复合载荷下圆薄膜大挠度问题的一般解
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摘要
薄膜结构已广泛应用于航空、航天、造船、化工、仪表等工业部门。本文的主要目的是求解轴对称圆薄膜在均布力和中心集中力联合作用(复合载荷)下的一般解。文中首先建立了轴对称圆薄膜在复合载荷作用下的大挠度基本方程,然后将基本方程无量纲化,推导了基本方程边界条件的无量纲表达式。推广了Hencky变换,采用幂级数法,得到了基本方程用无量纲集中力系数表示的一般解Ⅰ和用无量纲均布力系数表示的一般解Ⅱ。重点计算了两种载荷反向作用这一最具有实际意义的复合载荷情形,确定了一般解的待定常数及其收敛区域。最后运用得到的一般解,以圆薄膜的无量纲径向坐标x为变量,采用编程计算,得到以表格表示的弹性特征系数关于载荷比α和β以及材料泊松比ν之间的关系。
Circular Membrane structures play a very important role in engineering field which includes aviation, spaceflight, shipbuilding, civil engineering, meter, and so on. The fundamental objective in this dissertation lies in obtaining general solutions to large deflection of axis symmetrical circular membrane under the action of uniformly distributed load and concentrated force which are defined composed load. In this paper, firstly, the author constructs the basic equations of large deflection of axis symmetrical circular membrane under composed load, then, gains the dimensionless basic equations and its boundary conditions. Secondly, applying for the extended Hencky transformation and power progression method, two general solutions which were respectively described by uniformly distributed force and concentrated force were reasoned. The author focuses on the composed load that the uniformly distributed force and concentrated force are contrary, which is very significant in reality, and calculates the variable parameter and it’s constringency area. Finally, the author gains the elastic characters equations of axis symmetrical circular membrane under composed load, writes the calculation programs, and gains the relations of the elastic characters and the load ratio and the Poisson’s ratio. The relations are tabled so that we can inquire. Then, the author discussed the change trend of the general solutions with the variational physical parameters by drawings.
Circular Membrane structures play a very important role in engineering field which includes aviation, spaceflight, shipbuilding, civil engineering, meter, and so on. The fundamental objective in this dissertation lies in obtaining general solutions to large deflection of axis symmetrical circular membrane under the action of uniformly distributed load and concentrated force which are defined composed load. In this paper, firstly, the author constructs the basic equations of large deflection of axis symmetrical circular membrane under composed load, then, gains the dimensionless basic equations and its boundary conditions. Secondly, applying for the extended Hencky transformation and power progression method, two general solutions which were respectively described by uniformly distributed force and concentrated force were reasoned. The author focuses on the composed load that the uniformly distributed force and concentrated force are contrary, which is very significant in reality, and calculates the variable parameter and it’s constringency area. Finally, the author gains the elastic characters equations of axis symmetrical circular membrane under composed load, writes the calculation programs, and gains the relations of the elastic characters and the load ratio and the Poisson’s ratio. The relations are tabled so that we can inquire. Then, the author discussed the change trend of the general solutions with the variational physical parameters by drawings.
引文
[1] 李中立等. 国外膜结构在大跨度结构中的应用与发展趋势[J]. 空间结构,1996,2 (3):
[2] 叶开沅.柔韧构件研究的新进展.[J].力学进展,1987,17(1):1-10.(6),26-28.
[3] 高永祥等.轴对称环膜大变形计算分析及其机械特性参数的回归研究.[J].实验力学,1995,10(1):68-75. applicate , Vol. 3, No. 4, pp. 322-333,2001.
[4] YAO Peng-fei. The ellipticity of the elliptic membrane.[J].Acta analysis functionalis
[5] 王文蜂.浅谈薄膜理论解析法在散装货物罐车优化设计中的应用.[J]. 铁道车辆,1994 2005,41(2):99-103.
[6] 汝宇林等. 金属膜片联轴器圆环式膜片的应力分析.[J]. 兰州大学学报(自然科学版), 51-55.
[7] 林晓春等.圆片表面薄膜应力传感器设计[J].半导体光电,2004,25 (1):79-82.
[8] 曾富宝.薄膜基础理论[J]. 苏州城建环保学院学报,1994,7 (2):13-28.
[9] 马孜等. 光学薄膜表面形貌的原子力显微镜观察[J]. 电子显微学报,2000,19 (5): 704-708.
