混凝土路面板热稳定性分析
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摘要
本文从薄板小挠度理论出发，研究了混凝土路面板的热应力稳定问题。将实际工程中路面板在缩缝处的连接方式分别简化为固定铰支和旋转弹性铰支两种边界条件。由于沿板厚度方向的温差相对于容许温差可以忽略不计，因此可以不考虑沿板厚度方向的温度场对板弯曲的影响，从而将热应力稳定问题转化为等效热压力作用的稳定问题来研究。选择Poisson比和表征支座弹性的量β作为小参数，求得了两种边界条件下混凝土路面板的临界容许温差和挠度的摄动解。在求解过程中对于出现非齐次边界条件的摄动方程，结合最小二乘法求得近似解。
数值计算结果表明：混凝土路面板挠度的摄动解沿板宽方向关于板中心截面对称，有必要考虑挠度二级摄动解的影响；临界温差随厚度而递增，但几乎不随Poisson比和长宽比λ而变化；临界温差的二级与零级摄动解的相对偏差随表征支座弹性的小参量β线性递增。与Euler压杆理论的比较指出，压杆理论所得的临界温差值等于板理论所得零级摄动解且总小于板理论的值，这意味着压杆理论所得的临界温差作为临界值总是使路面设计偏于安全，由此表明可以应用压杆理论指导路面板铺设温度的设计。将临界热压力值摄动解与有限元解相比较得出，临界热压力的摄动解与有限元计算解之间的误差随着λ的增大而减小；对于固定铰支的混凝土路面板，当λ小于0.25时，两者的误差大于3%；当λ大于1时，两者间的误差小于1%并随着λ的增大逐步趋向于0，由此说明Ansys程序中的板壳单元不适用于模拟长宽比小于0.25的板的屈曲问题
In this paper, based on the thin plate theory of a small deflection, we discussed the thermal stress stability of concrete pavement in the engineering. We considered two kinds of boundary conditions (BC) as the style of connection at the contraction joints, one is called a fixed-hinged and the other is rotating flexible hinge. However, compared with the critical temperature variation, we can ignore the temperature variation along the thickness direction. So we didn’t consider the impact of the temperature field along the thickness direction of the bending plate. As a result, we considered the problem of the thermal stress stability as the stability under the equivalent thermal pressure. We chose the Poisson ratio and the parameterβwhich characterized the form of elastic supports in the BC as the perturbation arguments, and obtained perturbation solutions of the permissible critical temperature variations and the deflections of the concrete pavement in two kinds of boundary conditions. We get a approximate solution by means of the Least Square Method when the Non-homogeneous boundary conditions of the perturbation equations occured.
Numerical results show that: the perturbation solutions of the deflection of the concrete pavement plate present symmetry about the center section on the plate along the width direction, the second order perturbation solution of the deflection must be concerned; critical temperature variation increases with the thickness, but do a little change with the Poisson ratio, and aspect ratioλ; the ratio of the perturbation solution of second-order and the zero-order of the critical temperature variation has a linear increments with the parameterβ. Comparison with the theory of Euler struts, we obtain that the value of the critical temperature variation of is equal to the perturbation solution of zero-order which derived from the plate theory and it is less than the value of the plate theory, this means that the critical variable temperature of using Euler’s theory as a design method is inclined to safety, so Euler’s theory can be applied as a guidance in the pavement design. We also have a brief compare the perturbation solution of the critical heat stress with the finite element solution, it shows that, the error between the perturbation solution of the critical heat stress and the finite element solution reduced whenλincreases; as the fixed-hinged concrete pavement, whenλis less than 0.25, the error between the two is greater than 3%; whenλis larger than 1, the error is less than 1% and tends to 0 gradually asλincreases, this result implies that the solution abtained from Ansys can only be used whenλis higther than 0.25.
