采用自由参数摄动法求解板壳大挠度问题
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摘要
正则摄动法是一种在力学领域求解非线性微分方程的方法。尤其在求解板和壳体的大挠度问题上，正则摄动法得到了广泛的使用，并取得了比较理想的计算结果。但选择摄动参数却是正则摄动法求解过程中的一个比较困难的问题，现在对于这一问题仍然没有一个一般的原则，而主要是利用经验对该问题进行处理。
本文的主要目的是介绍一种求解轴对称板壳大挠度问题的新的正则摄动方法——自由参数摄动法。采用该方法求解板以及壳体的大挠度问题，研究者无须具体选择摄动解的摄动参数就能够得出问题的全部弹性特征，减少了在摄动过程中的先验因素，使得求解结果更趋于合理。
本文首先介绍了采用自由参数摄动法求解轴对称板壳大挠度问题的一般步骤及计算公式，然后采用该方法求解了均布和集中载荷单独及联合作用下圆板的大挠度问题，并在求解中通过与传统正则摄动法的计算结果以及计算原理的比较，证明了本方法的合理性。而后本文将自由参数摄动法与样条函数拟和法结合起来，研究了圆底扁球壳在均布载荷作用下的失稳问题。通过编制计算程序计算具体算例，并将其结果与半解析法与数值方法计算结果的比较，进一步验证了本方法的实用性以及准确性，同时也得出了一些具有理论意义和工程应用价值的结果。
Perturbation method of analysis is a prevalent method to solve nonlinear differential equation in mechanics field. Especially in the theory of plates and shells in large deflections, perturbation method of analysis was widely applied and gave relatively precise results. However, it is difficult to choose proper perturbation parameter in the process of applying this method. Even in present day, there have not a dependable norm to confirm certain parameter and researcher often search it through engineering experience.
This paper presents a new perturbation method of analysis to solve large deflections problems of plates and shells-Free-Parameter Perturbation Method (FPPM). By this method of analysis, researcher can get all of elastic characters without choosing proper perturbation parameter to create perturbation solution. Since this method of analysis reduce the experience factors in the process of analysis, the results of it can be more reasonable.
This paper firstly presents the general steps and formulas of FPPM to analysis the large-deflection problems of plate and shallow shell. After that, FPPM is applied to analysis large-deflection problems of circular plate under uniform pressure, concentrate load and compound load. The analysis process and results are compared to those of previous perturbation method and the applicability and rationality of FPPM are proved as well, In the following part, FPPM together with Spline-Method are applied to analysis the nonlinear stability problem of thin shallow spherical shell. Programmes are made to calculate specific example and the results are compared to other research's results. Through comparison, the advantages of FPPM are presented again. Besides that, some results of this paper are theoretically valuable and useful for engineering.
本文的主要目的是介绍一种求解轴对称板壳大挠度问题的新的正则摄动方法——自由参数摄动法。采用该方法求解板以及壳体的大挠度问题，研究者无须具体选择摄动解的摄动参数就能够得出问题的全部弹性特征，减少了在摄动过程中的先验因素，使得求解结果更趋于合理。
本文首先介绍了采用自由参数摄动法求解轴对称板壳大挠度问题的一般步骤及计算公式，然后采用该方法求解了均布和集中载荷单独及联合作用下圆板的大挠度问题，并在求解中通过与传统正则摄动法的计算结果以及计算原理的比较，证明了本方法的合理性。而后本文将自由参数摄动法与样条函数拟和法结合起来，研究了圆底扁球壳在均布载荷作用下的失稳问题。通过编制计算程序计算具体算例，并将其结果与半解析法与数值方法计算结果的比较，进一步验证了本方法的实用性以及准确性，同时也得出了一些具有理论意义和工程应用价值的结果。
Perturbation method of analysis is a prevalent method to solve nonlinear differential equation in mechanics field. Especially in the theory of plates and shells in large deflections, perturbation method of analysis was widely applied and gave relatively precise results. However, it is difficult to choose proper perturbation parameter in the process of applying this method. Even in present day, there have not a dependable norm to confirm certain parameter and researcher often search it through engineering experience.
This paper presents a new perturbation method of analysis to solve large deflections problems of plates and shells-Free-Parameter Perturbation Method (FPPM). By this method of analysis, researcher can get all of elastic characters without choosing proper perturbation parameter to create perturbation solution. Since this method of analysis reduce the experience factors in the process of analysis, the results of it can be more reasonable.
