# 统计代写|主成分分析代写Principal Component Analysis代考|Transforming Local Stiffness Matrix into Global Coordinate System

This text describes the process of directly transforming the local stiffness matrix of a bar element into the global coordinate system. The initial equation (3.17) expresses the equilibrium of a bar element in its own local (element) coordinate system. The goal is to convert these equations into the global coordinate system so that they can be assembled into a larger system for the entire structure.

Transformation of Local to Global Displacements: The relationship between the element axial displacements in the local and global coordinate systems is given by the transformation matrix [R] (Equation 3.22). It transforms the local displacements u(e_1) and u(e_2) into global displacements U(e_1), U(e_2), U(e_3), and U(e_4).

Transformation of Stiffness Matrix: To transform the local stiffness matrix into the global coordinate system, the matrix equation representing the element equilibrium is premultiplied by the transpose of [R]. This results in Equation 3.26, where [K(e)] is now the element stiffness matrix in the global coordinate system.

Calculation of Global Stiffness Matrix: The final form of the element stiffness matrix in the global system is derived in Equation 3.27. When simplified using the shorthand notation c = cos(θ) and s = sin(θ), the element stiffness matrix in global coordinates becomes a function of θ, the angle between the element axis and the global X-axis (Equation 3.28).

Properties from Nodal Coordinates: Practically, for a given bar element, the global stiffness matrix can be determined by knowing the nodal coordinates (Xi, Yi) and (Xj, Yj), which are used to calculate the element length L (Equation 3.29) and the orientation angle θ (Equations 3.30 to 3.32). Direction cosines provide the necessary trigonometric values to construct the transformation matrix.

By knowing the nodal coordinates, cross-sectional area, and modulus of elasticity, the global stiffness matrix for a single bar element can be fully characterized. This process can be extended to multiple elements, and their respective global stiffness matrices are assembled to form the global stiffness matrix for the entire structure, which allows for solving the system of equations to determine the global displacements and subsequently other structural responses like stresses and strains.

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