MIT2_094S11_lec14

MIT2_094S11_lec14 - 2.094 — Finite Element Analysis of...

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Unformatted text preview: 2.094 — Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 14- Total Lagrangian formulation, cont’d Prof. K.J. Bathe MIT OpenCourseWare Truss element. 2D and 3D solids. t +Δ t t +Δ t t +Δ t τ ij δ t +Δ t e ij d V = R (14.1) t +Δ t V t +Δ t t +Δ t t +Δ t S ij δ ij δ V = R (14.2) V ⏐ ⏐ linearization δ ij δ V + t S ij δ η ij δ V = t S ij δ V (14.3) V C ijrs e rs e V t +Δ t R − V ij δ e Note: t δ = δ e ij ij varying with respect to the configuration at time t . F.E. discretization k t t k t +Δ t t +Δ t k x i = h k x i x i = h k x i x i = h k x i (14.4a) k k k t t k t +Δ t t +Δ t k k u i = h k u i u i = h k u i u i = h k u i (14.4b) k k k (14.4) into (14.3) gives t K L + t K NL U = t +Δ t R − t F (14.5) 57 MIT 2.094 14. Total Lagrangian formulation, cont’d Truss Δ L 1 small strain assumption: L E A t K = L ⎤ ⎡ cos 2 θ cos θ sin θ − cos 2 θ − cos θ sin θ cos θ sin θ sin 2 θ − sin θ cos θ − sin 2 θ cos 2 θ sin θ cos...
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MIT2_094S11_lec14 - 2.094 — Finite Element Analysis of...

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