Basics of the Finite Element Method, Elements & Shape Functions

What the finite element method does

Most real structures — a bridge deck, a station box, a dam — are too complicated to solve with a closed-form formula. The finite element method (FEM) replaces the continuous structure by a finite number of small pieces (elements) joined at points (nodes). Inside each element the unknown (displacement) is approximated by simple functions of the nodal values. The result is a large set of algebraic equations that a computer solves.

In structural engineering the displacement-based method dominates:

$[K]$ = global stiffness matrix, $\{u\}$ = nodal displacements, $\{F\}$ = nodal forces. It is the matrix stiffness method of structural analysis, generalised to continuum problems.

Steps of an analysis

Shape functions

Displacement inside an element is written from the nodal values $u_i$:

The shape functions $N_i$ must satisfy:

• $N_i=1$ at node $i$ and 0 at every other node (the Kronecker property), so the interpolation returns the nodal values.
• $\sum N_i=1$ everywhere (so that a rigid-body translation gives zero strain).
• Continuity across element boundaries (compatibility) and the ability to represent constant strain (completeness).

For a two-node bar of length $L$ with local coordinate $x$: $N_1=1-x/L$, $N_2=x/L$ (linear). For higher accuracy, quadratic and cubic functions are used.

Element stiffness matrix

From the strain–displacement relation $\{\epsilon \}=[B]\{u\}$ and the material law $\{\sigma \}=[D]\{\epsilon \}$:

For the axial bar (two nodes, area $A$, modulus $E$):

For the plane frame / beam element in bending (Euler–Bernoulli, four degrees of freedom: deflection and rotation at each end):

The beam's shape functions are cubic (Hermite) polynomials, which ensure continuity of the deflection and slope.