MANE 4240

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

• Four-noded rectangular element

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Reading assignment Logan 10.2 Lecture notes

• Summary
• Computation of shape functions for 4-noded quad
• Special case rectangular element
• Properties of shape functions
• Computation of strain-displacement matrix
• Example problem
• Hint at how to generate shape functions of higher
  order (Lagrange) elements

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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
v3
3
px
4
3
u3
v4
2
v2
v
Element e
1
4
u
ST
u4
u2
2
v1
y
x
Su
u1
1
v
x
u
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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
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Constant Strain Triangle (CST) Simplest 2D
finite element

• 3 nodes per element
• 2 dofs per node (each node can move in x- and y-
  directions)
• Hence 6 dofs per element

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Formula for the shape functions are
where
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Approximation of displacements
Approximation of the strains
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Element stiffness matrix
t
Since B is constant
A
tthickness of the element Asurface area of the
element
Element nodal load vector
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Class exercise
For the CST shown below, compute the vector of
nodal loads due to surface traction
1
y
fS3y
fS2y
fS3x
fS2x
2
x
3
(0,0)
(1,0)
py-1
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Class exercise
The only nonzero nodal loads are
(can you derive this simpler?)
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Now compute
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4-noded rectangular element with edges parallel
to the coordinate axes

• 4 nodes per element
• 2 dofs per node (each node can move in x- and y-
  directions)
• 8 dofs per element

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Generation of N1
At node 1
has the property
Similarly
has the property
Hence choose the shape function at node 1 as
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Using similar arguments, choose
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Properties of the shape functions
1. The shape functions N1, N2 , N3 and N4 are
bilinear functions of x and y
2. Kronecker delta property
3. Completeness
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3. Along lines parallel to the x- or y-axes, the
shape functions are linear. But along any other
line they are nonlinear. 4. An element shape
function related to a specific nodal point is
zero along element boundaries not containing the
nodal point. 5. The displacement field is
continuous across elements 6. The strains and
stresses are not constant within an element nor
are they continuous across element boundaries.
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The strain-displacement relationship
Notice that the strains (and hence the stresses)
are NOT constant within an element
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Computation of the terms in the stiffness matrix
of 2D elements (recap)
    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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 Notice that these formulae are quite general
(apply to all kinds of finite elements, CST,
quadrilateral, etc) since we have not used any
specific shape functions for their derivation.  
 
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Example
1000 lb
300 psi
y
3
4
Thickness (t) 0.5 in E 30106 psi n0.25
2 in
2
1
x
3 in

1. Compute the unknown nodal displacements.
2. Compute the stresses in the two elements.

This is exactly the same problem that we solved
in last class, except now we have to use a single
4-noded element
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Realize that this is a plane stress problem and
therefore we need to use
Write down the shape functions
x y
0 0
3 0
3 2
0 2
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We have 4 nodes with 2 dofs per node8dofs.
However, 5 of these are fixed. The nonzero
displacements are
v3
u2
u3
Hence we need to solve
u2
u3
v3
Need to compute only the relevant terms in the
stiffness matrix
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Compute only the relevant columns of the B matrix
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Similarly compute the other terms
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How do we compute f3y
4
3
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How about a 9-noded rectangle?
Corner nodes
y
5
2
1
a
a
b
8
9
6
x
Midside nodes
b
7
4
3
Center node
Question Can you generate the shape functions of
a 16-noded rectangle? Note These elements, whose
shape functions are generated by multiplying the
shape functions of 1D elements, are said to
belong to the Lagrange family