水泥混凝土路面中厚板问题分析方法的研究
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摘要
我国对水泥混凝土路面的荷载应力分析,通常是基于Kirchhoff薄板理论,采用弹性地基上无限大板的解析解和Westergaard的板边和边角公式。然而,水泥混凝土路面是有限尺寸板,并且板厚有增加趋势,本文运用中厚板理论,利用有限积分变换法求中厚板问题的解析解。
由于中厚板问题的控制方程较为复杂,求其解析解是很困难的。目前通常采用半逆法求得弹性地基上有限尺寸中厚板的解析解,但这种方法具有局限性,它需要事先设定满足边界条件的位移函数,再确定未知常数。本文采用的有限积分变换法,则直接对中厚板问题的控制偏微分方程组进行有限积分变换,将偏微分方程组化为线性方程组,通过对线性方程组的求解,再施加有限积分逆变换即得到原问题的解。利用这种方法,本文分别详细地推导出了四边固支Mindlin矩形中厚板、改进的胡海昌矩形中厚板以及Winkler弹性地基上四边自由Reissner矩形中厚板弯曲问题的解析解。同时也分别得到了固支和自由两种边界条件下矩形中厚板挠度和内力的解析表达式。论文还分析了Reissner中厚板理论和Mindlin中厚板理论数值结果的差异,以四边固支Mindlin中厚板问题的解析解为例,还推导证明了这两种中厚板理论不能用于计算集中荷载问题。针对水泥混凝土路面,文中模拟双轮荷载作用板中和板边的情况,计算板内任意点的挠度及应力,并可推广到多轴多轮载作用于板内任意位置时求板内任意点的挠度和应力。
本文的计算不但验证了有限积分变换法求解中厚板问题的正确性,并求得水泥混凝土路面板设计所关心的弹性地基上四边自由有限尺寸中厚板问题的解析解。本文所采用的方法不但克服了常用解析法的一些弊端,而且编制的程序通用性较强,可以模拟任意多轴多轮载作用于同一块路面板,计算任意点的挠度和内力,具有一定的理论与实践指导意义。
The stress analysis of rigid pavement in our country adopts the theoretic solutions of infinite plates on elastic foundation which is based on Kirchhoff thin plates hypothesis. However, size of rigid pavement plates is limited and thickness of the plates have a trend of increasing, this thesis will discuss a new method called finite integral transform method and solve theoretic problem of finite size moderate thick plates on elastic foundation with it.
Basic equations of moderate thick plates are complex and it is hard to get the theoretic solutions, common method on solving this problem is semi-inverse method. But this method needs to be firstly selected deformation functions and make them satisfy the boundary condition, and then find out unknown coefficients which are contained in the deformation functions. When using the finite integral transform method which I will discuss in the thesis, solving the basic equations is more convenient. The high order partial differential basic equations are transformed to linear equations by finite transform method, through arriving at results easily of the linear equations we get solutions of the high order partial differential basic equations by applying inverse finite integral transform to them. With this method, this thesis will introduce process in detail on obtaining theoretic solutions of Mindlin rectangular moderate thick plates with completely camped support and theoretic solutions of Reissner rectangular moderate thick plates with four edges free on Winkler foundation. And then with these results, the thesis show the numerical results of moderate thick plates on both deflection and stress when load located on center and edge of plates, and discuss the numerical differences of Mindlin and Reissner moderate thick plates. At last, giving program that can simulate two-wheel load on any parts of rigid pavement plates and give results specially on two condition of loading on center and edge of plates, which is able to get deflections and stresses on any position of plates. Further more, the program is able to get results of multiply-wheel load locate on arbitrary position of a single plate.
由于中厚板问题的控制方程较为复杂,求其解析解是很困难的。目前通常采用半逆法求得弹性地基上有限尺寸中厚板的解析解,但这种方法具有局限性,它需要事先设定满足边界条件的位移函数,再确定未知常数。本文采用的有限积分变换法,则直接对中厚板问题的控制偏微分方程组进行有限积分变换,将偏微分方程组化为线性方程组,通过对线性方程组的求解,再施加有限积分逆变换即得到原问题的解。利用这种方法,本文分别详细地推导出了四边固支Mindlin矩形中厚板、改进的胡海昌矩形中厚板以及Winkler弹性地基上四边自由Reissner矩形中厚板弯曲问题的解析解。同时也分别得到了固支和自由两种边界条件下矩形中厚板挠度和内力的解析表达式。论文还分析了Reissner中厚板理论和Mindlin中厚板理论数值结果的差异,以四边固支Mindlin中厚板问题的解析解为例,还推导证明了这两种中厚板理论不能用于计算集中荷载问题。针对水泥混凝土路面,文中模拟双轮荷载作用板中和板边的情况,计算板内任意点的挠度及应力,并可推广到多轴多轮载作用于板内任意位置时求板内任意点的挠度和应力。
本文的计算不但验证了有限积分变换法求解中厚板问题的正确性,并求得水泥混凝土路面板设计所关心的弹性地基上四边自由有限尺寸中厚板问题的解析解。本文所采用的方法不但克服了常用解析法的一些弊端,而且编制的程序通用性较强,可以模拟任意多轴多轮载作用于同一块路面板,计算任意点的挠度和内力,具有一定的理论与实践指导意义。
The stress analysis of rigid pavement in our country adopts the theoretic solutions of infinite plates on elastic foundation which is based on Kirchhoff thin plates hypothesis. However, size of rigid pavement plates is limited and thickness of the plates have a trend of increasing, this thesis will discuss a new method called finite integral transform method and solve theoretic problem of finite size moderate thick plates on elastic foundation with it.
