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MIT2_094S11_hw_1_sol

# MIT2_094S11_hw_1_sol - 2.094 FINITE ELEMENT ANALYSIS OF...

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Page 1 of 5 2.094 F INITE E LEMENT A NALYSIS OF S OLIDS AND F LUIDS S PRING 2008 Homework 1 - Solution Instructor: Prof. K. J. Bathe Assigned: 02/07/2008 Due: 02/14/2008 Problem 1 (10 points): In this problem, the principle of virtual work reduces to 𝝉 ? ?? ? ? ?? ? 𝑽 ? 𝑽 ? = ? ? 𝑺 ? ? ? ? ? ? ? 𝑺 ? ? 𝑺 ? ? x 1 x 2 x 1 x 2 x 1 x 2 configuration at time t virtual displacement (a) (b) (c) Figure 1. Three simple independent virtual displacement patterns. (a) a simple tension in the x 1 direction; (b) a simple tension in the x 2 direction; (c) a simple shear

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Page 2 of 5 (a) A simple extension in ? ? direction; ? ? = ? ? ? − ? and ? ? = ? ? ? ?? = ? ? ? 𝑺 ? ? = ? Therefore, 𝝉 ? ?? ? ? ?? ? 𝑽 ? 𝑽 ? = ? ? ? 𝑺 ? ? ? ? ? ? ? 𝑺 ? ? 𝑺 ? ? = ? Check! (b) A simple extension in ? ? direction; ? ? = ? and ? ? = ? ? ? + ? ? ? ?? = ? ? ? 𝑺 ? ? = ? (on the top surface) and ? ? 𝑺 ? ? = ? (on the bottom surface) Therefore, 𝝉 ? ?? ? ? ?? ? 𝑽 ? 𝑽 ? = ?? × ? ? 𝑽 ? 𝑽 ? = ?? × 𝑽 ? = 𝟖? ? ? 𝑺 ? ? ? ? ? ? ? 𝑺 ? ? 𝑺 ? ? = ? × ?? ? 𝑺 ? ? 𝑺 ? ? = 𝟖? × 𝑺 ? ? = 𝟖? Check! (c) A simple shear; ? ? = ? and ? ? = ? − ? ? ? ? ? ?? = ? ? ? 𝑺 ? ? = ? − ? ? ? Therefore, 𝝉 ? ?? ? ? ?? ? 𝑽 ? 𝑽 ? = ? ? ? 𝑺 ? ? ? ? ? ? ? 𝑺 ? ? 𝑺 ? ? = ?? × ? − ? ? ? ? 𝑺 ? ? ? 𝑺 ? ? ? + −?? × ? − ? ? ? ? 𝑺 ?
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MIT2_094S11_hw_1_sol - 2.094 FINITE ELEMENT ANALYSIS OF...

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