软土流变模型及其工程应用
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摘要
软土是一种极其复杂的工程材料,其特殊的物质成分和复杂的结构决定了软土复杂的工程特性。在传统的土力学研究中,土被假定为弹性体或者弹塑性体,然而,无论是线性弹性体还是非线性的弹塑性体,其本构关系都是应力与应变两者之间的关系,与时间无关。而事实上,自然界土体的应力应变关系与时间紧密相关,因而在土体的本构关系中加入时间因素极为必要。这种包括应力、应变和时间三者在内的本构关系就是基于粘弹塑性理论而建立的土的流变力学。越来越多的工程实例表明,岩土很多工程的破坏都跟软土流变有关。因而,研究软土的流变特性具有重要的理论价值和实践意义。
正如土的本构关系在土力学中的重要性,土的流变本构关系也是土流变研究的关键所在。因此,本文首先总结了以往研究软土流变本构模型的各种理论,分析了理论模型和经验模型的构造及其优缺点。针对软土流变的非线性,建立一个简化的半理论半经验的流变本构组合模型,用理论模型描述流变的线性部分,用经验模型描述软土流变的非线性部分,并把它应用于上海淤泥质粘土的流变分析中。其次,引入经典的BP神经网络,利用人工神经网络超强的非线性映射能力和容错能力,根据广东江门软土蠕变试验结果,建立广东江门软土流变的网络本构模型,既克服了以往分别按各分级载荷下蠕变试验数据拟和出蠕变本构模型使用不便的缺点,又克服了蠕变本构关系式难以合理构造的困难,并通过上海软土蠕变试验的进一步验证,具有可考虑因素多、拟合精度高、泛化预测能力强、适应性好等特点。最后,针对目前路基沉降计算和预测存在的困难和缺陷,引入了考虑流变的有限元计算方法,通过MSC.Marc/Mentat有限元软件,对杭州绕城高速公路的路基沉降进行了考虑流变和不考虑流变两种方案的有限元分析。结果表明,考虑软土流变的计算方案更符合路基实际沉降规律,在软土地基沉降计算中,特别是工后沉降,考虑软土的流变特性是非常必要的。
Soft clay is an extremely complex engineering material, it has complex engineering properties for specific substance components and complex structure. In the traditional research of soil mechanics, soil is generally presumed to be elastic or elastic-plastic body. But whether to be linear elastic or to be nonlinear plastic, its constitutional relation is only the function of stress and strain and has nothing to do with the time. In fact, the stress and strain of soil in nature is closely related to time, so the time element is necessary to be included in the constitutional relation of soft clay. Based on the elastic-viscoplastic theory, the constitutional relation which contains stress, strain and time is called rheological mechanics of soil. An increasing number of projects show that many geotechnical engineering failures are related to rheology of soft clay, therefore it has great theoretical value and practical meaning to study the rheological properties of soft clay.
As the importance of constitutive relation of soil on studying soil mechanics, the rheological constitutive relation is also important for studying soil rheology. Therefore all kinds of traditional theories of studying the rheological constitutive model of soil are systematically summarized in the text firstly, then, the merits and demerits of theoretical model and empirical model analyses. According to rheological nonlinearity of soft clay, the text put forward a simplified rheological constitutive model of combining theories with experiences, the theoretical model is used to describe the linear part of rheology and the empirical model is used to describe the nonlinear part of rheology. The model is successfully used to describe the rheology of shanghai muddy clay later. Secondly, introducing the BP neural network and according to the creep data of laboratory direct shear experiment of jiangmen soil, the text build a network rheological constitutive model of jiangmen soil for its strong nonlinear mapping ability and fault-tolerance ability, which can avoid disadvantages of that traditional rheological constitutive models built under each level loading are impractical and a necessary complex constitutive equation is built in order to meeting the fact, then, the model is verified by the direct shear creep experiment of soil of shanghai. it has characteristics of considering more factors, high simulation precision and good generalization ability. Finally, according to difficulties and flaws of calculating and predicting the settlement of subgrade, the paper introduces finite element method which considers rheology of soil. By the MSC.Marc/Mentat software, the settlement of subgrade of loop freeway of Hangzhou is calculated under two schemes, one considers rheology of soft clay, another doesn’t consider rheology of soft clay. The result shows that the calculation scheme of considering rheology of soil is more applicable to practical settlement law of subgrade. In calculating settlement of soft subgrade, especially post-construction settlement, the rheological properties of soft clay is necessary to be considered.
