基于极限平衡理论的土质边坡空间效应研究及应用
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摘要
边坡空间效应是客观存在的,广义的空间效应包括滑体形态效应和坡面形态效应两部分,由于考虑空间效应的三维边坡稳定性分析方法及最危险滑面搜索技术尚不完善,对空间效应的研究比较零散,未能进行系统的研究；对于相关工程而言,忽略空间效应并非总是偏于安全。因此,课题围绕边坡空间效应问题,从案例统计、理论分析、试验研究和工程应用四个方面对其进行系统的研究。通过研究,取得如下主要研究成果：
(1)从案例统计分析的角度研究了边坡的空间效应的构成与显著度。分析了522个失稳边坡资料,着重对比分析了滑体形态、滑体宽高比及坡面形态的分布,结果表明：①其中82.4%的宽高比位于0.5-5.0之间,平均宽高比为3.08,而宽高比大于10的仅占3.4%；②坡面形态对边坡稳定性的影响比较显著。随机统计了300个失稳边坡的坡面形态,其中凸坡148个(占49.3%),平坡112个(占37.3%),凹坡40个(占13.4%),可见,凹坡稳定性较好,平坡次之,凸坡最差。
(2)提出了一种严格的二维极限平衡法并拓展至三维。通过类比经典土压力理论和Spencer的条间力假定,将离散后的条柱间作用力等效成滑面正应力,依据整个滑体的平衡条件,提出一种安全系数显式解答的严格二维极限平衡法;进而,将该方法拓展到三维,建立了满足4个平衡方程的非严格三维极限平衡法,可适用于一般空间形态滑面的安全系数计算。代表性算例验证了上述方法的合理性和可靠性,且精度较高。
(3)建立了均质边坡三维近似最危险滑面搜索方法和编制了三维安全系数计算曲线。对于均质边坡,假定三维最危险滑裂面为一族幂函数绕垂直于主滑面旋转轴的旋转曲面,通过优化幂函数中的参数n来确定近似的三维最危险滑裂面,算例验证了该方法的有效性和合理性,且实现过程也非常简便；考虑到三维安全系数求解过程复杂,为了便于快速获得均质边坡的三维安全系数,编制了一套三维安全系数计算曲线,并得到了已有成果的验证。
(4)研究了滑体形态效应的形成机理并编制了相应的修正曲线。依托三维边坡安全系数计算方法和近似三维最危险滑面搜索技术,对边坡滑体形态效应进行了详细分析,编制了滑体形态效应修正曲线,且该曲线得到了模型试验和数值计算成果的验证,通过算例进一步说明考虑滑体形态效应修正后的二维边坡安全系数能够达到三维分析方法的精度,且使用过程非常简便。同时,对影响滑体形态效应的各项参数(滑面形态、长高比、坡度、黏聚力和内摩擦角)进行了独立分析,总结了滑体形态效应的变化规律及影响幅度,在此基础上,从工程应用的角度建议了考虑滑体形态效应的界限标准。最后,对滑体形态效应的形成机理进行了深入讨论,并据此对滑体形态效应随各参数的变化规律进行了解释。
(5)建立了考虑滑体效应的滑带强度参数的三维反演模型。针对边坡二维反演分析确定的滑面强度参数偏高,给滑坡治理带来偏于不安全的风险,建议了一种考虑滑体形态效应的三维反演分析方法。该方法可以考虑三维滑体形态效应,得到的抗剪强度参数更精确、更符合实际情况。从两个算例来看,滑体形态效应对滑面参数反演结果的影响非常显著。
(6)从模型试验的角度研究了坡面形态效应。采用底抬升模型试验,研究了坡面形态对边坡稳定性的影响。试验结果表明：①同一坡度下,平坡破坏时模型槽倾角高于凸坡约2°-4°,但低于凹坡3°-5°。即在坡度一样的情况下,凹坡最稳定,平坡次之,凸坡稳定性最差。并从力学角度对这一现象进行了解释;②在主滑面上,坡面形态对滑面形态及位置影响不大,基本呈现圆弧滑面。受端部限制程度的不同,三维形态则出现较大的差别；③失稳滑体冲程呈现两端小、中间大的特点,最大冲程量随着坡度的增大而逐渐增大,且数据的离散程度较低,比较有规律可循。
(7)提出了圆形凸坡和凹坡的严格极限平衡法并探讨了坡面形态效应。基于严格极限平衡法建立了圆形凸坡和凹坡的安全系数理论解答,并据此分析了坡面形态效应。结果表明坡面形态(R/H)对安全系数有着重要的影响,其显著程度又受到c/(γHtanφ)和坡度β所制约。不论是凸坡还是凹坡,R/H越小,影响越大,且凹坡要比凸坡更显著;在R/H不变的情况下,无量纲指标c/(γHtanφ)和坡度β越大,坡形效应越明显。从工程应用的角度,编制了不同圆形坡面形态下的边坡安全系数速查曲线,便于定量评价相关工程边坡的稳定性。同时给出了是否需要考虑坡面形态效应的建议(适用于c/(γHtanφ)≤0.6时)：当R/H≥15时,可忽略坡面形态的影响;当15>R/H≥5时,应予以适当考虑坡面形态效应；当R/H<5时则必须考虑坡面形态对安全系数的影响。
(8)为了证明本文相关成果的工程实用性,将计算方法应用到金坪子Ⅱ区蠕滑斜坡和新桥硫铁矿露天高边坡的稳定性评价中,并取得了较理想的效果。
There is the three-dimensional effect in slope stability analysis, containing the shape effect of slip mass and the shape effect of slope topography. Since the3D slope stability analysis methods considering the three-dimensional effect and the algorithm of the3D critical slip surface are still not well developed, the three-dimensional effect can not yet be formed a systematical study and it is not always safe to take the three-dimensional effect out of account for a certain engineering case. Therefore, a comprehensive study on the three-dimensional effect of the slope stability analysis, employing statistical analysis, theoretical calculation, experimental test and engineering application, should be carried out. Several main conclusions can be drawn as following:
