ECIV 720 A
Advanced Structural Mechanics
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
(Discrete)
• A finite number of
state variables
describe solution
• Algebraic Equations
Continuous
• Differential Equations
Govern Response
Matrix Structural Analysis - Objectives
Use
Equations of Equilibrium
Constitutive Equations
Compatibility Conditions
Basic Equations
Form
[A]{x}={b}
Solve for Unknown Displacements/Forces
{x}= [A]-1{b}
FEM Analysis - Objective
Governing
Equations of
Mathematical
Model
FEM Procedures
System of
Algebraic
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
E, n
Isotropic Material:
Generalized Hooke’s Law
(a)
Equations (a) can be solved for s...
Stress-Strain Relations – Material Matrix
s x 
s 
 y
s z 
 
 yz 
 
 xz 
 xy 
Material Matrix
 x 
 
 y
 z 
 
 yz 
 
 xz 
 xy 
Special Cases
e.g. Two Dimensional – Plane Stress
Thin Planar Bodies subjected to in plane loading
Strain Energy
In the general state of stress for conservative systems
1 T
U   σ εdV
V
2
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

u 




2
1  a0  a1 1  a2 1
2
2  a0  a1 2  a2 2
  u1
n
   an 1
n
   an 2
  u2

un  


2
a0  a1 n  a2 n
  un
n
   an n
Shape Functions – A General Approach
Solve for ai
n
1 1
1   a1   u1 




n 
a
u
1





2
2
n

     2 

     


n 
 n  an  un 
1  n
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
Pascal Triangle
1
x
x2
x3
x4
x5
x 2y
x 3y
x 4y
…….
0
1
y2
xy2
x2y2
x3y2
Degree
y
xy
2
y3
x2y3
3
y4
xy3
xy4
n
4
y5
5
Shape Functions – Important Points
The continuous displacement field is expressed in
a discrete form as a linear combination of shape
functions.
u   
n

N i   ui
Or in matrix form
i 1
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
u   
n
 N   u
i
i
i 1
Representation at discrete nodes is exact
u   
n
 N   u
i
i 1
i
 ui
Solution of Continuous Systems –
Fundamental Concepts
Objective: Determine displacement u…
…satisfying equilibrium equations
s=f()
=g(u)
Governing Equations
2nd order PDE
Solution of Continuous Systems –
Fundamental Concepts
Exact solutions
limited to simple geometries and boundary & loading
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
WP
U
WP 
Strain Energy Density
U 1
u
 s
V 2
 u fdV
T
Body Forces
V
  uT TdV
Surface Loads
V
  u Pi
T
i
Point Loads
i
1 T
U   σ εdV
2 V
(conservative system)
Total Potential & Equilibrium
1 T
T
T
T
P   σ εdV   u fdV   u TdV   u i Pi
V
V
2 V
i
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
Min/Max:
P
 0 i=1,2… all admissible displ
ui
The Rayleigh-Ritz Method for Continua
1 T
P   σ εdV   uT fdV   uT TdV   uTi Pi
V
V
2 V
i
The displacement field appears in
work potential WP 
and strain energy
T
T
T
u
f
dV

u
T
dV

u
 i Pi


V
V
1 T
U   σ εdV
2 V
i
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

n

u x   N i x ui
i 1
u   N i  x, y , z  ui
v   N j  x, y , z  u j
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)
w.r.t each admissible displacement ai
P
0
a1
P
0
a2

P
0
an
Algebraic System of
n Equations and n unknowns
Weighted Residual Formulations
Consider a general representation of a
governing equation on a region V
Lu  P
L is a differential operator
eg. For Axial element
d 
du 
 EA   0
dx 
dx 
d
d
L  EA 
dx
dx

Weighted Residual Formulations
Lu  P
Assume approximate solution
(Recall shape functions)
n
~
u   N i ui
then
i 1
~
Lu  P'
Weighted Residual Formulations
Lu  P
~
Lu  P'
Exact
Approximate
n
u~   N i ui
i 1