This paper firstly presents the general steps and formulas of FPPM to analysis the large-deflection problems of plate and shallow shell. After that, FPPM is applied to analysis large-deflection problems of circular plate under uniform pressure, concentrate load and compound load. The analysis process and results are compared to those of previous perturbation method and the applicability and rationality of FPPM are proved as well, In the following part, FPPM together with Spline-Method are applied to analysis the nonlinear stability problem of thin shallow spherical shell. Programmes are made to calculate specific example and the results are compared to other research's results. Through comparison, the advantages of FPPM are presented again. Besides that, some results of this paper are theoretically valuable and useful for engineering.
引文
[1] 钱伟长．奇异摄动理论及其在力学中的应用[M]．(第一版)．北京：科学出版社，1981．8．
[2] A．H．奈弗．王辅俊等译．摄动方法[M]．(第一版)．上海：上海科学技术出版社，1984．11．
[3] 钱伟长，叶开沅．圆板的大挠度问题[J]．物理学报，1954，10(3)：209-238
[4] 胡海昌．在均布和集中载荷作用下的圆板大挠度问题[J]．物理学报，1954，10(4)：383-392．
[5] Nash, W. A., Cooley I. D.. Large deflection of a clamped elliptical plate subjected to uniform pressure, Transactions of the A. S. M. E.[J]. Journal of Applied Mechanics., 1959 , 26 (2): 291-293
[6] 黄黔．复合载荷下圆薄板的大挠度问题[J]．应用数学和力学．1983，4(5)：711-720
[7] Vincent, J. J.. The bending of a thin circular plate [J]. Phil. Mag., 1931, 12 (4): 185-196.
[8] Schmidt. R, and DaDeppo, D. A.. A new approach to the analysis of shells, plates and membranes with finite deflection. [J] Int. J of Non-linear Mech., 1974, 9 (5): 409-419.
[9] 陈山林，光积昌．圆薄板大挠度问题的摄动参数[J]．应用数学和力学，1981，2(1)：131-144
[10] 陈山林．圆板大挠度问题的钱伟长解及其渐进特性[J]．应用数学和力学，1982，3(4)：513-518
[11] von Kámán,Th and Chien Hsue-shen. The buckling of sphereical shells by external pressure[J]. J. Aero.Sci.. 1939, 7:48
[12] A．C．沃尔密尔著，卢文达等译．柔韧板与柔韧壳[M]．北京：科学出版社，1959
[13] 胡海昌．球面扁薄球壳的跳跃问题[J]．物理学报，1954，10(2)：105-131．
[14] 黄炎，黄瑞芳．薄壳稳定的变分原理[J]．应用数学和力学，1995，16(12)：1079-1086
[15] R. M. Simons. A power series solution of the nonlinear equations for axisymmetrical bending of shallow spherical shells[J]. J. Math. Physics, 1956, 35 (2): P164
[16] Kang, Shengliang. Singular perturbation solutions of the nonlinear stability of a truncated shallow spherical shell under linear distributed loads along the interior edge.[J] Applied Mathematics and Mechanics (English Edition), 1993, 14 (10): 931-944
[17] 叶开沅、顾淑贤．均布载荷作用下圆底扁球壳的非线性稳定性[C]．1980年全国计算力学会议论文集，1981：280-287
[18] 叶开沅、刘平．复合载荷下圆底扁球壳非线性稳定性[J]．应用数学和力学，1991，12(1)：11-17
[19] Yeh Kai-Yuan, F.P.J.Rimrott and Song Wei-Ping. Mathematical Treatment of Stability Problems for Thin Shallow Elastic Shells of Revolution[J]. International Journal of Nonlinear
Mechanics. 1994, 29 (4): 587-601.