(1) Based on the statistical analysis of522sets of failure slope data, the shape of slip mass, the aspect ratio of slip mass and the shape of slope topography are analysed emphatically in this study. Preliminary results show that about82.4%of the aspect ratio values are0.5-5.0, the average aspect ratio value is3.08and only3.4%of the aspect ratio values are higher than10. In addition, the shape of slope topography also has a great effect on the slope stability. A statistical analysis of300cases of failure slope shows that the concave slope is the most stable, the horizontal slope is the second and the convex slope is the worst since there are148sets of convex slope (49.3%).112sets of horizontal slope (37.3%) and40sets of concave slope (13.4%).
(2) On the analogy of the classical earth pressure theory and Spencer's assumption about inter-slice forces, the individual inter-slice forces are equivalent to the normal stresses over slip surface. Then, considering the equilibrium conditions of the whole sliding body, a rigorous two-dimensional limit equilibrium method for slope stability analysis is proposed, by which the safety factor can be explicitly solved. Furthermore, the proposed2D method will be developed to a3D non-rigorous limit equilibrium method which satisfies four equilibrium conditions and accommodates to slip surfaces of all shapes.Typical examples show the rationality, reliability and precision of the proposed2D and3D method.
(3) Regarding the slip surface of a homogeneous slope as a curved surface yielded from a family of power function rotating on y-axis, slip surfaces of different shapes are obtained by changing the values of the power function's power. Then, ascertain the simplified3D critical slip surface by optimizing the parameter of power function. Examples show the validity and rationality of the proposed method and the convenience of the procedure. Considering the complexity about the procedure of solving the3D safety factor, a set of calculation curves of the3D safety factor is drawn to obtain the3D safety factor more convenient and faster, which has been verified by other similar results.
