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### HW5

Course: CE 130, Fall 2009
School: Duke
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Word Count: 491

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University Duke Department of Civil and Environmental Engineering CE 130L. Uncertainty, Design, and Optimization Homework 5, Due Friday February 27, 2009 1. The simply-supported beam shown below is loaded with uniformly-distributed load w over its entire length. The coordinate system shown has its origin at the mid-span of the beam. Considering bending deformation only, the mid-span displacement vo corresponding...

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University Duke Department of Civil and Environmental Engineering CE 130L. Uncertainty, Design, and Optimization Homework 5, Due Friday February 27, 2009 1. The simply-supported beam shown below is loaded with uniformly-distributed load w over its entire length. The coordinate system shown has its origin at the mid-span of the beam. Considering bending deformation only, the mid-span displacement vo corresponding to equilibrated internal moments is (5/384)(wL4 /EI). (a) Write the strain energy function U of this system in terms of the bending deformation v(x). (Neglect shear deformation). Write the potential energy function of the external forces V of this system in terms of the load w and the bending deformation v(x). (b) Write the variation of the total potential energy function in terms of the expressions for U and V found above. (c) Find the approximate mid-span displacement that minimizes with respect to vo , (vo ) = vo = 0 , vo for the three approximations for the bending deformation shown below: i. v(x) = vo (1 (x/l)2 ) ii. v(x) = vo (1 (x/l)4 ) iii. v(x) = vo cos x 2l Note that each of these three expressions satisfy the end-conditions v(l) = 0 and v(l) = 0. What is the error of each approximation as a percentage of the equilibrated mid-span displacement? y w x 11111 00000 11111 00000 11111 00000 A x = l v(l)=0 vo L=2l v(x) 11111 00000 11111 00000 11111 00000 B x=l v(l)=0 1 2. The cantilever beam shown below is supported at point A and is loaded with a uniform load w over its entire length. Considering deformation bending only, the end displacement vo corresponding to equilibrated internal moments is (1/8)(wL4 /EI). Using your expressions for parts 1(a) and 1(b), nd the approximate end-displacement vo by minimizing the total potential energy under the assumption of transverse diplacements of the forms: (a) v(x) = vo (x/L)2 (b) v(x) = vo (x/L)3 (c) v(x) = vo (1 cos x 2L Note that each of these three expressions satisfy the end-conditions v(0) = 0, v (0) = 0, and v(L) = vo . What is the error of each approximation as a percentage of the equilibrated end-displacement? 3. The cantilever beam shown below is supported at point A and is loaded with a triangularly-distributed load w(x) over its entire length. Considering bending deformation only, the end displacement vo corresponding to equilibrated internal moments is (11/120)(qL4 /EI). Using your expressions for parts 1(a) and 1(b), nd the approximate end-displacement vo by minimizing the total potential energy under the assumption of transverse diplacements of the forms: (a) v(x) = vo (x/L)2 (b) v(x) = vo (x/L)3 (c) v(x) = vo (1 cos x 2L Note that each of these three expressions satisfy the end-conditions v(0) = 0, v (0) = 0, and v(L) = vo . What is the error of each approximation as a percentage of the equilibrated end-displacement? x B v(x) w L w(x) vo x = L v(L)=vo w(L)=q B x vo x = L v(L)=vo v(x) L A Problem 2. 11111 00000 11111 00000 11111 00000 x = 0 v(0)=0 v(0)=0 w(0)=0 A Problem 3. 11111 00000 11111 00000 11111 00000 x = 0 v(0)=0 v(0)=0 2
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