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MIT2_094S11_lec10

MIT2_094S11_lec10 - 2.094 Finite Element Analysis of Solids...

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2.094 Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 10 - F.E. large deformation/general nonlinear analysis Prof. K.J. Bathe MIT OpenCourseWare We developed t V t τ ij t e ij d V t = R t Reading: Ch. 6 (10.1) e t ij = 1 2 ∂u i x t j + ∂u j x t i (10.2) t V t τ ij δ t e ij d V t = R t (10.3) δ t e ij = 1 2 ( δu i ) t x j + ( δu j ) t x i ( t e ij ) (10.4) In FEA: t F = t R (10.5) In linear analysis t F = K t U KU = R (10.6) In general nonlinear analysis, we need to iterate. Assume the solution is known “at time t t 0 t x = x + u (10.7) Hence t F is known. Then we consider t t t t F = R (10.8) Consider the loads (applied external loads) to be deformation-independent, e.g. 41

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MIT 2.094 10. F.E. large deformation/general nonlinear analysis Then we can write t t F = t F + F (10.9) t t U = t U + U (10.10) where only t F and t U are known. = t K Δ U , t K = tangent stiffness matrix at time t (10.11) F From (10.8), t K Δ U = t t R t F (10.12) We use this to
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