[20] L. P. Khoroshun and A. S. Strel'chenko. Characteristics of Discrete Schemes Used in Stabilty Problem of Rods And Shells[J]. International Journal of Nonlinear Mechanics. 1981, 17 (3): 115-125
[21] Cho, Chahngmin (Agency for Defense Development) Park Hoon C., Lee Sung W.. Stability analysis using a geometrically nonlinear assumed strain solid shell element model[J]. Finite Elements in Analysis and Design, 1998, 29 (2): 121-135
[22] Mang, H.A. (Technical Univ of Vienna). On bounding properties of eigenvalues from linear initial FE stability analyses of thin, elastic shells with respect to stability limits from geometrically non-linear prebuckling analyses [J]. International Journal for Numerical Methods in Engineering, 1991, 31 (6): 1087-1111
[23] Montag, U., Kraetzig, W.B. & Soric, J.. Increasing solution stability for finite-element modeling of elasto-plastic shell response [J]. Advances in Engineering Software, 1999, 30 (9): 607-619
[24] 严圣平．扁球壳在均布压力作用下的非线性弯曲问题[J]．应用力学学报，1988，5(3)：21-29
[25] 叶开沅，宋卫平．中心分布载荷作用下圆底扁球壳的变形和稳定性[J]．应用数学和力学，1988，9(10)：857-863．
[26] 宋卫平．圆底扁锥壳在中心集中载荷作用下的轴对称非线性弯曲和稳定性[J]．应用力学学报，1988，5(1)：6-11．
[27] Liu Ren-Huai. Non-Linear Thermal Stability of Bimetallic Shallow Shells of Revolution[J]. International Journal of Nonlinear Mechanics, 1983, 18 (5): 409-429.
[28] Liu Ren-Huai. On The Nonlinear Stability of a Truncated .Shallow Spherical. Shell Under Axisymmetrically Distributed Load[y]. Solid Mechanics Archives, 1989, 2. (14): 81-102.
[29] Saleh, A. Stutzki, Ch.. Application of large deflection theory to the stability analysis of lattice shells[J]. Proceedings of the Asian Pacific Conference on Computational Mechanics, 1991, Part 1 (of2): 53
[30] Arbocz, Johann. Future directions and challenges in shell stability analysis, Collection of Technical Papers[J] - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, 1997, 3:1949-1962
[31] Amiro, I. Y., Zarutskii, V. A.. Experiment Study of the Stability of Stiffened Shells[J]. International Applied Mechanics, 1996.32 (9): 659
[32] 陈山林．自由参数摄动法[C]．钱伟长教授九十寿辰纪念文集，2002，上海
[33] 徐芝纶著．弹性力学(下)[M]．第二版．北京：高等教育出版社，1983．
[34] 李岳生，齐东旭著．样条函数方法[M]．第一版．北京：科学出版社，1979．6
[35] 秦荣著．结构力学中的样条函数方法。第一版．南宁：广西人民出版社，1980．12
[36] S. Timoshenko, S. Woinowsky-Krieger: Theory Of Plates And Shells (second edition).《板壳理论》翻译组 翻译．第一版．北京：科学出版社，1977．10
[2] A．H．奈弗．王辅俊等译．摄动方法[M]．(第一版)．上海：上海科学技术出版社，1984．11．
[3] 钱伟长，叶开沅．圆板的大挠度问题[J]．物理学报，1954，10(3)：209-238
[4] 胡海昌．在均布和集中载荷作用下的圆板大挠度问题[J]．物理学报，1954，10(4)：383-392．
[5] Nash, W. A., Cooley I. D.. Large deflection of a clamped elliptical plate subjected to uniform pressure, Transactions of the A. S. M. E.[J]. Journal of Applied Mechanics., 1959 , 26 (2): 291-293
[6] 黄黔．复合载荷下圆薄板的大挠度问题[J]．应用数学和力学．1983，4(5)：711-720
[7] Vincent, J. J.. The bending of a thin circular plate [J]. Phil. Mag., 1931, 12 (4): 185-196.
[8] Schmidt. R, and DaDeppo, D. A.. A new approach to the analysis of shells, plates and membranes with finite deflection. [J] Int. J of Non-linear Mech., 1974, 9 (5): 409-419.
