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ECIV 720 A Advanced Structural Mechanics and Analysis

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.

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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 7:Formulation Techniques: Variational Methods The Principle of Minimum Potential Energy and the Rayleigh-Ritz Method

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

  3. 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

  4. “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

  5. 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

  6. 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

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

  8. Total Potential & Equilibrium Min/Max: 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 i=1,2… all admissible displ

  9. For Example Min/Max:

  10. Example k1 F1 u1 k2 u2 u3 k4 k3 F3 1 2 3

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

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

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

  14. Recall…

  15. 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)

  16. 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 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 • Two segments and an educated assumption of displacement field

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