hw7 - (b a two-parameter trigonometric Ritz approximation...

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MAE 261A E NERGY AND V ARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS F ALL 2004 Homework Assignment No. 7 Due Friday, December 10, 2004. Problem 1 Examine the following problem for self-adjointness and positive definiteness using the normal inner product ( u,v ) = integraltext Ω uvdx . Lu = f, 1 x 2 u (1) = (2) = 0 L = x d 2 dx 2 + 2 x d dx and f = 6 x. Problem 2 Consider a simply-supported Euler-Bernoulli beam with a linear spring support at the middle, and sub- jected to a uniform distributed load q 0 . Use the principle of Minimum Potential Energy to approximate the maximum deflection with (a) a one-parameter trigonometric Ritz approximation (i.e., one free/unknown parameter), and
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Unformatted text preview: (b) a two-parameter trigonometric Ritz approximation. Problem 3 Consider the Ritz method applied to a Timoshenko beam fixed at both ends and subjected to a uniform distributed load. (a) What are the admissiblity requirements for the approximations of the deflection v ( x ) and the rotation ψ ( x ) ? In other words, what are the smoothness requirements, and what boundary conditions must be satisfied. (b) Use a one-parameter polynomial Ritz approximation for each of the unknowns v ( x ) and ψ ( x ) , and solve for the approximate deflection and rotation of the beam. 1...
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