EI\\,w''(x)=M(x),$$\u003C\u002Fp>\u003Cp>or with a minus sign according to the chosen sign convention.\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$E$\u003C\u002Fstrong> — Young's modulus;\u003C\u002Fli>\u003Cli>\u003Cstrong>$I$\u003C\u002Fstrong> — second moment of area;\u003C\u002Fli>\u003Cli>\u003Cstrong>$EI$\u003C\u002Fstrong> — flexural rigidity;\u003C\u002Fli>\u003Cli>\u003Cstrong>$w$\u003C\u002Fstrong> — transverse deflection;\u003C\u002Fli>\u003Cli>\u003Cstrong>$\\rho$\u003C\u002Fstrong> — radius of curvature.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>The calculator below uses the equivalent relation $M=EI\u002F\\rho$.\u003C\u002Fp>\u003Cdiv data-formula-calculator-config=\"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\">\u003C\u002Fdiv>\u003C\u002Fdiv>",[13,17,21,25],{"id":14,"name":15,"path":16},62,"Deflections in Bending","strength-of-materials\u002Fbending\u002Fdeflections-in-bending",{"id":18,"name":19,"path":20},58,"Bending","strength-of-materials\u002Fbending",{"id":22,"name":23,"path":24},63,"Differential Equation of the Elastic Curve","strength-of-materials\u002Fbending\u002Fdeflections-in-bending\u002Fdifferential-equation-elastic-curve",{"id":26,"name":27,"path":28},64,"Initial Parameter Method (Macaulay's Method)","strength-of-materials\u002Fbending\u002Fdeflections-in-bending\u002Finitial-parameter-method",{"en":30,"uk":31},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Fformulas\u002Fbeam-curvature-moment","https:\u002F\u002Fmechclassroom.com\u002Fformulas\u002Fbeam-curvature-moment",1787712543444]