Jäykkyysyhtälö

Jäykkyysyhtälö, also known as the stiffness matrix equation, is a fundamental concept in structural mechanics and finite element analysis. It represents the relationship between the external forces applied to a structure and the resulting displacements. The equation is typically expressed in matrix form as K * u = f, where K is the stiffness matrix, u is the vector of nodal displacements, and f is the vector of nodal forces.

The stiffness matrix K quantifies the elastic properties of the structure. Each element of the matrix represents the force required at one node to produce a unit displacement at another node, while all other displacements are held constant. The size of the stiffness matrix depends on the number of degrees of freedom in the structure. For a two-dimensional problem with n nodes, where each node has two degrees of freedom (translations in x and y directions), the stiffness matrix would be of size 2n x 2n.

The nodal displacement vector u contains the unknown displacements at each node. These are the primary unknowns that the analysis aims to solve for. The nodal force vector f represents the known external forces and applied loads acting on the structure at its nodes.

Solving the jäykkyysyhtälö allows engineers to determine the displacements and, consequently, the internal forces and stresses within the structure under a given set of loads. This is crucial for designing safe and efficient structures that can withstand expected operational conditions. The derivation of the stiffness matrix involves assembling element stiffness matrices from the material properties and geometric configurations of individual structural components.