Finite Element Method

(ME-349)

Air University Aerospace

and Aviation Campus 1
Kamra
 FINITE ELEMENT METHOD

Finite Element Formulation

• Shape Function

 1-D Linear Element

 1-D Quadratic Element

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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
The values of the field variable computed at the nodes are used to
approximate the values at non-nodal points (that is, in the
element interior) by interpolation of the nodal values.

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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
1-D Linear Element

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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
1-D Linear Element

u  a1  a 2 x (A)

u  u1 at x  0
u  u2 at x  L
u(x)
u1  a1
u  a1  a2 x 
u2  a1  a2 L
u2  u1
(A)  u  u1  x (1) (2)
L
 u  1 u1   u 2
x x
 L L 5
 FINITE ELEMENT METHOD
Finite Element Formulation
1-D Linear Element Shape Functions
u  1 u1   u 2
x x
 L L
u  N1(x)u1  N2 (x)u2
u1 
u  N1 N 2  
u2 

 x x u1 
u  1  
 L L u2 
u  [N]{d}
[N]  Shape Function Matrix
{d}  Nodal Displacement Vector 6
 FINITE ELEMENT METHOD

def lin_shape(x, l):

"""
Computes the shape functions for a 2-node linear element.

Parameters:
x : float
Local coordinate along the element length.
l : float
Length of the element.

Returns:
n1 : float
Shape function for node 1.
n2 : float
Shape function for node 2.
"""
n1 = 1 - (x / l)
n2 = x / l
return n1, n2
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 FINITE ELEMENT METHOD
Finite Element Formulation

1-D Linear Element

Shape Functions
Strain and Stress
du d
  [N]{d} u  [N]{d}
Stress
dx dx
x u1    E
  d 1 x  
dx  L L u2 
  EBd
1   u1 
   1  
 L L   u2   1 1 u1 
  E   
 L L u2 
  [B]{d}
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 FINITE ELEMENT METHOD

def strain_matrix_2node(L):
"""
Computes the derivatives of the shape functions for a 2-node linear
element.

Parameters:
L : float
Length of the element.

Returns:
b1 : float
Derivative of the shape function for node 1.
b2 : float
Derivative of the shape function for node 2.
"""
b1 = -1 / L
b2 = 1 / L
return b1, b2

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 FINITE ELEMENT METHOD
Finite Element Formulation

1-D Linear Element

Shape Functions
Example (Axial Member)
Determine the following for given end displacements;
1) Stiffness Matrix
2) Shape functions
3) Displacement equation
4) U, N1 and N2 at x = 20” 𝑥 = 𝑥 − 𝑋𝑖
5) Strain and stress

Given:
u1  0.003" u2  0.005"
E  3010 psi 6

Ae  1.2 in2

x1 15" x2  23"
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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
1-D Linear Element
Example (Axial Member)
Determine the following for given end displacements;
1) Stiffness Matrix (local & Global)
2) Shape functions
3) Displacement equation
4) U, N1 and N2 at x = 15”
5) Strain and stress

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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
Assignment-5.1
1-D Linear Element (Axial Member)
Determine the following for given end displacements;
1) Stiffness Matrix (local & Global)
2) Shape functions
3) Displacement equation
4) U, N1 and N2 at x = 21” at x = 15”
5) Strain and stress

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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
Assignment-5.2 1-D Linear Element

A two step bar subjected to loading condition as shown in figure. Find

(a) The displacement at the nodal points
(b) The stresses in each bar
(c) The reactions at supports

600 kN
300 kN

1 2

150 mm 150 mm 200 mm 200 mm 3.5 mm

A1 = 250 mm2 , A2 = 400 mm2 , E = 200 GPa 14

 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
1-D Linear Element

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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
1-D Linear Element 1-D Quadratic Element

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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
1-D Quadratic Element
Better approximation for the temperature profile could
be achieved if we use parabolic arcs.
The function T(x) would therefore be quadratic in x
within each element and is of the form;

We now have three parameters to determine and

hence we need the temperature at one more point in
addition to two end points of an element. We choose
the mid-point in addition to the end values to get the
following equations for the temperature at these three
locations,
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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
1-D Quadratic Element

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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
1-D Quadratic Element

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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions
1-D Quadratic Element
First derivative of temperature can now be written as;

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 FINITE ELEMENT METHOD

function [ n1,n2,n3 ] = QuadShape( x,l )

n1= 1-((3*x)/l)+((2*x^2)/(l^2));

n2= (4*x/l)-(4*x^2/(l^2));

n3= (2*x^2/(l^2) - x/l);

end

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 FINITE ELEMENT METHOD
function [ b1,b2,b3 ] = StrainMatrix3Node( x,L )

b1= 4*x/(L^2) - 3/L;

b2= 4/L - 8*x/(L^2);

b3= 4*x/(L^2) - 1/L;

end

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 FINITE ELEMENT METHOD

1-D Quadratic Element

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 FINITE ELEMENT METHOD

1-D Quadratic Element

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 FINITE ELEMENT METHOD
Finite Element Formulation
Shape Functions 1-D Quadratic Element
Assignment-6.1
Example (Axial Member)
Determine the following for given end displacements, using
1-D quadratic element;
1) Stiffness Matrix
2) Shape functions
3) Displacement equation
4) U, N1 N2 and N3 at x = 20”
5) Strain and stress X=L/2

Given:
E  3010 6 psi u1  0.003" u2  0.001" u3  0.005"

Ae  1.2 in2

x1 15" mran26
 FINITE ELEMENT METHOD
Finite Element Formulation

Assignment-2
Shape Functions 1-D Quadratic Element
Consider a solid circular bars structure with steel (Gs = 12 msi)
and aluminum (Ga = 4 msi) shafts shown in figure (do not create
any node where rotation is required);
a) Rotation at 0.25 ft from ‘C’ by 1-D linear element.
b) Rotation at 0.25 ft from ‘C’ by 1-D quadratic element. (create
a node in the middle of ‘Al’ bar for third point )

T G 
 
J L r

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FINITE ELEMENT METHOD

Thank you

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