Basic equations of moderate thick plates are complex and it is hard to get the theoretic solutions, common method on solving this problem is semi-inverse method. But this method needs to be firstly selected deformation functions and make them satisfy the boundary condition, and then find out unknown coefficients which are contained in the deformation functions. When using the finite integral transform method which I will discuss in the thesis, solving the basic equations is more convenient. The high order partial differential basic equations are transformed to linear equations by finite transform method, through arriving at results easily of the linear equations we get solutions of the high order partial differential basic equations by applying inverse finite integral transform to them. With this method, this thesis will introduce process in detail on obtaining theoretic solutions of Mindlin rectangular moderate thick plates with completely camped support and theoretic solutions of Reissner rectangular moderate thick plates with four edges free on Winkler foundation. And then with these results, the thesis show the numerical results of moderate thick plates on both deflection and stress when load located on center and edge of plates, and discuss the numerical differences of Mindlin and Reissner moderate thick plates. At last, giving program that can simulate two-wheel load on any parts of rigid pavement plates and give results specially on two condition of loading on center and edge of plates, which is able to get deflections and stresses on any position of plates. Further more, the program is able to get results of multiply-wheel load locate on arbitrary position of a single plate.
引文
[1]邓学均,陈荣生.刚性路面设计[M].北京:人民交通出版社,2005.
[2]唐伯明,蒙华,刘志军.欧美水泥混凝土路面设计使用现状综述[J].公路.2003,10(10):37-39.
[3]郑木莲,王秉纲,陈拴发.推进水泥混凝土路面在中国的发展[J].公路交通科技,2007,4(4):5-7.
[4]交通部水泥混凝土路面推广组.水泥混凝土路面研究[M].北京:人民交通出版社,1995.
[5]交通部公路规划设计院,同济大学.水泥混凝土路面设计理论、方法和参数研究[M].人民交通出版社,1986.
[6]姚祖康,王秉纲.采用有限元法分析水泥混凝土路面的荷载应力[J].华东公路,1979,(1):1-21.
[7]姚祖康,桓腊元,冷培义等.水泥混凝土路面荷载应力计算理论的试验验证[J].同济大学学报,1981,1(1):68-77.
[8]王秉纲.弹性半空间地基混凝土路(地)面的承载力分析[J].西安公路学院学报,1986,4(1):1-14.
[9]房英武,王秉纲.弹性地基上矩形厚板的荷载应力分析[J].西安公路学院学报,1987,5(2):150-160.
[10]郑健龙,张起森.刚性路面设计理论新探[J].长沙交通学院院报,1989,5(2):54-64.
[11]郑健龙,张起森.叠加法在刚性路面应力分析中的应用[J].长沙交通学院院报,1988,4(1):81-92.
[12]王虎.Winkler地基板的积分变换解[J].中南公路工程,2000,25(4):5-6.
[13]黄义,何芳社.弹性地基上的梁、板、壳[M].科学出版社,2005.
[14]生跃,黄义.双参数弹性地基上自由边矩形板[J].应用数学和力学,1987,8(4):317-329.
[15]孙家驷,高建平.道路设计资料集<4>路面设计[M].人民交通出版社,2003.
[16]姚祖康.公路设计手册-路面[M].人民交通出版社,2003.
[17]Civalek(o|¨).Three-dimensional Vibration,Buckling and Bending Analyses of Thick Rectangular Plates based on Discrete Singular Convolution Method[J].International Journal of Mechanical Sciences,2007,49,752-765.
[18]Henwood D J.Finite Difference Solution of a System of First-order Partial Differential Equations[J].Int J Numerical Methods of Engineering,1981,17(9):1385-1395.
[19]Henwood D J.Fourier Series Solution for a Rectangular Thick Plate with Free Edges on an Elastic Foundation[J].Int J Numerical Methods Engineering,1982,18(12):1801-1820.