数值计算结果表明：混凝土路面板挠度的摄动解沿板宽方向关于板中心截面对称，有必要考虑挠度二级摄动解的影响；临界温差随厚度而递增，但几乎不随Poisson比和长宽比λ而变化；临界温差的二级与零级摄动解的相对偏差随表征支座弹性的小参量β线性递增。与Euler压杆理论的比较指出，压杆理论所得的临界温差值等于板理论所得零级摄动解且总小于板理论的值，这意味着压杆理论所得的临界温差作为临界值总是使路面设计偏于安全，由此表明可以应用压杆理论指导路面板铺设温度的设计。将临界热压力值摄动解与有限元解相比较得出，临界热压力的摄动解与有限元计算解之间的误差随着λ的增大而减小；对于固定铰支的混凝土路面板，当λ小于0.25时，两者的误差大于3%；当λ大于1时，两者间的误差小于1%并随着λ的增大逐步趋向于0，由此说明Ansys程序中的板壳单元不适用于模拟长宽比小于0.25的板的屈曲问题
In this paper, based on the thin plate theory of a small deflection, we discussed the thermal stress stability of concrete pavement in the engineering. We considered two kinds of boundary conditions (BC) as the style of connection at the contraction joints, one is called a fixed-hinged and the other is rotating flexible hinge. However, compared with the critical temperature variation, we can ignore the temperature variation along the thickness direction. So we didn’t consider the impact of the temperature field along the thickness direction of the bending plate. As a result, we considered the problem of the thermal stress stability as the stability under the equivalent thermal pressure. We chose the Poisson ratio and the parameterβwhich characterized the form of elastic supports in the BC as the perturbation arguments, and obtained perturbation solutions of the permissible critical temperature variations and the deflections of the concrete pavement in two kinds of boundary conditions. We get a approximate solution by means of the Least Square Method when the Non-homogeneous boundary conditions of the perturbation equations occured.
Numerical results show that: the perturbation solutions of the deflection of the concrete pavement plate present symmetry about the center section on the plate along the width direction, the second order perturbation solution of the deflection must be concerned; critical temperature variation increases with the thickness, but do a little change with the Poisson ratio, and aspect ratioλ; the ratio of the perturbation solution of second-order and the zero-order of the critical temperature variation has a linear increments with the parameterβ. Comparison with the theory of Euler struts, we obtain that the value of the critical temperature variation of is equal to the perturbation solution of zero-order which derived from the plate theory and it is less than the value of the plate theory, this means that the critical variable temperature of using Euler’s theory as a design method is inclined to safety, so Euler’s theory can be applied as a guidance in the pavement design. We also have a brief compare the perturbation solution of the critical heat stress with the finite element solution, it shows that, the error between the perturbation solution of the critical heat stress and the finite element solution reduced whenλincreases; as the fixed-hinged concrete pavement, whenλis less than 0.25, the error between the two is greater than 3%; whenλis larger than 1, the error is less than 1% and tends to 0 gradually asλincreases, this result implies that the solution abtained from Ansys can only be used whenλis higther than 0.25.
引文
[1]陈铁云,沈惠申.结构的屈曲.上海科学技术文献出版社.1993年.
[2] Timoshenko S P. History of strength of materials.McGraw-Hill,New York.1953.
[3] Donnell L H. A new theory for the buckling of thin cylinders under axial compression and bending.Trans ASME.56,12(1934),795～806.
[4] Von Karman T, Tsien H S. The buckling of spherical shells on external pressure.J.Aero Sci.7 (1939),43～50.
[5] Von Karman T, Tsien H S. The buckling of thin cylindrical shells under axial compression.J. Aero Sci 8.(1941),303～312.
[6] Friedrichs K O, Stoker J J. The non-linear boundary value problem of the buckled plate.. Proc.N at Acad.Sci U S A.25(1939),532～540.
[7] Koiter W T, Elastic stability and postbuckling behaviorNonliner Problems,pp257～275,Univ of Wisconsin Press.1963.
[8] Reissner E.The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics, 12, 69-77 (1945).
[9]钱伟长.圆板在均布重压下所生之挠曲.物理学报.第2期. 1947: 102～113.
[10] Goodier J N. Cylindrical buckling of sandwich plates.Journal of Applied Mechanics,13(4), 1946.
[11] Reissner E. Finite deflections of sandwich plates.Journal of Aeronautical Science.15 July,1948.
[12] Hoff N J. Bending and buckling of rectangular sandwich plates. NASA Technical Note 2225,Nov. 1950.
[13]杜庆华.矩形三合板的稳定问题,物理学报.第11卷第三期. 1955:239～258.
[14] Donnell L H,Wan C C.Effect of imperfections on buckling of thin cylinders and columus under axial compression.J.Appl.March.17(1950),73～83.
