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ECIV Advanced Structural Mechanics and Analysis

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ECIV Advanced Structural Mechanics and Analysis Powered By Docstoc
					           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  ui

               
         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  ui          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  ak
   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  ak
 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

				
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posted:7/15/2011
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