一边支承矩形板弯曲精确解法
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摘要
弹性薄板广泛应用于工程各领域。研究薄板弯曲的理论解法,特别是研究一种适合于各类边界条件、各种荷载及不同位移作用下的统一解法或精确解法具有很高的理论意义及一定实用价值。本文所研究的一边支承矩形板弯曲精确解法就是该研究方向的一个子课题。
论文综述了各种经典解法及其局限性,特别详细阐述了矩形板弯曲统一求解法的基本思想。该解法由板角点求解条件的完备性将矩形板弯曲划分为广义静定弯曲和广义超静定弯曲:有柱支角点且其柱支反力无法由静力平衡条件确定的为广义超静定弯曲,其它均为广义静定弯曲。广义静定弯曲可以由板的边界条件直接求解,广义超静定弯曲要由叠加法求解。在求解静定弯曲时,挠度表达式有通解和特解两部分,通解W_1的一般形式是包含8个待定常数的双向单三角级数,以反映板的双向弯曲变形并与板的8个边界条件相对应。三角级数应是完整的正交三角函数族,其形式要切合板边界所能激发出的弯曲变形形态。通解W_1还要满足柱支角点位移条件、自由角点力为零条件,要反映支承边端角点发生位移时而导致边界线性位移特征。由边界条件的相近性,广义静定弯曲可以分为7种类型矩形板。弯曲特解W_2仅考虑板面荷载作用下及自由角点作用集中力时相应的特解。在板面荷载作用下,特解可以采用双重三角级数形式或多项式形式;在角点集中力作用下,特解只能采用多项式形式。
统一解法在求解思想体系上及求解方法上比各种经典解法都有了很大的改进,但仍然存在一定的不足。在统一解法的基础上,精确解法建立了一套全新的求解框架体系:首先,精确解法采用更加科学的理念来划分静定弯曲和超静定弯曲。它认为板弯曲平衡微分方程是综合力的平衡条件、几何方程以及物理方程而成,边界上力的边界条件与位移边界条件均是等效的。对固定边、简支边、自由边而言,每条边界上均有4个参数(2个边界力和2个边界位移),其中2个是已知的,2个是未知的;即未知量数等于已知条件数,这些边界求解条件是完备的。而对于柱支座来说,虽然柱支座处的位移是已知的,但柱支反力只能通过力的平衡条件确定,与几何方程及物理方程无关,即柱支反力不能通过柱支座处的位移确定。因此当柱支反力可以由力的平衡条件确定时,求解条件是完备的,否则是不完备的,前者称为静定弯曲,后者称为超静定弯曲。这种分类方法最大的优点是它不限定统一解法中的角柱支座,柱支座可以在板的角点上、在板的边界上或板面上的任意点处。其次,精确解法严格遵守以下规则:通解W_1中所采用的三角级数波形必须切合板边界条件所能激发出的变形形态。从而将一边固定矩形板与一边简支一柱支座支承的矩形板弯曲合并处理。此外,精确解法认为板弯曲平衡微分方程是以挠度为参数表示的板法向力的平衡方程,因此所有已知的边
The elastic thin plate is widely used in each domain of engineering, It has high theoretical significance and certain practice to research a theoretical solution to the plate bending, especially an unified solution or an accurate solution which adapts to various boundary conditions, load conditions and displacement of the boundary. In this paper, we studied the accurate bending solution to rectangular plates with one supported edge, which is a sub-item of the accurate solution of rectangular plate bending.Various classic solutions and their limitations were summarized in this thesis, and especially expounded the basic thought of the unified solution. According to the completeness on calculating conditions of the corner point, it divided the plate bending into the generalized statically determinate bending and indeterminate bending. If the plate has supported corner point and the reverse force can be determined by the static equilibrium equation, it belongs to the generalized statically determinate bending; the others belong to the generalized indeterminate bending. As the calculating condition is complete, the plate bending can be solved directly by using the equilibrium differential equation and the boundary conditions. For the latter the superposition method can be used. When solving the generalized statically determinate bending, the deflection expression consists of a homogeneous part W_1 and a particular part W _2, which represent the homogeneous and particular solution respectively. The common form of W_1 is a bi-directional single trigonometric series that contains eight undetermined constants to reflect the bi-directional bending deformation. The trigonometric series should be complete and be suitable to the deformation resulting from the boundary displacement. The homogeneous solution W_1 should also satisfy the displacement conditions of the column support, the concentrated force condition at the column support, and reflect the boundary's linear displacement character that resulted form the displacement at endpoint of supported edge. The generalized statically determinate bending can be divided into seven kinds of rectangular plates according to the boundary's approximation. The particular solution W2 reflects the loads on the plate and concentrated fore at the free corner point. The form of particular solution may use duple trigonometric series or multinomial when loads operate on the plate, it can only use multinomial when concentrated force operates on the corner point.Though the unified solution has made a big improvement than those classic solutions, it still has some drawbacks. The accurate solution founded one set of completely new solution system based on the unified one. Firstly, it adopted more scientific reason thought to divide the generalized statically determinate bending and indeterminate bending. It considers that the equilibrium differential equation of the
