MIT2_094S11_lec13

MIT2_094S11_lec13 - 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 13- Total Lagrangian formulation, cont’d Prof. K.J. Bathe MIT OpenCourseWare Example truss element. Recall: Principle of virtual displacements applied at some time t + Δ t : t +Δ t t +Δ t t +Δ t τ ij δ t +Δ t e ij d V = R (13.1) t +Δ t V t +Δ t t +Δ t t +Δ t S δ δ V = R (13.2) ij ij V t +Δ t S ij = t S ij + S ij (13.3) t +Δ t ij = t ij + ij (13.4) ij = e ij + η ij (13.5) where t S ij and t ij are known, but S ij and ij are not. 1 e ij = 2 u i,j + u j,i + t u k,i u k,j + t u k,j u k,i (13.6) 1 η ij = u k,i u k,j (13.7) 2 Substitute into (13.2) and linearize to obtain t +Δ t δ e C e rs + ij δ η ij d V = δ e ij ij d V V ij ijrs d V V t S R − V t S (13.8) F.E. discretization gives t K L + t K NL Δ U = t +Δ t R − t F (13.9) 53 MIT 2.094 13. Total Lagrangian formulation, cont’d t K t B T t B L = L C L d V (13.10) V T t K = t B t S t B NL d V (13.11) NL NL V matrix T t F = t B t S ˆ d V (13.12)(13....
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This note was uploaded on 12/29/2011 for the course ENGINEERIN 2.094 taught by Professor Prof.klaus-jürgenbathe during the Spring '11 term at MIT.

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MIT2_094S11_lec13 - 2.094 — Finite Element Analysis of...

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