HW8.pdf - 1 5.1 A beam of square cross section a u00d7 a...

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1 5.1. A beam of square cross section a × a is bent about a diagonal axis by a moment M , as shown in Figure P5.1. Find the relation between M and the radius of curvature R , if the material obeys the constitutive law σ zz = E parenleftbigg e zz + e 3 zz e 2 0 parenrightbigg . M a a y O x Figure P5.1 Using equations (5.17), we have σ zz = E parenleftbigg y R + y 3 R 3 e 2 0 parenrightbigg . To perform the integral in the second of (5.16), we first note that the width of the section, w ( y ) in Figure P5.1.1, varies linearly from 2 a at y = 0 to zero at y = a / 2 and hence w ( y ) = 2 a parenleftBigg 1 - 2 y a parenrightBigg = 2 a - 2 y .
2 a a y O w(y) Figure P5.1.1 It follows that M = M x = integraldisplay integraldisplay A σ zz ydA = 2 integraldisplay a / 2 0 σ zz w ( y ) ydy = 2 E integraldisplay a / 2 0 parenleftbigg y R + y 3 R 3 e 2 0 parenrightbigg ( 2 a - 2 y ) ydy , where we have used the symmetry of the section about the x -axis to perform the integral only for the upper triangle and then multiply by 2.

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