MANE%204240%20

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MANE 4240 CIVL 4240Introduction to Finite
Elements
Prof. Suvranu De

• FEM Discretization of 2D Elasticity

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Reading assignment Lecture notes

• Summary
• FEM Formulation of 2D elasticity (plane
  stress/strain)
• Displacement approximation
• Strain and stress approximation
• Derivation of element stiffness matrix and nodal
  load vector
• Assembling the global stiffness matrix
• Application of boundary conditions
• Physical interpretation of the stiffness matrix

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Recap 2D Elasticity
Su Portion of the boundary on which
displacements are prescribed (zero or
nonzero) ST Portion of the boundary on which
tractions are prescribed (zero or nonzero)
Examples concept of displacement field
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Example
y
2
1
2
2
4
3
x
For the square block shown above, determine u and
v for the following displacements
Case 2 Pure shear
y
Case 1 Stretch
y
1/2
2
4
2
1
x
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Solution
Case 1 Stretch
Check that the new coordinates (in the deformed
configuration)
Case 2 Pure shear
Check that the new coordinates (in the deformed
configuration)
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Recap 2D Elasticity
For plane stress (3 nonzero stress components)
For plane strain (3 nonzero strain components)
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Strong formulation
Equilibrium equations
Boundary conditions
1. Displacement boundary conditions
Displacements are specified on portion Su of the
boundary
2. Traction (force) boundary conditions
Tractions are specified on portion ST of the
boundary Now, how do I express this
mathematically?
But in finite element analysis we DO NOT work
with the strong formulation (why?), instead we
use an equivalent Principle of Minimum Potential
Energy
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Principle of Minimum Potential Energy (2D)
Definition For a linear elastic body subjected
to body forces XXa,XbT and surface tractions
TSpx,pyT, causing displacements uu,vT and
strains e and stresses s, the potential energy P
is defined as the strain energy minus the
potential energy of the loads (X and TS)
PU-W
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Strain energy of the elastic body
Using the stress-strain law
In 2D plane stress/plane strain
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Principle of minimum potential energy Among all
admissible displacement fields the one that
satisfies the equilibrium equations also render
the potential energy P a minimum. admissible
displacement field 1. first derivative of the
displacement components exist 2. satisfies the
boundary conditions on Su
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Finite element formulation for 2D Step 1
Divide the body into finite elements connected to
each other through special points (nodes)
py
3
px
4
2
v
Element e
1
u
ST
y
x
Su
x
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Total potential energy
Potential energy of element e
This term may or may not be present depending on
whether the element is actually on ST
Total potential energy sum of potential
energies of the elements
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Step 2 Describe the behavior of each element
(i.e., derive the stiffness matrix of each
element and the nodal load vector). Inside the
element e
v3
Displacement at any point x(x,y)
3
(x3,y3)
u3
v4
v2
(x4,y4)
Nodal displacement vector
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u4
u2
v
2
v1
where u1u(x1,y1) v1v(x1,y1) etc
u
(x2,y2)
u1
1
(x1,y1)
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Recall
If we knew u then we could compute the strains
and stresses within the element. But I DO NOT
KNOW u!! Hence we need to approximate u first
(using shape functions) and then obtain the
approximations for e and s (recall the case of a
1D bar) This is accomplished in the following 3
Tasks in the next slide
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TASK 1 APPROXIMATE THE DISPLACEMENTS WITHIN
EACH ELEMENT TASK 2 APPROXIMATE THE STRAIN
and STRESS WITHIN EACH ELEMENT TASK 3 DERIVE
THE STIFFNESS MATRIX OF EACH ELEMENT USING THE
PRINCIPLE OF MIN. POT ENERGY Well see these
for a generic element in 2D today and then derive
expressions for specific finite elements in the
next few classes
Displacement approximation in terms of shape
functions
Strain approximation
Stress approximation
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TASK 1 APPROXIMATE THE DISPLACEMENTS WITHIN EACH
ELEMENT
Displacement approximation in terms of shape
functions
Displacement approximation within element e
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Well derive specific expressions of the shape
functions for different finite elements later
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TASK 2 APPROXIMATE THE STRAIN and STRESS WITHIN
EACH ELEMENT
Approximation of the strain in element e
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Compact approach to derive the B matrix
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Stress approximation within the element e
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TASK 3 DERIVE THE STIFFNESS MATRIX OF EACH
ELEMENT USING THE PRINCIPLE OF MININUM POTENTIAL
ENERGY
Potential energy of element e
Lets plug in the approximations
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Rearranging
From the Principle of Minimum Potential Energy
Discrete equilibrium equation for element e
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Element stiffness matrix for element e
For a 2D element, the size of the k matrix is 2 x
number of nodes of the element
Question If there are n nodes per element,
then what is the size of the stiffness matrix of
that element?
Element nodal load vector
STe
e
Due to body force
Due to surface traction
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If the element is of thickness t
For a 2D element, the size of the k matrix is 2 x
number of nodes of the element
dA
dVtdA
Element nodal load vector
t
Due to body force
Due to surface traction
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The properties of the element stiffness matrix
1. The element stiffness matrix is singular and
is therefore non-invertible 2. The stiffness
matrix is symmetric 3. Sum of any row (or column)
of the stiffness matrix is zero! (why?)
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Computation of the terms in the stiffness matrix
of 2D elements
    The B-matrix (strain-displacement)
corresponding to this element is                  
  We will denote the columns of the B-matrix
as              
 
