ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 7: Formulation Techniques: Variational Methods The Principle of Minimum Potential Energy and the Rayleigh-Ritz Method

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

We have talked about Elements, Nodes, Degrees of Freedom Interpolation Element Stiffness Matrix Structural Stiffness Matrix Superposition Element & Structure Load Vectors Boundary Conditions Stiffness Equations of Structure & Solution

“FEM Procedures” The FEM Procedures we have considered so far are limited to direct physical argument or the Principle of Virtual Work. “FEM Procedures” are more general than this… General “FEM Procedures” are based on Functionals and statement of the mathematical model in a weak sense

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 This integral expression is called a functional e.g. Total Potential Energy A mathematical model is stated by an integral expression that implicitly contains the governing equations and boundary conditions. Problem is stated in a weak form G.E. and B.C. are satisfied in an average sense

Potential Energy   = Strain Energy - Work Potential U Strain Energy Density WP (conservative system) Body Forces Surface Loads Point Loads

Total Potential & Equilibrium Principle of Minimum Potential Energy For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium extremize the total potential energy. If the extremum condition is minimum, the equilibrium state is stable Min/Max: i=1,2… all admissible displ

For Example Min/Max:

k1k1 k2k2 k3k3 k4k Example F1F1 F3F3 u1u1 u2u2 u3u3

The Rayleigh-Ritz Method for Continua The displacement field appears in work potential and strain energy

The Rayleigh-Ritz Method for Continua Before we evaluate , an assumed displacement field needs to be constructed Recall Shape Functions For 1-D For 3-D

The Rayleigh-Ritz Method for Continua Before we evaluate , an assumed displacement field needs to be constructed For 3-D Generalized Displacements OR

Recall…

The Rayleigh-Ritz Method for Continua Interpolation introduces n discrete independent displacements (dof) a 1, a 2, …, a n. (u 1, u 2, …, u n ) u= u(a 1, a 2, …, a n ) and  =  (a 1, a 2, …, a n ) Thus u= u (u 1, u 2, …, u n )  =  (u 1, u 2, …, u n )

The Rayleigh-Ritz Method for Continua For Equilibrium we minimize the total potential  (u,v,w) =  (a 1, a 2, …, a n ) w.r.t each admissible displacement a i Algebraic System of n Equations and n unknowns

Example x y 11 2 A=1E=1 Calculate Displacements and Stresses using 1)A single segment between supports and quadratic interpolation of displacement field 2)Two segments and an educated assumption of displacement field