Rainfall-induced slope failure mechanism and reliability analyses based on observation information
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摘要: 目前降雨诱发滑坡机理及可靠度研究不仅很少同时考虑土体水力参数和抗剪强度参数空间变异性的影响，而且忽略了“天然工况下斜坡基本上不会发生失稳破坏”这一客观事实。以无限长斜坡模型为例，融合观测信息提前对空间变异抗剪强度参数进行概率反演分析，再建立非平稳随机场模型模拟土体渗透系数的空间变异性及非平稳分布特征，在蒙特卡洛模拟框架下评估不同降雨历时下斜坡失效概率及最危险滑动面分布特征。在此基础上，探讨同时考虑土体水力参数和抗剪强度参数空间变异性的降雨诱发斜坡失稳机理。结果表明：利用概率反演获得的抗剪强度参数后验信息，计算的斜坡失效概率由先验的28.1%降至7.2%。不同降雨阶段斜坡失稳的诱因不同，如果忽略“天然工况下斜坡不会失稳破坏”这一观测信息，将会造成对降雨诱发斜坡失稳机理和失效概率的错误估计，特别是在降雨初期。Abstract: The rainfall-induced slope failure mechanism and reliability analyses rarely consider the spatial variability of hydraulic and shear strength parameters at the same time and ignore a fact that the slopes always keep stable under the natural condition. An infinite slope model is taken as an example to conduct probabilistic back analyses of spatially varying shear strength parameters using the observation information in advance. Then, a non-stationary random field model is established to simulate the spatial variability and non-stationary distribution feature of the hydraulic conductivity. The probabilities of slope failure and distributions of the critical slip surface under different rainfall durations are evaluated within the framework of Monte-Carlo simulation. Based on these, the rainfall-induced slope failure mechanisms considering the spatial variability of hydraulic and shear strength parameters simultaneously are investigated. The results indicate that the probability of slope failure evaluated based on the posterior information of shear strength parameters obtained from the probabilistic back analyses is reduced from 28.1% (prior) to 7.2%. It is found that the triggering factors for the slope instability are different for different rainfall stages. The rainfall-induced slope failure mechanism and probability of failure will be erroneously evaluated, especially at the initial stage of rainfall, if the field observation information is ignored.
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0. 引言
北疆供水渠道在进行渠道设计时，不仅仅要考虑经济最优和保证流量最优的问题，还需要考虑渠道的冬季冻胀破坏。梯形渠道占中国输水渠道的比例较大，应用较广。因此，梯形渠道的冻胀问题受到了北方寒区广泛关注。王正中等[1-2]对梯形渠道冻胀模型进行了理论分析，将梯形渠道考虑为在切向冻结力、法向冻胀力的以及衬砌板约束下的简支梁，王正中等[3]、刘旭东等[4]、李爽等[5]用数值模拟的方法模拟梯形断面形式下冻胀的规律，得到了温度场，位移场沿着渠道分布情况。
本文主要研究不同梯形断面情况下，以北疆某供水工程总干渠退水渠段梯形渠道为例，对该渠道用comsol进行数值模拟，对不同的断面渠道进行参数化模拟，得到最佳参数范围，并结合水力最佳断面，得到双优断面形式，为寒区渠道设计提供参考。
