VIEWS: 5 PAGES: 17 POSTED ON: 2/21/2012 Public Domain
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 System of Differential “FEM Procedures” Algebraic Equations of Equations Mathematical Model 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 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 Potential Energy P P = Strain Energy - Work Potential U WP WP u fdV T Body Forces V Strain Energy Density uT TdV Surface Loads V U 1 u Pi u T V 2 Point Loads i i (conservative system) 1 T U σ εdV 2 V Total Potential & Equilibrium 1 T P σ εdV u fdV u TdV u i Pi T T T 2 V V V i 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 P Min/Max: 0 i=1,2… all admissible displ ui For Example 1 T P σ εdV 2 V ui Pi i T P Min/Max: 0 ui Example k1 1 F1 k2 u1 2 u2 k3 u3 k4 3 F3 The Rayleigh-Ritz Method for Continua 1 T P σ εdV uT fdV uT TdV uT Pi i 2 V V V i The displacement field appears in work potential WP uT fdV uT TdV uT Pi i V V i 1 T and strain energy U σ εdV 2 V The Rayleigh-Ritz Method for Continua Before we evaluate P, an assumed displacement field needs to be constructed Recall Shape Functions For 3-D For 1-D u N i x, y , z u i n u x N i x ui v N j x, y , z u j i 1 w N k x, y , z u k The Rayleigh-Ritz Method for Continua Before we evaluate P, an assumed displacement field needs to be constructed For 3-D Generalized Displacements u N i x, y , z u i u i x, y, z ai OR v N j x, y , z u j v j x, y , z a j w N k x, y , z u k w k x, y , z a k Recall… u1 x u2 Alt ernatively… x1 x2 A,E,L u x a bx Linear Variat ion u x a b x u 1 1 1 1 x1 a u1 u x a b x u 2 2 2 1 x 2 b u2 Solve for a and b u i x, y, z ai v j x, y , z a j w k x, y , z a k The Rayleigh-Ritz Method for Continua Interpolation introduces n discrete independent displacements (dof) a1, a2, …, an. (u1, u2, …, un) Thus u= u(a1, a2, …, an) u= u (u1, u2, …, un) and P= P(a1, a2, …, an) P= P (u1, u2, …, un) The Rayleigh-Ritz Method for Continua For Equilibrium we minimize the total potential P(u,v,w) = P(a1, a2, …, an) w.r.t each admissible displacement ai P 0 a1 Algebraic System of P n Equations and n unknowns 0 a2 P 0 an y Example A=1 E=1 x 2 1 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