高土石坝三维动力稳定分析研究
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摘要
能源问题一直是制约国民经济又好又快发展的一个主要问题。水电能源作为一种清洁能源在我国发展很快。随着西部大开发、西电东送以及南水北调等国家级大工程项目的开工建设,高坝大库的建设越来越多。我国水电能源集中在西部和西南地区,这些地区恰是我国的强震区。在高坝大库中,土石坝占了相当大的比重。而土石坝的边坡和坝基的稳定是大坝安全的基本保证。
虽然极限平衡法有很多优点,但在处理三维稳定问题时,还很不成熟,而有限元强度折减法处理三维问题则要方便很多。本文以FLAC3D为计算平台,选取面板堆石坝和黏土心墙坝两种坝型,运用强度折减法计算不同工况下坝坡的稳定安全系数,并分析坝体最危险滑裂面的位置,具体工作如下：
(1)最危险滑裂面的确定。强度折减计算只能直接给出稳定安全系数,而最危险滑面的准确位置需要进一步分析确定。在现有的一种基于等效塑性应变二维临界滑裂面搜索方法基础上,推广出三维滑裂面的搜索方法。采用强度折减法计算边坡稳定时,处于临界破坏状态下的边坡,其潜在滑裂面是由塑性区中沿竖直方向上等效塑性应变最大的点所组成的。通过这一原理,寻找到这些三维数据点,然后运用NURBS曲面法和最小二乘法对这些数据点进行三维拟合,得到最危险滑裂面。
(2)对均质面板坝运用拟静力法计算其在不同坝高、坝坡、地震烈度以及河谷系数下的稳定安全系数,并分析坝体滑裂面的位置。计算结果表明：坝越高,坝坡越陡、地震烈度越大、岸坡越宽,安全系数越小；安全系数与坝坡高宽比值呈线性关系,与坝高、地震烈度系数、河谷系数等呈非线性关系。根据计算结果,文中给出了这些因素与安全系数之间的经验计算公式。同时发现地震烈度的大小对滑裂面位置有较大的影响,随着地震烈度的增加,滑裂面滑出点位置逐渐向坝顶靠近。
(3)对黏土心墙坝运用拟静力法计算其在不同坝高、坝坡及地震烈度下的稳定安全系数,并分析最危险滑裂面的位置。计算结果表明：坝高、坝坡及地震烈度对安全系数影响规律与面板坝类似,但其滑裂面的滑出点位置大部分都在坝高1／5的位置,滑弧较深。
Energy has been a main problem that restricts the rapid development of the national economy and society. As a kind of clean energy, hydropower develops rapidly in China. Along with the development of the West Regions, West-East electricity transmission project, South-to-North Water Transfer project are under way, there are more and more constructions of high dam and giant reservoir coming up. Our country's hydropower energy is centered mainly in western and southwestern regions, where are coincidentally meizoseismal areas in our country. The rockfill dam takes up a great proportion in high dam. What's more, the basis of dam security is stability analysis of slope.
Although the limit equilibrium method has many advantageous, it is still not mature when dealing with three dimensional stability analysis problems. Compared with it, the finite element-strength reduction technique is more convenient when doing so. FLAC3D analysis software is used as the main calculating tool in this thesis. The strength reduction technique is applied to analyze the factor of safety and the location of the critical slip surface in two kinds of dams, which are concrete faced rockfill dam and clay core wall dam. The work of this thesis is as follows:
(1) The location of the critical slip surface. With the strength reduction technique, only a factor of safety can be obtained directly, and the critical slip surface is usually determined through more analysis. There is a new approach for searching the critical slip surface based on equivalent plastic strain in the two-dimensional slope stability analysis, and this thesis extends this method into three-dimensional case. Once the slope is led to the limit equilibrium state by means of the strength reduction technique, a plastic zone will go through the slope from the toe to the top. Along each of the vertical lines in some planes, the point with the maximum equivalent plastic strain is sought out in this study. All of these points form a functional data. Lastly, these control points could constitute a smooth surface by the NURBS method and the least square method, and then the critical slip surface will be built.