1. 力学模型的建立
1.1 基本假设
渠基土冻结时，土体、水和冰之间的相互作用的微观结构及动态过程相当复杂，目前很难准确模拟整个冻胀过程。为了便于分析，对其进行适当的简化，以便抓住影响冻结过程中及冻胀变形的主要特征，主要假设如下[6]：①根据现场及室内试验研究，假设冻土是均匀连续各向同性体；②尽管土的冻胀与其温度、水分、土质密切相关，对具体工程当水分及土壤条件确定时土体最终冻胀主要取决于温度，将水分迁移对冻土的体积影响，以线膨胀系数表达；③根据试验研究假定相变温度在同一种土中和同种外力条件下为常值，即暂取相变温度为0℃[7]；④由于渠道为细长结构，不考虑长方向上土颗粒对温度的影响，选取平面应变问题进行模拟。
1.2 热传导方程
根据以上假设，将温度热传递视为二维瞬态热传递的过程，建立二维热传导控制方程：
(λx∂2T∂x2)+(λy∂2T∂y2)=ρc∂T∂t。 (1) 式中T为温度；λx，λy分别是冻土沿x，y方向的导热系数；ρ为土体的密度；c为土体的比热容；t为时间。
1.3 本构方程
冻土属于冷胀热缩材料，冻土在冻结过程中水冻结成冰，除了原位水冻结体积膨胀，还有从未冻结区向冻结区迁移的水分冻结成冰，本文将原位水以及迁移水冻结成冰的体积膨胀以关于温度T的函数的线膨胀系数表示。因此，土体的本构方程可以表示为
ϵx=σxE−μσyE+α(T−Tref) ，ϵy=σyE−μσxE+α(T−Tref) ，γxy=2(1+μ)Eτxy 。} (2) 式中εx，εy为正应变；γxy为剪应变；σx、σy为正应力；τxy为剪应力；E为弹性模量；μ为泊松比，α为混凝土或者冻土自由冻胀时的热膨胀系数；T为计算温度；Tref为参考温度。
2. 计算模型
2.1 原型概况
本文选择北疆地区某输水渠道退水渠段的梯形渠道（图 1）为研究对象，原型渠道的基本情况如表 1，2所示，并对该模型渠道进行了冻胀模拟分析计算，因原型渠道各部位坡向，土质，水分不同，在模型边界设置时对阴阳破和渠堤分别对温度和土体冻胀率赋值。
表 1 渠道各部位的表面温度以及冻结期Table 1. Surface temperatures and freezing periods of various parts of canal部位 月平均表面温度/(℃) 冻结期、（月-日） 12月 1月 2月 阴坡 -14.92 -18.85 -10.72 11-27—02-27 渠底 -14.56 -16.22 -9.15 11-27—02-26 阳坡 -12.55 -14.75 -10.54 11-27—02-27 表 2 原型渠道基本情况Table 2. Basic information of prototype canal部位 渠床
土质基土干密度/(g·cm-3) 冻深h/cm 冻胀量Δh/cm 冻胀率η/% 阴坡 砂砾石 1.80 171 5.0 2.92 渠底 159 4.4 2.77 阳坡 146 3.7 2.53 2.2 有限元模型
有限元计算模型为：①原型梯形断面渠道冻胀数值模拟；②梯形断面渠道冻胀“参数化”断面渠道数值模拟。
有限元模型如图 2所示，选取各表面温度接近原型渠道，渠道两边设置为热绝缘边界，下边界为固定温度边界取10℃，阴坡边界温度取冻结期平均值-14.8℃，渠底温度取-13.3℃，阳坡温度为-12.6℃。几何模型如图 2所示，渠底长2.0 m，渠坡的横向长度为3.75 m，坡比为1.5，基础底向下取2.5 m，左右边界取0.75 m。模型将衬砌板与冻土作为一个整体进行数值模拟，力学的边界设置时，左右边界设置为辊支撑，底部设置指定位移0。
2.3 参数取值
参考文献[5]中线膨胀系数按照η/Tmin取值，η为冻胀率，Tmin为相应部位月平均表面温度的最小值。由式（2），将T-Tref看成1℃，那么可以将线膨胀系数α考虑跟温度Tmin相关的函数，数值上与冻胀率相等。未冻土与混凝土的导热系数参考文献[3]，冻土的导热系数与土体的未冻水含量相关，冻土融土导热系数比与未冻水含量相关[9]，模型渠道的未冻水含量选取19.4%，因此冻土的导热系数为未冻土导热系数的0.9倍。
3. 计算结果分析
3.1 温度场
图 3可以看出：接近于渠底的温度分布接近于一组平行的直线，0℃等温线以上，由于未冻水冻结导致剧烈相变，该区域导热系数小，因此等温线较密集，温度梯度比较小；0℃等温线以下，导热系数较大，等温线较稀疏。阴坡冻深为162 cm，渠底冻深为146 cm，阳坡的冻深为141 cm。与表 3模型试验较符合，最大相对误差为8.2%。
表 3 材料热力学参数Table 3. Thermodynamic parameters of materials介质 弹性模量E/Pa 泊松比ν 导热系数λ/(W·m-1·℃) 线膨胀系数α/℃-1 混凝土 2.4×1010 0.167 1.580 1.1×10-5 冻土 4.6×107 0.330 1.188[9] 阴坡2.92%、渠底2.77%、阳坡2.53% 未冻土 1.5×107 0.375 1.320 0 3.2 位移场
如图 4所示的是模拟得到的结果与试验结果进行对比，模型试验结果冻胀量边坡较大，渠底较小。模型试验的冻胀量整体比模拟冻胀量大，因为模型试验所用的衬砌板受到边坡的约束较大，且衬砌板厚度小于模拟设定厚度。模拟得到渠底冻胀量比较平缓而试验得到的渠底中间部位冻胀量大而两边小，模拟设定的衬砌板是整体现浇型板，在冻胀力的作用下容易产生整体变形；而模型试验的衬砌板分布较为离散。
