1. Dr. A. S. Sayyad
Professor & Head
Department of Structural Engineering
Sanjivani College of Engineering, Kopargaon 423603.
(An Autonomous Institute, Affiliated to Savitribai Phule Pune University, Pune)
Finite Element Method In Civil Engineering
Shape functions for three
nodded CST element
1 2 3 1 2 3 1 2 3
1 2 3
2 2 2
a a x a y b b x b y c c x c y
N N N
A A A
2. Shape functions for three nodded CST element
Let us consider three nodded constant strain
triangular (CST) element as shown in figure.
(x1, y1), (x2, y2), (x3, y3) are the Cartesian coordinates
of nodes 1, 2, 3 respectively.
(u1, v1), (u2, v2), (u3, v3) are the displacements of
nodes 1, 2, 3 respectively.
u, v are the displacements of any point on the
element having Cartesian coordinates (x, y).
I) Displacement function
1 2 3
u x y
3. In matrix form
1
2
3
1
u x y
u P
------------------ (1)
where, [P] = Parametric matrix
II) Displacement function in-terms of nodal displacements
Express displacement function in terms of nodal displacements using the
coordinates of nodes (x1, y1), (x2, y2), (x3, y3).
4. 1 1 1 1
2 2 2 2
3 3 3 3
1
1
1
u x y
u x y
u x y
e
x A
where, [A] = Connectivity matrix
Obtained from Eq. (2) and put into the Eq. (1), we get
------------------ (2)
1
e
u P A x
e
u N x
5. III) Shape functions
Considering displacements in x-direction only (i.e. u, same shape functions are
applicable to displacement in y-direction i.e. v)
1
1 1
1
2 2
3
1
1 1
1
x y
N P A x y x y
x y
1 1 1
2 2 2
3 3 3
1
1
2
a b c
N x y a b c
A
a b c
1 2 3 1 2 3 1 2 3
1 2 3
2 2 2
a a x a y b b x b y c c x c y
N N N
A A A
6. where
1 2 3 3 2 2 2 3 3 3 2
1 3 1 1 3 2 3 1 3 1 3
1 1 2 2 1 2 1 2 3 2 1
a x y x y a y y a x x
b x y x y b y y b x x
c x y x y c y y c x x
