Stiffness Matrix Formulation

The stiffness matrix relates nodal displacements to equivalent nodal forces. At the element level, the stiffness matrix for a linear elastic element is generally written as:

K_e = integral( B^T D B dV )

where:

• B is the strain-displacement matrix.
• D is the constitutive matrix.
• dV represents integration over the element volume.

Numerical integration is commonly used to evaluate this integral. The element stiffness matrix must be transformed and assembled into the global coordinate system when required.

The stiffness matrix should be symmetric and positive definite for a properly constrained linear elastic model without mechanisms or rigid body modes.