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ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 9: Solution of Continuous Systems – Fundamental Concepts • Rayleigh-Ritz Method and the Principle of Minimum Potential Energy • Galerkin’s Method and the Principle of Virtual Work Objective Governing System of Differential “FEM Procedures” Algebraic Equations of Equations Mathematical Model Solution of Continuous Systems – Fundamental Concepts Exact solutions limited to simple geometries and boundary & loading conditions Approximate Solutions Reduce the continuous-system mathematical model to a discrete idealization Variational Weighted Residual Methods Rayleigh Ritz Method Galerkin Least Square Collocation Subdomain Strong Form of Problem Statement A mathematical model is stated by the governing equations and a set of boundary conditions e.g. Axial Element du Governing Equation: AE P(x ) dx Boundary Conditions: u(0) a Problem is stated in a strong form G.E. and B.C. are satisfied at every point Weak Form of Problem Statement A mathematical model is stated by an integral expression that implicitly contains the governing equations and boundary conditions. This integral expression is called a functional e.g. Total Potential Energy Problem is stated in a weak form G.E. and B.C. are satisfied in an average sense Solution of Continuous Systems – Fundamental Concepts Approximate Solutions Reduce the continuous-system mathematical model to a discrete idealization Weighted Residual Methods For linear elasticity Galerkin Principle of Virtual Least Square Work Collocation Subdomain Weighted Residual Formulations Consider a general representation of a governing equation on a region V Lu P L is a differential operator d du eg. For Axial element EA 0 dx dx L EA d d dx dx Weighted Residual Formulations Lu P Assume approximate solution ~ u then ~ P' Lu Weighted Residual Formulations Exact Approximate ERROR Lu P ~ Objective: Define ~ u so that weighted average of Error vanishes NOT THE ERROR ITSELF !! Weighted Residual Formulations Objective: Define ~ u so that weighted average of Error vanishes Set Error relative to a weighting function f ~ P dV 0 f Lu V Weighted Residual Formulations f Lu P dV 0 ~ V f1 f ERROR Weighted Residual Formulations f Lu P dV 0 ~ V f1 f ERROR Weighted Residual Formulations f Lu P dV 0 ~ V f ERROR Weighted Residual Formulations Assumption for approximate solution (Recall shape functions) n u i i ~ Nu n ERROR L N i ui P i 1 i 1 Assumption for weighting function n f N ifi i 1 GALERKIN FORMULATION Weighted Residual Formulations n f Lu ~ P dV 0 f N ifi V i 1 ~ P dVf N Lu P dVf N1 Lu V 1 2 ~ V 2 N n Lu P dVfn 0 ~ V fi are arbitrary and 0 Galerkin Formulation N1 Lu P dV 0 ~ V Algebraic System of N 2 Lu P dV 0 n Equations and n unknowns V ~ N n Lu P dV 0 ~ V n u N i ui ~ i 1 y Example A=1 E=1 x 2 1 1 Calculate Displacements and Stresses using a single segment between supports and quadratic interpolation of displacement field Galerkin’s Method in Elasticity Governing equations Interpolated Displ Field Interpolated Weighting Function u N i x, y , z u i f x N i x, y , z f x i v N j x, y , z u j f y N j x, y , z f y j w N k x, y , z u k f z N k x, y , z f z k Galerkin’s Method in Elasticity f Lu P dV 0 ~ V x xy xz x y z V f x f x xy y yz x f y f y y z xz zy z x y z f z f z dV 0 Integrate by part… Galerkin’s Method in Elasticity Virtual Work Virtual Total Potential Energy Compare to Total Potential Energy 1 T σ εdV u fdV u TdS u i Pi T T T 2 V V S i Galerkin’s Formulation •More general method •Operated directly on Governing Equation •Variational Form can be applied to other governing equations •Preffered to Rayleigh-Ritz method especially when function to be minimized is not available.

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posted: | 10/9/2011 |

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