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Unformatted text preview: 16.21 Techniques of Structural Analysis and Design Spring 2005 Unit #10 - Principle of minimum potential energy and Castigliano’s First Theorem Ra´ul Radovitzky April 4, 2005 Principle of minimum potential energy The principle of virtual displacements applies regardless of the constitutive law. Restrict attention to elastic materials (possibly nonlinear). Start from the PVD: ¯ ¯ u/ ¯ σ ij ¯ ij dV = t i u i dS + f i u i dV, ∀ ¯ u = 0 on S u (1) V S V Replacing the expression for the stresses for elastic materials: ∂U σ ij = ∂ ij and assuming that the virtual displacement field is a variation of the equili- brated displacement field u = δu , ¯ ¯ ij = δ ij . ∂U 0 δ ij dV = t i δu i dS + f i δu i dV V ∂ ij V S 1 The expression over the brace is the variation of the strain energy density δU : ∂U δU 0 = δ ij ∂ ij Using the properties of calculus of variations δ () = δ (): δU dV = δ U dV = δU = δ t i u i dS + f i u i dV = δ ( − V ) S V where V is the potential of the external loads. is the potential of the external loads....
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This note was uploaded on 11/07/2011 for the course AERO 16.21 taught by Professor Dffs during the Fall '10 term at MIT.
- Fall '10