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final_quiz - 2.081J/16.230J Plates and Shells Quiz Monday...

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2.081J/16.230J Plates and Shells Quiz Monday, March 24 Problem 1 [Must be completed in the class.] A simply supported square elastic plate ( a × a ) is loaded by a transervese load q ( x, y ) . The origin of the coordinate system is at the center of the plate. The shape of the de ected plate was precisely measured and then fi ted by the following function: w x, y ¯) = w 0 ¡ 1 x ¯ 2 ¢ ¡ 1 y ¯ 2 ¢ where x y x ¯ = ; y ¯ = a/ 2 a/ 2 and w 0 is the central de ectioin of the plate w 0 = w (0 , 0) . a 0 1 x 1 0 x a 1 1 y y (a) Check if the kinematic boundary conditions are satis fi ed. (b) Calculate the components of the bending moment tensor M αβ and e ff ec- tive shear force vector V α . (c) Check if the static boundary conditions are satis fi ed (both moments and e ff ective shears). 1
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(d) Determine the magnitude and the distribution of the transverse loading q x, y ¯) that equilibrates the plate. (e) Is the plate resistance derived from the direct bending moments M xx and M yy or from the twisting moment M xy ?
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