MIT2_094S11_lec2

# MIT2_094S11_lec2 - 2.094 Finite Element Analysis of Solids...

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2.094 Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 2 - Finite element formulation of solids and structures Prof. K.J. Bathe MIT OpenCourseWare Reading: Ch. 1, Sec. Assume that on t S u the displacements are zero (and t S u is constant). Need to satisfy at time t : 6.1-6.2 Equilibrium of Cauchy stresses t τ ij with applied loads t τ T = t τ 11 t τ 22 t τ 33 t τ 12 t τ 23 t τ 31 (2.1) (For i = 1 , 2 , 3) t τ ij,j + t f i B = 0 in t V (sum over j ) (2.2) t τ ij t n j = t f i S f on t S f (sum over j ) (2.3) (e.g. t f i S f = t τ i 1 t n 1 + t τ i 2 t n 2 + t τ i 3 t n 3 ) (2.4) And: t τ 11 t n 1 + t τ 12 t n 2 = t f 1 S f Compatibility The displacements t u i need to be continuous and zero on t S u . Stress-Strain law t τ ij = function t u j (2.5) 7

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MIT 2.094 2. Finite element formulation of solids and structures 2.1 Principle of Virtual Work t V t τ ij t e ij d t V = t V t f B i u i d t V + t S f t f S f i u S f i d t S f (2.6) where t e ij with u i t S u = = 1 2 ∂u i t x j 0 + ∂u j t x i
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