For a linear structural finite element, nodal forces and nodal displacements are related by an element stiffness matrix:

$$\mathbf f=\mathbf k\mathbf u.$$

The matrix $\mathbf k$ contains the element's material, geometry and interpolation behaviour in algebraic form.

From elements to a structure

Each element shares some nodes with neighboring elements. Their stiffness contributions are therefore assembled into a global stiffness matrix $\mathbf K$ according to this connectivity.

The complete model takes the form

$$\mathbf F=\mathbf K\mathbf U,$$

where $\mathbf U$ contains the global nodal degrees of freedom and $\mathbf F$ contains the corresponding applied nodal loads.

Boundary conditions

Known displacements or rotations impose constraints on components of $\mathbf U$. Loads define known components of $\mathbf F$.

Without enough displacement constraints, a structural model can contain rigid-body motion and the algebraic system cannot determine a unique equilibrium configuration.

Local rules, global behaviour

The global model is built by repeatedly applying the same element-level relation and combining contributions at shared degrees of freedom. This assembly is one of the central ideas of FEM.