# ECIV 720 A Advanced Structural Mechanics and Analysis(2)

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```					          ECIV 720 A
and Analysis

Final Review
Final Review

Fundamental Concepts
Stresses and Equilibrium
Boundary Conditions
Strain Displacement Relations
Stress-Strain Relations
Interpolation
Potential Energy and Equilibrium
Rayleigh Ritz Method
Galerkin’s Method
Engineering Systems

Lumped Parameter
Continuous
(Discrete)

• A finite number of      • Differential Equations
state variables           Govern Response
describe solution
• Algebraic Equations
Matrix Structural Analysis - Objectives

Use
Equations of Equilibrium
Basic Equations
Constitutive Equations
Compatibility Conditions

Form
[A]{x}={b}

Solve for Unknown Displacements/Forces
{x}= [A]-1{b}
FEM Analysis - Objective

Governing
Equations of                                  System of
Mathematical                                  Algebraic
FEM Procedures
Model                                         Equations

Differential
Equations

Where is it used?
Analysis of Solids and Structures
Soil and Rock Mechanics
Heat Transfer
Fluids
Virtually in any field of engineering analysis…
Assumptions

• Linear Strain-Displacement Relationship

• Small Deformations

• Equilibrium Pertains to Undeformed Configuration
FEM Process of Analysis

Reliability of Solution
depends on choice
of Mathematical
Model

Accurate
Approximations
of Solutions
Basic Relationships of Elasticity Theory
Equilibrium

Equilibrium
Write Equations of
Equilibrium

SFx=0
SFy=0
SFz=0
Boundary Conditions
Equilibrium at Surface

Prescribed Displacements
Strain-Displacement Relations

Assumption
Small Deformations
Stress-Strain Relations

Isotropic Material:         E, n
Generalized Hooke’s Law

(a)

Equations (a) can be solved for s...
Stress-Strain Relations – Material Matrix

s x                                             x 
s                                               
 y                                              y
s z 
                                                z 
 
                                               
 yz                                            yz 
                                               
 xz                                             xz 
 xy 
                                                xy 
 

Material Matrix
Special Cases

e.g. Two Dimensional – Plane Stress
Strain Energy

In the general state of stress for conservative systems

1 T
U   σ εdV
2 V
Function Interpolation/Approximation

Interpolation or approximation of state
variables is a key operation in FEM
procedures

Common Approach:
Interpolation of State Variables Using
Polynomials
Shape Functions – A General Approach
Assume a polynomial

For example for 1-D

u   a0  a1  a2  an
2          n
Shape Functions – A General Approach
Enforce Boundary Conditions

u             
1  a0  a1 1  a2 1
2
  u1
   an 1n

u             
2  a0  a1 2  a2 2
2
                u2
   an 2n



un           
a0  a1 n  a2 n
2
             un
   an n
n
Shape Functions – A General Approach

Solve for ai

1 1    1   a1   u1 
n

          n      u 
1 2     n  a2   2 
                
     

          n 
1  n
         n  an  un 
       
Shape Functions – A General Approach

Substitute in
u   a0  a1  a2  an2            n

u   N1  u1  N 2  u2    N n  un

Shape Functions
Shape Functions – A General Approach

Substitute in
u   a0  a1  a2  an2            n

u   N1  u1  N 2  u2    N n  un

Degrees of Freedom
(nodal values)
2-D The Pascal Triangle
ux, y   a0  a1 x  a2 y  a3 xy an x                      n

Pascal Triangle                            Degree
1                                      0
x        y                            1
x2         xy         y2                     2
x3       x 2y        xy2        y3                3
x4     x 3y      x2y2         xy3          y4          4
x5   x 4y      x3y2        x2y3           xy4        y5     5

…….
Shape Functions – Important Points

The continuous displacement field is expressed
in a discrete form as a linear combination of
shape functions.
n
u      
i 1
N i   ui   Or in matrix form

u1 
u 
 2
u    N1 , N 2  N n    Nu
 
un 
 
Shape Functions – A generalized concept

• Shape Functions are not necessarily
Lagrange Polynomials

• Shape functions may be any functions
that approximate the variation of state
variables and satisfy Boundary
Conditions
• e.g. trigonometric, logarithmic, incomplete
polynomials, etc
Shape Functions – Important Points

Depending on choice of Shape Functions
the representation is exact or approximate at
intermediate points
n
u       N   u
i 1
i     i

