均布和静水压力作用下固支矩形薄板力学特性
|
摘要
利用重三角级数构造了均布和静水压力两种载荷作用下四边固支弹性矩形薄板的挠度函数,并依据最小势能原理,求解了构造挠度函数的系数。由于选取了完全满足边界条件的重三角级数作为四边固支矩形薄板的挠度函数,不需要采用叠加法或简化其边界条件,使得四边固支矩形薄板弯曲问题的求解过程思路清晰、计算简便,简化了现有解析方法求解的繁琐过程。最后,采用弹性薄板理论,对两种载荷作用下矩形薄板的应力、内力等力学特性进行了对比分析。结果表明:1两种不同载荷作用下,固支矩形薄板最大拉应力、压应力、剪应力、横向剪力的位置明显不同;2均布载荷作用下固支矩形薄板在其两条长边中点的位置拉应力较大,中部位置压应力较大,而在(0.5,2 m)和(1.5,2 m)的位置横向剪力最大;3静水压力载荷作用下固支矩形薄板仅其一条长边中点的位置拉应力较大,中部偏下的位置压应力较大,而在(0.80,2 m)和(1.69,2 m)的位置横向剪力最大;4两种载荷作用下,固支矩形薄板长边中点的位置容易出现拉破坏而导致其失稳,剪应力较大的位置则容易出现剪切破坏。
Using the double Fourier series, the deflection function of a rectangular thin plate with four edges clamped under uniform load and hydrostatic pressure is proposed, and the coefficient of the proposed deflection function is solved based on the principle of minimum potential energy. Due to the complete satisfaction with the boundary conditions for the double Fourier series that selected as the deflection function during solving process, and do not need to adopt the superposition method or simplify the boundary conditions, the solving process is simple in form and calculation process. Compared with the analysis of existing methods, the computational complexity of the used method is reasonable, which can simplify the complex solving process on the bending problem of rectangular thin plate with four edges clamped. Finally, the mechanical properties of the clamped rectangular thin plate under these two kinds of loads are compared by using the elastic thin plate theory, including stress, internal force and so on. Results obtained indicate that the position of maximum tension stress, compressive stress, shear stress and transverse shear stress is obviously different for these two kinds of loads. Under the uniform load, the tensile tress in the long side midpoint of the four edges clamped rectangular thin plate is larger and the compressive stress in the middle of the clamped rectangular thin plate is larger, and themaximum transverse shear stress is in the position of(0.5, 2m) and(1.5, 2m). While under the hydrostatic pressure, the tensile tress appeared only in one of the long side midpoint of the four edges clamped rectangular thin plate and the compressive stress in the downward position of the middle part of the clamped rectangular thin plate is larger as well as the maximum transverse shear stress is in the position of(0.80, 2m) and(1.69, 2m). For these two kinds of loads, the position of the long side midpoint of the clamped rectangular thin plate will more easily fail due to tensile yield, and the position with large shear stress will also fail due to shear yield.
Using the double Fourier series, the deflection function of a rectangular thin plate with four edges clamped under uniform load and hydrostatic pressure is proposed, and the coefficient of the proposed deflection function is solved based on the principle of minimum potential energy. Due to the complete satisfaction with the boundary conditions for the double Fourier series that selected as the deflection function during solving process, and do not need to adopt the superposition method or simplify the boundary conditions, the solving process is simple in form and calculation process. Compared with the analysis of existing methods, the computational complexity of the used method is reasonable, which can simplify the complex solving process on the bending problem of rectangular thin plate with four edges clamped. Finally, the mechanical properties of the clamped rectangular thin plate under these two kinds of loads are compared by using the elastic thin plate theory, including stress, internal force and so on. Results obtained indicate that the position of maximum tension stress, compressive stress, shear stress and transverse shear stress is obviously different for these two kinds of loads. Under the uniform load, the tensile tress in the long side midpoint of the four edges clamped rectangular thin plate is larger and the compressive stress in the middle of the clamped rectangular thin plate is larger, and themaximum transverse shear stress is in the position of(0.5, 2m) and(1.5, 2m). While under the hydrostatic pressure, the tensile tress appeared only in one of the long side midpoint of the four edges clamped rectangular thin plate and the compressive stress in the downward position of the middle part of the clamped rectangular thin plate is larger as well as the maximum transverse shear stress is in the position of(0.80, 2m) and(1.69, 2m). For these two kinds of loads, the position of the long side midpoint of the clamped rectangular thin plate will more easily fail due to tensile yield, and the position with large shear stress will also fail due to shear yield.
引文
[1]阿加雷A C.板壳应力[M].范钦珊,译.北京:中国建筑工业出版社,1986.(Ugural A C.Plates and shells stress[M].Fan Qinshan,translated.Beijing:Chinese Building Industry Press,1986(in Chinese)).