~
Lu   P  L  N i ui   P
 i 1

n
Error =
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
~
 Wi Lu  P dV  0
i  1,.., n
V
If we choose shape functions as weighting functions
GALERKIN FORMULATION
Galerkin Formulation
Let function Wi=Ni
~  P dV  0

N
L
u
 1
V
~  P dV  0

N
L
u
 2
V

~  P dV  0

N
L
u
 n
V
Algebraic System of
n Equations and n unknowns
Galerkin Formulation
Set Error relative to a weighting function Wi=0
~


W
L
u

P
dV

0
i

i  1,.., n
V
n
In General define
   N ii
i 1
~



L
u

P
dV

0

V
Principle of Virtual Work
A body is in equilibrium if the internal virtual
work equals the external virtual work for every
kinematically admissible displacement field
Galerkin’s Method in Elasticity
Virtual Work
Virtual Total Potential Energy
Compare to Total Potential Energy
1 T
T
T
T
P   σ εdV   u fdV   u TdS   u i Pi
V
S
2 V
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
T
T
T
P   σ εdV   u fdV   u TdS   u i Pi
V
S
2 V
i
P
0
ui
Galerkin
0
 σ εφdV   φ fdV   φ TdS   φ P
T
V
T
V
T
S
T
i
i
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
~  N u  Nu
u
 i i
e
φ   N i φ i Nφ e
For Each Element
x  Nxe
 = B ue
dN
B
dx
εφ  Bφe
s = E B ue
Introduce into variational forms
Rayleigh Ritz
1 T
T
T
T
P   σ εdV   u fdV   u TdS   u i Pi
V
S
2 V
i
1 T
P   u e k eu e   uTe f e   uTe Te  uTe P
e 2
e
e
Galerkin
 σ εφdV   φ fdV   φ TdS   φ
0   φ k u   φ f   φ T  φP
0
T
T
V
V
T
e
e
T
e
S
T
e e
e
e
T
e
e
i
e
T
i
Pi
Assemble…
u1
u  u1 u2  un 
T
u2
u3
φ  1  2   n 
P4
P  P1 P2  Pn 
Pi
T
T
F
  f e  Te   P
K
  k e
e
e
P
0
ui
1 T
T
P  u Ku  u F
2
0  φ Ku  φ F  φ Ku  F 
T
T
T
Pn
u4
ui
ui+1
un-1
un
0  Ku  F
Boundary Conditions – Elimination Approach
kii kij kik kil kim
ui
Pi
kji K
kjj kjk kK
jl kjm
ff
fs
kki kkj kkk kkl kkm
uujf
PPfj
kli klj klk kll klm
Ksf
Kss
kli klj klk kll klm
ul
us
u
Pl
Ps
P
Kffuf+ Kfsus=Pf
Ksfuf+ Kssus=Ps
uk = Pk
uf =
m
m
-1
Kff (Pf +
Kfsus)
Ksfuf+ Kssus=Ps
Elements
•
•
•
•
•
•
•
Axial Element – Linear & Higher Order
2D Constant Strain Triangle
2D 4-node Quadrilateral
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
h
2

Element Stiffness Matrix ke
For Example 2-D Plane Strain
3
1 T
U e   ε σDdV
2 Ve
dV  tdA
 = B qe
Ue 
1 T T
q e  B DBtdA q e
le
2

1 T
T
 q e t  B DBdAq e
2  A

s = D B qe
ke
8x8 matrix
Element Stiffness Matrix ke
Furthermore
dV  tdA  t det Jddh
and
Jacobian
k e  t  B DBdA 
T
A
1 1
t   B DB det Jddh
T
1 1
Numerical
Integration
Jacobian of Transformation
u u x u y