(2) The factor of safety is calculated by means of pseudo-static method under circumstance of different height of dam, gradient of slope, earthquake intensity as well as shape of valley. Of course, the location of critical slip surface is analyzed as well. Through analysis, a result is concluded that the higher dam is, the steeper slope is, the greater earthquake intensity is, the wider valley is and the smaller safety factor is. What's more, the factor of safety has a linear relation with the gradient of slope and a non-linear relation with the other elements. Through the result, an empirical correlation is provided in this thesis. And it's found that earthquake intensity has a great effect on the location of critical slip surface. The location of critical slip surface ascends gradually to dam crest with the increase of earthquake intensity.
(3) A pseudo-static method is also used to study the dynamic stability of clay core wall dam under different circumstances, and the location of critical slip surface is also analyzed. By analyzing the stability of slope, it is known that the effect of height of dam, gradient of slope and earthquake intensity on safety factor is similar to concrete faced rockfill dam, but the location of critical slip surface is deeper, and about 1/5 height of the dam.
虽然极限平衡法有很多优点,但在处理三维稳定问题时,还很不成熟,而有限元强度折减法处理三维问题则要方便很多。本文以FLAC3D为计算平台,选取面板堆石坝和黏土心墙坝两种坝型,运用强度折减法计算不同工况下坝坡的稳定安全系数,并分析坝体最危险滑裂面的位置,具体工作如下：
(1)最危险滑裂面的确定。强度折减计算只能直接给出稳定安全系数,而最危险滑面的准确位置需要进一步分析确定。在现有的一种基于等效塑性应变二维临界滑裂面搜索方法基础上,推广出三维滑裂面的搜索方法。采用强度折减法计算边坡稳定时,处于临界破坏状态下的边坡,其潜在滑裂面是由塑性区中沿竖直方向上等效塑性应变最大的点所组成的。通过这一原理,寻找到这些三维数据点,然后运用NURBS曲面法和最小二乘法对这些数据点进行三维拟合,得到最危险滑裂面。
(2)对均质面板坝运用拟静力法计算其在不同坝高、坝坡、地震烈度以及河谷系数下的稳定安全系数,并分析坝体滑裂面的位置。计算结果表明：坝越高,坝坡越陡、地震烈度越大、岸坡越宽,安全系数越小；安全系数与坝坡高宽比值呈线性关系,与坝高、地震烈度系数、河谷系数等呈非线性关系。根据计算结果,文中给出了这些因素与安全系数之间的经验计算公式。同时发现地震烈度的大小对滑裂面位置有较大的影响,随着地震烈度的增加,滑裂面滑出点位置逐渐向坝顶靠近。
(3)对黏土心墙坝运用拟静力法计算其在不同坝高、坝坡及地震烈度下的稳定安全系数,并分析最危险滑裂面的位置。计算结果表明：坝高、坝坡及地震烈度对安全系数影响规律与面板坝类似,但其滑裂面的滑出点位置大部分都在坝高1／5的位置,滑弧较深。
Energy has been a main problem that restricts the rapid development of the national economy and society. As a kind of clean energy, hydropower develops rapidly in China. Along with the development of the West Regions, West-East electricity transmission project, South-to-North Water Transfer project are under way, there are more and more constructions of high dam and giant reservoir coming up. Our country's hydropower energy is centered mainly in western and southwestern regions, where are coincidentally meizoseismal areas in our country. The rockfill dam takes up a great proportion in high dam. What's more, the basis of dam security is stability analysis of slope.