正如土的本构关系在土力学中的重要性,土的流变本构关系也是土流变研究的关键所在。因此,本文首先总结了以往研究软土流变本构模型的各种理论,分析了理论模型和经验模型的构造及其优缺点。针对软土流变的非线性,建立一个简化的半理论半经验的流变本构组合模型,用理论模型描述流变的线性部分,用经验模型描述软土流变的非线性部分,并把它应用于上海淤泥质粘土的流变分析中。其次,引入经典的BP神经网络,利用人工神经网络超强的非线性映射能力和容错能力,根据广东江门软土蠕变试验结果,建立广东江门软土流变的网络本构模型,既克服了以往分别按各分级载荷下蠕变试验数据拟和出蠕变本构模型使用不便的缺点,又克服了蠕变本构关系式难以合理构造的困难,并通过上海软土蠕变试验的进一步验证,具有可考虑因素多、拟合精度高、泛化预测能力强、适应性好等特点。最后,针对目前路基沉降计算和预测存在的困难和缺陷,引入了考虑流变的有限元计算方法,通过MSC.Marc/Mentat有限元软件,对杭州绕城高速公路的路基沉降进行了考虑流变和不考虑流变两种方案的有限元分析。结果表明,考虑软土流变的计算方案更符合路基实际沉降规律,在软土地基沉降计算中,特别是工后沉降,考虑软土的流变特性是非常必要的。
Soft clay is an extremely complex engineering material, it has complex engineering properties for specific substance components and complex structure. In the traditional research of soil mechanics, soil is generally presumed to be elastic or elastic-plastic body. But whether to be linear elastic or to be nonlinear plastic, its constitutional relation is only the function of stress and strain and has nothing to do with the time. In fact, the stress and strain of soil in nature is closely related to time, so the time element is necessary to be included in the constitutional relation of soft clay. Based on the elastic-viscoplastic theory, the constitutional relation which contains stress, strain and time is called rheological mechanics of soil. An increasing number of projects show that many geotechnical engineering failures are related to rheology of soft clay, therefore it has great theoretical value and practical meaning to study the rheological properties of soft clay.
As the importance of constitutive relation of soil on studying soil mechanics, the rheological constitutive relation is also important for studying soil rheology. Therefore all kinds of traditional theories of studying the rheological constitutive model of soil are systematically summarized in the text firstly, then, the merits and demerits of theoretical model and empirical model analyses. According to rheological nonlinearity of soft clay, the text put forward a simplified rheological constitutive model of combining theories with experiences, the theoretical model is used to describe the linear part of rheology and the empirical model is used to describe the nonlinear part of rheology. The model is successfully used to describe the rheology of shanghai muddy clay later. Secondly, introducing the BP neural network and according to the creep data of laboratory direct shear experiment of jiangmen soil, the text build a network rheological constitutive model of jiangmen soil for its strong nonlinear mapping ability and fault-tolerance ability, which can avoid disadvantages of that traditional rheological constitutive models built under each level loading are impractical and a necessary complex constitutive equation is built in order to meeting the fact, then, the model is verified by the direct shear creep experiment of soil of shanghai. it has characteristics of considering more factors, high simulation precision and good generalization ability. Finally, according to difficulties and flaws of calculating and predicting the settlement of subgrade, the paper introduces finite element method which considers rheology of soil. By the MSC.Marc/Mentat software, the settlement of subgrade of loop freeway of Hangzhou is calculated under two schemes, one considers rheology of soft clay, another doesn’t consider rheology of soft clay. The result shows that the calculation scheme of considering rheology of soil is more applicable to practical settlement law of subgrade. In calculating settlement of soft subgrade, especially post-construction settlement, the rheological properties of soft clay is necessary to be considered.
