Lujuusmatriisi

Lujuusmatriisi, or stiffness matrix in English, is a fundamental concept in structural analysis and mechanical engineering. It is a mathematical representation of how a structure or a mechanical component deforms under applied loads. Specifically, the stiffness matrix relates the forces and moments acting on a system to the displacements and rotations it experiences.

In the context of finite element analysis (FEA), the stiffness matrix is a core component. A structure is discretized into smaller elements, and for each element, a local stiffness matrix is defined. This matrix describes the element's resistance to deformation. These local matrices are then assembled into a global stiffness matrix that represents the entire structure. The global stiffness matrix, often denoted as K, is used in the equilibrium equation K * u = f, where u is the vector of nodal displacements and f is the vector of nodal forces. Solving this equation allows engineers to determine how the structure will deform under various loading conditions.

The entries of the stiffness matrix are derived from the material properties (like Young's modulus and Poisson's ratio) and the geometry (like cross-sectional area and moment of inertia) of the structure or element. A higher value in the stiffness matrix indicates a greater resistance to deformation for a given load. Understanding and accurately calculating the stiffness matrix is crucial for predicting a structure's behavior, ensuring its safety, and optimizing its design.