![]() ![]() Boundary conditions are applied to the equation by "zeroing out" the rows in the matrices corresponding to applied constraints. The following equation relates the forces and displacements in the overall structure: Once the global stiffness matrix and the applied force vector are built, the nodal displacements can be solved for. The force vector is assembled by including the applied forces on each degree of freedom on each node in the mesh: The applied force vector will be an n x 1 vector, where n is 3 times the number of nodes in the mesh. For any elements which share a node, the stiffness contributions of that node will be summed from each element. When assembling the global stiffness matrix, the stiffness terms for each node in the elemental stiffness matrix are positioned in the corresponding location in the global matrix. The global stiffness matrix will be a square n x n matrix, where n is 3 times the number of nodes in the mesh (since each node has 3 degrees of freedom). ![]() ![]() Where the matrix is the local stiffness matrix of the i th element. At a high level, the global stiffness matrix is created by summing the local stiffness matrices: The global stiffness matrix for the overall structure is assembled based on the combination of the local stiffness matrices. Because each of the nodes in the beam element have 3 degrees of freedom, a 6 x 6 matrix can completely describe the stiffness of the element. it determines the displacement of each node in each degree of freedom under a given load). This matrix represents the stiffness of each node in the element in a specific degree of freedom (i.e. ![]() The element stiffness matrix for an Euler-Bernoulli beam element is shown below. The element has 2 nodes, each of which has 3 degrees of freedom: translation in x, translation in y, and rotation.Ī figure illustrating this element is shown below: This analysis uses beam elements which are based on Euler-Bernoulli beam theory. Structure - whereas the physical structure is continuous, the model consists of discrete elements. In the first step, a mathematical model of the structure is composed. Apply boundary conditions and solve for the nodal displacements.Assemble a global stiffness matrix for the overall structure based on the combination of the local stiffness matrices.Determine a local stiffness matrix for each element.These elements are connected to one another via nodes. Discretize the structure into elements.There are several basic steps in the finite element method: The theory of Finite Element Analysis (FEA) essentially involves solving the spring equation, F = kδ, at a large scale. NOTE: This page relies on JavaScript to format equations for proper display. ![]()
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