# 统计代写|主成分分析代写Principal Component Analysis代考|Converting Local Element Properties to Global Coordinate System in 2D Truss Analysis

The passage above explains the process of converting the local element properties to a global coordinate system when dealing with a two-dimensional truss structure. Here’s a condensed version:

A simple two-dimensional truss consisting of two bar elements joined by pin connections is analyzed under external loads. The equilibrium conditions are derived based on rigid body mechanics for each node (Equations 3.1 to 3.3).

The goal is to transform the problem from one based on forces to one based on displacements. The global displacements are defined for each node in the global X and Y directions, and the orientation angles for each element are specified.

For a bar element in a truss, the element nodal displacements must match the joint displacements due to the pin joint connections, meaning that there will be rotations along with axial motion. The relationship between element and global displacements is established through transformation equations (Equations 3.4a and 3.4b).

The axial deformation of the element and the resulting net axial force are calculated in terms of the global displacements using Equation 3.5 and Equation 3.6. These equations are then applied to specific elements (Equations 3.7 and 3.8), where compatibility of displacements at shared nodes is maintained.

Substituting the expressions for element forces into the nodal equilibrium equations leads to a set of six algebraic equations (Equations 3.9 to 3.14), which can be rewritten in matrix form (Equation 3.15):

[K]{U} = {F}

Where [K] is the 6×6 global stiffness matrix, {U} is the vector of nodal displacements, and {F} is the vector of applied nodal forces.

The global stiffness matrix represents the equilibrium conditions for the entire truss system and incorporates the effect of the stiffness of each element and its orientation relative to the global coordinate system. By solving this system of equations, taking into account any boundary conditions, the nodal displacements are found. Post-solving, these displacements are used to compute secondary variables such as strain, stress, and reaction forces.

The benefit of the displacement-based formulation in the finite element method (FEM) is that it handles statically indeterminate systems efficiently, requiring fewer assumptions about compatibility, and provides a consistent framework for solving a wide range of structural problems. Additionally, the formulation naturally accounts for displacement compatibility at joints, which is critical for physically realistic solutions.

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