 
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The stiffness matrix corresponding to this
element is
which has the following form
The individual entries of the stiffness matrix
may be computed as follows  
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Step 3 Assemble the element stiffness matrices
into the global stiffness matrix of the entire
structure
For this create a node-element connectivity chart
exactly as in 1D
v3
Element 1
3
u3
ELEMENT Node 1 Node 2 Node 3
1 1 2 3
2 2 3 4
v1
v4
1
u1
u4
4
v2
Element 2
u2
2
v
u
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Stiffness matrix of element 1
Stiffness matrix of element 2
u2
v2
u3
v3
u4
v4
u2
v2
u3
v3
u4
v4
There are 6 degrees of freedom (dof) per element
(2 per node)
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Global stiffness matrix
u1
v1
u2
v2
v4
v3
u3
u4
u1
ù
é
ú
ê
v1
ú
ê
u2
ú
ê
v2

K
ú
ê
u3
ú
ê
v3
ú
ê
u4
ú
ê
ú
ê
v4
û
ë

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How do you incorporate boundary
conditions? Exactly as in 1D
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Finally, solve the system equations taking care
of the displacement boundary conditions.

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Physical interpretation of the stiffness matrix
Consider a single triangular element. The six
corresponding equilibrium equations ( 2
equilibrium equations in the x- and y-directions
at each node times the number of nodes) can be
written symbolically as
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Choose u1 1 and rest of the nodal
displacements 0
u11
1
3
y
2
x
Hence, the first column of the stiffness matrix
represents the nodal loads when u11 and all
other dofs are fixed. This is the physical
interpretation of the first column of the
stiffness matrix. Similar interpretations exist
for the other columns
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Now consider the ith row of the matrix equation
This is the equation of equilibrium at the ith dof
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Consistent and Lumped nodal loads
Recall that the nodal loads due to body forces
and surface tractions
These are known as consistent nodal loads 1.
They are derived in a consistent manner using the
Principle of Minimum Potential Energy 2. The same
shape functions used in the computation of the
stiffness matrix are employed to compute these
vectors
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p per unit area
Example
y
Traction distribution on the 1-2-3 edge px p py
0
1
b
2
x
b
3
Well see later that
N1
N2
N3
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The consistent nodal loads are
y
1
pb/3
b
2
4pb/3
x
b
3
pb/3
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The lumped nodal loads are
y
1
pb/2
b
2
pb
x
b
3
pb/2
Lumping produces poor results and will not be
pursued further
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Summary For each element
Displacement approximation in terms of shape
functions
Strain approximation in terms of
strain-displacement matrix
Stress approximation
Element stiffness matrix
Element nodal load vector