MIT2_094S11_hw_6_sol

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 tt F = ˆ R 1/2 2 1 ˆˆ 1( 12 s i n 1 5 t FR U UU ⎧⎫ ⎪⎪ == + ° ⎨⎬ ⎡⎤ −° + ⎣⎦ ⎩⎭ s i n 1 5 ) The tangent stiffness can be calculated by ( ) ( ) (1 ) ) ) tt i FU K ε ΔΔ Δ ++ +− = with a sufficiently small . Then the Newton-Raphson iteration is performed ) ) () ( 1 ) i ii KU R F Δ Δ + =− =+ i U Δ with the initial conditions (0) t Δ + = , t K K Δ + = , t FF 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 you used for the calculation of the tangent stiffness. Page 1 of 3
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Note that this problem has multiple solutions as shown in the graph in the textbook. Therefore if your stiffness is not correct, then you may get another solution which is far from the initial equilibrium state. Make sure that the
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.094 taught by Professor Klaus-jürgenbathe during the Spring '11 term at MIT.

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MIT2_094S11_hw_6_sol - 2.094 FINITE ELEMENT ANALYSIS OF...

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