# 统计代写|主成分分析代写Principal Component Analysis代考|Analyzing 3-D Truss Structures using Bar Elements in Finite Element Framework

The passage above outlines the process of analyzing a three-dimensional truss structure using bar elements in a finite element framework. Here’s a summary of the main points and the application to the specific example:

In a 3-D truss, member axes are represented by unit vectors in the global coordinate system using Equation 3.53 or 3.54, which depends on the relative positions of the end nodes. The element displacements are expressed in terms of the global displacements at those nodes.

Similar to the 2-D case, there’s a transformation matrix [R] that maps the one-dimensional element displacements to the three-dimensional global coordinates (Equation 3.57).

The element stiffness matrix in the global coordinate system is derived from the local stiffness matrix by applying the transformation (Equation 3.58), leading to Equation 3.59, where cx, cy, and cz are the cosine values of the angles between the element axis and the global axes.

The assembly of the global stiffness matrix involves summing the transformed element stiffness matrices, taking into account the nodal connectivity. Since nodes 1-3 are fixed in the given example, only the equations associated with the displacements at node 4 are considered.

The three-element truss in Figure 3.8a has a load of -5000 lb applied to node 4 in the negative Y direction. All members have a stiffness of 3 million pounds per inch (lb/in).

Element stiffness matrices for each member are calculated after transforming them into the global coordinate system. For instance, for Element 1, the orientation angle results in (cx, cy, cz) = (0.8, 0, -0.6) and the transformed stiffness matrix is computed accordingly.

By assembling the pertinent parts of the global stiffness matrix using the displacement correspondence table, a system of equations is formed that relates the unknown displacements at node 4 (U10, U11, and U12) to the applied load.

Solving this system of linear equations leads to the following displacements at node 4:

U10 = 0.01736 in U11 = -0.06944 in U12 = 0 Although the full analysis is not carried out in detail, it’s noted that once the displacements are found, other quantities like reaction forces, strains, and stresses can be computed by extending the procedures outlined for the 2-D case to the 3-D scenario. These secondary variables are calculated using the displacement interpolation functions and the assembled stiffness matrices.

### MATLAB代写

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