| 薄板理论的正交关系及其变分原理 |
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| 引用本文： | 罗建辉,龙驭球,刘光栋.薄板理论的正交关系及其变分原理[J].力学学报,2004,36(5):527-532. |
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| 作者姓名： | 罗建辉 龙驭球 刘光栋 |
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| 作者单位： | 湖南长沙，湖南大学土木工程学院 |
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| 基金项目： | 国家自然科学基金(10272063)，教育部高等学校博士点基金(20020003044)，清华大学基础研究基金(JC2002003)，高等学校全国优秀博士论文作者专项基金(200242)资助项目.~~ |
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| 摘 要： | 利用平面弹性与板弯曲的相似性理论，将弹性力学新正交关系中构造对偶向量的思路推广到
各向同性薄板弹性弯曲问题，由混合变量求解法直接得到对偶微分方程并推导了对应的变分
原理. 所导出的对偶微分矩阵具有主对角子矩阵为零矩阵的特点. 发现了两个独立的、对称
的正交关系，利用薄板弹性弯曲理论的积分形式证明了这种正交关系的成立. 在恰当选择对
偶向量后，弹性力学的新正交关系可以推广到各向同性薄板弹性弯曲理论.
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| 关 键 词： | 弹性力学 薄板理论 对偶向量 正交关系 变分原理 |
| 修稿时间： | 2004年2月11日 |
An orthogonality relationship for thin plate theory and its variational principle |
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| Luo Jianhui Long Yuqiu Liu Guangdong
College of Civil Engineering,Hunan University,Changsha ,China.An orthogonality relationship for thin plate theory and its variational principle[J].chinese journal of theoretical and applied mechanics,2004,36(5):527-532. |
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| Authors: | Luo Jianhui Long Yuqiu Liu Guangdong
College of Civil Engineering Hunan University Changsha China |
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| Affiliation: | Luo Jianhui Long Yuqiu Liu Guangdong
College of Civil Engineering,Hunan University,Changsha 410082,China Department of Civil Engineering,Tsinghua University,Beijing 100084,China |
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| Abstract: | While Hamiltonian system was led to solution of elastic theory a new systematic methodology for theory of elasticity was established and a symplectic orthogonality relationship was presented (Zhong Wanxie, 1995). For two-dimensional theory of elasticity a new dual vector and a new dual differential matrix were presented by putting the old dual vector in a new order. It was discovered for isotropic materials that the symplectic orthogonality relationship may be decomposed into 2 independently and symmetrically orthogonal sub-relationships (Luo Jianhui et al., 2002). The new orthogonality relationship includes the symplectic orthogonality relationship. The new orthogonal relationship was generalized into three-dimensional elasticity problems in which a direction of coordinate is an orthogonal direction of materials (Luo Jianhui et al., 2003). The research of a systematic methodology for bending theory of thin and thick plate has also been noticed. Some conclusion of the systematic methodology for bending theory of Reissner-Mindlin thick plate was obtained (Luo Jianhui et al., 2004). Firstly, the Hamiltonian dual differential equations for thick plates were derived. Then, the functional expressions of Hamiltonian variational principle were obtained by using the variable substitution and multiplier method. At last, the new orthogonality relationship of thick plate theory was proposed. But the new orthogonality relationship of thick plate theory can not be degenerated into thin plate theory. Therefore it is necessary to research the new orthogonality relationship of thin plate theory. Based on the analogy between plate bending problems and plane elasticity problems Hamiltonian system was applied to thin plate bending problems and its symplectic orthogonality relationship was presented (Zhong Wanxie et al., 1999). For thin plate bending theory a new dual vector is presented while the dual vectors based on the analogy are put in a new order. A variational principle based on the new dual vector is proposed and also demonstrated by a new method. The principal diagonal sub-matrixes of the dual differential matrix are zero matrixes. As a result of the peculiarity of the dual differential matrix it is discovered that the orthogonality relationship of thin plate bending theory based on the analogy may be decomposed into 2 orthogonal sub-relationships. Based on the integral form (Luo Jianhui et al., 2002) of the systematic methodology for elasticity, the new orthogonal relationship is demonstrated. The new orthogonality relationship of theory of elasticity is generalized into anisotropic thin plate bending theory. The theoretical achievements of the Hamiltonian system for thin plates provide new effective tools for the research on analytical and finite element solutions of thin plates. |
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| Keywords: | theory of elasticity thin plate theory dual vectors orthogonality relationship variational principle |
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