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ECIV 720 A Advanced Structural Mechanics and Analysis

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.

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ECIV 720 A Advanced Structural Mechanics and Analysis

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  1. 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

  2. Objective Governing Differential Equations of Mathematical Model System of Algebraic Equations “FEM Procedures”

  3. 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 • Weighted Residual Methods • Galerkin • Least Square • Collocation • Subdomain • Variational • Rayleigh Ritz Method

  4. Strong Form of Problem Statement Governing Equation: Boundary Conditions: A mathematical model is stated by the governing equations and a set of boundary conditions e.g. Axial Element Problem is stated in a strong form G.E. and B.C. are satisfied at every point

  5. 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

  6. Solution of Continuous Systems – Fundamental Concepts For linear elasticity Principle of Virtual Work Approximate Solutions Reduce the continuous-system mathematical model to a discrete idealization • Weighted Residual Methods • Galerkin • Least Square • Collocation • Subdomain

  7. Weighted Residual Formulations eg. For Axial element Consider a general representation of a governing equation on a region V L is a differential operator

  8. Weighted Residual Formulations Assume approximate solution then

  9. Weighted Residual Formulations Exact Approximate Objective: Define so that weighted average of Error vanishes NOT THE ERROR ITSELF !!

  10. Weighted Residual Formulations Set Error relative to a weighting function f Objective: Define so that weighted average of Error vanishes

  11. Weighted Residual Formulations f = 1 f ERROR

  12. Weighted Residual Formulations f = 1 f ERROR

  13. Weighted Residual Formulations f ERROR

  14. Weighted Residual Formulations Assumption for approximate solution (Recall shape functions) Assumption for weighting function GALERKIN FORMULATION

  15. Weighted Residual Formulations fi are arbitrary and  0

  16. Galerkin Formulation Algebraic System of n Equations and n unknowns

  17. Example y x 2 1 1 A=1 E=1 Calculate Displacements and Stresses using a single segment between supports and quadratic interpolation of displacement field

  18. Galerkin’s Method in Elasticity Governing equations Interpolated Displ Field Interpolated Weighting Function

  19. Galerkin’s Method in Elasticity Integrate by part…

  20. Galerkin’s Method in Elasticity Virtual Work Virtual Total Potential Energy Compare to Total Potential Energy

  21. 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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