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SHAPE FUNCTIONS AND HIGHER ORDER
FORMULATIONS
Relation between type of plane triangular element and
polynomial coefficients based on a Pascal triangle
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LST Stiffness Determination
• The triangle is of base dimension b and height
h, with midside nodes.
u ( x, y )  a1  a2 x  a3 y  a4 x 2  a5 xy  a6 y 2
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Rewrite the above equations in terms of
a1, a2, a3, a4, a5 and a6
Sub a1, a2….a6 in equation u(x,y)
u ( x, y )  a1  a2 x  a3 y  a4 x 2  a5 xy  a6 y 2
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CST vs LST
CST ELEMENT
LST ELEMENT
It is a two dimensional linear
element (simplex element)
It is a two dimensional non-linear
element (complex element)
It has only three primary nodes at
the corners
It has three primary nodes at the
corners and three secondary
nodes at the midsides
The strain is constant throughout
the element
The strain is varying linearly
inside the element
The polynominal functions for
CST element are
u  a1  a2 x  a3 y
The polynominal functions for
LST element are
v  a4  a5 x  a6 y
u  a1  a2 x  a3 y  a4 x 2  a5 xy  a6 y 2
v  a7  a8 x  a9 y  a10 x 2  a11 xy  a12 y 2
2D SCALAR VARIABLE
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Derivation of shape function for 2D
linear element
ø3
ø1
ø2
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2D VECTOR VARIABLE
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