Local Stiffness Matrix of a Structural Element

A local stiffness matrix describes the mechanical behavior of one member element in coordinates aligned with its own axis. It provides a linear relation between nodal displacements and member-end forces.

Basic relation

For an element without initial strains or member loading,

$$\{q\}=[k]\{u\},$$

where $\{u\}$ is the local nodal-displacement vector, $\{q\}$ is the local member-end force vector, and $[k]$ is the local stiffness matrix.

Truss member

For a two-node bar carrying axial deformation only, the local degrees of freedom are the axial end displacements and

$$[k]=\frac{EA}{L}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}.$$

The factor $EA/L$ represents the axial stiffness associated with relative displacement of the member ends.

Beam element

For a planar bending element, typical local coordinates are transverse displacement and rotation at each end. Its stiffness coefficients contain combinations of $EI/L^3$, $EI/L^2$, and $EI/L$, representing flexural resistance.

Plane-frame element

A plane-frame element combines axial and bending behavior. Each node commonly has three local degrees of freedom: axial translation, transverse translation, and rotation, giving a full $6\times6$ local matrix.

Properties

For a linear-elastic conservative element, $[k]$ is symmetric. Before sufficient boundary conditions are imposed it may be singular because rigid-body motions cause no deformation and therefore no internal force.