Matrix Structural Analysis

Matrix structural analysis systematizes the displacement method and provides a general algorithm for framed structures with many elements and degrees of freedom. Its principal form is the matrix stiffness method.

Governing equation

After numbering the nodal degrees of freedom, structural equilibrium is written as

$$[K]\{d\}=\{F\},$$

where $[K]$ is the global stiffness matrix, $\{d\}$ is the nodal displacement vector, and $\{F\}$ is the equivalent nodal load vector.

Element approach

The structure is divided into member elements. Each element is first described by local degrees of freedom and a local stiffness matrix $[k]$, relating its local nodal displacements to member-end forces.

Coordinate transformation

Because members may have different orientations, their local matrices are transformed to a common global coordinate system. Contributions from all elements are then assembled into the global matrix $[K]$.

Boundary conditions

Supports prescribe selected nodal displacements, usually zero. After these conditions are imposed, the reduced system is solved for unknown displacements and support reactions are recovered.

Member-force recovery

The global end displacements of each element are transformed back to local coordinates. Local stiffness relations then provide member-end forces and moments, from which internal-force diagrams can be constructed.

Connection to finite elements

For framed structures, the matrix stiffness method is a natural foundation of the finite element method: a model is composed of elements with their own matrices and degrees of freedom, and the global problem is obtained by assembly.