# ECIV 720 AA dvanced Structural Mechanics and Analysis by Yf66SYP5

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```									           ECIV 720 A
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

•Elements, Nodes, Degrees of Freedom
•Interpolation
•Element Stiffness Matrix
•Structural Stiffness Matrix
•Superposition
•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
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
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  u i

              
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  u i         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  a k
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  a k
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)

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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