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3-11 - CHAP 3 Weighted Residual and Energy Methods 185 11...

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CHAP 3 Weighted Residual and Energy Methods 185 11. The boundary value problem for a cantilevered beam can be written as 4 4 2 3 2 3 ( ) 0, 0 1 (0) (0) 0, (1) 1, (1) 1: boundary condtions d w p x x dx dw d w d w w dx dx dx   Assume ( ) p x x . Assuming the approximate deflection in the form 2 3 1 1 2 2 1 2 ( ) ( ) ( ) w x c x c x c x c x . Solve for the boundary value problem using the Galerkin method. Compare the approximate solution to the exact solution by plotting the solutions on a graph. Solution: (a) Exact solution: By integrating the governing differential equation four times and applying four boundary conditions, we can obtain the exact solution, as 5 3 2 1 1 7 ( ) 120 4 6 w x x x x (b) Galerkin method Using the second derivatives of the two trial functions, 1 2 2, 6 x   , we can calculate the coefficient matrix and the vector on the RHS, as 1 2 11 1 0 1 2 22 2 0 1 12 21 1 2 0 1 13 1 1 1 1 1 1 4 0 1 21 2 2 2 2 2 2 5 0 ( ) 4 ( ) 12 6 (1) (1) (0) (0) (1) (1) (0) (0) (1) (1) (0) (0) (1) (1) (0) (0) K dx K dx K K dx F x dx w w w w F x dx w w w w  
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