Although the limit equilibrium method has many advantageous, it is still not mature when dealing with three dimensional stability analysis problems. Compared with it, the finite element-strength reduction technique is more convenient when doing so. FLAC3D analysis software is used as the main calculating tool in this thesis. The strength reduction technique is applied to analyze the factor of safety and the location of the critical slip surface in two kinds of dams, which are concrete faced rockfill dam and clay core wall dam. The work of this thesis is as follows:
(1) The location of the critical slip surface. With the strength reduction technique, only a factor of safety can be obtained directly, and the critical slip surface is usually determined through more analysis. There is a new approach for searching the critical slip surface based on equivalent plastic strain in the two-dimensional slope stability analysis, and this thesis extends this method into three-dimensional case. Once the slope is led to the limit equilibrium state by means of the strength reduction technique, a plastic zone will go through the slope from the toe to the top. Along each of the vertical lines in some planes, the point with the maximum equivalent plastic strain is sought out in this study. All of these points form a functional data. Lastly, these control points could constitute a smooth surface by the NURBS method and the least square method, and then the critical slip surface will be built.
(2) The factor of safety is calculated by means of pseudo-static method under circumstance of different height of dam, gradient of slope, earthquake intensity as well as shape of valley. Of course, the location of critical slip surface is analyzed as well. Through analysis, a result is concluded that the higher dam is, the steeper slope is, the greater earthquake intensity is, the wider valley is and the smaller safety factor is. What's more, the factor of safety has a linear relation with the gradient of slope and a non-linear relation with the other elements. Through the result, an empirical correlation is provided in this thesis. And it's found that earthquake intensity has a great effect on the location of critical slip surface. The location of critical slip surface ascends gradually to dam crest with the increase of earthquake intensity.
(3) A pseudo-static method is also used to study the dynamic stability of clay core wall dam under different circumstances, and the location of critical slip surface is also analyzed. By analyzing the stability of slope, it is known that the effect of height of dam, gradient of slope and earthquake intensity on safety factor is similar to concrete faced rockfill dam, but the location of critical slip surface is deeper, and about 1/5 height of the dam.
引文
[1]贾金生,袁玉兰,李铁洁.2003年中国及世界大坝情况[J].中国水利,2004(13)：25-33.
[2]祁庆和.水工建筑物(第三版)[M],北京：中国水利水电出版社.1997.
[3]杨振怀.中国水利百科全书[M],北京：水利电力出版社.1991.
[4]陈祖煜.土质边坡稳定分析——原理·方法·程序[M].北京：中国水利水电出版社.2003.
[5]孙涛,顾波.边坡稳定性分析方法评述[J].岩土工程界,2002,5(11)：48-50.
[6]周红祖,刘晓辉.边坡稳定分析的原理和方法[J].高等教育研究,2008,25(1)：91-93.
[7]Fellenius W. Calculation of stability of earth dams[C]. Trans 2nd international congress of large dams,1936, (4):445.
[8]Bishop A W. The use of the slip circle in the stability analysis of slopes[J].Geotechnique,1955,5(1):7-17.
[9]Janbu N. Application of composite slip surfaces for stability analysis[C]. Proceedings of European Conference on Stability of Earth Slopes, Sweden,1954,3:43-49.
[10]Spencer E. A method of analysis of the stability of embankments assuming parallel inter-slice forces[J]. Geotechnique,1967,17:11-26.
[11]Morgenstern N R, Price V. The analysis of the stability of general slip surfaces[J].Geotechnique,1965,15(1):79-93.
[12]赵杰.边坡稳定有限元分析方法中若干应用问题研究[D]：(博士学位论文).大连：大连理工大学.2006.
[13]Fredlund D G, Krahn J. Comparison of slope stability method of analysis[J], Canadian Geotechnical Journal,1977,14(3):429-439.
[14]肖武.基于强度折减法和容量增加法的边坡稳定分析及工程研究[D]：(硕士学位论文).南京：河海大学,2005.
[15]Zienkiewicz 0 C, Humpheson C, Lewis R W. Associated and non-associated viscoplasticity and plasticity in soil mechanics[J]. Geotechnique,1975,25(4):671-689.