引文
[1] 郭增玉,张朝鹏,夏旺民.高湿度 Q2 黄土的非线性流变本构模型及参数.岩石力学与工程学报,2000,19(6):780-784
[2] 韩爱果,聂德新,任光明等.大型滑坡滑带土剪切流变特性研究.工程地质学报,2001,9(4):345-348
[3] 沈珠江,赵魁芝.堆石坝流变变形的反馈分析.水利学报,1998,(6):1-6
[4] 梁军.堆石坝体流变特性浅析.四川水利,1998,19(2):9-13
[5] Perzyna P. Fundamental problems in viscoplasticity. Advancdes in Appl Mech, 1966, (9):243-248
[6] Singh A and Mitchell J K .General stress-strain-time function for soils. Journal of the Soil Mechanics and Foundations Division, ASCE,1968, 94(1):21-46
[7] 张忠亭,王宏,陶振宇.岩石蠕变特性研究进展概况.长江科学院院报,1996,13(增刊):1-5
[8] 孙钧.岩土材料流变及其工程应用.第一版.北京:中国建筑工业出版社,1999, 10-14
[9] 沈珠江.鲁布革心墙堆石坝变形的反馈分析.岩土工程学报.1994,16(3):l-13
[10] 谢宁,姚海明.土流变研究综述.云南工业大学学报.1999,15(1):52-56
[11] 陈宗基. Structure Mechanics of Clay.中国科学,1959,8(1):41-45
[12] 汤岳飞.非线性流变与固结的藕合分析以及在地下工程中的应用:[同济大学硕士学位论文].上海:同济大学,1999,3-5
[13] 夏耀明,孙逸明,王大龄.饱和软粘土固结、蠕变变形和应力松弛规律.同济大学学报,1989,17(3):4-8
[14] 李希元,周俞明.岩土流变理论中 Boltzmann 积分的一般表达形式.岩土工程师,1996,8(4):1-4
[15] 郑榕明,陆浩亮,孙钧.软土工程中的非线性流变分析.岩土工程学报,1996,18 (5):1-13
[16] Dafalias Y F. On Rate Dependence and Anisotropy in Soil Constitutive Modeling. Paris: Results of the Int. Workshops on Constitutive Relations for Soil,1982, 79-83
[17] 陈冬元.上海地区饱和软粘土蠕变微观机理及其流变模型与工程计算的研究:[同济大学硕士学位论文].上海:同济大学,1993,1-10
[18] Findley W N, Lai J S , Onaran K.Creep and Relaxation of Nonlinear ViscoelasticMaterials.New York: Holland Publishing Company ,1976,164-189
[19] 詹美礼 ,钱家欢 .软土流变特性实验及流变模型 .岩土土程学报 ,1993,15(3): 55-62
[20] Keedwell J K. Rheology and Soil Mechanics. London:Elsevier Applied Science Publishingers ,1984,91-92
[21] Vyalov S S. Rheological Fundamentals of Soil Mechanics. London:Elsevier Applied Science Publishingers ,1986,231-232
[22] Bazant Z P, Ansal A M, Krizek R J. Viscoplasticity of Transversely Isotropic Clays. Journal of Engineering Mechanics Division,ASCE,1979,105(4):1024-1028
[23] 周维垣.高等岩石力学.北京:水利电力出版社,1990,97-105
[24] 范镜泓,高芝晖.非线性连续介质力学基础.重庆:重庆大学出版社,1987,54-62
[25] Cristescu N. A Procedure for Determing the Constitutive Equations for Materials Exhibiting Both Time - dependent and Time – independent plasticity.Int.J .Solids Structures ,1972,(6) :32-36
[26] 黄文熙.土的工程性质.北京:水利电力出版社,1996,2-10
[27] 赵维炳,施建勇.软土固结与流变.南京:河海大学出版社,1996,5-8
[28] 范广勤.岩土工程流变力学.北京:煤炭工业出版社,1992,1-10
[29] [俄] C.C.维亚洛夫.土力学的流变原理.杜余培译.北京:科学出版社,1987,54-60
[30] 王祥秋.软粘土非线性流变分析与模拟理论研究:[中南大学硕士学位论文].长沙:中南大学,1998,54-56
[31] 钱家欢,殷宗泽.土工原理与计算.北京:中国水利水电出版社,1996,44-50
[32] 孙钧,汪炳见.地下结构有限元法解析.上海:同济大学出版社,1988,85-92
