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 “FEM Procedures”
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
Variational
Rayleigh Ritz Method
Weighted Residual Methods
Galerkin
Least Square
Collocation
Subdomain

4. 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

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
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

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

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

11. Weighted Residual Formulations
ERROR

12. Weighted Residual Formulations
ERROR

13. Weighted Residual Formulations
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. n Equations and n unknowns
Galerkin Formulation
Algebraic System of
n Equations and n unknowns

17. 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

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.