plate bending synthesized the force's equilibrium condition, geometric equation and physical equation, and that the boundary conditions of the force and displacement are equivalent. As far as the fixed edge, the simply supported edge and the free edge, there are four parameters on each boundary. Because the unknown quantity number is equaled to the known condition number, the calculating condition is complete. Though the displacement of the column support is known, the reverse force can only be determined by way of the force's equilibrium condition, having nothing to do with the geometric and physical equation. Therefore the calculating condition is complete when the reverse force can be determined by way of the force's equilibrium condition, otherwise is incomplete. The former is called the generalized statically determinate bending, and the latter is called indeterminate bending. The largest merit of this kind of classification method is that the point of column support can be at the any place on the plate. Secondly, the accurate solution rigorously abides by the following rules: the trigonometric series waveform that adopted in the homogeneous solution must suit the deformation form that the boundary conditions can arouse out. Thus the plate supported by a fixed edge or by a simple edge and a column was incorporated. In addition, the accurate solution considers that the equilibrium differential equation of the bending plate is expressed by the deflection parameter. Therefore the all known boundary deflections and normal direction loads should have corresponding particular solutions. The accurate solution adopted composite particular solution, which can all together satisfy the governing differential equation, the deflection along the supported edge, the shear force condition along the free edge, the concentrated force condition at the column support, and raised the precision of the solution.The rectangular plate bending with a supported edge combined the cantilever rectangular plate with the rectangular plate supported by a simple edge and a column, and cancelled the limit that the column must be on the comer point. The solution of this kind of plate made a great difference with the unified solution because of the variety of boundary condition. The very method has the advantage of high precision and has been proved by inverse analysis examples.
论文综述了各种经典解法及其局限性,特别详细阐述了矩形板弯曲统一求解法的基本思想。该解法由板角点求解条件的完备性将矩形板弯曲划分为广义静定弯曲和广义超静定弯曲:有柱支角点且其柱支反力无法由静力平衡条件确定的为广义超静定弯曲,其它均为广义静定弯曲。广义静定弯曲可以由板的边界条件直接求解,广义超静定弯曲要由叠加法求解。在求解静定弯曲时,挠度表达式有通解和特解两部分,通解W_1的一般形式是包含8个待定常数的双向单三角级数,以反映板的双向弯曲变形并与板的8个边界条件相对应。三角级数应是完整的正交三角函数族,其形式要切合板边界所能激发出的弯曲变形形态。通解W_1还要满足柱支角点位移条件、自由角点力为零条件,要反映支承边端角点发生位移时而导致边界线性位移特征。由边界条件的相近性,广义静定弯曲可以分为7种类型矩形板。弯曲特解W_2仅考虑板面荷载作用下及自由角点作用集中力时相应的特解。在板面荷载作用下,特解可以采用双重三角级数形式或多项式形式;在角点集中力作用下,特解只能采用多项式形式。
统一解法在求解思想体系上及求解方法上比各种经典解法都有了很大的改进,但仍然存在一定的不足。在统一解法的基础上,精确解法建立了一套全新的求解框架体系:首先,精确解法采用更加科学的理念来划分静定弯曲和超静定弯曲。它认为板弯曲平衡微分方程是综合力的平衡条件、几何方程以及物理方程而成,边界上力的边界条件与位移边界条件均是等效的。对固定边、简支边、自由边而言,每条边界上均有4个参数(2个边界力和2个边界位移),其中2个是已知的,2个是未知的;即未知量数等于已知条件数,这些边界求解条件是完备的。而对于柱支座来说,虽然柱支座处的位移是已知的,但柱支反力只能通过力的平衡条件确定,与几何方程及物理方程无关,即柱支反力不能通过柱支座处的位移确定。因此当柱支反力可以由力的平衡条件确定时,求解条件是完备的,否则是不完备的,前者称为静定弯曲,后者称为超静定弯曲。这种分类方法最大的优点是它不限定统一解法中的角柱支座,柱支座可以在板的角点上、在板的边界上或板面上的任意点处。其次,精确解法严格遵守以下规则:通解W_1中所采用的三角级数波形必须切合板边界条件所能激发出的变形形态。从而将一边固定矩形板与一边简支一柱支座支承的矩形板弯曲合并处理。此外,精确解法认为板弯曲平衡微分方程是以挠度为参数表示的板法向力的平衡方程,因此所有已知的边
The elastic thin plate is widely used in each domain of engineering, It has high theoretical significance and certain practice to research a theoretical solution to the plate bending, especially an unified solution or an accurate solution which adapts to various boundary conditions, load conditions and displacement of the boundary. In this paper, we studied the accurate bending solution to rectangular plates with one supported edge, which is a sub-item of the accurate solution of rectangular plate bending.Various classic solutions and their limitations were summarized in this thesis, and especially expounded the basic thought of the unified solution. According to the completeness on calculating conditions of the corner point, it divided the plate bending into the generalized statically determinate bending and indeterminate bending. If the plate has supported corner point and the reverse force can be determined by the static equilibrium equation, it belongs to the generalized statically determinate bending; the others belong to the