hw6_sol - EECS 414 HW 6 1. Given a polysilicon beam with...

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1 EECS 414 Fall 2005 HW 6 1. Given a polysilicon beam with the following properties: Length, L 100 μm Width, w 20 μm Thickness, t 1 μm Young’s Modulus, E 160 GPa Poisson’s Ratio, v 0.23 (a) The beam is fixed on one end and the other end is free to move (cantilever). A force of 5 μN is applied at the very end of the beam (on the free end). Calculate the deflection at the tip of the beam. (b) The beam is fixed at both ends. Derive an equation for the maximum deflection. The deflection is maximum if the applied force, F, (assumed to be constant through the width of the beam) is applied at the very center of the beam. The equation should be in terms of the parameters given above (i.e. F, L, w, etc). (c) Using your result in part (b), find the maximum deflection of a beam fixed at both ends if a force of 100 μN is applied at the center of the beam. 30 points Solution: (a) For a point load, deflection is given by: ) 3 ( 6 ) ( 3 2 x L x EI F x y = For maximum deflection, x=L and in general, I is: 12 23 2 ) 6 1 )( 6 20 ( 12 1 12 1 3 3 = = = e e e wt I Plugging this into the equation for y(x) and solving gives the maximum deflection as 6.25 um. (b) For a beam fixed at both ends and a point force, the deflection is given by: ) 4 3 ( 48 ) ( 2 x Lx EI Fx x y = For maximum deflection, x=L/2 and I is: 3 12 1 wt I = Substituting the above in the equation for deflection gives: 3 3 2 2 3 2 3 max 16 ) 2 3 ( 8 ) 4 4 2 3 ( 48 ) 2 / ( 12 Ewt FL L L Ewt FL L L L Ewt L F y = = = (c) Plugging F=5e-6, L=100e-6, E=160e9, w=20e-6 and t=1e-6 into the equation for y max and solving gives the maximum deflections as 1.95 uN. 2. A 1x1x1μm cube is subject to equal normal stresses in the x, y, z direction: σ X = σ Y = σ Z =200MPa. There are no shear stresses. The cube is made of single crystal silicon,
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2 with anisotropic elastic constants C 11 =166GPa, C 12 =64GPa, C 44 =80GPa. Calculate the strains ε X , ε Y , ε Z . 10 points Solution: With no shear stresses, the stress-strain matrix reduces to: = Z Y X Z Y X C C C C C C C C C ε σ 11 12 12 12 11 12 12 12 11 We are given
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This note was uploaded on 04/30/2008 for the course EECS 414 taught by Professor Maharbiz during the Fall '06 term at University of Michigan.

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hw6_sol - EECS 414 HW 6 1. Given a polysilicon beam with...

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