[10] 谭伟等.表面力仪的改进及超薄膜流变特性的实验研究[J]. 摩擦学学报,2002,22 (5): 381-385.
[11] 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.
[12] 徐芝纶著. 弹性力学(下)[M]. 第二版. 北京: 高等教育出版社,1983.
[13] 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
[14] S. Timoshenko, J.Gere: Theory Of Elastic Stability (second edition). 张福范 翻译. 北京: 科学出版社,1965.
[15] 钱伟长,叶开沅. 圆板的大挠度问题[J]. 物理学报,1954,10(3):209-238
[16] 郑晓静. 圆薄板大挠度理论及应用[M].(第一版). 吉林:吉林科学技术出版社,1990.4.
[17] Euler L. De motu vibratorio tympanorum, Novi commentary Acad. Petropolit. 10(1766)243-260.
[18] Bernoulli J. Essai théorique sur les vibrations dè plaques élastiques rectangulaires et libers, Nova Acta Petroplit. 5(1789)197-219.
[19] Todhunter I. and Pearson K. A history of the theory of elasticity, Vois.1 and 2, Dover Publications, Inc. New York, 1960.
[20] Kirchhoff G.,Vorlesungen über mathematische physik, V.1, B.G. Teubner, Leipzig, 1876.
[21] F?ppl A. Vorlesungen über technische mechanic, Vols.3 and 5, 8th and 3th ed.B.G. Teubner, Leipzig, 1923.
[22] von Kármán Th. Festigkeits probleme im Maschinenbau, Encycl. Der math.wiss.4(1910)348-351.
[23] Hencky H. Uber den spannungszustand in Kreisunden platten mit verschwindender Biegungssteifigkeit [J ] . Zeit F Math U Phyzik , 1915 , (63) : 311 - 317.
[24] Алексеев C A. Κοльпеобразная упругая мембрана под действuем попереиной счлы, приноженной кжесткому центрально расположенному диску [A] . Инженерный Сборник [C] ,10 ,1951 ,71 —80.
[25] 钱伟长,王志忠,徐尹格,等. 圆薄膜中心部分受均布载荷产生的对称变形[J ] . 应用数学和力学,1981 ,2 (6) :599 —612.
[26] 陈山林,郑周练.圆薄膜在集中力作用下的大变形[J]. 应用数学和力学,2003 ,24 (1) :25 —28.
[27] Sherbourne A N , lennox , WC. Elastic large deflections of annular membranes [J ] . J Engng Mech Divis , Proc ASCE , 1966 ,92 ( EM2) : 75 —99.
[28] Kao R , Perrone N. Large deflections of axisymmetric circular membranes [ J ] . Internat J Solids Structures , 1971 , (7) 12 : 1601 —1612.
[29] Dadeppo A , Schmidt R. New analysis of an unprestretched annular membrane with restrained edges [J ] . J Indus Math Soc , 1972 ,24 (1) :49 —58.
[30] 曾又林.关于圆薄膜的 Hencky 解[J]. 上海力学,1995,16 (2):164-165.
[31] 邓长根.大挠度弯曲薄板的实用计算方法[J].上海力学,1995,16 (1):73-76.
[32] 邓长根.弹性常数对薄板大挠度弯曲的影响[J].上海力学,1993,14 (4):76-81.
[33] L.D.AKULENKO. Control of the motions of a membrane by means of boundary forces.[J].Appl Maths Mechs, Vol. 59, No. 5, pp. 701-710,1995.
[34] E. K. LAVROVSKII and A. M. FORMAL’SKII. Stabilization of the position of a circular membrane.[J].Appl Maths Mechs, Vol. 61, No. 3, pp. 443-450,1997.
[35] S. Reese a,*, T. Raible b, P. Wriggers b. Finite element modeling of orthotropic material behavior in pneumatic membranes.[J].International Journal of Solids and Structure 38 (2001) 9525-9544.
[36] CHEN Shan-lin and ZHENG Zhou-lian. Large deformation of circular membrane under the concentrated force.[J].Applied mathematics and mechanics(English edtion), Vol. 24, No. 1, pp. 28-31,2003.
[37] G. G. DENOSOV and V. V. NOVIKOV. Membranes deformation by a moving load.[J].Appl. Maths Mechs, Vol.61,No.4,pp.627-632,1997.
