#### 期刊菜单

The Effectiveness and Limitation of Support Translation Method
DOI: 10.12677/IJM.2023.124012, PDF, HTML, XML, 下载: 112  浏览: 158  科研立项经费支持

Abstract: Based on the geometric invariance of hinged triangle, a series of discriminant rules can be derived, which is the main teaching content of geometric composition analysis in existing textbooks. However, the practical problems are complex and changeable, and the geometric method may be difficult to deal with. Therefore, other kinds of analysis methods emerge as the times require. As the classical method of Structural Mechanics, the support translation method is often used in geometric composition analysis. According to the typical example, the support translation method and the analytical method are used to analyze the geometric composition of the system, and the analysis results are compared with the calculation results of the structural mechanics solver. The results show that the translation of the support along the axial direction of the hinged bar does not change the nature of the constraint, and the translation is effective. Here, the geometric composition of the system can be analyzed by the support translation method. In other cases, the translation of the support may lead to wrong results. Therefore, the support translation method has applicability and limitations, which should be used conditionally in the teaching of Structural Mechanics.

1. 引言

2. 支座平移法

Figure 1. Plane bar system

Figure 2. Simplified system

1处的水平支座链杆平移到5点，2处的竖向支座链杆平移到6点，3处的水平支座链杆平移到7点，4处的竖向支座链杆平移到8点。4根支座链杆均没有沿链杆作用线方向平移，或许改变了约束的性质，“无多余约束的几何不变体”的结论有待验证。

3. 解析法

Figure 3. Number of plane bar system

$\begin{array}{c}\left(1\right)\\ \left(2\right)\\ \left(3\right)\\ \left(4\right)\end{array}\left[\begin{array}{cc}1& 6\\ 7& 2\\ 8& 3\\ 4& 5\end{array}\right]\begin{array}{c}\end{array}\left[\begin{array}{cccc}0& 1& 2& 3\\ 4& 3& 2& 0\\ 4& 1& 0& 3\\ 4& 0& 2& 1\end{array}\right]$ (1)

${\Delta }_{1}\mathrm{cos}{\theta }_{1}=-{\Delta }_{2}\mathrm{sin}{\theta }_{1}+{\Delta }_{3}\mathrm{cos}{\theta }_{1}$ (2)

${\Delta }_{2}\mathrm{cos}{\theta }_{2}={\Delta }_{3}\mathrm{sin}{\theta }_{2}+{\Delta }_{4}\mathrm{cos}{\theta }_{2}$ (3)

${\Delta }_{1}\mathrm{cos}{\theta }_{3}-{\Delta }_{4}\mathrm{sin}{\theta }_{3}\text{=}{\Delta }_{3}\mathrm{cos}{\theta }_{3}$ (4)

${\Delta }_{4}\mathrm{cos}{\theta }_{4}\text{=}{\Delta }_{1}\mathrm{sin}{\theta }_{4}+{\Delta }_{2}\mathrm{cos}{\theta }_{4}$ (5)

$\left[\begin{array}{cccc}\mathrm{cos}{\theta }_{1}& \mathrm{sin}{\theta }_{1}& -\mathrm{cos}{\theta }_{1}& 0\\ 0& \mathrm{cos}{\theta }_{2}& -\mathrm{sin}{\theta }_{2}& -\mathrm{cos}{\theta }_{2}\\ \mathrm{cos}{\theta }_{3}& 0& -\mathrm{cos}{\theta }_{3}& -\mathrm{sin}{\theta }_{3}\\ \mathrm{sin}{\theta }_{4}& \mathrm{cos}{\theta }_{4}& 0& -\mathrm{cos}{\theta }_{4}\end{array}\right]\cdot \left[\begin{array}{c}{\Delta }_{1}\\ {\Delta }_{2}\\ {\Delta }_{3}\\ {\Delta }_{4}\end{array}\right]=0$ (6)

