# ECIV 720 A Advanced Structural Mechanics and Analysis

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```					           ECIV 720 A
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
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
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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Jun Wang Dr
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