 x  y 
u u x u y


h x h y h
 u   x
    
 u    x
  
 h   h
J
y   u 
   x 
  u 
y 
 
h   y 
Jacobian of Transformation
v v x v y


 x  y 
v v x v y


h x h y h
 v   x
    
 v    x
  
 h   h
J
y   v 
   x 
  v 
y 
 
h   y 
Gaussian Quadrature
Karl Friedriech Gauss discovered that by a
special placement of nodes the accuracy of the
numerical integration could be greatly increased
Gaussian Quadrature
Theorem on Gaussian nodes
Let q be a polynomial of degree n such that
b
 qx x dx  0
k
k  0,1,..., n - 1
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 
i
a
i
1
1
2
2
n
n
i
with xi’s as nodes is exact for all polynomials of
degree 2n-1.
Gaussian Quadrature 2-point
W2=1
W1=1
-1
 f x dx  1 f 
1
f()
f(1)
1   1 3
1
f(2)
2  1 3
  1 3
1 3 1 f
1
Gauss Points and Weights
2-Dimensional Integration
Gaussian Quadrature
1 1

f  ,h ddh 
1 1


wi f  i ,h  dh 
1 
i 1

1
n
 w w f  ,h 
n
n
j 1 i 1
j
i
i
j
2-D Integration 2-point formula
h
h2  1 3
w1w2  1
w1  1
w2 w2  1
w2  1

h1   1 3
w1w1  1
w2 w1  1
1   1 3
2  1 3
2-D Integration 2-point formula
h
h1   1 3

h1   1 3
1 1
1   1 3
2  1 3
  f  ,h ddh 
1 1
w1w1 f 1 ,h1   w2 w1 f 2 ,h1   w1w2 f 1 ,h2   w2 w2 f 2 ,h2 
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:
Rule of Thumb
Ratio of sides <2
Larger ratios
may be used
with caution
Angular Distortion
Internal Angle < 180o
Modeling Issues: Degenerate Quadrilaterals
Coincident Corner Nodes
4
4
3
x
x
x
x
1
x
x
1
2
x
x
2
3
Integration Bias
Less accurate
Modeling Issues: Degenerate Quadrilaterals
Three nodes collinear
4
Integration Bias
3
x
3
x
x
4
x
1
x
x
x
1
x
2
2
Less accurate
Modeling Issues: Degenerate Quadrilaterals
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)
h
(6,4)

4
Parent
1
(3,2)
2
J singular
(9,2)
All interior angles < 180
x
Another NoNo Situation
h

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
4-node
8-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
must accommodate
•RIGID BODY MOTION
•CONSTANT STRAIN STATE
n
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
x2
x 2y
y
y2
xy
xy2
Behavior accurately predicted
Effects of Element Distortion
Angular Distortion
1
x
x2
x 2y
y
y2
xy
xy2
Predictability is lost for all quadratic terms
Effects of Element Distortion
Quadratic Curved Edge Distortion
1
x
x2
x 2y
y
y2
xy
xy2
Predictability is lost for all quadratic terms
Effects of Element Distortion
The advantage (reduced #of dof)
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
x2
x 2y
y
xy
y2
xy2
x2y2
Behavior accurately predicted
Effects of Element Distortion
9-node
Angular Distortion
1
x
x2
x 2y
y
y2
xy
xy2
Behavior predicted better than 8-node
Effects of Element Distortion
9-node
Quadratic Curved Edge Distortion
1
x
x2
x 2y
y
y2
xy
xy2
Predictability is lost for high order terms
Effects of Element Distortion
The advantage (reduced #of dof)
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
3
1
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
  d11   d12   d1n 
      
d 21   d 22  d 2 n 

D      
         
      
 d n1  d n 2   d nn 
KD  DΛ
1



2

Λ





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
E=1
v=0.3
1
8 degrees of freedom
Solve Eigenproblem
8 modes
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
Flexural Mode
5  0.495
Flexural Mode
Displacement Modes & Stiffness Matrix
6  0.769
Shear Mode
Displacement Modes & Stiffness Matrix
7  0.769
8  1.43
Stretching Mode
Uniform Extension 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
1
F  s x Ht 
2
Compute results of FE
patch
If
(computed sx) = (assumed sx)
test passed