$G=\left[\begin{array}{cccc}\mathrm{cos}{\theta }_{1}& \mathrm{sin}{\theta }_{1}& -\mathrm{cos}{\theta }_{1}& 0\\ 0& \mathrm{cos}{\theta }_{2}& -\mathrm{sin}{\theta }_{2}& -\mathrm{cos}{\theta }_{2}\\ \mathrm{cos}{\theta }_{3}& 0& -\mathrm{cos}{\theta }_{3}& -\mathrm{sin}{\theta }_{3}\\ \mathrm{sin}{\theta }_{4}& \mathrm{cos}{\theta }_{4}& 0& -\mathrm{cos}{\theta }_{4}\end{array}\right]$ (7)

$|G|\ne 0$ ，则 ${\Delta }_{1}={\Delta }_{2}={\Delta }_{3}={\Delta }_{4}=0$ ，体系几何不变，且无多余约束；若 $|G|=0$ ，则体系几何可变。

4. 支座平移法的有效性

Figure 4. Number of modified plane bar system

${\Delta }_{1}\mathrm{sin}{\theta }_{1}=0$ (8)

${\Delta }_{2}\mathrm{sin}{\theta }_{2}=0$ (9)

${\Delta }_{3}\mathrm{sin}{\theta }_{3}\text{=}0$ (10)

${\Delta }_{4}\mathrm{sin}{\theta }_{4}\text{=}0$ (11)

$\left[\begin{array}{cccc}\mathrm{sin}{\theta }_{1}& 0& 0& 0\\ 0& \mathrm{sin}{\theta }_{2}& 0& 0\\ 0& 0& \mathrm{sin}{\theta }_{3}& 0\\ 0& 0& 0& \mathrm{sin}{\theta }_{4}\end{array}\right]\cdot \left[\begin{array}{c}{\Delta }_{1}\\ {\Delta }_{2}\\ {\Delta }_{3}\\ {\Delta }_{4}\end{array}\right]=0$ (12)

$G=\left[\begin{array}{cccc}\mathrm{sin}{\theta }_{1}& 0& 0& 0\\ 0& \mathrm{sin}{\theta }_{2}& 0& 0\\ 0& 0& \mathrm{sin}{\theta }_{3}& 0\\ 0& 0& 0& \mathrm{sin}{\theta }_{4}\end{array}\right]$ (13)

Figure 5. Modified simplified system

5. 结论

 [1] 朱慈勉, 张伟平. 结构力学(上册) [M]. 第3版. 北京: 高等教育出版社, 2016. [2] 龙驭球, 包世华, 袁驷. 结构力学I: 基础教程[M]. 第4版. 北京: 高等教育出版社, 2018. [3] 杨茀康, 李家宝, 范洪文, 汪梦甫. 结构力学(上册) [M]. 第6版. 北京: 高等教育出版社, 2016. [4] 吴子明. 谈平面体系的几何组成分析[J]. 南方冶金学院学报, 1989, 10(4): 81-85. [5] 刘永军. 结构几何构造分析中的四个辅助规则及其应用[J]. 力学与实践, 2022, 44(1): 197-202. [6] 于苏民. 铰结三角形代换法作平面杆系几何组成分析[J]. 力学与实践, 2005, 27(2): 72-73. [7] 樊友景, 樊大为. 几何构造分析中的等效变换[J]. 力学与实践, 2012, 34(2): 77-78. [8] 郇筱林, 王崇革. 平面体系几何组成规则的理解和简化分析技巧[J]. 力学与实践, 2018, 40(6): 696-699. [9] 蔡长青, 汪大洋, 张永山, 孙静, 刘东滢, 朱勇. 复杂平面体系几何组成分析的等价思想及其应用[J]. 力学与实践, 2022, 44(6): 1416-1421. [10] 吴耀鹏, 吴耀欢. 平面体系几何组成分析的解析法研究[J]. 力学与实践, 2012, 34(6): 62-64. [11] 袁驷. 结构力学求解器, 2.0版[M]. 北京: 高等教育出版社, 2004.