3.3 “参数化”位移场模拟结果
根据梯形断面的设计规范要求，应满足：①渠道边坡系数应不小于允许最小边坡系数，应不大于允许最大边坡系数，1≤m≤2。②渠道的宽深比应满足规范给定的要求，0.8 < b/h < 3.5。
根据要求，“参数化”分析模拟了宽深比β一定时，不同边坡比的渠道断面冻胀量情况；和边坡比一定时，不同宽深比的渠道断面冻胀量情况。
由图 5，宽深比一定时，边坡比越小，渠坡的法向冻胀量越大；m=2时，左侧阴坡法向冻胀量最大为5.2 cm，阳坡为1.7 cm。m=1时，阴坡法向冻胀量最大值为3.3 cm。由图 6，边坡比一定时，宽深比对法向冻胀量影响不大。
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图 5 不同坡比法向冻胀量比较Figure 5. Comparison of normal frost-heave quantity under different slopes![]()
图 6 不同宽深比法向冻胀量比较Figure 6. Comparison of normal frost-heave quantity under different breadth depth ratios4. 水力最佳断面
从设计角度考虑水力最佳断面通常是指过水流量一定，所要求的过水断面面积最小，也就是所需要的材料最少，施工且最方便[12]。或者是过水断面面积一定，所通过的流量最大。
考虑最小冻胀量，所以采用边坡比为2的冻胀断面。水力学中考虑流量一定，过水断面面积最小可采用以下公式计算宽深比[13]：
β=2（√1+m2−1)。 (3) 将相关参数代入式（3），可得到考虑水力冻胀最优宽深比为2.472。
5. 结语
本文利用comsol有限元软件，对北疆供水渠道以及渠基土冻胀破坏规律进行有限元分析，与试验数据对比，基本与试验数据相似。对不同坡比以及不同宽深比的渠道进行有限元模拟分析，发现坡比影响冻胀量的大小，坡比越大，渠道冻胀量越小。理论分析受力情况易得坡比越大越能释放变形，符合实际。再结合水力最佳断面的公式，可以得到最佳宽深比。最终的断面形式能有效的防止冻胀，且满足水力断面要求。为北疆地区渠道防冻胀断面设计提供了理论支撑。
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图 1 无限长斜坡模型及单位滑块随机场离散
Figure 1. Diagram of an infinite slope model and random fields discretization for a unit soil column
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图 2 天然工况下斜坡安全系数频率直方图及累积分布函数
Figure 2. Histogram and cumulative distribution of factor of safety under natural condition
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图 3 沿埋深方向有效黏聚力先验与后验统计特征的比较
Figure 3. Comparison of prior and posterior statistics of effective cohesions along depth
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图 4 沿埋深方向有效内摩擦角先验与后验统计特征的比较
Figure 4. Comparison of prior and posterior statistics of effective friction angles along depth
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图 5 基于参数后验样本计算的斜坡安全系数频率直方图及累积分布函数
Figure 5. Histogram and cumulative distribution of factor of safety based on posterior samples of shear strength parameters
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图 6 斜坡失效概率随降雨历时的变化关系曲线
Figure 6. Variation of probability of slope failure with rainfall durations
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图 7 基于抗剪强度参数后验信息计算的斜坡最危险滑动面深度频率直方图的比较
Figure 7. Comparison of histogram of depth of critical slip surface based on posterior information of shear strength parameters
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