generalized indeterminate bending. As the calculating condition is complete, the plate bending can be solved directly by using the equilibrium differential equation and the boundary conditions. For the latter the superposition method can be used. When solving the generalized statically determinate bending, the deflection expression consists of a homogeneous part W_1 and a particular part W _2, which represent the homogeneous and particular solution respectively. The common form of W_1 is a bi-directional single trigonometric series that contains eight undetermined constants to reflect the bi-directional bending deformation. The trigonometric series should be complete and be suitable to the deformation resulting from the boundary displacement. The homogeneous solution W_1 should also satisfy the displacement conditions of the column support, the concentrated force condition at the column support, and reflect the boundary's linear displacement character that resulted form the displacement at endpoint of supported edge. The generalized statically determinate bending can be divided into seven kinds of rectangular plates according to the boundary's approximation. The particular solution W2 reflects the loads on the plate and concentrated fore at the free corner point. The form of particular solution may use duple trigonometric series or multinomial when loads operate on the plate, it can only use multinomial when concentrated force operates on the corner point.Though the unified solution has made a big improvement than those classic solutions, it still has some drawbacks. The accurate solution founded one set of completely new solution system based on the unified one. Firstly, it adopted more scientific reason thought to divide the generalized statically determinate bending and indeterminate bending. It considers that the equilibrium differential equation of the
plate bending synthesized the force's equilibrium condition, geometric equation and physical equation, and that the boundary conditions of the force and displacement are equivalent. As far as the fixed edge, the simply supported edge and the free edge, there are four parameters on each boundary. Because the unknown quantity number is equaled to the known condition number, the calculating condition is complete. Though the displacement of the column support is known, the reverse force can only be determined by way of the force's equilibrium condition, having nothing to do with the geometric and physical equation. Therefore the calculating condition is complete when the reverse force can be determined by way of the force's equilibrium condition, otherwise is incomplete. The former is called the generalized statically determinate bending, and the latter is called indeterminate bending. The largest merit of this kind of classification method is that the point of column support can be at the any place on the plate. Secondly, the accurate solution rigorously abides by the following rules: the trigonometric series waveform that adopted in the homogeneous solution must suit the deformation form that the boundary conditions can arouse out. Thus the plate supported by a fixed edge or by a simple edge and a column was incorporated. In addition, the accurate solution considers that the equilibrium differential equation of the bending plate is expressed by the deflection parameter. Therefore the all known boundary deflections and normal direction loads should have corresponding particular solutions. The accurate solution adopted composite particular solution, which can all together satisfy the governing differential equation, the deflection along the supported edge, the shear force condition along the free edge, the concentrated force condition at the column support, and raised the precision of the solution.The rectangular plate bending with a supported edge combined the cantilever rectangular plate with the rectangular plate supported by a simple edge and a column, and cancelled the limit that the column must be on the comer point. The solution of this kind of plate made a great difference with the unified solution because of the variety of boundary condition. The very method has the advantage of high precision and has been proved by inverse analysis examples.
