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MIT2_094S11_hw_6_sol - 2.094 FINITE ELEMENT ANALYSIS OF...

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2.094 F INITE E LEMENT A NALYSIS OF S OLIDS AND F LUIDS S PRING 2008 Homework 6 - Solution Instructor: Prof. K. J. Bathe Assigned: 03/13/2008 Due: 03/20/2008 Problem 1 (20 points): Let’s define ˆ 2 t t R R kL = , ˆ 2 t t F F kL = and t t U L Δ = . Since at equilibrium, ˆ t t F = ˆ R 1/2 2 1 ˆ ˆ 1 ( 1 2 sin15 t t t t t F R U U U = = − + ° − ° + sin15 ) The tangent stiffness can be calculated by ( ) ( ) ( 1) ( 1) ( 1) ˆ ˆ t t t t i t t t t i t t i F U F U K ε ε Δ Δ Δ Δ Δ + + + + + + = with a sufficiently small ε . Then the Newton-Raphson iteration is performed ( 1) ( 1) ( ) ( 1) ( ) ( ) ˆ ˆ t t i t t t t i t t i i i t t K U R F U U Δ Δ Δ Δ Δ Δ + + + + + = = + i U Δ with the initial conditions (0) t t t U U Δ + = , (0) t t t K K Δ + = , (0) ˆ ˆ t t t F F Δ + = The calculated values in each step are listed in Table 1. The number of iterations and the calculated values can be different from those in Table 1 depending on which value of ε you used for the calculation of the tangent stiffness.
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