[2]张福范.弹性平板[M].第二版.北京:科学出版社,1984.(Zhang Fufan.Elasticity plates[M].2nd ed.Beijing:Science Press,1984(in Chinese)).
[3]龚永庆.四边简支矩形薄板集中力作用中心弯矩影响面[J].山西建筑,2010,36(2):84-85.(Gong Yongqing.The central bending moment influence surface of the concentrated force of the rectangular thin plate[J].Shanxi Architecture,2010,36(2):84-85(in Chinese)).
[4]王效民,刘新东,汤翔.重三角级数解一对边简支一对边自由的矩形板[J].山西建筑,2007,33(29):12-13.(Wang Xiaomin,Liu Xindong,Tang Xiang.Rectangular board of heavy triangle grade solution with one side simple and another free[J].Shanxi Architecture,2007,33(29):12-13(in Chinese)).
[5]岳建勇,曲庆璋.三边固定一边自由矩形板的精确解[J].青岛建筑工程学院学报,1999,20(1):16-21.(Yue Jianyong,Qu Qingzhang.The precise solution of rectangular plate with three edges built in and the fourth edge free[J].Journal of Qingdao Institute of Architecture and Engineering,1999,20(1):16-21(in Chinese)).
[6]岳建勇,曲庆璋.两对边固定两对边自由矩形板的精确解[J].青岛建筑工程学院学报,2000,21(2):12-31.(Yue Jianyong,Qu Qingzhang.Exact solution of rectangular thin plates with two opposite edges clamped and the other two edges free[J].Journal of Qingdao Institute of Architecture and Engineering,2000,21(2):12-31(in Chinese)).
[7]周叮.任意载荷作用下矩形弹性薄板横向弯曲的一个高精度近似解[J].应用数学和力学,1993,14(3):225-230.(Zhou Ding.An approximate solution with high accuracy of transverse bending of thin rectangular plates under arbitrarily distributed loads[J].Applied Mathematics and Mechanics,1993,14(3):225-230(in Chinese)).
[8]钟阳,张永山.弹性地基上四边自由矩形薄板的解析解[J].应用数学和力学,2006,27(6):735-740.(Zhong Yang,Zhang Yongshan.Theoretic solution of rectangular thin plate on foundation with four edges free by symplectic geometry method[J].Applied Mathematics and Mechanics,2006,27(6):735-740(in Chinese)).
[9]钟阳,张永山.四边任意支承条件下弹性矩形薄板弯曲问题的解析解[J].应用力学学报,2005,22(2):293-297.(Zhong Yang,Zhang Yongshan.Theoretic solution for rectangular thin plate with arbitrarily boundary conditions by symplectic geometry method[J].Chinese Journal of Applied Mechanics,2005,22(2):293-297(in Chinese)).
[10]Zhong Y,Liu H.Theoretical solution for thick plate resting on Pasternak foundation by symplectic geometry method[J].Journal of Applied Mechanics-Transactions of the ASME,2014,81(3):31007(4).
[11]Zhong Y,Li R,Liu Y M,et al.On new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported[J].International Journal of Solids and Structures,2009,46(11/12):2506-2513.
[12]Li R,Tian B,Zhong Y.Analytical bending solutions of free orthotropic rectangular thin plates under arbitrary loading[J].Meccanica,2013,48(10):2497-2510.
[13]Li R,Zhong Y,Tian B,et al.On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates[J].Applied Mathematics Letters,2009,22(12):1821-1827.
[14]Timoshenko S P,Woinowsky-Krieger S W.Theory of plates and shells[M].New York:Mc Graw-Hill,1959.
[15]徐芝纶.弹性力学[M].第四版.北京:高等教育出版社,2006.(Xu Zhilun.Elasticity mechanics[M].4th ed.Beijing:Higher Education Press,2006(in Chinese)).
[16]李元媛,蔡睿贤.矩形薄板弯曲的严格简明解析解[J].机械工程学报,2008,44(10):72-76.(Li Yuanyuan,Cai Ruixian.Some concise exact analytical solutions of rectangular plate bending[J].Chinese Journal of Mechanical Engineering,2008,44(10):72-76(in Chinese)).
[17]钟阳,李锐,刘月梅.四边固支矩形弹性薄板的精确解析解[J].力学季刊,2009,30(2):297-302.(Zhong Yang,Li Rui,Liu Yuemei.Exact analytic solution of rectangular thin plate with four edges clamped[J].Chinese Quarterly of Mechanics,2009,30(2):297-302(in Chinese)).
[18]吴洪洋.均布荷载作用下四边固支板的一种逼近解法[J].贵州工业大学学报,2000,29(1):32-35.(Wu Hongyang.An asymptotic approximating solution of the clamped rectangular plate under the average loads action[J].Journal of Guizhou University of Technology,2000,29(1):32-35(in Chinese)).
