A matrix method of structural analysis fundamental to the 有限元素 method (FEM). It models a structure as an assembly of elements connected at nodes. The method relates nodal forces [latex]{R}[/latex] to nodal displacements [latex]{D}[/latex] through a global stiffness matrix [latex][K][/latex], expressed as [latex][K]{D} = {R}[/latex]. Solving this system of linear equations yields the unknown nodal displacements.
The Direct Stiffness 方法 operationalized structural analysis for computers. The process involves three main stages. First, the structure is discretized into a finite number of elements (e.g., beams, shells, solids), each with a defined stiffness matrix [latex][k][/latex] in a local coordinate system. This element stiffness matrix relates forces and displacements at the element’s nodes.
Second, a global stiffness matrix [latex][K][/latex] for the entire structure is assembled by systematically combining the individual element stiffness matrices. This assembly process enforces compatibility (displacements must match at shared nodes) and equilibrium (forces must balance at each node). Boundary conditions, such as fixed supports where displacements are zero, are then applied to modify the global system of equations.
Finally, the resulting system of linear algebraic equations, [latex][K]{D} = {R}[/latex], is solved for the unknown nodal displacements [latex]{D}[/latex]. Once the displacements are known, they can be used to calculate the internal strains and stresses within each element, providing a complete picture of the structure’s response to the applied loads.
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