(4) Based on the3D slope stability analysis method and simplified search algorithm of the3D critical slip surface, the shape effect of the slip mass is studied in detail and several modified curves for shape effect of the slip mass are also drawn, which are verified by experiment and numerical calculation. Several examples show that the two-dimensional slope safety factor modified by considering the shape effect of the slip mass is quite close to those3D analysis methods and the proposed method is more convenient. In addition, independently parameters, containing the shape of slip surface, the length to height, slope gradient, cohesions and the internal friction angles, are analyzed and change rules and influence amplitude on the shape effect of the slip mass are summarized. Furthermore, a limit whether or not to consider the shape effect of the slip mass is suggested. Finally, the formation mechanism on the shape effect of the slip mass is discussed and change relationships between the shape effect of slip mass and parameters are explained.
(5) Since the2D back analysis often overestimates the shear strength parameters of slip soil and has some unsafe risks in the slope treatment, a3D back analysis model considering the shape effect of the slip mass is established. The proposed method can take the shape effect of3D landslide mass into account, thus the shear strength parameters calculated is more reasonable. Two case studies show that the shape effect of the slip mass has a significant effect on the back analysis result.
(6) The slope topography effect is studied by tilting the model box. The results indicate that the ultimate stable of horizontal slope angle is2°～4°higher than convex slope and3°～5°lower than concave slope under a given gradient. That is to say, the concave slope is the most stable, the horizontal slope is the second and the convex slope is the worst. This phenomenon was explained from mechanical viewpoint. The slope topography has little effect on the shape and location of the main slip surface, which basically shows a circular slip surface. The difference about the three-dimensional configuration is changed obviously with the limitation of the slip mass ends. The runout is big in the middle and small at both ends, of which the maximum value is increased with the gradient.
(7) Based on the rigorous limit equilibrium methods, the theoretical solution procedure of the circular convex slope and concave slope is deduced in this study. Furthermore analytical results show that the shape effect of the slope topography(R/H) has a significant influence on the factor of safety and the amplitude is restricted by c/(γHtanφ) and slope gradient β. The amplitude for the convex slope or concave slope is increased with decreasing R/H and the concave slope is more obvious than the convex slope. The larger value of c/(yHtanR/H≥5, the shape effect of the slope topography should be considered.Finally, while R/H<5, the shape effect of the slope topography must be considered.
(8) Some project cases are employed to check the engineering practicability of the methods suggested in this thesis. The results indicate that these methods are of value in project cases.