[16]Griffiths D V, Lane P A. Slope stability analysis by finite elements[J]. Geotechnique 1999,49(3):387-403.
[17]胡柳青,李夕兵,温世游.边坡稳定性研究及其发展趋势[J].矿业研究与开发,2000,20(5)：7-8.
[18]胡军.高土石坝动力稳定分析及加固措施研究[D]：(博士学位论文).大连：大连理工大学.2008.
[19]赵尚毅,郑颖人,时卫民等.用有限元强度折减法求边坡稳定安全系数[J].岩土工程学报,2002,24(3)：343～346.
[20]彭文斌.FLAC 3D实用教程[M].北京：机械工业出版社,2007.
[21]刘波,韩彦辉.FLAC原理、实例与应用指南[M].北京：人民交通出版社,2005.
[22]陈育民,徐鼎平.FLAC/FLAC3D基础与工程实例[M].北京：中国水利水电出版社,2008.
[23]Dawson E M, Roth W H, Drescher A. A slope stability analysis by strength reduction[J].Geotechnique,1999,49(6):835-840.
[24]Ugai K. A Method of Calculation of Total Factor of Safety of Slopes by Elasto-Plastic FEM[J]. Soils and Foundations JGS,1989,29(2):190-195.
[25]赵尚毅,郑颖人,张玉芳.极限分析有限元法讲座——Ⅱ有限元强度折减法中边坡失稳的判据探讨[J].岩土力学,2005,26(2)：332-336.
[26]宋二祥.土工结构安全系数的有限元计算[J].岩上工程学报,1997,19(2)：1-7.
[27]栾茂田,武亚军,年延凯.强度折减有限元法中边坡失稳的塑性区判据及其应用[J].防灾减灾工程学报,2003,23(3)：1-8.
[28]连镇营,韩国城,孔宪京.强度折减有限元法研究开挖边坡的稳定性[J].岩土工程学报,2001,23(4)：407-411.
[29]吕擎峰,殷宗泽,王叔华,等.拟静力法边坡稳定分析的改进[J].岩土力学,2005,26(21)：35-38.
[30]栾茂田,李湛,范庆来.土石坝拟静力抗震稳定性分析与坝坡地震滑移量估算[J].岩土力学,2007,28(2)：224-230／236.
[31]左秀泓.关于拟静力法分析土石坝地震稳定性的问题[J].水利水电技术,1991(7)：11-16.
[32]中华人民共和国电力行业标准.水工建筑物抗震设计规范(DL5073-2000)[S].2000.
[33]钱家欢,殷宗泽.土工原理与计算[M].北京：中国水利水电出版社,1995.
[34]夏元友,李梅.边坡稳定性评价方法研究及发展趋势[J].岩石力学与工程学报,2002,21(7)：34-38.
[35]张鲁渝,欧阳小秀,郑颖人.国内岩土边坡稳定分析软件面临的问题及几点思考[J].岩石力学与工程学报,2003,22(1)：166-169.
[36]吕擎峰,殷宗泽.非线性强度参数对高土石坝坝坡稳定性的影响[J].岩石力学与工程学报,2004,23(16)：2708-2711.
[37]陈立宏,陈祖煜.堆石非线性强度特性对高土石坝稳定性的影响[J].岩土力学,2007(28)：1807-1810.
[38]张启岳.用大型三轴仪测定砂砾石料和堆石料的抗剪强度[J].水利水运科学研究,1980，(1)：25-38.
[39]Indraratna B, Wijewaedena L S S, Balasubramaniam A S. Large-scale triaxial testing of grewacke rockfill[J]. Geotechnique,1993,43(1):37-51.
[40]柏树田,崔亦吴.堆石的力学性质[J].水利发电学报,1997,3：21-30.
