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

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 Rayleigh Ritz Method Weighted Residual Methods 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 Governing Equation: Boundary Conditions: 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 Galerkin Least Square Collocation Subdomain For linear elasticity Principle of Virtual Work

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

Weighted Residual Formulations Assume approximate solution then

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

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

Weighted Residual Formulations ERROR

Weighted Residual Formulations ERROR

Weighted Residual Formulations ERROR

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

Weighted Residual Formulations fi are arbitrary and  0

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

Example x y 1 2 A=1 E=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

Galerkin’s Method in Elasticity Integrate by part…

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

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.