Representation at discrete nodes is exact
n
u        N   u
i 1
i   i      ui
Solution of Continuous Systems –
Fundamental Concepts
Objective: Determine displacement u…

…satisfying equilibrium equations

Governing Equations
s=f()   =g(u)
2nd order PDE
Solution of Continuous Systems –
Fundamental Concepts

Exact solutions
conditions
Approximate Solutions
Reduce the continuous-system mathematical model to a
discrete idealization

Rayleigh Ritz Method
Galerkin Method
Potential Energy P & Rayleigh Ritz Method

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    s                              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
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

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

Thus

u= u (u1, u2, …, un)

and

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
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
(Recall shape functions)
n
u  i i
~ Nu              then
i 1

~  P'
Lu
Weighted Residual Formulations

Lu  P          ~  P'                  n
Lu                  u   N i ui
~
i 1

Exact          Approximate

~   P  L N u   P
n
Error =     Lu           i i 
 i 1 
Objective:
Define ui so that weighted average of Error vanishes
Weighted Residual Formulations
Objective:
Define ui so that weighted average of Error vanishes

Set Error relative to a weighting function Wi=0

~  P dV  0
 Wi Lu
V
i  1,.., n

If we choose shape functions as weighting functions
GALERKIN FORMULATION
Galerkin Formulation

Let function Wi=Ni

 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
Galerkin Formulation

Set Error relative to a weighting function Wi=0

      ~  P dV  0
Wi Lu                         i  1,.., n
V

n
In General define      N ii
i 1

     ~  P dV  0
 Lu
V
Principle of Virtual Work

A body is in equilibrium if the internal virtual
work equals the external virtual work for every
Galerkin’s Method in Elasticity
Virtual Work

Virtual Total Potential Energy

Compare to Total Potential Energy
1 T
P   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.
FEM General Procedure
Finite Element Formulation

Rayleigh-Ritz
1 T
P   σ εdV   u fdV   u TdS   u i Pi
T         T         T

2 V         V         S
i
P
0
ui

Galerkin

0    σ εφdV   φ fdV   φ TdS   φ
T               T         T          T
i   Pi
V               V         S
i
Discretization

Define a number of nodes and
elements
Interpolate real and virtual displacement
field within each element by same type
of interpolating functions Ni

u   N i u i  Nue
~                          φ   N i φ i Nφ e
For Each Element

x  Nxe       = B ue         s = E B ue

B
dN
εφ  Bφe
dx
Introduce into variational forms

Rayleigh Ritz
1 T
P   σ εdV   u fdV   u TdS   u i Pi
T         T         T

2 V         V         S
i
1 T
P   u e k eu e   uT f e   uT Te  uT P
e          e       e
e 2            e          e

Galerkin
0   σ εφdV   φ fdV   φ TdS   φ
T                   T                   T       T
i   Pi
V                   V               S
i

0   φ k u   φ f   φ T  φP
T
e   e   e
T
e e
T
e       e
e                   e           e
Assemble…
u1
u  u1 u2  un 
T
u2
u3
φ  1  2   n 
T
P4
u4
P  P P2  Pn 
T
1                                  Pi   ui

K  ke          F   f e  Te   P
ui+1
               
un-1
e                e              Pn   un
1 T                    P
P  u Ku  u F
T
0
2                      ui
0  Ku  F
0  φ Ku  φ F  φ Ku  F 
T       T        T
Boundary Conditions – Elimination Approach

kii kij kik kil kim    ui   Pi
k
kji Kjj kjk kjl kjm   u
ujf   P
Pfj
ff    Kfs
kki kkj kkk kkl kkm    uk = Pk
kli klj klk kll klm   ul    Pl
Ksf        Kss     us    Ps
kli klj klk kll klm   u m   P m

-1
Kffuf+ Kfsus=Pf       uf =   Kff (Pf +   Kfsus)
Ksfuf+ Kssus=Ps         Ksfuf+ Kssus=Ps
Elements

• Axial Element – Linear & Higher Order
• 2D Constant Strain Triangle
• 2D Higher Order Elements
• 3D Solid Elements
• Beam Elements with and without Shear Deformation
• Plate/Shell Elements with and without Shear
Deformation
• Special Elements (rigid links, elastic supports, infinite
boundaries, etc)
1
4   Element Stiffness Matrix ke

