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 Differential Equations of Mathematical Model
System of Algebraic Equations
“FEM Procedures”
Solution of Continuous Systems – Fundamental Concepts
Exact solutions
Approximate Solutions
Variational
Rayleigh Ritz Method
Weighted Residual Methods
Galerkin
Least Square
Collocation
Subdomain
limited to simple geometries and boundary & loading conditions
Reduce the continuous-system mathematical model to a discrete idealization
Strong Form of Problem Statement
A mathematical model is stated by the governing equations and a set of boundary conditions
e.g. Axial Element
Governing Equation: )(xPdx
duAE
Boundary Conditions: au )(0
Problem is stated in a strong form
G.E. and B.C. are satisfied at every point
Weak Form of Problem Statement
This integral expression is called a functional e.g. Total Potential Energy
A mathematical model is stated by an integral expression that implicitly contains the governing equations and boundary conditions.
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
Weighted Residual Methods
Galerkin
Least Square
Collocation
Subdomain
Reduce the continuous-system mathematical model to a discrete idealization
For linear elasticity
Principle of Virtual Work
Weighted Residual Formulations
Consider a general representation of a governing equation on a region V
PLu L is a differential operator
0
dx
duEA
dx
deg. For Axial element
dx
dEA
dx
dL
Weighted Residual Formulations
Exact Approximate
PuL ~ ERROR
Objective:
Define so that weighted average of Error vanishesu~
NOT THE ERROR ITSELF !!
Weighted Residual Formulations
Set Error relative to a weighting function
0~ V
dVPuL
Objective:
Define so that weighted average of Error vanishesu~
Weighted Residual Formulations
Assumption for approximate solution
(Recall shape functions)
n
iiiuNu
1
~PuNL
n
iii
1
ERROR
Assumption for weighting function
n
iiiN
1
GALERKIN FORMULATION
Weighted Residual Formulations
0~
~~2211
n
V
n
VV
dVPuLN
dVPuLNdVPuLN
0~ V
dVPuL
n
iiiN
1
i are arbitrary and 0
Galerkin Formulation
Algebraic System of
n Equations and n unknowns
0~
1 V
dVPuLN
0~2
V
dVPuLN
0~ V
n dVPuLN
n
iiiuNu
1
~
Example
x
y
1 1
2
A=1 E=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
ii uzyxNu ,,
jj uzyxNv ,,
kk uzyxNw ,,
Interpolated Weighting Function
ixix zyxN ,,
jyjy zyxN ,,
kzkz zyxN ,,
Galerkin’s Method in Elasticity
0
dVfzyx
fzyx
fzyx
zzzzyxz
yyyzyxy
V
xxxzxyx
Integrate by part…
0~ V
dVPuL
Galerkin’s Method in Elasticity Virtual Work
i
iTiS
T
V
T
V
T dSdVdV PuTufuεσ2
1
Compare to Total Potential Energy
Virtual Total Potential Energy