引文
[1] S.铁摩辛柯,S.沃诺斯基,板壳理论,科学出版社,1997年
[2] 黄克智等,板壳理论,清华大学出版社,1987年
[3] 寿楠椿,弹性薄板弯曲,高等教育出版社,1984年
[4] 刘鸿文,板壳理论,浙江大学出版社,1987年
[5] 张福范,弹性薄板(第二版),科学出版社,1984年
[6] [美]A.C阿加雷,板壳应力,中国建工出版社,1986年
[7] 王俊奎,张志民,板壳的弯曲与稳定,国防工业出版社,1980年
[8] 薛大为,板壳理论,北京工业学院出版社,1988年
[9] [美]R.Szilard,板的理论和分析,中国铁道出版社,1984年
[10] 黄炎,弹性薄板理论,国防科技大学出版社,1982年
[11] 徐芝纶,弹性力学简明教程(第二版),高等教育出版社
[12] 徐芝纶,弹性力学(第二版),高等教育出版社
[13] 日鹫津久一郎,能量原理,中国建筑工业出版社,1977年
[14] 施振东,韩耀新,弹性力学教程,北京航空学院出版社,1987年
[15] 龙驭球,有限元法概论,人民教育出版社,1978年
[16] R.D库克,有限元分析的概念与应用,科学出版社,1989年
[17] 钱伟长,叶开沅,弹性力学,科学出版社,1957年
[18] 杨耀乾,平板理论,中国铁道出版社,1980年
[19] C. E. TURNER, Introduction to Plate and Shell Theory, 1965年
[20] Stanislaw Lukasicwicz, Local loads in plates and shells, 1997年
[21] David McFarland, Bert L. Smith, Walter D. Bernhart, Analysis of Plates, 1975年
[22] S.铁摩辛柯,J.盖莱著,张福范译,弹性稳定理论(第二版),科学出版社,1965年
[23] 周叮,任意荷载作用下矩形弹性薄板横向弯曲的一个高精度近似解,应用数学与力学,1980年,第11卷,第四期
[24] 朱雁滨,付宝连,再论在一集中荷载作用下悬臂矩形板的弯曲,应用数学与力学,1986年,第7卷,第10期
[25] 王磊,李家宝,矩形薄板弯曲问题的近似解法——康托洛维奇迦辽金法,应用数学与力学,1986年,第7卷,第1期
[26] 成祥生,悬臂矩形板在对称边界荷载下的屈曲,应用数学与力学,1980年,第11卷,第4期
[27] Reissner, E., On the theory of bending of elastic plates, J. Math. Phy., Vol. 23, 1994.
[28] Reissner, E., The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech., Vol. 12 1945.
[29] Reissner, E., On bending of elastic plates, Quart. Appl. Math., Vol. 5, 1947.
