考虑水平摩阻的Pasternak路基基层及路面变形分析
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摘要
为了分析实际工程中路面结构在各类情况下的变形特性,采用Timoshenko梁模型模拟路面及基层特性,Pasternak地基模型模拟路基特性,考虑路面基层间的水平摩阻、路面和基层梁单元的形心轴向位移、基层材料与路基土的非线性应力-应变关系,根据弹性体的虚功原理列出虚功方程,离散后得到单元切向刚度矩阵,运用有限元方法进行组装并采用迭代法进行求解,得到路面结构在不同参数作用下的变形结果。分析基层厚度,模拟水平制动荷载、水平摩阻系数及其他参数对路面结构变形的影响。计算结果表明:如果不考虑基层作用和水平摩阻,采用线性Winkler地基模型计算所得结果与已有文献的计算结果非常接近,验证了该模型与计算方法的正确性;加厚基层厚度能抑制路面沉降及弯曲变形,路面与基层间的摩阻能减小路面沉降,两者都会增大形心轴向位移,水平摩阻的影响不能忽略;模拟水平制动荷载会引起路面微小的转动并从而引起路面结构的变形;增大地基极限承载力能减小路面沉降,一定程度上能抑制路面的弯曲及轴向变形,但效果不显著;增大基层中的剪切作用能有效减小路面沉降及弯曲变形,基层中最终剪应力的大小对路面形心轴向位移的影响尤为明显。
To analyze pavement deformation under different working conditions, the pavement and subbase were simplified as finite-length Timoshenko beams. Considering the horizontal friction, axial displacements, and nonlinear stress-strain relationships, the virtual work equation tangent stiffness matrix was deduced based on the virtual work principle. The finite element and Newton-Raphson iteration methods were used to solve the equation. If horizontal friction, nonlinear relationships, and the subbase were ignored, the foundation was degenerated into a classical Winkler foundation. A very good agreement was achieved by comparing the results of this study and reference 11. Then pavement deformations caused by different subbase thicknesses, horizontal loads, horizontal friction, and other parameters were analyzed. The results show that deflection and bending deformation of a pavement are reduced by increasing the subbase thickness, while friction and axial displacement increase. Moreover, horizontal friction could not be ignored. Rotations of the cross sections and deflection of pavement are caused by the horizontal load. A decrease in deflection is caused by increasing the ultimate bearing capacity of the foundation. Axial displacement and bending deformation of the pavement are also reduced, but it is not as apparent. The shearing stress in the subbase has a great effect on pavement deformation, and the effect of ultimate shear resistance on axial displacement is especially obvious.
To analyze pavement deformation under different working conditions, the pavement and subbase were simplified as finite-length Timoshenko beams. Considering the horizontal friction, axial displacements, and nonlinear stress-strain relationships, the virtual work equation tangent stiffness matrix was deduced based on the virtual work principle. The finite element and Newton-Raphson iteration methods were used to solve the equation. If horizontal friction, nonlinear relationships, and the subbase were ignored, the foundation was degenerated into a classical Winkler foundation. A very good agreement was achieved by comparing the results of this study and reference 11. Then pavement deformations caused by different subbase thicknesses, horizontal loads, horizontal friction, and other parameters were analyzed. The results show that deflection and bending deformation of a pavement are reduced by increasing the subbase thickness, while friction and axial displacement increase. Moreover, horizontal friction could not be ignored. Rotations of the cross sections and deflection of pavement are caused by the horizontal load. A decrease in deflection is caused by increasing the ultimate bearing capacity of the foundation. Axial displacement and bending deformation of the pavement are also reduced, but it is not as apparent. The shearing stress in the subbase has a great effect on pavement deformation, and the effect of ultimate shear resistance on axial displacement is especially obvious.
引文
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[15] 赵明华,张玲,马缤辉,等.考虑水平摩阻力的弹性地基梁非线性分析[J].岩土工程学报,2009,31(7):985-990.ZHAO Ming-hua,ZHANG Ling,MA Bin-hui,et al.Nonlinear Analysis of Beams on Elastic Foundation with Consideration of Horizontal Resistance [J].Chinese Journal of Geotechnical Engineering,2009,31 (7):985-990.
[16] 周慧,罗松南,孙丹.考虑水平摩阻力的弹性地基梁大变形弯曲分析[J].工程力学,2011,28(1):43-47,54.ZHOU Hui,LUO Song-nan,SUN Dan.The Bending Analysis of a Beam on Elastic Foundation with Large Deflection Including the Effects of Horizontal Friction [J].Engineering Mechanics,2011,28 (1):43-47,54.
[17] LANZONI L,RAD E.A Loaded Timoshenko Beam Bonded to An Elastic Half Plane [J].International Journal of Solids and Structures,2016,92-93:76-90.
[18] GHOSH C,MADHA V.Settlement Response of a Reinforced Shallow Earth Bed[J].Geotextiles and Geomembranes,1994,13 (9):643-656.
[19] KONDNER R L.Hyperbolic Stress-strain Response:Cohesive Soils [J].Journal of Soil Mechanics and Foundation Engineering,1963,89 (SM1):115-143.
[20] 李晓伟,何光辉.Pasternak地基中Levinson梁的静力响应与动力特性[J].力学季刊,2016,37(4):710-720.LI Xiao-wei,HE Guang-hui.Static Response and Dynamic Characteristics of Levinson Beam on Pasternak Foundation [J].Chinese Quarterly of Mechanics,2016,37 (4):710-720.
