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Shape functions

Shape functions describe how displacement is interpolated between the nodal values inside an element. For a linear element on $[0,1]$ , $N_1$ and $N_2$ are straight lines, each equal to 1 at its own node and 0 at the other. The sum of the shape functions is always 1, a "partition of unity".

N_1(\xi) = 1-\xi,\quad N_2(\xi) = \xi

linear shape functions on

[0,1]

u(\xi) = N_1u_1 + N_2u_2

interpolated displacement

Symbols

$\xi$	local coordinate
$N_1,\ N_2$	shape functions
$u_1,\ u_2$	nodal values

Example

$u_1 = 2$ , $u_2 = 8$ , $\xi = 0.25$ :

$u = 2(1-0.25) + 8\cdot 0.25 = 3.5$ .

The sum

N_1 + N_2 = 1

everywhere – this lets the element reproduce a constant displacement.

Practise weak form and shape functions for free →

← Boundary conditions and Ku = f · Weak form →

Part of Finite Element Method: Weak form and shape functions.