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 Differential Equations of Mathematical Model

System of Algebraic Equations

“FEM Procedures”

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

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

Potential Energy

= Strain Energy - Work Potential

U

dVUV

T εσ2

1

2

1

V

Uu

Strain Energy Density

WP

ii

Ti

V

T

V

T

dV

dVWP

Pu

Tu

fu

(conservative system)

Body Forces

Surface Loads

Point Loads

Total Potential & Equilibrium

i

iTiV

T

V

T

V

T dVdVdV PuTufuεσ2

1

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

0

iuMin/Max: i=1,2… all admissible displ

For Example

ii

Ti

V

T dV

Pu

εσ2

1

0

iuMin/Max:

k1

k2

k3 k4

1

2

3

Example

F1

F3

u1

u2

u3

The Rayleigh-Ritz Method for Continua

i

iTiV

T

V

T

V

T dVdVdV PuTufuεσ2

1

The displacement field appears in

work potential i

iTiV

T

V

T dVdVWP PuTufu

and strain energy dVUV

T εσ2

1

The Rayleigh-Ritz Method for Continua

Before we evaluate , an assumed displacement field needs to be constructed

Recall Shape Functions

n

iii uxNxu

1

For 1-D ii uzyxNu ,,

jj uzyxNv ,,

kk uzyxNw ,,

For 3-D

The Rayleigh-Ritz Method for Continua

Before we evaluate , an assumed displacement field needs to be constructed

ii uzyxNu ,,

jj uzyxNv ,,

kk uzyxNw ,,

For 3-D

kk azyxw ,,

ii azyxu ,,

jj azyxv ,,

Generalized Displacements

OR

Recall…

111 uxbaxu

222 uxbaxu

u1 u2

A,E,L

x

x1 x2

Alternatively…

xbaxu

2

1

2

1

1

1

u

u

b

a

x

x

Solve for a and b

Linear Variation

111 uxbaxu

222 uxbaxu

u1 u2

A,E,L

x

x1 x2

Alternatively…

xbaxu

2

1

2

1

1

1

u

u

b

a

x

x

Solve for a and b

Linear Variation

u1 u2

A,E,L

x

x1 x2

u1 u2

A,E,L

x

x1 x2

Alternatively…

xbaxu

2

1

2

1

1

1

u

u

b

a

x

x

2

1

2

1

1

1

u

u

b

a

x

x

Solve for a and b

Linear Variation

kk azyxw ,,

ii azyxu ,,

jj azyxv ,,

The Rayleigh-Ritz Method for Continua

Interpolation introduces n discrete independent displacements (dof) a1, a2, …, an. (u1, u2, …, un)

u= u(a1, a2, …, an)

and

= (a1, a2, …, an)

Thus

u= u (u1, u2, …, un)

= (u1, u2, …, un)

The Rayleigh-Ritz Method for Continua

For Equilibrium we minimize the total potential

(u,v,w) = (a1, a2, …, an)

w.r.t each admissible displacement ai

01

a

02

a

0

na

Algebraic System of

n Equations and n unknowns

Example

x

y

1 1

2

A=1 E=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