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MIT2_094S11_lec3

# MIT2_094S11_lec3 - 2.094 Finite Element Analysis of Solids...

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2.094 Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 3 - Finite element formulation for solids and structures Prof. K.J. Bathe MIT OpenCourseWare Reading: Sec. 6.1-6.2 We need to satisfy at time t : Equilibrium t ∂ τ ij + t f B = 0 ( i = 1 , 2 , 3) in t V (3.1) t i ∂ x j t τ ij n t j = f t S f i ( i = 1 , 2 , 3) on t S f (3.2) Compatibility Stress-strain law(s) Principle of virtual displacements t V t τ ij t e ij d V t = t V u i t f B i d V t + t S f u i | t S f f t S f i d S t f (3.3) t e ij = 2 1 t u x i j + ∂u t x j i (3.4) If (3.3) holds for any continuous virtual displacement (zero on t S ), then (3.1) and (3.2) hold and u vice versa. Refer to Ex. 4.2 in the textbook. 10

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MIT 2.094 3. Finite element formulation for solids and structures Major steps I. Take (3.1) and weigh with u i : t τ ij,j + t f i B u i = 0 . (3.5a) II. Integrate (3.5a) over volume t V : t τ ij,j + t f i B u i d t V = 0 (3.5b) t V III. Use divergence theorem. Obtain a boundary term of stresses times virtual displacements on t S = t S u
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MIT2_094S11_lec3 - 2.094 Finite Element Analysis of Solids...

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