[2]张福范.弹性平板[M].第二版.北京:科学出版社,1984.(Zhang Fufan.Elasticity plates[M].2nd ed.Beijing:Science Press,1984(in Chinese)).
[3]龚永庆.四边简支矩形薄板集中力作用中心弯矩影响面[J].山西建筑,2010,36(2):84-85.(Gong Yongqing.The central bending moment influence surface of the concentrated force of the rectangular thin plate[J].Shanxi Architecture,2010,36(2):84-85(in Chinese)).
[4]王效民,刘新东,汤翔.重三角级数解一对边简支一对边自由的矩形板[J].山西建筑,2007,33(29):12-13.(Wang Xiaomin,Liu Xindong,Tang Xiang.Rectangular board of heavy triangle grade solution with one side simple and another free[J].Shanxi Architecture,2007,33(29):12-13(in Chinese)).
[5]岳建勇,曲庆璋.三边固定一边自由矩形板的精确解[J].青岛建筑工程学院学报,1999,20(1):16-21.(Yue Jianyong,Qu Qingzhang.The precise solution of rectangular plate with three edges built in and the fourth edge free[J].Journal of Qingdao Institute of Architecture and Engineering,1999,20(1):16-21(in Chinese)).
[6]岳建勇,曲庆璋.两对边固定两对边自由矩形板的精确解[J].青岛建筑工程学院学报,2000,21(2):12-31.(Yue Jianyong,Qu Qingzhang.Exact solution of rectangular thin plates with two opposite edges clamped and the other two edges free[J].Journal of Qingdao Institute of Architecture and Engineering,2000,21(2):12-31(in Chinese)).
[7]周叮.任意载荷作用下矩形弹性薄板横向弯曲的一个高精度近似解[J].应用数学和力学,1993,14(3):225-230.(Zhou Ding.An approximate solution with high accuracy of transverse bending of thin rectangular plates under arbitrarily distributed loads[J].Applied Mathematics and Mechanics,1993,14(3):225-230(in Chinese)).
[8]钟阳,张永山.弹性地基上四边自由矩形薄板的解析解[J].应用数学和力学,2006,27(6):735-740.(Zhong Yang,Zhang Yongshan.Theoretic solution of rectangular thin plate on foundation with four edges free by symplectic geometry method[J].Applied Mathematics and Mechanics,2006,27(6):735-740(in Chinese)).
[9]钟阳,张永山.四边任意支承条件下弹性矩形薄板弯曲问题的解析解[J].应用力学学报,2005,22(2):293-297.(Zhong Yang,Zhang Yongshan.Theoretic solution for rectangular thin plate with arbitrarily boundary conditions by symplectic geometry method[J].Chinese Journal of Applied Mechanics,2005,22(2):293-297(in Chinese)).
[10]Zhong Y,Liu H.Theoretical solution for thick plate resting on Pasternak foundation by symplectic geometry method[J].Journal of Applied Mechanics-Transactions of the ASME,2014,81(3):31007(4).
[11]Zhong Y,Li R,Liu Y M,et al.On new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported[J].International Journal of Solids and Structures,2009,46(11/12):2506-2513.
[12]Li R,Tian B,Zhong Y.Analytical bending solutions of free orthotropic rectangular thin plates under arbitrary loading[J].Meccanica,2013,48(10):2497-2510.
[13]Li R,Zhong Y,Tian B,et al.On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates[J].Applied Mathematics Letters,2009,22(12):1821-1827.
[14]Timoshenko S P,Woinowsky-Krieger S W.Theory of plates and shells[M].New York:Mc Graw-Hill,1959.
[15]徐芝纶.弹性力学[M].第四版.北京:高等教育出版社,2006.(Xu Zhilun.Elasticity mechanics[M].4th ed.Beijing:Higher Education Press,2006(in Chinese)).
[16]李元媛,蔡睿贤.矩形薄板弯曲的严格简明解析解[J].机械工程学报,2008,44(10):72-76.(Li Yuanyuan,Cai Ruixian.Some concise exact analytical solutions of rectangular plate bending[J].Chinese Journal of Mechanical Engineering,2008,44(10):72-76(in Chinese)).
[17]钟阳,李锐,刘月梅.四边固支矩形弹性薄板的精确解析解[J].力学季刊,2009,30(2):297-302.(Zhong Yang,Li Rui,Liu Yuemei.Exact analytic solution of rectangular thin plate with four edges clamped[J].Chinese Quarterly of Mechanics,2009,30(2):297-302(in Chinese)).
[18]吴洪洋.均布荷载作用下四边固支板的一种逼近解法[J].贵州工业大学学报,2000,29(1):32-35.(Wu Hongyang.An asymptotic approximating solution of the clamped rectangular plate under the average loads action[J].Journal of Guizhou University of Technology,2000,29(1):32-35(in Chinese)).