[15] Fung Y C,Wittrick W H.A boundary layer phenomenon in the large deflection of thin plate.Quart,J Mech Appl Math.8,Part 2(1955),191～210.
[16] Stein M.The effect on the buckling of perfect cylinders of prebuckling deformations and stresses induced by edge support.NASA TN D1510,1962
[17]罗祖道,吴连元.弹性圆柱薄壳的一般稳定性.力学学报.第5卷第1期.1962:39～57.
[18]施振东.用弹性薄板广义变分原理解边界支承矩形板在均匀压力作用下的屈曲问题.力学学报第7卷第1期.1964: 39～57.
[19] B.Sekhar Reddy,R.S.Alwar.Post-buckling analysis of orthotropic annular plates. Acta Mechanica 39,289～296(1981).
[20]朱正佑,程昌钧.关于开孔薄板大挠度问题的一般数学理论.力学学报第18卷第2期.1986: 123～135.
[21]沈惠申,张建武.单向压缩简支矩形板后屈曲分析摄动分析.应用力学学报.第9卷第8期.1988: 741～752.
[22] Jean-Pierre Puel,Annie Raoult.Buckling for an elastoplastic plate with an incremental constitutive relation.Appl Math Optim 19:157～185(1989).
[23]陈建康,王汝鹏.刚性路面板热胀屈曲的理论分析.上海力学.第13卷第2期.1992: 65～72.
[24]黄保和,朱照宏.水泥混凝土路面的热胀屈曲稳定性分析.同济大学学报.1984:62～75.
[25]王汝鹏,陈建康.刚性路面热胀屈曲的实验分析.上海力学.第12卷第3期.1991:55～62.
[26]陈建康,王汝鹏.无胀缝坡道路面板热胀屈曲的摄动解.应用数学和力学第14卷第3期, 1993: 259～266.
[27]陈建康,杨鼎久,陆永泉.刚性路面板单体性热屈曲的几何非线性分析.江苏农学院学报.第16卷第4期.1995: 47～51.
[28]沈惠申,林忠钦.弹性基础上矩形板热后屈曲分析.第13卷第1期.1996: 66～74.
[29]程选生等.四边简支钢筋混凝土矩形薄板等热屈曲.工程力学.第24卷第4期.2007: 1～6.
[30]刘鸿文.板壳理论.浙江大学出版社,1987年.
[31]徐芝纶.弹性力学(下).第四版.高等教育出版社.2006年.
[32]邓学钧等编.路面设计原理与方法.人民交通出版社.2001年.
[33]徐次达.加权残数法解固体力学问题.力学与实践.第2卷第4期.1980:12～20.
[34]丁浩江,童晓华,范本隽,谢贻权,用最小二乘法解弹性力学平面问题.合肥工业大学学报.1983年第三期:125～131.
[35]朱照宏等.路面力学计算.人民交通出版社.1985.
[36] [苏]H.A.梅特万特柯娃,B.H.勒杨浦辛.路面设计.人民交通出版社.1987年.
[37] Timoshenko S P , Gere J M.Theory of elastic stability.2ed,McGraw-Hill,New York.1961;346～429.
[38] Timoshenko S P,Goodier J N.Theory of elasticity.3ed,McGraw-Hill,New York.1970.
[39]仇定良,刘麟闻,陈江瑛.混凝土路面板热屈曲的临界变温.宁波大学学报(理工版),第22卷第1期,2009: 120～126.
[2] Timoshenko S P. History of strength of materials.McGraw-Hill,New York.1953.
[3] Donnell L H. A new theory for the buckling of thin cylinders under axial compression and bending.Trans ASME.56,12(1934),795～806.
[4] Von Karman T, Tsien H S. The buckling of spherical shells on external pressure.J.Aero Sci.7 (1939),43～50.
[5] Von Karman T, Tsien H S. The buckling of thin cylindrical shells under axial compression.J. Aero Sci 8.(1941),303～312.
[6] Friedrichs K O, Stoker J J. The non-linear boundary value problem of the buckled plate.. Proc.N at Acad.Sci U S A.25(1939),532～540.
[7] Koiter W T, Elastic stability and postbuckling behaviorNonliner Problems,pp257～275,Univ of Wisconsin Press.1963.