[33] 肖燕.软土蠕变特性研究及其在桥台桩基工程中的应用:[湖南大学硕士学位论文].长沙:湖南大学,2004,20-34
[34] 何振亚.神经智能-认知科学中若干重大问题的研究.长沙:湖南科学技术出版社,1997,1-6
[35] 贺清碧.BP 神经网络及应用:[重庆交通学院硕士学位论文].重庆:重庆交通学院工程管理系,2004,1-4
[36] 袁曾任.人工神经网络及其应用.北京:清华大学出版社,1999,32-36
[37] 周开利,康耀红.神经网络模型及其 MATLAB 仿真程序设计.北京:清华大学出版社,2005,6-8
[38] 张立明.人工神经网络的模型及其应用.上海:复旦大学出版社,1993,30-35
[39] 李建中,彭芳乐.黏土的蠕变特性试验研究.岩土力学,2006,27(2):214-218
[40] 飞思科技产品研发中心.神经网络理论与 MATLAB 7 实现.北京:电子工业出版社,2005,100-103
[41] 阎平凡,张长水.人工神经网络与模拟进化计算.北京:清华大学出版社,2000, 80-85
[42] 吴凌云.BP 神经网络学习算法的改进及其应用.信息技术,2003,27(7):42-44
[43] 丛爽.面向 MATLAB 工具箱的神经网络理论与应用.合肥:中国科学技术大学出版社,1998,55-58
[44] 王永骥,涂健.神经元网络控制.北京:机械工业出版社,1998,57-59
[45] 吴建昱,何小荣.用于过程建模的 BP 网络的 M 训练法及其改进.计算机与应用化学,2001,18(5):473-476
[46] Morris P H. Analytical solutions of linear finite-strain one-dimensional consolidation. Journal of Geotechnical and Geoenvironmental Engineering ASCE,2002,128(4):319-326
[47] 殷宗泽.土体沉降与固结.北京:中国电力出版社,1998,15-33
[48] 赵维炳,施建勇.软土固结与流变.南京:河海大学出版社,1966,183-187
[49] Y. K. Chow. Settlement analysis of sand compaction pile, Soils and Foundations, 1996,36(1):111-113
[50] 陈仲颐,周景星,王洪瑾.土力学.北京:清华大学出版社,1994,81-90
[51] 孙更生,郑大同.软土地基与地下工程.北京:中国建筑工业出版社,1984,20-25
[52] Skempton A W and Bjerrum L. A Contribution to the Settlement Analysis of Foundation on Clay. Geotechnique,1957,7 (4):168-178
[53] 陈国荣,姜弘道,高谦.高速公路路基性态反分析及沉降预报.工程地质学报,1998,6(4):340-343
[54] Dennes T B, Casan L S, Balasubramaniam A S. Inverse analysis of geotechnical parameters on improved soft Bangkok clay. Journal of Geotechnical Engineering, 1992,118(7):1012-1029
[55] Arai K, Ohta H, Kojima K. Application of back-analysis to several test embankments on soft clay deposits. Soil and Foundation,1986,26(2):60-72
[56] 朱百里,沈珠江.计算土力学.上海:上海科学技术出版社,1990:42-49
[57] 王志亮.软基路堤沉降预测和计算:[河海大学博士学位论文].南京:河海大学,2004,3-6
[58] AsaokoA. Observational procedure of settlement prediction. Soil and Foundation, 1978,18(4):87-91
[59] 张仪萍,张土桥,龚晓南.沉降的灰色预测.工业建筑,1999,29(4):45-48
[60] 韩汝才,李亮,付鹤林.利用灰色理论对地基沉降进行不等时距预测.探矿工程,2000,(2):4-5
[61] 增超,肖峰,唐仲华.应用灰色模型(GM)预测软土路基沉降量.安全和环境工程,2002,19(1):20-22
[62] 石 世 云 . 多 变 量 灰 色 模 型 MGM(1,n) 在 变 形 预 测 中 的 应 用 . 测 绘 通 报 , 1998,(10):9-12
[63] 谢宁,孙钧.土体非线性流变的有限元解析及其工程应用.岩土工程学报,1995, 17(4):95-99
[64] 曹文贵.多步开挖工程蠕变损伤大变形有限元模拟及工程应用:[北京科技大学博士学位论文].北京:北京科技大学资源与环境工程学院,1998,58-78
[65] 陈火红.Marc 有限元实例分析教程.北京:机械工业出版社,2002,166-169
[66] 俞亚南.粉喷桩加固软土路基试验研究与沉降分析:[南开大学硕士论文].南京:南开大学,2003,15-27
