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# 数学代写|有限元方法代写finite differences method代考|ENGR7961 Equations in Global Coordinate System

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## 数学代写|有限元代写Finite Element Method代考|Equations in Global Coordinate System

Having known the element matrices in the local coordinate system, the next thing to do is to transform the element matrices into the global coordinate system to account for the differences in orientation of all the local coordinate systems that are attached on individual frame members.

Assume that the local nodes 1 and 2 of the element correspond to global nodes $i$ and $j$, respectively. The displacement at a local node should have three translational components in the $x, y$ and $z$ directions, and three rotational components with respect to the $x, y$ and $z$-axes. They are numbered sequentially by $d_1-d_{12}$ corresponding to the physical deformations as defined by Eq. (6.16). The displacement at a global node should also have three translational components in the $X, Y$ and $Z$ directions, and three rotational components with respect to the $X, Y$ and $Z$ axes. They are numbered sequentially by $D_{6 i-5}, D_{6 i-4}, \ldots$, and $D_{6 i}$ for the $i$ th node, as shown in Figure 6.5. The same sign convention applies to node $j$. The coordinate transformation gives the relationship between the displacement vector $\mathbf{d}_e$ based on the local coordinate system and the displacement vector $\mathbf{D}_e$ for the same element but

based on the global coordinate system:
$$\mathbf{d}e=\mathbf{T D}_e$$ where $$\mathbf{D}_e=\left{\begin{array}{c} D{6 i-5} \ D_{6 i-4} \ D_{6 i-3} \ D_{6 i-2} \ D_{6 i-1} \ D_{6 i} \ D_{6 j-5} \ D_{6 j-4} \ D_{6 j-3} \ D_{6 j-2} \ D_{6 j-1} \ D_{6 j} \end{array}\right}$$
and $\mathbf{T}$ is the transformation matrix for the truss element given by
$$\mathbf{T}=\left[\begin{array}{cccc} \mathbf{T}_3 & \mathbf{0} & \mathbf{0} & \mathbf{0} \ \mathbf{0} & \mathbf{T}_3 & \mathbf{0} & \mathbf{0} \ \mathbf{0} & \mathbf{0} & \mathbf{T}_3 & \mathbf{0} \ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{T}_3 \end{array}\right]$$

## 数学代写|有限元代写Finite Element Method代考|REMARKS

In the formulation of the matrices for the frame element in this chapter, the superposition of the truss element and the beam element has been used. This technique assumes that the axial effects are not coupled with the bending effects in the element. What this means simply is that the axial forces applied on the element will not result in any bending deformation, and the bending forces will not result in any axial deformation. Frame elements can also be used for frame structures with curved members. In such cases, the coupling effects can exist even in the elemental level. Therefore, depending on the curvature of the member, the meshing of the structure can be very important. For example, if the curvature is very large resulting in a significant coupling effect, a finer mesh is required to provide the necessary accuracy.
In practical structures, it is very rare to have beam structures subjected to purely transverse loading. Most skeletal structures are either trusses or frames that carry both axial and transverse loads. It can now be seen that the beam element, developed in Chapter 5, as well as the truss element, developed in Chapter 4, are simply specific cases of the frame element. Therefore, in most commercial software packages, including ABAQUS, the frame element is just known generally as the beam element.

The beam element formulated in Chapter 5 , or general beam element formulated in this chapter, is based on so-called Euler-Bernoulli beam theory that is suitable for thin beams with a small thickness to pan ratio $(<1 / 20)$. For thick or deep beams of a large thickness to pan ratio, corresponding beam theories should be used to develop thick beam elements. The procedure of developing thick beams is very similar to that of developing thick plates, to be discussed in Chapter 8. Most commercial software packages also offer thick beam elements, and the use of these elements is much the same as the thin beam elements.

## 数学代写|有限元代写Finite Element Method代考|Equations in Global

Coordinate System 的方向差异。

$$\mathbf{d} e=\mathbf{T D}_e$$

〈left 缺少或无法识别的分隔符

$$\mathbf{T}=\left[\begin{array}{llllllllllllllll} \mathbf{T}_3 & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{T}_3 & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{T}_3 & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{T}_3 \end{array}\right]$$

## 数学代写|有限元代写Finite Element Method代考|REMARKS

. 对于厚梁比大的厚梁或深梁，应使用相应的梁理论来开发厚梁单元。开发厚梁的过程与开发厚板的过程非常相似，将在第 8 章中讨论。大多数商业软件包也提供厚梁单元，这些单元的使用与薄梁单元非常相似。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。