(1)从案例统计分析的角度研究了边坡的空间效应的构成与显著度。分析了522个失稳边坡资料,着重对比分析了滑体形态、滑体宽高比及坡面形态的分布,结果表明：①其中82.4%的宽高比位于0.5-5.0之间,平均宽高比为3.08,而宽高比大于10的仅占3.4%；②坡面形态对边坡稳定性的影响比较显著。随机统计了300个失稳边坡的坡面形态,其中凸坡148个(占49.3%),平坡112个(占37.3%),凹坡40个(占13.4%),可见,凹坡稳定性较好,平坡次之,凸坡最差。
(2)提出了一种严格的二维极限平衡法并拓展至三维。通过类比经典土压力理论和Spencer的条间力假定,将离散后的条柱间作用力等效成滑面正应力,依据整个滑体的平衡条件,提出一种安全系数显式解答的严格二维极限平衡法;进而,将该方法拓展到三维,建立了满足4个平衡方程的非严格三维极限平衡法,可适用于一般空间形态滑面的安全系数计算。代表性算例验证了上述方法的合理性和可靠性,且精度较高。
(3)建立了均质边坡三维近似最危险滑面搜索方法和编制了三维安全系数计算曲线。对于均质边坡,假定三维最危险滑裂面为一族幂函数绕垂直于主滑面旋转轴的旋转曲面,通过优化幂函数中的参数n来确定近似的三维最危险滑裂面,算例验证了该方法的有效性和合理性,且实现过程也非常简便；考虑到三维安全系数求解过程复杂,为了便于快速获得均质边坡的三维安全系数,编制了一套三维安全系数计算曲线,并得到了已有成果的验证。
(4)研究了滑体形态效应的形成机理并编制了相应的修正曲线。依托三维边坡安全系数计算方法和近似三维最危险滑面搜索技术,对边坡滑体形态效应进行了详细分析,编制了滑体形态效应修正曲线,且该曲线得到了模型试验和数值计算成果的验证,通过算例进一步说明考虑滑体形态效应修正后的二维边坡安全系数能够达到三维分析方法的精度,且使用过程非常简便。同时,对影响滑体形态效应的各项参数(滑面形态、长高比、坡度、黏聚力和内摩擦角)进行了独立分析,总结了滑体形态效应的变化规律及影响幅度,在此基础上,从工程应用的角度建议了考虑滑体形态效应的界限标准。最后,对滑体形态效应的形成机理进行了深入讨论,并据此对滑体形态效应随各参数的变化规律进行了解释。
(5)建立了考虑滑体效应的滑带强度参数的三维反演模型。针对边坡二维反演分析确定的滑面强度参数偏高,给滑坡治理带来偏于不安全的风险,建议了一种考虑滑体形态效应的三维反演分析方法。该方法可以考虑三维滑体形态效应,得到的抗剪强度参数更精确、更符合实际情况。从两个算例来看,滑体形态效应对滑面参数反演结果的影响非常显著。
(6)从模型试验的角度研究了坡面形态效应。采用底抬升模型试验,研究了坡面形态对边坡稳定性的影响。试验结果表明：①同一坡度下,平坡破坏时模型槽倾角高于凸坡约2°-4°,但低于凹坡3°-5°。即在坡度一样的情况下,凹坡最稳定,平坡次之,凸坡稳定性最差。并从力学角度对这一现象进行了解释;②在主滑面上,坡面形态对滑面形态及位置影响不大,基本呈现圆弧滑面。受端部限制程度的不同,三维形态则出现较大的差别；③失稳滑体冲程呈现两端小、中间大的特点,最大冲程量随着坡度的增大而逐渐增大,且数据的离散程度较低,比较有规律可循。
(7)提出了圆形凸坡和凹坡的严格极限平衡法并探讨了坡面形态效应。基于严格极限平衡法建立了圆形凸坡和凹坡的安全系数理论解答,并据此分析了坡面形态效应。结果表明坡面形态(R/H)对安全系数有着重要的影响,其显著程度又受到c/(γHtanφ)和坡度β所制约。不论是凸坡还是凹坡,R/H越小,影响越大,且凹坡要比凸坡更显著;在R/H不变的情况下,无量纲指标c/(γHtanφ)和坡度β越大,坡形效应越明显。从工程应用的角度,编制了不同圆形坡面形态下的边坡安全系数速查曲线,便于定量评价相关工程边坡的稳定性。同时给出了是否需要考虑坡面形态效应的建议(适用于c/(γHtanφ)≤0.6时)：当R/H≥15时,可忽略坡面形态的影响;当15>R/H≥5时,应予以适当考虑坡面形态效应；当R/H<5时则必须考虑坡面形态对安全系数的影响。
(8)为了证明本文相关成果的工程实用性,将计算方法应用到金坪子Ⅱ区蠕滑斜坡和新桥硫铁矿露天高边坡的稳定性评价中,并取得了较理想的效果。
There is the three-dimensional effect in slope stability analysis, containing the shape effect of slip mass and the shape effect of slope topography. Since the3D slope stability analysis methods considering the three-dimensional effect and the algorithm of the3D critical slip surface are still not well developed, the three-dimensional effect can not yet be formed a systematical study and it is not always safe to take the three-dimensional effect out of account for a certain engineering case. Therefore, a comprehensive study on the three-dimensional effect of the slope stability analysis, employing statistical analysis, theoretical calculation, experimental test and engineering application, should be carried out. Several main conclusions can be drawn as following:
(1) Based on the statistical analysis of522sets of failure slope data, the shape of slip mass, the aspect ratio of slip mass and the shape of slope topography are analysed emphatically in this study. Preliminary results show that about82.4%of the aspect ratio values are0.5-5.0, the average aspect ratio value is3.08and only3.4%of the aspect ratio values are higher than10. In addition, the shape of slope topography also has a great effect on the slope stability. A statistical analysis of300cases of failure slope shows that the concave slope is the most stable, the horizontal slope is the second and the convex slope is the worst since there are148sets of convex slope (49.3%).112sets of horizontal slope (37.3%) and40sets of concave slope (13.4%).