2                   For Example 2-D Plane Strain
3


1 T                   Ue 
U e   ε σDdV               1 T T
2 Ve                      q e  B DBtdA q e
2      le

dV  tdA               1 T             
 q e t  B DBdAq e
T

 = B qe               2  A            
s = D B qe                          ke
8x8 matrix
Element Stiffness Matrix ke

Furthermore
dV  tdA  t det Jdd

and                    Jacobian

k e  t  B DBdA 
T

A
1 1
Numerical
t   B DB det Jdd
T
Integration
1 1
Jacobian of Transformation

u u x u y       u   x     y   u 
     
 x  y           
                 x 
 
u u x u y
     
 u    x
  
    
  
J        u 
y 
 
   y 
 
 x  y 
Jacobian of Transformation
v v x v y       v   x     y   v 
                    
 x  y                        x 
 

v v x v y
     
 x  y 
  
    
  
J
 v    x        v 
y 
 
   y 
 

Karl Friedriech Gauss discovered that by a

special placement of nodes the accuracy of the

numerical integration could be greatly increased
Theorem on Gaussian nodes
Let q be a polynomial of degree n such that
b

 qx x dx  0          k  0,1,..., n - 1
k

a

Let x1,x2,…,xn be the roots of q(x). Then
b

 f x dx   w f x   w f x   w f x     w f x 
a              i
i        i   1   1     2    2         n   n

with xi’s as nodes is exact for all polynomials of
degree 2n-1.

W2=1   f(2)   f()
W1=1      f(1)

1   1 3                2  1 3
-1                                          1

 f x dx  1 f         1 3
1
1 3 1 f
1
Gauss Points and Weights
2-Dimensional Integration

1 1

      f  , dd 
1 1

               
1     n

1  wi f i ,  d 
  i 1           

 w w f  , 
n      n

j   i     i    j
j 1 i 1
2-D Integration 2-point formula

w1  1
w1w2  1       w2 w2  1    w2  1
2  1 3


w1w1  1       w2 w1  1
1   1 3

1   1 3   2  1 3
2-D Integration 2-point formula



1   1 3


1   1 3                                 1 1

1   1 3   2  1 3
  f  , dd 
1 1

w1w1 f 1 ,1   w2 w1 f 2 ,1   w1w2 f 1 ,2   w2 w2 f 2 ,2 
Choices in Numerical Integration

• Numerical Integration cannot produce exact
results
• Accuracy of Integration is increased by using
more integration points.
• Accuracy of computed FE solution DOES
NOT necessarily increase by using more
integration points.
Modeling Issues: Element Shape

Square : Optimum Shape
Not always possible to use

Rectangles:         Larger ratios
Rule of Thumb       may be used
Ratio of sides <2   with caution

Angular Distortion
Internal Angle < 180o
Coincident Corner Nodes
4                                   4

x             3
x
x

x                                   x       x
x                                   x
1                                      1

2                                   2   3

Integration Bias

Less accurate
Three nodes collinear
4
Integration Bias
x           3                                    3
x
4           x
x
x                               x
x                                x
1                                 1

2                                2

Less accurate

2 nodes

Use only as necessary to improve representation of
geometry
Do not use in place of triangular elements
A NoNo Situation
y
3   (7,9)



(6,4)                            
4

Parent

1                               2
J singular
(3,2)                        (9,2)

x
All interior angles < 180
Another NoNo Situation





x, y
not uniquely
defined
Convergence Considerations

For monotonic convergence of solution

Requirements

Elements (mesh) must be compatible

Elements must be complete
Monotonic Convergence

FEM Solution

Exact Solution

No of Elements

For monotonic convergence the elements must be
complete and the mesh must be compatible
Mixed Order Elements

Consider the following Mesh

4-node

8-node

Incompatible Elements…
Mixed Order Elements

We can derive a mixed order element for grading

8-node                                4-node

7-node

By blending shape functions appropriately
Convergence Considerations

For monotonic convergence of solution

Requirements

Elements (mesh) must be compatible

Elements must be complete
Element Completeness

For an element to be complete

Assumption for displacement field

ux, y   a0  a1 x  a2 y  a3 xy an x   n

must accommodate

•RIGID BODY MOTION
•CONSTANT STRAIN STATE
Element Completeness

Consider
ux, y   a1  a2 x  a3 y

This is not a complete polynomial

However,
Element Completeness
Assume displacement field
ux, y   a1  a2 x  a3 y
The Computed nodal displacement
corresponding to this field
ui  a1  a2 xi  a3 yi      i=1,…,#of nodes

