stivhedsmatrix

Stivhedsmatrix, or stiffness matrix, is a mathematical construct used in structural analysis to relate nodal forces to nodal displacements. In linear elastic finite element analysis, the global relationship is written as F = K u, where F is the vector of external nodal forces, u is the vector of nodal displacements, and K is the stiffness matrix of the structure.

Construction and interpretation: The stiffness matrix encapsulates how the structure resists deformation. It is assembled from the stiffness matrices of individual elements by summing their contributions according to the connectivity of the finite element mesh. Each element stiffness depends on material properties (for example Young’s modulus and Poisson’s ratio), geometry, and the element type (truss, beam, shell, solid). For simple elements, explicit formulas exist; for continuum elements, K is obtained through material stiffness D and a strain–displacement matrix B integrated over the element volume: K_e = ∫ B^T D B dV. The global K collects all element contributions in the chosen coordinate system and has a size equal to the total number of degrees of freedom in the model.

Properties and use: The stiffness matrix is typically symmetric and positive definite for well-posed linear elastic problems, and it is usually sparse in large models. After applying boundary conditions (constraints) and external loads, solving Ku = F yields the nodal displacements, from which reactions and internal member forces can be derived. The stiffness matrix is central to static analysis, natural frequency analyses (via eigenproblems), and many nonlinear extensions where K is updated to reflect changing stiffness.

Limitations: The standard stiffness matrix assumes linear elastic behavior and small strains; nonlinear materials, large deformations, or contact problems require updated or incremental formulations.