[41]陈立宏,陈祖煜.堆石非线性强度特性对高土石坝稳定性的影响[J].岩土力学,2007(28)：1807-1810.
[42]Duncan J M, Byme P M, Wong K S. Strength, stress-strain and bulk modulus parameters for finite element-analysis of stress and movements in soil masses[R]. Report. No. VCB/GT/78-02, Berkeley:University of Califomia,1978.
[43]中华人民共和国行业标准.碾压式土石坝设计规范(SL274-2001)[S].2001.
[44]Liu Jun, Xu Chunlei. Parameter Statistic and Reliability Analysis of Rock-fill Dam's Slope Stability[C]. The 3rd International Geotechnical Engineering Symposium on Disaster Prevention and Reduction, IGS2009, July 22-25, Harbin, China, PN232.
[45]颜庆津.数值分析(第3版)[M],北京：北京航空航天大学出版社,2006.
[46]Cavounidis S. On the ratio of factors of safety in slope stability analyses[J]. Geotechnique,1987,37(2):207-210.
[47]Naylor D. J. Finite elements and slope stability[C]. Proceedings of the NATO Advanced Study Institute, Lisbon, Portugal. Numer. Methods Geomech.1982:229-244.
[48]Donald I.B., Giam S. K. Application of the nodal displacement method to slope stability analysis[C]. In:Proceedings of the 5th Australia-New Zealand conference on geomechanics, Sydney, Australia,1988:456-460.
[49]Matsui T, San K. C. Finite element slope stability analysis by shear strength reduction technique [J]. Soils Foundations,1992,32(1):59-70.
[50]Ugai K, Leschinsky D. Three-dimensional limit equilibrium and finite element analyses:a comparison of results[J]. Soils Foundations,1995,29(4):1-7.
[51]Song E. Finite element analysis of safety factor for soil structures [J]. Chinese J Geotech Eng,1997,19(2):1-7.
[52]Hong Zheng, Guanhua Sun, Defu Liu. A practical procedure for searching critical slip surfaces of slopes based on the strength reduction technique" [J]. Comput & Geotech,2009,36:1-5.
[53]徐士良.C常用算法程序集(第三版)[M],北京：清华大学出版社,2004.
[54]Cheng Y. M., Liu H. T.,Wei W. B., Au S. K. Location of critical three-dimensional nonspherical failure surface by nurbs functions and ellipsoid with applications to highway slopes[J]. Comput Geotech,2005,32:387-399.
[55]Les P, Wayne T. The NURBS book. second ed[M]. Berlin:Springer Verlag,1997.
[56]ZHANG Xing. Three-dimensional stability analysis of concave slopes in plan view[J]. Journal of Geotechnical Engineering,1988,114(6):658-671.
[57]孔宪京,邹德高,邓学晶,刘莹光.高土石坝综合抗震措施及其效果的验算[J].水利学报,2006,37(12)：1489-1495.
[58]毛文娟.强震区高土石坝三维动力反应分析[D]：(硕士学位论文).大连：大连理工大学.2008.
[2]祁庆和.水工建筑物(第三版)[M],北京：中国水利水电出版社.1997.
[3]杨振怀.中国水利百科全书[M],北京：水利电力出版社.1991.
[4]陈祖煜.土质边坡稳定分析——原理·方法·程序[M].北京：中国水利水电出版社.2003.
[5]孙涛,顾波.边坡稳定性分析方法评述[J].岩土工程界,2002,5(11)：48-50.
[6]周红祖,刘晓辉.边坡稳定分析的原理和方法[J].高等教育研究,2008,25(1)：91-93.
[7]Fellenius W. Calculation of stability of earth dams[C]. Trans 2nd international congress of large dams,1936, (4):445.
[8]Bishop A W. The use of the slip circle in the stability analysis of slopes[J].Geotechnique,1955,5(1):7-17.
[9]Janbu N. Application of composite slip surfaces for stability analysis[C]. Proceedings of European Conference on Stability of Earth Slopes, Sweden,1954,3:43-49.
