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

PLu

'~ PuL then

Assume approximate solution u~

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

ERROR

0~ V

dVPuL

Weighted Residual Formulations

ERROR

0~ V

dVPuL

Weighted Residual Formulations

ERROR

0~ V

dVPuL

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

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.