PHYS 598 hw2 - 1 Elastic Rods The elastic energy per unit...

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Physics 498/MMA Handout 2 Fall 2007 Mathematical Methods in Physics I http://w3.physics.uiuc.edu/ m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Elastic Rods. The elastic energy per unit length of a bent steel rod is given by 1 2 Y I/R 2 . Here R is the radius of curvature due to the bending, Y is the Young’s modulus of the steel and I = RR y 2 dxdy is the moment of inertia of the rod’s cross section about an axis through its centroid and perpendicular to the plane in which the rod is bent. If the rod is only slightly bent into the yz plane and lies close to the z axis, show that this elastic energy can be approximated as U [ y ] = Z L 0 1 2 Y I ( y 00 ) 2 dz, where the prime denotes differentiation with respect to z and L is the length of the rod. We will use this approximate energy functional to discuss two practical problems. L Mg Mg a) b) a) Euler’s problem: The buckling of a slender column . The rod is used as a column which supports a compressive load Mg directed along the z axis (which is vertical). Show
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