[9] 陈山林，光积昌．圆薄板大挠度问题的摄动参数[J]．应用数学和力学，1981，2(1)：131-144
[10] 陈山林．圆板大挠度问题的钱伟长解及其渐进特性[J]．应用数学和力学，1982，3(4)：513-518
[11] von Kámán,Th and Chien Hsue-shen. The buckling of sphereical shells by external pressure[J]. J. Aero.Sci.. 1939, 7:48
[12] A．C．沃尔密尔著，卢文达等译．柔韧板与柔韧壳[M]．北京：科学出版社，1959
[13] 胡海昌．球面扁薄球壳的跳跃问题[J]．物理学报，1954，10(2)：105-131．
[14] 黄炎，黄瑞芳．薄壳稳定的变分原理[J]．应用数学和力学，1995，16(12)：1079-1086
[15] R. M. Simons. A power series solution of the nonlinear equations for axisymmetrical bending of shallow spherical shells[J]. J. Math. Physics, 1956, 35 (2): P164
[16] Kang, Shengliang. Singular perturbation solutions of the nonlinear stability of a truncated shallow spherical shell under linear distributed loads along the interior edge.[J] Applied Mathematics and Mechanics (English Edition), 1993, 14 (10): 931-944
[17] 叶开沅、顾淑贤．均布载荷作用下圆底扁球壳的非线性稳定性[C]．1980年全国计算力学会议论文集，1981：280-287
[18] 叶开沅、刘平．复合载荷下圆底扁球壳非线性稳定性[J]．应用数学和力学，1991，12(1)：11-17
[19] Yeh Kai-Yuan, F.P.J.Rimrott and Song Wei-Ping. Mathematical Treatment of Stability Problems for Thin Shallow Elastic Shells of Revolution[J]. International Journal of Nonlinear
Mechanics. 1994, 29 (4): 587-601.
[20] L. P. Khoroshun and A. S. Strel'chenko. Characteristics of Discrete Schemes Used in Stabilty Problem of Rods And Shells[J]. International Journal of Nonlinear Mechanics. 1981, 17 (3): 115-125
[21] Cho, Chahngmin (Agency for Defense Development) Park Hoon C., Lee Sung W.. Stability analysis using a geometrically nonlinear assumed strain solid shell element model[J]. Finite Elements in Analysis and Design, 1998, 29 (2): 121-135
[22] Mang, H.A. (Technical Univ of Vienna). On bounding properties of eigenvalues from linear initial FE stability analyses of thin, elastic shells with respect to stability limits from geometrically non-linear prebuckling analyses [J]. International Journal for Numerical Methods in Engineering, 1991, 31 (6): 1087-1111
[23] Montag, U., Kraetzig, W.B. & Soric, J.. Increasing solution stability for finite-element modeling of elasto-plastic shell response [J]. Advances in Engineering Software, 1999, 30 (9): 607-619
[24] 严圣平．扁球壳在均布压力作用下的非线性弯曲问题[J]．应用力学学报，1988，5(3)：21-29
[25] 叶开沅，宋卫平．中心分布载荷作用下圆底扁球壳的变形和稳定性[J]．应用数学和力学，1988，9(10)：857-863．
[26] 宋卫平．圆底扁锥壳在中心集中载荷作用下的轴对称非线性弯曲和稳定性[J]．应用力学学报，1988，5(1)：6-11．
[27] Liu Ren-Huai. Non-Linear Thermal Stability of Bimetallic Shallow Shells of Revolution[J]. International Journal of Nonlinear Mechanics, 1983, 18 (5): 409-429.
[28] Liu Ren-Huai. On The Nonlinear Stability of a Truncated .Shallow Spherical. Shell Under Axisymmetrically Distributed Load[y]. Solid Mechanics Archives, 1989, 2. (14): 81-102.
[29] Saleh, A. Stutzki, Ch.. Application of large deflection theory to the stability analysis of lattice shells[J]. Proceedings of the Asian Pacific Conference on Computational Mechanics, 1991, Part 1 (of2): 53
[30] Arbocz, Johann. Future directions and challenges in shell stability analysis, Collection of Technical Papers[J] - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, 1997, 3:1949-1962
[31] Amiro, I. Y., Zarutskii, V. A.. Experiment Study of the Stability of Stiffened Shells[J]. International Applied Mechanics, 1996.32 (9): 659
[32] 陈山林．自由参数摄动法[C]．钱伟长教授九十寿辰纪念文集，2002，上海
[33] 徐芝纶著．弹性力学(下)[M]．第二版．北京：高等教育出版社，1983．
[34] 李岳生，齐东旭著．样条函数方法[M]．第一版．北京：科学出版社，1979．6
[35] 秦荣著．结构力学中的样条函数方法。第一版．南宁：广西人民出版社，1980．12
[36] S. Timoshenko, S. Woinowsky-Krieger: Theory Of Plates And Shells (second edition).《板壳理论》翻译组 翻译．第一版．北京：科学出版社，1977．10