Assembly of the Global Equation System

Assembly combines the stiffness contributions of individual elements into the global equation system of the complete structure. It is based on compatibility of nodal displacements and equilibrium of nodal forces.

Global numbering

Each independent structural degree of freedom is assigned a global number. Every element then has a connectivity map relating its element coordinates to these global degree-of-freedom numbers.

Adding element contributions

Each global coefficient $K_{IJ}$ is the sum of contributions from all elements containing the corresponding global degrees of freedom $I$ and $J$. Schematically,

$$[K]=\sum_e[A_e]^T[k_e^{(g)}][A_e],$$

where $[A_e]$ maps element degrees of freedom to global coordinates.

Load vector

Concentrated nodal forces and moments are added directly to $\{F\}$. Distributed and other member loads are represented by equivalent nodal forces according to the adopted element formulation.

Global system

Assembly produces

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

Before supports are imposed, the matrix may be singular because rigid-body motion is possible. Correct boundary conditions remove the corresponding kinematic modes in a stable structure.

Matrix structure

The global matrix is normally sparse because an element directly couples only the degrees of freedom of its own nodes. For a linear-elastic conservative model, $[K]$ is symmetric.

Model checks

Unexpected singularity after applying supports often indicates a mechanism, insufficient restraint, connectivity error, incorrect member-end release, or duplicated independent degrees of freedom.