# 统计代写|主成分分析代写Principal Component Analysis代考|”Enhancing Beam Elements for 3D Deformations”

The passage describes the extension of a beam-axial element to account for two-plane bending and torsional behavior, thereby creating a general three-dimensional beam element. This upgraded element can capture all three types of deformations: axial, bending in two orthogonal planes, and torsion.

For bending about the z-axis, the stiffness matrix is given by Equation 4.48, and for bending about the y-axis (xz plane bending), the stiffness matrix is provided as Equation 4.66. This matrix differs from the xy plane bending stiffness matrix due to sign changes and dependence on the area moment of inertia IyI_yI y ​ .

By concatenating the stiffness matrices for axial loading ([kaxial][k_{axial}][k axial ​ ]), xy plane bending ([kbending]xy[k_{bending}]_{xy}[k bending ​ ] xy ​ ), and xz plane bending ([kbending]xz[k_{bending}]_{xz}[k bending ​ ] xz ​ ), we get a 10×10 element stiffness matrix representing a two-plane bending element with axial stiffness (Equation 4.68). The nodal loads due to distributed loads are calculated based on work equivalence principles, as exemplified by Equation 4.69.

The addition of torsional behavior involves considering a torsional finite element. The torsional stiffness matrix, given by Equation 4.74, is derived from the relationship between the twisting moment and the angle of twist for a uniform circular cylinder. The torsional stiffness kTk_Tk T ​ is proportional to the polar moment of inertia JJJ, the shear modulus GGG, and the length LLL of the element.

The final element stiffness matrix for the general 3-D beam element (Equation 4.75) is a 12×12 symmetric matrix that combines the individual stiffness matrices for axial loading, two-plane bending, and torsion. In practice, when using such elements in a finite element analysis of three-dimensional frame structures, it is essential to transform the local element stiffness matrices into the global coordinate system, which follows a similar process to lower-dimensional elements but with added complexity due to the larger size of the matrix and orientation considerations.

### MATLAB代写

MATLAB 是一款高性能的技术计算语言，集成了计算、可视化和编程环境于一体，以熟悉的数学符号表达问题和解决方案。MATLAB 的基本数据元素是一个不需要维度的数组，使得能够快速解决带有矩阵和向量公式的多种技术计算问题，相比使用 C 或 Fortran 等标量非交互式语言编写的程序，效率大大提高。MATLAB 名称源自“矩阵实验室”（Matrix Laboratory）。最初开发 MATLAB 的目标是为了提供对 LINPACK 和 EISPACK 项目的矩阵软件的便捷访问，这两个项目代表了当时矩阵计算软件的先进技术。经过长期发展和众多用户的贡献，MATLAB 已成为数学、工程和科学入门及高级课程的标准教学工具，在工业界，MATLAB 是高效研究、开发和分析的理想选择。MATLAB 提供了一系列名为工具箱的特定应用解决方案集，这对广大 MATLAB 用户至关重要，因为它们极大地扩展了 MATLAB 环境，使其能够解决特定类别问题。工具箱包含了针对特定应用领域的 MATLAB 函数（M 文件），涵盖信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等诸多领域。