(2) On the analogy of the classical earth pressure theory and Spencer's assumption about inter-slice forces, the individual inter-slice forces are equivalent to the normal stresses over slip surface. Then, considering the equilibrium conditions of the whole sliding body, a rigorous two-dimensional limit equilibrium method for slope stability analysis is proposed, by which the safety factor can be explicitly solved. Furthermore, the proposed2D method will be developed to a3D non-rigorous limit equilibrium method which satisfies four equilibrium conditions and accommodates to slip surfaces of all shapes.Typical examples show the rationality, reliability and precision of the proposed2D and3D method.
(3) Regarding the slip surface of a homogeneous slope as a curved surface yielded from a family of power function rotating on y-axis, slip surfaces of different shapes are obtained by changing the values of the power function's power. Then, ascertain the simplified3D critical slip surface by optimizing the parameter of power function. Examples show the validity and rationality of the proposed method and the convenience of the procedure. Considering the complexity about the procedure of solving the3D safety factor, a set of calculation curves of the3D safety factor is drawn to obtain the3D safety factor more convenient and faster, which has been verified by other similar results.
(4) Based on the3D slope stability analysis method and simplified search algorithm of the3D critical slip surface, the shape effect of the slip mass is studied in detail and several modified curves for shape effect of the slip mass are also drawn, which are verified by experiment and numerical calculation. Several examples show that the two-dimensional slope safety factor modified by considering the shape effect of the slip mass is quite close to those3D analysis methods and the proposed method is more convenient. In addition, independently parameters, containing the shape of slip surface, the length to height, slope gradient, cohesions and the internal friction angles, are analyzed and change rules and influence amplitude on the shape effect of the slip mass are summarized. Furthermore, a limit whether or not to consider the shape effect of the slip mass is suggested. Finally, the formation mechanism on the shape effect of the slip mass is discussed and change relationships between the shape effect of slip mass and parameters are explained.
(5) Since the2D back analysis often overestimates the shear strength parameters of slip soil and has some unsafe risks in the slope treatment, a3D back analysis model considering the shape effect of the slip mass is established. The proposed method can take the shape effect of3D landslide mass into account, thus the shear strength parameters calculated is more reasonable. Two case studies show that the shape effect of the slip mass has a significant effect on the back analysis result.
(6) The slope topography effect is studied by tilting the model box. The results indicate that the ultimate stable of horizontal slope angle is2°～4°higher than convex slope and3°～5°lower than concave slope under a given gradient. That is to say, the concave slope is the most stable, the horizontal slope is the second and the convex slope is the worst. This phenomenon was explained from mechanical viewpoint. The slope topography has little effect on the shape and location of the main slip surface, which basically shows a circular slip surface. The difference about the three-dimensional configuration is changed obviously with the limitation of the slip mass ends. The runout is big in the middle and small at both ends, of which the maximum value is increased with the gradient.
(7) Based on the rigorous limit equilibrium methods, the theoretical solution procedure of the circular convex slope and concave slope is deduced in this study. Furthermore analytical results show that the shape effect of the slope topography(R/H) has a significant influence on the factor of safety and the amplitude is restricted by c/(γHtanφ) and slope gradient β. The amplitude for the convex slope or concave slope is increased with decreasing R/H and the concave slope is more obvious than the convex slope. The larger value of c/(yHtan
(8) Some project cases are employed to check the engineering practicability of the methods suggested in this thesis. The results indicate that these methods are of value in project cases.
引文
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