[10]Spencer E. A method of analysis of the stability of embankments assuming parallel inter-slice forces[J]. Geotechnique,1967,17:11-26.
[11]Morgenstern N R, Price V. The analysis of the stability of general slip surfaces[J].Geotechnique,1965,15(1):79-93.
[12]赵杰.边坡稳定有限元分析方法中若干应用问题研究[D]：(博士学位论文).大连：大连理工大学.2006.
[13]Fredlund D G, Krahn J. Comparison of slope stability method of analysis[J], Canadian Geotechnical Journal,1977,14(3):429-439.
[14]肖武.基于强度折减法和容量增加法的边坡稳定分析及工程研究[D]：(硕士学位论文).南京：河海大学,2005.
[15]Zienkiewicz 0 C, Humpheson C, Lewis R W. Associated and non-associated viscoplasticity and plasticity in soil mechanics[J]. Geotechnique,1975,25(4):671-689.
[16]Griffiths D V, Lane P A. Slope stability analysis by finite elements[J]. Geotechnique 1999,49(3):387-403.
[17]胡柳青,李夕兵,温世游.边坡稳定性研究及其发展趋势[J].矿业研究与开发,2000,20(5)：7-8.
[18]胡军.高土石坝动力稳定分析及加固措施研究[D]：(博士学位论文).大连：大连理工大学.2008.
[19]赵尚毅,郑颖人,时卫民等.用有限元强度折减法求边坡稳定安全系数[J].岩土工程学报,2002,24(3)：343～346.
[20]彭文斌.FLAC 3D实用教程[M].北京：机械工业出版社,2007.
[21]刘波,韩彦辉.FLAC原理、实例与应用指南[M].北京：人民交通出版社,2005.
[22]陈育民,徐鼎平.FLAC/FLAC3D基础与工程实例[M].北京：中国水利水电出版社,2008.
[23]Dawson E M, Roth W H, Drescher A. A slope stability analysis by strength reduction[J].Geotechnique,1999,49(6):835-840.
[24]Ugai K. A Method of Calculation of Total Factor of Safety of Slopes by Elasto-Plastic FEM[J]. Soils and Foundations JGS,1989,29(2):190-195.
[25]赵尚毅,郑颖人,张玉芳.极限分析有限元法讲座——Ⅱ有限元强度折减法中边坡失稳的判据探讨[J].岩土力学,2005,26(2)：332-336.
[26]宋二祥.土工结构安全系数的有限元计算[J].岩上工程学报,1997,19(2)：1-7.
[27]栾茂田,武亚军,年延凯.强度折减有限元法中边坡失稳的塑性区判据及其应用[J].防灾减灾工程学报,2003,23(3)：1-8.
[28]连镇营,韩国城,孔宪京.强度折减有限元法研究开挖边坡的稳定性[J].岩土工程学报,2001,23(4)：407-411.
[29]吕擎峰,殷宗泽,王叔华,等.拟静力法边坡稳定分析的改进[J].岩土力学,2005,26(21)：35-38.
[30]栾茂田,李湛,范庆来.土石坝拟静力抗震稳定性分析与坝坡地震滑移量估算[J].岩土力学,2007,28(2)：224-230／236.
[31]左秀泓.关于拟静力法分析土石坝地震稳定性的问题[J].水利水电技术,1991(7)：11-16.
[32]中华人民共和国电力行业标准.水工建筑物抗震设计规范(DL5073-2000)[S].2000.
[33]钱家欢,殷宗泽.土工原理与计算[M].北京：中国水利水电出版社,1995.
[34]夏元友,李梅.边坡稳定性评价方法研究及发展趋势[J].岩石力学与工程学报,2002,21(7)：34-38.
[35]张鲁渝,欧阳小秀,郑颖人.国内岩土边坡稳定分析软件面临的问题及几点思考[J].岩石力学与工程学报,2003,22(1)：166-169.
