# 3-21 - 208 Finite Element Analysis and Design 21 Consider...

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208 Finite Element Analysis and Design 21. Consider the tapered bar in Problem 16. Use the Rayleigh-Ritz method to solve the same problem. Assume the displacement in the form of 2 1 2 ( ) ( 1)( ) u x x c x c x . Solution: The assumed displacements satisfy the essential BCs u (0) = u (1) = 0. The assumed displacements take the form: 2 1 2 ( ) ( 1)( ) u x x c x c x Then, the strain is given by 2 1 2 (2 1) (3 2 ) xx du c x c x x dx The strain energy in the bar becomes 1 2 2 1 2 0 ( ) (2 1) (3 2 ) 2 E U A x c x c x x dx where A ( x ) is the area of cross-section defined as 2 2 ( ) 0.05 (1 0.8 ) A x x The potential energy of the distributed load f = 10,000 N/m is 1 1 2 1 2 0 0 ( ) 10,000 ( 1)( ) V fu x dx x c x c x dx     The total potential energy 1 2 1 2 1 2 ( , ) ( , ) ( , ) c c U c c V c c . The principle of minimum potential energy requires

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