[30] 杨骁,宁建军,程思钧,考虑横向剪切效应的悬臂板的弯曲,应用数学与力学,1992年,第13卷,第1期
[31] 钱伟长,不用假定的Kirchhoff-Love三维弹性板近似理论及其边界条件,应用数学与力学,1995年,第16卷,第13期
[32] 施军,一种高精度的简单矩形板弯曲元素,上海力学,1984年,第1期
[33] 黄晓梅,钱雪英,孙汝,陈季筠,一边固定,对边由多个点支承的矩形板,上海力学,1984年,第1期
[34] 范建中,贾宝范,均布荷载下悬臂矩形中厚板的弯曲问题,上海力学,1989年,第10卷,第1期
[35] 姬鸿恩,支座位移作用下矩形薄板弯曲统一求解方法,郑州大学硕士学位论文,2001年
[36] 许琪楼,杨卫忠,一对边简支矩形薄板在集中荷载作用下的弯曲解,第三届全国结构工程学术会议论文集,1994年
[37] 许琪楼,姬同庚,二邻边简支其余边自由的矩形板在均布荷载作用下的弯曲解,土木工程学报,1995年,第三期
[38] 许琪楼,姜锐等,四边支承矩形板弯曲统一求解方法,工程力学,1999年第16卷,第3期
[39] 许琪楼等,三边支承一边自由的矩形板弯曲统一求解方法,东南大学学报,1999年,第二期
[40] 许琪楼,均布荷载作用下无自由角点的矩形板弯曲通用解法,建筑环境与结构工程最新发展,浙江大学出版社,1995年
[41] 许琪楼,姬同庚,一边固定及一角点支承的矩形板在均布荷载作用下的弯曲解,应用数学与力学,1996年,第17卷,第12期
[42] 许琪楼,姜锐,唐国明,一边固定一角点或二角点支承的矩形板弯曲统一求解方法,计算力学学报,1999年,第16卷,第2期
[43] 许琪楼,弹性矩形薄板弯曲通用求解理论初探,工程力学增刊,1995年
[44] 许琪楼,一边固定,对边简支矩形板在均布荷载作用下的弯曲,工程力学增刊,1995年
[45] 许琪楼,三边支承一边自由矩形板弯曲,郑州工业大学学报,1997年,第3期
[2] 黄克智等,板壳理论,清华大学出版社,1987年
[3] 寿楠椿,弹性薄板弯曲,高等教育出版社,1984年
[4] 刘鸿文,板壳理论,浙江大学出版社,1987年
[5] 张福范,弹性薄板(第二版),科学出版社,1984年
[6] [美]A.C阿加雷,板壳应力,中国建工出版社,1986年
[7] 王俊奎,张志民,板壳的弯曲与稳定,国防工业出版社,1980年
[8] 薛大为,板壳理论,北京工业学院出版社,1988年
[9] [美]R.Szilard,板的理论和分析,中国铁道出版社,1984年
[10] 黄炎,弹性薄板理论,国防科技大学出版社,1982年
[11] 徐芝纶,弹性力学简明教程(第二版),高等教育出版社
[12] 徐芝纶,弹性力学(第二版),高等教育出版社
[13] 日鹫津久一郎,能量原理,中国建筑工业出版社,1977年
[14] 施振东,韩耀新,弹性力学教程,北京航空学院出版社,1987年
[15] 龙驭球,有限元法概论,人民教育出版社,1978年
[16] R.D库克,有限元分析的概念与应用,科学出版社,1989年
[17] 钱伟长,叶开沅,弹性力学,科学出版社,1957年
[18] 杨耀乾,平板理论,中国铁道出版社,1980年
[19] C. E. TURNER, Introduction to Plate and Shell Theory, 1965年
[20] Stanislaw Lukasicwicz, Local loads in plates and shells, 1997年
[21] David McFarland, Bert L. Smith, Walter D. Bernhart, Analysis of Plates, 1975年
[22] S.铁摩辛柯,J.盖莱著,张福范译,弹性稳定理论(第二版),科学出版社,1965年
[23] 周叮,任意荷载作用下矩形弹性薄板横向弯曲的一个高精度近似解,应用数学与力学,1980年,第11卷,第四期
[24] 朱雁滨,付宝连,再论在一集中荷载作用下悬臂矩形板的弯曲,应用数学与力学,1986年,第7卷,第10期
[25] 王磊,李家宝,矩形薄板弯曲问题的近似解法——康托洛维奇迦辽金法,应用数学与力学,1986年,第7卷,第1期
[26] 成祥生,悬臂矩形板在对称边界荷载下的屈曲,应用数学与力学,1980年,第11卷,第4期
[27] Reissner, E., On the theory of bending of elastic plates, J. Math. Phy., Vol. 23, 1994.
[28] Reissner, E., The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech., Vol. 12 1945.
[29] Reissner, E., On bending of elastic plates, Quart. Appl. Math., Vol. 5, 1947.
[30] 杨骁,宁建军,程思钧,考虑横向剪切效应的悬臂板的弯曲,应用数学与力学,1992年,第13卷,第1期
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