[36]吕擎峰,殷宗泽.非线性强度参数对高土石坝坝坡稳定性的影响[J].岩石力学与工程学报,2004,23(16)：2708-2711.
[37]陈立宏,陈祖煜.堆石非线性强度特性对高土石坝稳定性的影响[J].岩土力学,2007(28)：1807-1810.
[38]张启岳.用大型三轴仪测定砂砾石料和堆石料的抗剪强度[J].水利水运科学研究,1980，(1)：25-38.
[39]Indraratna B, Wijewaedena L S S, Balasubramaniam A S. Large-scale triaxial testing of grewacke rockfill[J]. Geotechnique,1993,43(1):37-51.
[40]柏树田,崔亦吴.堆石的力学性质[J].水利发电学报,1997,3：21-30.
[41]陈立宏,陈祖煜.堆石非线性强度特性对高土石坝稳定性的影响[J].岩土力学,2007(28)：1807-1810.
[42]Duncan J M, Byme P M, Wong K S. Strength, stress-strain and bulk modulus parameters for finite element-analysis of stress and movements in soil masses[R]. Report. No. VCB/GT/78-02, Berkeley:University of Califomia,1978.
[43]中华人民共和国行业标准.碾压式土石坝设计规范(SL274-2001)[S].2001.
[44]Liu Jun, Xu Chunlei. Parameter Statistic and Reliability Analysis of Rock-fill Dam's Slope Stability[C]. The 3rd International Geotechnical Engineering Symposium on Disaster Prevention and Reduction, IGS2009, July 22-25, Harbin, China, PN232.
[45]颜庆津.数值分析(第3版)[M],北京：北京航空航天大学出版社,2006.
[46]Cavounidis S. On the ratio of factors of safety in slope stability analyses[J]. Geotechnique,1987,37(2):207-210.
[47]Naylor D. J. Finite elements and slope stability[C]. Proceedings of the NATO Advanced Study Institute, Lisbon, Portugal. Numer. Methods Geomech.1982:229-244.
[48]Donald I.B., Giam S. K. Application of the nodal displacement method to slope stability analysis[C]. In:Proceedings of the 5th Australia-New Zealand conference on geomechanics, Sydney, Australia,1988:456-460.
[49]Matsui T, San K. C. Finite element slope stability analysis by shear strength reduction technique [J]. Soils Foundations,1992,32(1):59-70.
[50]Ugai K, Leschinsky D. Three-dimensional limit equilibrium and finite element analyses:a comparison of results[J]. Soils Foundations,1995,29(4):1-7.
[51]Song E. Finite element analysis of safety factor for soil structures [J]. Chinese J Geotech Eng,1997,19(2):1-7.
[52]Hong Zheng, Guanhua Sun, Defu Liu. A practical procedure for searching critical slip surfaces of slopes based on the strength reduction technique" [J]. Comput & Geotech,2009,36:1-5.
[53]徐士良.C常用算法程序集(第三版)[M],北京：清华大学出版社,2004.
[54]Cheng Y. M., Liu H. T.,Wei W. B., Au S. K. Location of critical three-dimensional nonspherical failure surface by nurbs functions and ellipsoid with applications to highway slopes[J]. Comput Geotech,2005,32:387-399.
[55]Les P, Wayne T. The NURBS book. second ed[M]. Berlin:Springer Verlag,1997.
[56]ZHANG Xing. Three-dimensional stability analysis of concave slopes in plan view[J]. Journal of Geotechnical Engineering,1988,114(6):658-671.
[57]孔宪京,邹德高,邓学晶,刘莹光.高土石坝综合抗震措施及其效果的验算[J].水利学报,2006,37(12)：1489-1495.
[58]毛文娟.强震区高土石坝三维动力反应分析[D]：(硕士学位论文).大连：大连理工大学.2008.