[67] 张敏江,张丽萍,张树标等.结构性软土非线性流变本构关系模型的研究.吉林大学学报,2004,34(2):242-246
[2] 韩爱果,聂德新,任光明等.大型滑坡滑带土剪切流变特性研究.工程地质学报,2001,9(4):345-348
[3] 沈珠江,赵魁芝.堆石坝流变变形的反馈分析.水利学报,1998,(6):1-6
[4] 梁军.堆石坝体流变特性浅析.四川水利,1998,19(2):9-13
[5] Perzyna P. Fundamental problems in viscoplasticity. Advancdes in Appl Mech, 1966, (9):243-248
[6] Singh A and Mitchell J K .General stress-strain-time function for soils. Journal of the Soil Mechanics and Foundations Division, ASCE,1968, 94(1):21-46
[7] 张忠亭,王宏,陶振宇.岩石蠕变特性研究进展概况.长江科学院院报,1996,13(增刊):1-5
[8] 孙钧.岩土材料流变及其工程应用.第一版.北京:中国建筑工业出版社,1999, 10-14
[9] 沈珠江.鲁布革心墙堆石坝变形的反馈分析.岩土工程学报.1994,16(3):l-13
[10] 谢宁,姚海明.土流变研究综述.云南工业大学学报.1999,15(1):52-56
[11] 陈宗基. Structure Mechanics of Clay.中国科学,1959,8(1):41-45
[12] 汤岳飞.非线性流变与固结的藕合分析以及在地下工程中的应用:[同济大学硕士学位论文].上海:同济大学,1999,3-5
[13] 夏耀明,孙逸明,王大龄.饱和软粘土固结、蠕变变形和应力松弛规律.同济大学学报,1989,17(3):4-8
[14] 李希元,周俞明.岩土流变理论中 Boltzmann 积分的一般表达形式.岩土工程师,1996,8(4):1-4
[15] 郑榕明,陆浩亮,孙钧.软土工程中的非线性流变分析.岩土工程学报,1996,18 (5):1-13
[16] Dafalias Y F. On Rate Dependence and Anisotropy in Soil Constitutive Modeling. Paris: Results of the Int. Workshops on Constitutive Relations for Soil,1982, 79-83
[17] 陈冬元.上海地区饱和软粘土蠕变微观机理及其流变模型与工程计算的研究:[同济大学硕士学位论文].上海:同济大学,1993,1-10
[18] Findley W N, Lai J S , Onaran K.Creep and Relaxation of Nonlinear ViscoelasticMaterials.New York: Holland Publishing Company ,1976,164-189
[19] 詹美礼 ,钱家欢 .软土流变特性实验及流变模型 .岩土土程学报 ,1993,15(3): 55-62
[20] Keedwell J K. Rheology and Soil Mechanics. London:Elsevier Applied Science Publishingers ,1984,91-92
[21] Vyalov S S. Rheological Fundamentals of Soil Mechanics. London:Elsevier Applied Science Publishingers ,1986,231-232
[22] Bazant Z P, Ansal A M, Krizek R J. Viscoplasticity of Transversely Isotropic Clays. Journal of Engineering Mechanics Division,ASCE,1979,105(4):1024-1028
[23] 周维垣.高等岩石力学.北京:水利电力出版社,1990,97-105
[24] 范镜泓,高芝晖.非线性连续介质力学基础.重庆:重庆大学出版社,1987,54-62
[25] Cristescu N. A Procedure for Determing the Constitutive Equations for Materials Exhibiting Both Time - dependent and Time – independent plasticity.Int.J .Solids Structures ,1972,(6) :32-36
[26] 黄文熙.土的工程性质.北京:水利电力出版社,1996,2-10
[27] 赵维炳,施建勇.软土固结与流变.南京:河海大学出版社,1996,5-8
[28] 范广勤.岩土工程流变力学.北京:煤炭工业出版社,1992,1-10
[29] [俄] C.C.维亚洛夫.土力学的流变原理.杜余培译.北京:科学出版社,1987,54-60
[30] 王祥秋.软粘土非线性流变分析与模拟理论研究:[中南大学硕士学位论文].长沙:中南大学,1998,54-56
[31] 钱家欢,殷宗泽.土工原理与计算.北京:中国水利水电出版社,1996,44-50
[32] 孙钧,汪炳见.地下结构有限元法解析.上海:同济大学出版社,1988,85-92
