In FEA, trial function having order of the polynomial more than two.

1. Higher Order Elements
• In this section, we will develop a higher-
order triangular element, called the Linear-
Strain Triangular (LST).
• This element has many advantages over
the CST because it has 6 nodes and 12
displacements DOF.
• The displacement function of the triangle
is quadratic.
Steps in the formulation of LST element
stiffness equations
The procedure to derive LST element stiffness and element
equations is identical to that used for the CST element.
x
2
3
1
2L = 1
y
L3= 1
L1= 1
L1= 1/2
L1=
0
L2= 0
L2= 1/2
L3= 0 L3= 1/2
4
5
6
By
R.Sakthivel Murugan,
Assistant Professor/Mech,
KCET.

2. Step 1 : Discretize and Select element types
Consider the triangular element shown below: Each node is defined by
nodes numbers. Each node has 2 DOFs (displacements in x-,y- directions)
x
y
u1
v1
1
u4
v2
2 u2
6
5
v6
v3
3 u3
v4
4
u6
u5
v5
Step 2 : Select Displacement Functions
The variation of the displacements over the element may be expressed as
The displacement compatibility among the adjoining elements is
satisfied because the 3 nodes defining adjacent sides define a unique
parabola.
1
y5
xy4
x3
y2
x2
y3
x4
yx5
y4
x2
y2
x3
yx4
x3
y2
y3
xy3
x 2
xy
x2
y xy2
yx
PASCAL TRIANGLE
The CST and LST are variations of the Pascal Triangle.

3. Step 3 : Define the Strain-Displacement and Stress-Strain Relationships
Elemental Strains: The strains over a 2D element are:
Observe that the strains are linear!! over the triangular element and that is
why it’s called a linear-strain triangle (LST).
Write
The above equation may be written in matrix form as:

4. Step 4: Derive the Element Stiffness Matrix and Equations using the Total
Potential Energy Approach
Step 5,6, and 7: Assembling the global stiffness matrix, determining the
global displacements and calculating the stresses, are identical to the
procedures used for CST elements.
1
(0,0)
6
(0.5b,0)
2
(b,0)
4
(0.5b,0.5h)
3
(0,h)
5
(0,0.5h)
The triangle has a base dimension of b and a height h, with mid-side nodes.
We can calculate the coefficients a1 through a6 by evaluating the displacement
u at each node.

5. Solving for the a’s gives
The u displacement equation is

6. Comparison of Elements
Observations
• For a given number of nodes, a better representation of true stress and displacement
is obtained using LST elements than is obtained using the same no. of nodes a finer
subdivision of CST elements.
• The larger the no. of degrees of freedom for a given type of triangular element, the
closer the solution converges to the exact one.
• Although the CST is poor in modeling bending, we observe from Table 2 that the
element can be used to model a beam in bending if sufficient no. of elements is
used through the depth of the beam.
• In general, both LST and CST analyses yield sufficient results for most plane
stress/strain problems provided a sufficient number of elements are used (LST
model might be preferred over the CST for plane stress when a relatively small no.
of nodes is used).
• Most commercial programs incorporate the use of CST and/or LST elements for
plane stress/strain problems although these elements are used primarily as
transition elements (usually during mesh generation).

7. some other elements
4-noded rectangle
y
a a 1
4
2
b
b
3
In local coordinate system
4 ab
4
3
2
1
4 ab
( a  x )( b  y )
N 
N 
4 ab
( a  x )( b  y )
4 ab
( a  x )( b  y )
x N 
( a  x )( b  y )
N 
1
x5
x4
y x3
y2
x2
y3
xy4
y5
x4
x3
x 2
y xy 2
y3
x3
y x 2
y 2
xy3
y 4
xyx2
y2
x y
4 node;p=2
Corner nodes
3 4
1 2
2 2 4ab 2 24ab
2 2 4ab 2 24ab
N 
(a  x)(b  y)

N6

N7
N 
(a  x)(b  y)

N8

N7
N 
(a  x)(b  y)

N5

N8
N 
(a  x)(b  y)

N5

N6
x
y
a a 1
4
2
b
b
3
5
6
7
8
Midside nodes
  
  
2 2
8
2 2
2 2
6
2 2
b2  2a 
(a x) b y       
 N  2b 
7  a2
a  x  (b y)
N 
 b2  2a 
N  (a x)  b y 
 2b 
5  a2 
a  x  (b y)
N 
8-noded rectangle
4 node; p=2
8 node; p=3
5 4 3 2 2 3 4 5
x x y x y x y xy y
x 3
x 2
y xy 2
y 3
x 4
x 3
y x 2
y 2
xy 3
y 4
xyx 2
y 2
1
x y
STATIC CONDENSATION