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 N_i are the shape functions and U_i 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 S_i is the slope.
In the formulas below,

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

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

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

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

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)