[33] 肖燕.软土蠕变特性研究及其在桥台桩基工程中的应用:[湖南大学硕士学位论文].长沙:湖南大学,2004,20-34
[34] 何振亚.神经智能-认知科学中若干重大问题的研究.长沙:湖南科学技术出版社,1997,1-6
[35] 贺清碧.BP 神经网络及应用:[重庆交通学院硕士学位论文].重庆:重庆交通学院工程管理系,2004,1-4
[36] 袁曾任.人工神经网络及其应用.北京:清华大学出版社,1999,32-36
[37] 周开利,康耀红.神经网络模型及其 MATLAB 仿真程序设计.北京:清华大学出版社,2005,6-8
[38] 张立明.人工神经网络的模型及其应用.上海:复旦大学出版社,1993,30-35
[39] 李建中,彭芳乐.黏土的蠕变特性试验研究.岩土力学,2006,27(2):214-218
[40] 飞思科技产品研发中心.神经网络理论与 MATLAB 7 实现.北京:电子工业出版社,2005,100-103
[41] 阎平凡,张长水.人工神经网络与模拟进化计算.北京:清华大学出版社,2000, 80-85
[42] 吴凌云.BP 神经网络学习算法的改进及其应用.信息技术,2003,27(7):42-44
[43] 丛爽.面向 MATLAB 工具箱的神经网络理论与应用.合肥:中国科学技术大学出版社,1998,55-58
[44] 王永骥,涂健.神经元网络控制.北京:机械工业出版社,1998,57-59
[45] 吴建昱,何小荣.用于过程建模的 BP 网络的 M 训练法及其改进.计算机与应用化学,2001,18(5):473-476
[46] Morris P H. Analytical solutions of linear finite-strain one-dimensional consolidation. Journal of Geotechnical and Geoenvironmental Engineering ASCE,2002,128(4):319-326
[47] 殷宗泽.土体沉降与固结.北京:中国电力出版社,1998,15-33
[48] 赵维炳,施建勇.软土固结与流变.南京:河海大学出版社,1966,183-187
[49] Y. K. Chow. Settlement analysis of sand compaction pile, Soils and Foundations, 1996,36(1):111-113
[50] 陈仲颐,周景星,王洪瑾.土力学.北京:清华大学出版社,1994,81-90
[51] 孙更生,郑大同.软土地基与地下工程.北京:中国建筑工业出版社,1984,20-25
[52] Skempton A W and Bjerrum L. A Contribution to the Settlement Analysis of Foundation on Clay. Geotechnique,1957,7 (4):168-178
[53] 陈国荣,姜弘道,高谦.高速公路路基性态反分析及沉降预报.工程地质学报,1998,6(4):340-343
[54] Dennes T B, Casan L S, Balasubramaniam A S. Inverse analysis of geotechnical parameters on improved soft Bangkok clay. Journal of Geotechnical Engineering, 1992,118(7):1012-1029
[55] Arai K, Ohta H, Kojima K. Application of back-analysis to several test embankments on soft clay deposits. Soil and Foundation,1986,26(2):60-72
[56] 朱百里,沈珠江.计算土力学.上海:上海科学技术出版社,1990:42-49
[57] 王志亮.软基路堤沉降预测和计算:[河海大学博士学位论文].南京:河海大学,2004,3-6
[58] AsaokoA. Observational procedure of settlement prediction. Soil and Foundation, 1978,18(4):87-91
[59] 张仪萍,张土桥,龚晓南.沉降的灰色预测.工业建筑,1999,29(4):45-48
[60] 韩汝才,李亮,付鹤林.利用灰色理论对地基沉降进行不等时距预测.探矿工程,2000,(2):4-5
[61] 增超,肖峰,唐仲华.应用灰色模型(GM)预测软土路基沉降量.安全和环境工程,2002,19(1):20-22
[62] 石 世 云 . 多 变 量 灰 色 模 型 MGM(1,n) 在 变 形 预 测 中 的 应 用 . 测 绘 通 报 , 1998,(10):9-12
[63] 谢宁,孙钧.土体非线性流变的有限元解析及其工程应用.岩土工程学报,1995, 17(4):95-99
[64] 曹文贵.多步开挖工程蠕变损伤大变形有限元模拟及工程应用:[北京科技大学博士学位论文].北京:北京科技大学资源与环境工程学院,1998,58-78
[65] 陈火红.Marc 有限元实例分析教程.北京:机械工业出版社,2002,166-169
[66] 俞亚南.粉喷桩加固软土路基试验研究与沉降分析:[南开大学硕士论文].南京:南开大学,2003,15-27
[67] 张敏江,张丽萍,张树标等.结构性软土非线性流变本构关系模型的研究.吉林大学学报,2004,34(2):242-246