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Unformatted text preview: 2.092/2.093 — Finite Element Analysis of Solids & Fluids I Fall ‘09 Lecture 5- The Finite Element Formulation Prof. K. J. Bathe MIT OpenCourseWare In this system, (X, Y, Z) is the global coordinate system, and (x, y, z) is the local coordinate system for the element i . We want to satisfy the following equations: τ ij,j + f i B = 0 in V Equilibrium Conditions τ ij n j = f i S f on S f → u i = u i S u Compatibility Conditions (A) S u → τ ij = f ( ε kl ) Stress-strain Relations → Then we have the exact solution. Principle of Virtual Displacements ε T Cε dV = u T f B dV + u S f T f S f dS f (B) V V S f Here, real stresses ( Cε ) are in equilibrium with the external forces ( f B , f S f ). Note that Eq. (B) is equivalent to Eq. (A). Recall that we defined ε T = ε xx ε yy ε zz γ xy γ yz γ zx ∂u ε T = ε ¯ xx ε ¯ yy ε ¯ zz γ ¯ xy γ ¯ yz γ ¯ zx = . . . ∂x Basic assumptions: ⎡ ⎤ ( m ) u ( x, y, z ) u ( m ) = ⎣ v ( x, y, z ) ⎦ = H ( m ) u ˆ (1) w...
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.092 taught by Professor Klaus-jürgenbathe during the Fall '09 term at MIT.
- Fall '09
- Finite Element Analysis