[8] Reissner E.The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics, 12, 69-77 (1945).
[9]钱伟长.圆板在均布重压下所生之挠曲.物理学报.第2期. 1947: 102～113.
[10] Goodier J N. Cylindrical buckling of sandwich plates.Journal of Applied Mechanics,13(4), 1946.
[11] Reissner E. Finite deflections of sandwich plates.Journal of Aeronautical Science.15 July,1948.
[12] Hoff N J. Bending and buckling of rectangular sandwich plates. NASA Technical Note 2225,Nov. 1950.
[13]杜庆华.矩形三合板的稳定问题,物理学报.第11卷第三期. 1955:239～258.
[14] Donnell L H,Wan C C.Effect of imperfections on buckling of thin cylinders and columus under axial compression.J.Appl.March.17(1950),73～83.
[15] Fung Y C,Wittrick W H.A boundary layer phenomenon in the large deflection of thin plate.Quart,J Mech Appl Math.8,Part 2(1955),191～210.
[16] Stein M.The effect on the buckling of perfect cylinders of prebuckling deformations and stresses induced by edge support.NASA TN D1510,1962
[17]罗祖道,吴连元.弹性圆柱薄壳的一般稳定性.力学学报.第5卷第1期.1962:39～57.
[18]施振东.用弹性薄板广义变分原理解边界支承矩形板在均匀压力作用下的屈曲问题.力学学报第7卷第1期.1964: 39～57.
[19] B.Sekhar Reddy,R.S.Alwar.Post-buckling analysis of orthotropic annular plates. Acta Mechanica 39,289～296(1981).
[20]朱正佑,程昌钧.关于开孔薄板大挠度问题的一般数学理论.力学学报第18卷第2期.1986: 123～135.
[21]沈惠申,张建武.单向压缩简支矩形板后屈曲分析摄动分析.应用力学学报.第9卷第8期.1988: 741～752.
[22] Jean-Pierre Puel,Annie Raoult.Buckling for an elastoplastic plate with an incremental constitutive relation.Appl Math Optim 19:157～185(1989).
[23]陈建康,王汝鹏.刚性路面板热胀屈曲的理论分析.上海力学.第13卷第2期.1992: 65～72.
[24]黄保和,朱照宏.水泥混凝土路面的热胀屈曲稳定性分析.同济大学学报.1984:62～75.
[25]王汝鹏,陈建康.刚性路面热胀屈曲的实验分析.上海力学.第12卷第3期.1991:55～62.
[26]陈建康,王汝鹏.无胀缝坡道路面板热胀屈曲的摄动解.应用数学和力学第14卷第3期, 1993: 259～266.
[27]陈建康,杨鼎久,陆永泉.刚性路面板单体性热屈曲的几何非线性分析.江苏农学院学报.第16卷第4期.1995: 47～51.
[28]沈惠申,林忠钦.弹性基础上矩形板热后屈曲分析.第13卷第1期.1996: 66～74.
[29]程选生等.四边简支钢筋混凝土矩形薄板等热屈曲.工程力学.第24卷第4期.2007: 1～6.
[30]刘鸿文.板壳理论.浙江大学出版社,1987年.
[31]徐芝纶.弹性力学(下).第四版.高等教育出版社.2006年.
[32]邓学钧等编.路面设计原理与方法.人民交通出版社.2001年.
[33]徐次达.加权残数法解固体力学问题.力学与实践.第2卷第4期.1980:12～20.
[34]丁浩江,童晓华,范本隽,谢贻权,用最小二乘法解弹性力学平面问题.合肥工业大学学报.1983年第三期:125～131.
[35]朱照宏等.路面力学计算.人民交通出版社.1985.
[36] [苏]H.A.梅特万特柯娃,B.H.勒杨浦辛.路面设计.人民交通出版社.1987年.
[37] Timoshenko S P , Gere J M.Theory of elastic stability.2ed,McGraw-Hill,New York.1961;346～429.
[38] Timoshenko S P,Goodier J N.Theory of elasticity.3ed,McGraw-Hill,New York.1970.
[39]仇定良,刘麟闻,陈江瑛.混凝土路面板热屈曲的临界变温.宁波大学学报(理工版),第22卷第1期,2009: 120～126.