[20]张福范.弹性薄板[M].北京:科学出版社,1984
[21]Timoshenko.Theory of Plates and Shells[M].New York:McGraw-Hill,1959:126-130.
[22]何福保,沈亚鹏.板壳理论[M].西安交通大学出版社,1993.
[23]Reissner E.On Bending of Elastic Plates[J].Quarterly of Applied Mathematics,1947,5:55-68.
[24]Reissner E.The Effects of Transverse Shear Deformation on the Bending of Elastic Plates[J].Journal of Applied Mechanics,1945,12:A69-A77.
[25]Vlazov,V.Z,Leontiev.Beams,Plates and Shells on Elastic Foundation[M].Israel Program for Scientific Translations.1966.
[26]Mindlin R D.Influence of Rotatory Inertia and Shear on Flexural Motion of Isotropic Elastic Plates[J].Journal of Applied Mechanics,1951,18:31-38.
[27]胡海昌.弹性力学的变分原理及其应用[M].北京:科学出版社,1981.
[28]Reissner E.On the theory of bending of elastic plates[J].Journal of Mathematics and Physics 1944;23:184-191.
[29]王克林,黄义.弹性地基上四边自由的矩形厚板[J].固体力学学报,1986,31(1):36-49.
[30]钟阳,孙爱民,周福霖等.弹性地基上四边自由矩形薄板分析的有限积分变换法[J].岩土工程学报,2006,28(11):2019-2022.
[31]钟阳,王国新,孙爱民.弹性地基上四边自由矩形薄板振动分析有限积分变换法[J].大连理工大学学报,2007,47(1):73-77.
[32]石小平,姚祖康.集装箱堆场水泥混凝土铺面的结构计算[J].土木工程学报,1985,18(3):61-73.
[33]Yoo H H.Modal Characteristic of a Rotating Rectangular Cantilever Plate[J].JSV,2003,259(1):81-96.
[34]曲乃泗,孙焕纯.厚板的振动分析[J].工程力学,1987,4(2):27-46.
[35]汪克让.弹性地基上自由边矩形厚板[J].应用数学与力学,1990,11(9):317-326.
[36]宋启根,林洋,徐培.双参数地基Reissner板局部荷载解[J].岩土工程学报,1993,15(5):48-58.
[37]石小平,姚祖康.Pasternak基础上四边自由矩形厚板的解[J].同济大学学报,1989,17(2):173-184.
[38]陈颂英,曲延鹏,刘燕等.Reissner-Mindlin板弯曲理论解得差异[J].山东大学学报,2003,33(2):147-149.
[49]邓安福.弹性地基上厚板的计算分析[J].岩土工程学报,2003,25(6):653-657.
[40]Yao W A,Sui Y F.Symplectic Solution System for Reissner Plate Bending[J].Applied Mathematics and Mechanics,2004,25(2):178-185.
[41]钟阳,胡波.四边固支矩形厚板的有限积分变换法[J].土木建筑与环境工程.2009,31(3):1-5.
[42]孙卫明,杨光松,张承宗.双参数地基上弹性厚板弯曲的一般解析解[J].工程力学,1999,16(2):71-78.
[43]尤占平,夏永旭,胡长顺等.过路构造物上水泥混凝土路面结构分析研究[C].中国公路学会,99学术交流论文集.北京:人民交通出版社,1999.
[44]Sneddon I H.The Use of Integral Transforms[M].McGraw-Bill Inc,U S A.1972:169-182.
[45]中科院北京力学研究所.夹层板壳的弯曲稳定和振动[M].北京:科学出版社,1977.
[2]唐伯明,蒙华,刘志军.欧美水泥混凝土路面设计使用现状综述[J].公路.2003,10(10):37-39.
[3]郑木莲,王秉纲,陈拴发.推进水泥混凝土路面在中国的发展[J].公路交通科技,2007,4(4):5-7.
[4]交通部水泥混凝土路面推广组.水泥混凝土路面研究[M].北京:人民交通出版社,1995.
[5]交通部公路规划设计院,同济大学.水泥混凝土路面设计理论、方法和参数研究[M].人民交通出版社,1986.
[6]姚祖康,王秉纲.采用有限元法分析水泥混凝土路面的荷载应力[J].华东公路,1979,(1):1-21.
[7]姚祖康,桓腊元,冷培义等.水泥混凝土路面荷载应力计算理论的试验验证[J].同济大学学报,1981,1(1):68-77.
[8]王秉纲.弹性半空间地基混凝土路(地)面的承载力分析[J].西安公路学院学报,1986,4(1):1-14.
[9]房英武,王秉纲.弹性地基上矩形厚板的荷载应力分析[J].西安公路学院学报,1987,5(2):150-160.
[10]郑健龙,张起森.刚性路面设计理论新探[J].长沙交通学院院报,1989,5(2):54-64.
