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Shape Functions

The following graphs of shape functions are used to solve problems by finite element analysis. The first three are Lagrange functions and the last shows Hermite cubic polynomials. Notice that each function is unity at its own node and zero at the other nodes. Furthermore, the Lagrange shape functions sum to unity everywhere. For the Hermite polynomials H1 and H3 sum to unity.

The shape functions are used to find the field variable U from known values at other locations. The formula is

        U  = N1 U1 + N2 U2 + ...
where Ni are the shape functions and Ui are known values. The Hermite shape functions are used for beam analysis where both the deflection and slope of adjacent elements must be the same at each node. H1 and H3 are the deflection while H2 and H4 are the slope.
        D  = H1 D1 + H2 S1 + H3 D2 + H4 S2
where D is the deflection and Si is the slope.

In the formulas below,

  • L is the length
  • S is X - X1
  • r is S / L

Lagrange Linear Shape functions
Figure 1. The Lagrange linear shape functions
N1 = 1 - S / L,  N2 = S / L,  L = X2 - X1,  S = X - X1

Lagrange Quadratic Shape functions
Figure 2. The Lagrange quadratic shape functions
N1 = (r-1)(2r-1),  N2 = 4r (1 - r),  N3 = r (2r -1)

Lagrange Cubic Shape functions
Figure 3. The Lagrange cubic shape functions
N1 = (1 - r) (2 - 3 r) (1 - 3 r) / 2,  N2 = 9 r (1 - r) (2 - 3 r) / 2,
N3 = 9 r (1 - r) (3 r - 1) / 2,  N4 = r (2 - 3 r) (1 - 3 r) / 2

Hermite Cubic Shape functions
Figure 4. The Hermite Cubic Shape functions
H1 = 1 - 3 r 2 + 2 r 3,  H2 = L ( r - 2 r 2 + r 3),
H3 = 3 r 2 - 2 r 3,  H4 = L ( - r 2 + r 3)

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