[2] 艾智勇,蔡建邦.层状地基上弹性地基梁的有限元-边界元耦合分析[J].岩土力学,2015,36(增2):685-688.AI Zhi-yong,CAI Jian-bang.FEM/BEM Coupling Analysis of Elastic Foundation Beam on Layered Soils[J].Rock and Soil Mechanics,2015,36 (S2):685-688.
[3] JANG T S,HONG G S.A New Semi-analytical Method for the Non-linear Static Analysis of an Infinite Beam on a Non-linear Elastic Foundation [J].International Journal of Non-linear Mechanics,2012,47 (4):132-139.
[4] MAHESHWARI P,VILADKAR M N.A Mathematical Model for Beams on Geosynthetic Reinforced Earth Beds Under Strip Loading [J].Applied Mathematical Modelling,2009,33 (4):1803-1814.
[5] MAHESHWARI P,SHUBHA K.Nonlinear Analysis of Infinite Beams on Granular Bed-stone Column-Reinforced Earth Beds Under Moving Loads [J].Soils and Foundations,2012,52 (1):114-125.
[6] 冯又全,杨敏,陈俊岭.线性分布基床系数弹性地基梁有限单元法改进[J].岩土力学,2014,35(10):3027-3034.FENG You-quan,YANG Min,CHEN Jun-ling.Improvement of FEM for Beam on Elastic Foundation with Linearly Distributed Coefficient of Subgrade Reaction [J].Rock and Soil Mechanics,2014,35 (10):3027-3034.
[7] TEZZON E,TULLINI N,LANZONI L.A Coupled FE-BIE Model for the Static Analysis of Timoshenko Beams Bonded to an Orthotropic Elastic Half-plane[J].Engineering Analysis with Boundary Elements,2016,71:112-128.
[8] AI Z Y,CAI J B.Static Interaction Analysis Between a Timoshenko Beam and Layered Soils by Analytical Layer Element/Boundary Element Method Coupling[J].Applied Mathematical Modelling,2016,40 (21/22):9485-9499.
[9] 李潇,王宏志,李世萍,等.解析型Winkler弹性地基梁单元构造[J].工程力学,2015,32(3):66-72.LI Xiao,WANG Hong-zhi,LI Shi-ping,et al.Element for Beam on Winkler Elastic Foundation Based on Analytical Trial Functions [J].Engineering Mechanics,2015,32 (3):66-72.
[10] SHIRIMA L M,GIGER W M.Timoshenko Beam Element Resting on Two-parameter Elastic Foundation [J].Journal of Engineering Mechanics,1992,118 (2):180-195.
[11] 夏桂云,俞茂宏,李传习,等.考虑地基水平摩阻的Winkler地基Timoshenko梁分析[J].土木工程学报,2011,44(6):98-104.XIA Gui-yun,YU Mao-hong,LI Chuan-xi,et al.Analysis of Timoshenko Beam on Winkler Foundation with Foundation Horizontal Friction [J].China Civil Engineering Journal,2011,44 (6):98-104.
[12] 何芳社,郭春霞.考虑地基摩阻时Timoshenko梁弯曲的Fourier级数解答[J].力学季刊,2012,33(3):487-491.HE Fang-she,GUO Chun-xia.The Fourier Series Solutions of Timoshenko Beam on Elastic Foundation Considering Longitudinal Friction Resistances[J].Chinese Quarterly of Mechanics,2012,33 (3):487-491.
[13] DENG H,CHEN K D,CHENG W,et al.Vibration and Buckling Analysis of Double-functionally Graded Timoshenko Beam System on Winkler-Pasternak Elastic Foundation [J].Composite Structures,2016,160:152-168.
[14] MORFIDIS K.Vibration of Timoshenko Beams on Three-parameter Elastic Foundation [J].Computers and Structures,2010,88 (5/6):294-308.
[15] 赵明华,张玲,马缤辉,等.考虑水平摩阻力的弹性地基梁非线性分析[J].岩土工程学报,2009,31(7):985-990.ZHAO Ming-hua,ZHANG Ling,MA Bin-hui,et al.Nonlinear Analysis of Beams on Elastic Foundation with Consideration of Horizontal Resistance [J].Chinese Journal of Geotechnical Engineering,2009,31 (7):985-990.
[16] 周慧,罗松南,孙丹.考虑水平摩阻力的弹性地基梁大变形弯曲分析[J].工程力学,2011,28(1):43-47,54.ZHOU Hui,LUO Song-nan,SUN Dan.The Bending Analysis of a Beam on Elastic Foundation with Large Deflection Including the Effects of Horizontal Friction [J].Engineering Mechanics,2011,28 (1):43-47,54.
[17] LANZONI L,RAD E.A Loaded Timoshenko Beam Bonded to An Elastic Half Plane [J].International Journal of Solids and Structures,2016,92-93:76-90.
[18] GHOSH C,MADHA V.Settlement Response of a Reinforced Shallow Earth Bed[J].Geotextiles and Geomembranes,1994,13 (9):643-656.
[19] KONDNER R L.Hyperbolic Stress-strain Response:Cohesive Soils [J].Journal of Soil Mechanics and Foundation Engineering,1963,89 (SM1):115-143.
[20] 李晓伟,何光辉.Pasternak地基中Levinson梁的静力响应与动力特性[J].力学季刊,2016,37(4):710-720.LI Xiao-wei,HE Guang-hui.Static Response and Dynamic Characteristics of Levinson Beam on Pasternak Foundation [J].Chinese Quarterly of Mechanics,2016,37 (4):710-720.