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ECIV 720 A Advanced Structural Mechanics and Analysis

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ECIV 720 A Advanced Structural Mechanics and Analysis Powered By Docstoc
					           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




Governing                         System of
Differential   “FEM Procedures”   Algebraic
Equations of                      Equations
Mathematical
Model
        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                Weighted Residual Methods
   Rayleigh Ritz Method      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
                            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
       Solution of Continuous Systems –
             Fundamental Concepts

              Approximate Solutions
 Reduce the continuous-system mathematical model to a
                  discrete idealization


Weighted Residual Methods
                                  For linear elasticity
  Galerkin
                                  Principle of Virtual
  Least Square                          Work
  Collocation
  Subdomain
  Weighted Residual Formulations

 Consider a general representation of a
   governing equation on a region V


            Lu  P
     L is a differential operator
                        d     du 
eg. For Axial element       EA   0
                        dx    dx 

             L  EA       
                d  d
                dx dx
   Weighted Residual Formulations


           Lu  P
Assume approximate solution   ~
                              u
             then

          ~  P'
         Lu
         Weighted Residual Formulations




              Exact          Approximate


                ERROR  Lu   P
                          ~

Objective:
Define   ~
         u   so that weighted average of Error vanishes

             NOT THE ERROR ITSELF !!
         Weighted Residual Formulations

Objective:
Define   ~
         u   so that weighted average of Error vanishes



  Set Error relative to a weighting function f


                    ~  P dV  0
                 f Lu
               V
        Weighted Residual Formulations

               f Lu  P dV  0
                    ~
              V
                    f1
   f




ERROR
        Weighted Residual Formulations

               f Lu  P dV  0
                    ~
              V
                    f1
   f




ERROR
        Weighted Residual Formulations

               f Lu  P dV  0
                    ~
              V

   f




ERROR
    Weighted Residual Formulations


 Assumption for approximate solution
      (Recall shape functions)
    n
u  i i
~ Nu                                    
                                   n
                      ERROR  L  N i ui   P
   i 1                         i 1     
   Assumption for weighting function
                  n
            f   N ifi
                 i 1

          GALERKIN FORMULATION
        Weighted Residual Formulations
                                            n

      f Lu
          ~  P dV  0                f   N ifi
V
                                          i 1



       ~  P dVf  N Lu  P dVf
 N1 Lu
V
                 1  2  ~
                          V
                                   2



    N n Lu  P dVfn  0
              ~
        V

            fi are arbitrary and  0
                   Galerkin Formulation




 N1 Lu  P dV  0
       ~
V
                              Algebraic System of

  N 2 Lu  P dV  0
                          n Equations and n unknowns

V
        ~

    


 N n Lu  P dV  0
        ~
V
             n
        u   N i ui
        ~
            i 1
       y            Example

                                A=1 E=1
                                     x
                          2


            1               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
u   N i  x, y , z  u i
                                 f x   N i  x, y , z  f x i
v   N j  x, y , z  u j
                                f y   N j  x, y , z  f y j
w   N k  x, y , z  u k
                                f z   N k  x, y , z  f z k
       Galerkin’s Method in Elasticity

            f Lu  P dV  0
                 ~
          V
    x  xy  xz       
  x y z
  
V 
                     f x f x 
                            
                            
     xy  y  yz       
   
    x              f y f y 
                            
           y   z         
    xz  zy  z    
   x  y  z  f z f z  dV  0
                      
                       
         Integrate by part…
        Galerkin’s Method in Elasticity
                 Virtual Work

 Virtual Total Potential Energy




Compare to Total Potential Energy
   1 T
   σ εdV   u fdV   u TdS   u i Pi
                 T         T         T

   2 V         V         S
                                  i
             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.

				
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posted:10/9/2011
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Jun Wang Jun Wang Dr
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