[11]郑健龙,张起森.叠加法在刚性路面应力分析中的应用[J].长沙交通学院院报,1988,4(1):81-92.
[12]王虎.Winkler地基板的积分变换解[J].中南公路工程,2000,25(4):5-6.
[13]黄义,何芳社.弹性地基上的梁、板、壳[M].科学出版社,2005.
[14]生跃,黄义.双参数弹性地基上自由边矩形板[J].应用数学和力学,1987,8(4):317-329.
[15]孙家驷,高建平.道路设计资料集<4>路面设计[M].人民交通出版社,2003.
[16]姚祖康.公路设计手册-路面[M].人民交通出版社,2003.
[17]Civalek(o|¨).Three-dimensional Vibration,Buckling and Bending Analyses of Thick Rectangular Plates based on Discrete Singular Convolution Method[J].International Journal of Mechanical Sciences,2007,49,752-765.
[18]Henwood D J.Finite Difference Solution of a System of First-order Partial Differential Equations[J].Int J Numerical Methods of Engineering,1981,17(9):1385-1395.
[19]Henwood D J.Fourier Series Solution for a Rectangular Thick Plate with Free Edges on an Elastic Foundation[J].Int J Numerical Methods Engineering,1982,18(12):1801-1820.
[20]张福范.弹性薄板[M].北京:科学出版社,1984
[21]Timoshenko.Theory of Plates and Shells[M].New York:McGraw-Hill,1959:126-130.
[22]何福保,沈亚鹏.板壳理论[M].西安交通大学出版社,1993.
[23]Reissner E.On Bending of Elastic Plates[J].Quarterly of Applied Mathematics,1947,5:55-68.
[24]Reissner E.The Effects of Transverse Shear Deformation on the Bending of Elastic Plates[J].Journal of Applied Mechanics,1945,12:A69-A77.
[25]Vlazov,V.Z,Leontiev.Beams,Plates and Shells on Elastic Foundation[M].Israel Program for Scientific Translations.1966.
[26]Mindlin R D.Influence of Rotatory Inertia and Shear on Flexural Motion of Isotropic Elastic Plates[J].Journal of Applied Mechanics,1951,18:31-38.
[27]胡海昌.弹性力学的变分原理及其应用[M].北京:科学出版社,1981.
[28]Reissner E.On the theory of bending of elastic plates[J].Journal of Mathematics and Physics 1944;23:184-191.
[29]王克林,黄义.弹性地基上四边自由的矩形厚板[J].固体力学学报,1986,31(1):36-49.
[30]钟阳,孙爱民,周福霖等.弹性地基上四边自由矩形薄板分析的有限积分变换法[J].岩土工程学报,2006,28(11):2019-2022.
[31]钟阳,王国新,孙爱民.弹性地基上四边自由矩形薄板振动分析有限积分变换法[J].大连理工大学学报,2007,47(1):73-77.
[32]石小平,姚祖康.集装箱堆场水泥混凝土铺面的结构计算[J].土木工程学报,1985,18(3):61-73.
[33]Yoo H H.Modal Characteristic of a Rotating Rectangular Cantilever Plate[J].JSV,2003,259(1):81-96.
[34]曲乃泗,孙焕纯.厚板的振动分析[J].工程力学,1987,4(2):27-46.
[35]汪克让.弹性地基上自由边矩形厚板[J].应用数学与力学,1990,11(9):317-326.
[36]宋启根,林洋,徐培.双参数地基Reissner板局部荷载解[J].岩土工程学报,1993,15(5):48-58.
[37]石小平,姚祖康.Pasternak基础上四边自由矩形厚板的解[J].同济大学学报,1989,17(2):173-184.
[38]陈颂英,曲延鹏,刘燕等.Reissner-Mindlin板弯曲理论解得差异[J].山东大学学报,2003,33(2):147-149.
[49]邓安福.弹性地基上厚板的计算分析[J].岩土工程学报,2003,25(6):653-657.
[40]Yao W A,Sui Y F.Symplectic Solution System for Reissner Plate Bending[J].Applied Mathematics and Mechanics,2004,25(2):178-185.
[41]钟阳,胡波.四边固支矩形厚板的有限积分变换法[J].土木建筑与环境工程.2009,31(3):1-5.
[42]孙卫明,杨光松,张承宗.双参数地基上弹性厚板弯曲的一般解析解[J].工程力学,1999,16(2):71-78.
[43]尤占平,夏永旭,胡长顺等.过路构造物上水泥混凝土路面结构分析研究[C].中国公路学会,99学术交流论文集.北京:人民交通出版社,1999.
[44]Sneddon I H.The Use of Integral Transforms[M].McGraw-Bill Inc,U S A.1972:169-182.
[45]中科院北京力学研究所.夹层板壳的弯曲稳定和振动[M].北京:科学出版社,1977.