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After developing the element stiffness matrix in the global coordinate system, they must be merged into a single “master” or “global” stiffness matrix. When merging these matrices together there are two rules that must be followed: compatibility of displacements and force equilibrium at each node.
The full stiffness matrix A is the sum of the element stiffness matrices. In particular, for basis functions that are only supported locally, the stiffness matrix is sparse . For many standard choices of basis functions, i.e. piecewise linear basis functions on triangles, there are simple formulas for the element stiffness matrices.
The stiffness matrix components corresponding to each degree of freedom are determined by assuming a unit displacement in the studied direction and by determining forces at the centroid of each element. The 2D element stiffness matrix size is 6 × 6; the components of the upper left quarter of the stiffness matrix are shown below:
Examples of Galerkin methods are: the Galerkin method of weighted residuals, the most common method of calculating the global stiffness matrix in the finite element method, [3] [4] the boundary element method for solving integral equations, Krylov subspace methods. [5]
That is, the modulus is an intensive property of the material; stiffness, on the other hand, is an extensive property of the solid body that is dependent on the material and its shape and boundary conditions. For example, for an element in tension or compression, the axial stiffness is = where
The origin of finite method can be traced to the matrix analysis of structures [1] [2] where the concept of a displacement or stiffness matrix approach was introduced. Finite element concepts were developed based on engineering methods in 1950s.
The finite element method has been the tool of choice since civil engineer Ray W. Clough in 1940 derived the stiffness matrix of a 3-node triangular finite element (and coined the name). The precursors of FEM were elements built-up from bars (Hrennikoff, Argyris, Turner) and a conceptual variation approach suggested by R. Courant.
Elastic constants are specific parameters that quantify the stiffness of a material in response to applied stresses and are fundamental in defining the elastic properties of materials. These constants form the elements of the stiffness matrix in tensor notation, which relates stress to strain through linear equations in anisotropic materials.