hw3 - MAE 261A E NERGY AND VARIATIONAL P RINCIPLES IN S...

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MAE 261A E NERGY AND V ARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS F ALL 2004 Homework Assignment No. 3 Due Wednesday, November 3, 2004. Problem 1 Take the first variation of the functionals listed below, derive the Euler-Lagrange equations, and identify essential and natural boundary conditions. a. I [ u ( x )] = 1 2 R b a { ( u 00 ) 2 + 2 u 0 u 00 + ( u 0 ) 2 - 2 u } dx b. Π[ u ( x ) , w ( x )] = R L 0 EA 2 [ u 0 + 1 2 ( w 0 ) 2 ] 2 + EI 2 ( w 00 ) 2 + qw dx Problem 2 Consider the problem of maximizing the functional I [ u ( x )] = Z b a udx, u ( a ) = u a , u ( b ) = u b , subject to the constraint G [ u 0 ] = Z b a p 1 + ( u 0 ) 2 dx = L = const . Use the method of Lagrange Multipliers to derive the Euler equation. Problem 3 Consider the problem of minimizing the functional Π[ w, κ ] = Z L 0 EI 2 κ 2 - qwdx subject to the constraint κ - w 00 = 0 , where w is the transverse deflection, q is the distributed load, EI is the flexural stiffness, and L is the length of a beam. Use the method of Lagrange Multipliers to derive the Euler equation.
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Unformatted text preview: Problem 4 Consider an elastic body B , with boundary ∂ B = ∂ B u ∪ ∂ B t , displacements u = ¯ u specified on ∂ B u , conservative tractions t applied on ∂ B t , and conservative body forces f applied throughout B . As discussed in class, the total potential energy of the body is Π[ u ] = Z B wdV + Z B GdV + Z ∂ B gdS, where w is the strain energy density and f i =-∂G ∂u i and t i =-∂g ∂u i . Complete the derivation outlined in class, to show that stationarity of Π requires that equilibium of the body and the appropriate traction and displacement conditions on the boundary be satisfied. 1...
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• Fall '04
• Klug
• Energy, Trigraph, Calculus of variations, Euler–Lagrange equation, Joseph Louis Lagrange

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