MIT2_092F09_lec15

MIT2_092F09_lec15 - 2.092/2.093 — Finite Element Analysis...

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Unformatted text preview: 2.092/2.093 — Finite Element Analysis of Solids & Fluids I Fall ‘09 Lecture 15- Solution of Dynamic Equilibrium Equations Prof. K. J. Bathe MIT OpenCourseWare In the last lecture, we described a physical setup that demonstrates the technique of Gauss elimination. We used clamps on each DOF and removed one clamp for one step of Gauss elimination. ⎤ ⎡ ⎤ ⎡ ⊗ × × ⊗ × × u 1 × × ⎢ ⎢ ⎣ ⎢ ⎢ ⎣ ⎥ ⎥ ⎦ ⎥ ⎥ ⎦ u 2 u 3 × × ⊗ × × × × ⊗ u 4 ⊗ should be positive, and should remain positive. Our rule: Remove clamps one at a time, in the order we would perform Gauss elimination. If there is “a” clamp “seeing” no more stiffness after having removed some clamp(s), the structure is unstable. Example ⎤ ⎡ K = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ × × × ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ × × × All diagonal terms are positive. However, there will be a zero diagonal entry after Gauss elimination has been performed for the 3rd DOF....
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MIT2_092F09_lec15 - 2.092/2.093 — Finite Element Analysis...

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