[38] D.CHEN and S. CHENG. Non-liner analysis of prestretched circular membrane and a modified iteration technique.[J].Solids structures, Vol. 33.No.4,pp.545-553,1996.
[39] Leyland G. Young, Suresh Ramanathan, Jiazhu Hu, P. Frank Pai. Numerical and experimental dynamic characteristics of thin-film membranes.[J]. International Journal of Solids and Structures 42 (2005) 3001–3025
[40] Arthur W. Lessia*.Singularity considerations in membrane, plate and shell behaviors.[J].International Journal of Solids and structures, 38(2001) 3341-3353.
[41] Bernard Maurin and René Motro. The surface stress density method as a form-finding tool for tensile membranes.[J].Engineering Structures,Vol.20,No.8,pp.712-719.1998.
[42] Adnan Ibrahimbegovi?1.Finite elastoplastic deformation of space-cured membranes.[J].Comput. Methods Appl. Mech. Engrg. 119 (1994) 371-394.
[43] J. Bonet a,*, R.D. Wood a, J. Mahaney b, P. Heywood c. Finite element analysis of air supported membrane structures.[J]. Comput. Methods Appl. Mech. Engrg. 190 (2000) 579-595.
[44] Shiro Kato a,*, Tatsuya Yoshino b, Hirokazu Minami c. Formulation of constitutive equations for fabric membranes based on the concept of fabric lattice model.[J]. Engineering Structures 21 (1999) 691–708.
[45] 钱能著. C++程序设计教程[M]. 第一版. 北京:清华大学出版社,1999.
[46] 姚东等著. MATLAB 命令大全[M]. 第一版. 北京:人民邮电出版社,2000.
[2] 叶开沅.柔韧构件研究的新进展.[J].力学进展,1987,17(1):1-10.(6),26-28.
[3] 高永祥等.轴对称环膜大变形计算分析及其机械特性参数的回归研究.[J].实验力学,1995,10(1):68-75. applicate , Vol. 3, No. 4, pp. 322-333,2001.
[4] YAO Peng-fei. The ellipticity of the elliptic membrane.[J].Acta analysis functionalis
[5] 王文蜂.浅谈薄膜理论解析法在散装货物罐车优化设计中的应用.[J]. 铁道车辆,1994 2005,41(2):99-103.
[6] 汝宇林等. 金属膜片联轴器圆环式膜片的应力分析.[J]. 兰州大学学报(自然科学版), 51-55.
[7] 林晓春等.圆片表面薄膜应力传感器设计[J].半导体光电,2004,25 (1):79-82.
[8] 曾富宝.薄膜基础理论[J]. 苏州城建环保学院学报,1994,7 (2):13-28.
[9] 马孜等. 光学薄膜表面形貌的原子力显微镜观察[J]. 电子显微学报,2000,19 (5): 704-708.
[10] 谭伟等.表面力仪的改进及超薄膜流变特性的实验研究[J]. 摩擦学学报,2002,22 (5): 381-385.
[11] 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.
[12] 徐芝纶著. 弹性力学(下)[M]. 第二版. 北京: 高等教育出版社,1983.
[13] 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
[14] S. Timoshenko, J.Gere: Theory Of Elastic Stability (second edition). 张福范 翻译. 北京: 科学出版社,1965.
[15] 钱伟长,叶开沅. 圆板的大挠度问题[J]. 物理学报,1954,10(3):209-238
[16] 郑晓静. 圆薄板大挠度理论及应用[M].(第一版). 吉林:吉林科学技术出版社,1990.4.
[17] Euler L. De motu vibratorio tympanorum, Novi commentary Acad. Petropolit. 10(1766)243-260.
[18] Bernoulli J. Essai théorique sur les vibrations dè plaques élastiques rectangulaires et libers, Nova Acta Petroplit. 5(1789)197-219.
[19] Todhunter I. and Pearson K. A history of the theory of elasticity, Vois.1 and 2, Dover Publications, Inc. New York, 1960.
[20] Kirchhoff G.,Vorlesungen über mathematische physik, V.1, B.G. Teubner, Leipzig, 1876.
[21] F?ppl A. Vorlesungen über technische mechanic, Vols.3 and 5, 8th and 3th ed.B.G. Teubner, Leipzig, 1923.
[22] von Kármán Th. Festigkeits probleme im Maschinenbau, Encycl. Der math.wiss.4(1910)348-351.