Test for ELEMENT completeness
Isoparametric Formulation

u  N1u1  N 2u2  N 3u3   N i ui
Element Completeness

u  a1  N i a2  N i xi a3  N i yi
Isoparametric Formulation

x   N i xi           y   N i yi

Thus, computed displacement field

u  a1  N i a2 x  a3 y
Element Completeness

u  a1  N i a2 x  a3 y       Computed

ux, y   a1  a2 x  a3 y     Assumed

In order for the computed displacements to be the
assumed ones we must satisfy

N   i   1   Condition for element completeness
Effects of Element Distortion

Loss of predictive capability of isoparametric
element

No Distortion
1
x        y
x2       xy         y2
x 2y        xy2

Behavior accurately predicted
Effects of Element Distortion

Angular Distortion
1
x        y
x2       xy         y2
x 2y         xy2

Predictability is lost for all quadratic terms
Effects of Element Distortion

1
x        y
x2       xy         y2
x 2y         xy2

Predictability is lost for all quadratic terms
Effects of Element Distortion

of using 8-node higher order element
based on an incomplete polynomial is lost
when high element distortions are present
Effects of Element Distortion

Loss of predictive capability of isoparametric
element
No Distortion
9-node
1
x        y
x2       xy       y2
x 2y      xy2
x2y2

Behavior accurately predicted
Effects of Element Distortion

9-node
Angular Distortion
1
x        y
x2       xy         y2
x 2y        xy2

Behavior predicted better than 8-node
Effects of Element Distortion

1
x        y
x2       xy         y2
x 2y        xy2

Predictability is lost for high order terms
Effects of Element Distortion

of using higher order element
based on an incomplete polynomial is lost
when high element distortions are present

For angular distortion 9-node element
shows better behavior

For Curved edge distortion all elements
give low order prediction
Polynomial Element Predictability
Tests of Element Quality

Eigenvalue Test
Identify Element Deficiencies

Patch Test
Convergence of Solutions
Eigenvalue Test

4
1                   3

2

Apply loads –{r} in proportion to displacements

 r  d

kd  r
Eigenvalue Test

 r  d

kd  r
 k  I d  0

Eigenproblem
As many eigenvalues  as dof

For each  there is a solution for {d}
Displacement Modes & Stiffness Matrix

For all eigenvalues and modes

Kd  d                          KD  DΛ

  d11   d12   d1n         1         
                                     
 d 21   d 22  d 2 n          2
D                           Λ           
                                 
                                     
 d n1  d n 2  d nn                n 
Eigenvalue Test

Scale {d} so that

d d  1
T

then

d k d  d d  
T                        T

 2U
Eigenvalue Test

d k d  d d  
T                     T

 2U
Rigid Body Motion => System is not strained => U=0

System is strained => U=0
Rigid Body Motion

Rigid Body Modes

+    Element Straining Modes

Total Number of Element Displacement Modes
(=number of degrees of freedom)
Displacement Modes & Stiffness Matrix
Consider the 4-node plane stress element
1

t=1
1                             E=1
v=0.3

8 degrees of freedom            8 modes

Solve Eigenproblem
Displacement Modes & Stiffness Matrix

1  0
Rigid Body Mode           2  0
Rigid Body Mode
Displacement Modes & Stiffness Matrix

3  0
Rigid Body Mode
Displacement Modes & Stiffness Matrix

4  0.495                5  0.495
Flexural Mode             Flexural Mode
Displacement Modes & Stiffness Matrix

6  0.769
Shear Mode
Displacement Modes & Stiffness Matrix

7  0.769             8  1.43
Uniform Extension Mode
Stretching Mode
(breathing)
Displacement Modes & Stiffness Matrix

The eigenvalues of the stiffness matrix display
directly how stiff the element is in the
corresponding displacement mode

  2U
Patch Test

Objective
Examine solution convergence for displacements,
stresses and strains in a particular element
type with mesh refinement
Patch Test - Procedure

Build a simple FE
model

Consists of a Patch of
Elements

At least one internal
node

Load by nodal equivalent forces consistent with
state of constant stress

Internal Node is unloaded and unsupported
Patch Test - Procedure

F  s x Ht 
1
2
Compute results of FE
patch

If
(computed sx) = (assumed sx)
test passed

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Jun Wang Dr
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