[23] Hencky H. Uber den spannungszustand in Kreisunden platten mit verschwindender Biegungssteifigkeit [J ] . Zeit F Math U Phyzik , 1915 , (63) : 311 - 317.
[24] Алексеев C A. Κοльпеобразная упругая мембрана под действuем попереиной счлы, приноженной кжесткому центрально расположенному диску [A] . Инженерный Сборник [C] ,10 ,1951 ,71 —80.
[25] 钱伟长,王志忠,徐尹格,等. 圆薄膜中心部分受均布载荷产生的对称变形[J ] . 应用数学和力学,1981 ,2 (6) :599 —612.
[26] 陈山林,郑周练.圆薄膜在集中力作用下的大变形[J]. 应用数学和力学,2003 ,24 (1) :25 —28.
[27] Sherbourne A N , lennox , WC. Elastic large deflections of annular membranes [J ] . J Engng Mech Divis , Proc ASCE , 1966 ,92 ( EM2) : 75 —99.
[28] Kao R , Perrone N. Large deflections of axisymmetric circular membranes [ J ] . Internat J Solids Structures , 1971 , (7) 12 : 1601 —1612.
[29] Dadeppo A , Schmidt R. New analysis of an unprestretched annular membrane with restrained edges [J ] . J Indus Math Soc , 1972 ,24 (1) :49 —58.
[30] 曾又林.关于圆薄膜的 Hencky 解[J]. 上海力学,1995,16 (2):164-165.
[31] 邓长根.大挠度弯曲薄板的实用计算方法[J].上海力学,1995,16 (1):73-76.
[32] 邓长根.弹性常数对薄板大挠度弯曲的影响[J].上海力学,1993,14 (4):76-81.
[33] L.D.AKULENKO. Control of the motions of a membrane by means of boundary forces.[J].Appl Maths Mechs, Vol. 59, No. 5, pp. 701-710,1995.
[34] E. K. LAVROVSKII and A. M. FORMAL’SKII. Stabilization of the position of a circular membrane.[J].Appl Maths Mechs, Vol. 61, No. 3, pp. 443-450,1997.
[35] S. Reese a,*, T. Raible b, P. Wriggers b. Finite element modeling of orthotropic material behavior in pneumatic membranes.[J].International Journal of Solids and Structure 38 (2001) 9525-9544.
[36] CHEN Shan-lin and ZHENG Zhou-lian. Large deformation of circular membrane under the concentrated force.[J].Applied mathematics and mechanics(English edtion), Vol. 24, No. 1, pp. 28-31,2003.
[37] G. G. DENOSOV and V. V. NOVIKOV. Membranes deformation by a moving load.[J].Appl. Maths Mechs, Vol.61,No.4,pp.627-632,1997.
[38] D.CHEN and S. CHENG. Non-liner analysis of prestretched circular membrane and a modified iteration technique.[J].Solids structures, Vol. 33.No.4,pp.545-553,1996.
[39] Leyland G. Young, Suresh Ramanathan, Jiazhu Hu, P. Frank Pai. Numerical and experimental dynamic characteristics of thin-film membranes.[J]. International Journal of Solids and Structures 42 (2005) 3001–3025
[40] Arthur W. Lessia*.Singularity considerations in membrane, plate and shell behaviors.[J].International Journal of Solids and structures, 38(2001) 3341-3353.
[41] Bernard Maurin and René Motro. The surface stress density method as a form-finding tool for tensile membranes.[J].Engineering Structures,Vol.20,No.8,pp.712-719.1998.
[42] Adnan Ibrahimbegovi?1.Finite elastoplastic deformation of space-cured membranes.[J].Comput. Methods Appl. Mech. Engrg. 119 (1994) 371-394.
[43] J. Bonet a,*, R.D. Wood a, J. Mahaney b, P. Heywood c. Finite element analysis of air supported membrane structures.[J]. Comput. Methods Appl. Mech. Engrg. 190 (2000) 579-595.
[44] Shiro Kato a,*, Tatsuya Yoshino b, Hirokazu Minami c. Formulation of constitutive equations for fabric membranes based on the concept of fabric lattice model.[J]. Engineering Structures 21 (1999) 691–708.
[45] 钱能著. C++程序设计教程[M]. 第一版. 北京:清华大学出版社,1999.
[46] 姚东等著. MATLAB 命令大全[M]. 第一版. 北京:人民邮电出版社,2000.