修改图2均质边坡稳定性极限曲线法.doc

均质边坡稳定性极限曲线法方宏伟1，李长洪2，李波1(1辽宁交通高等专科学校 道桥系，沈阳 110122；2北京科技大学 土木与环境工程学院，北京 100083)摘 要：基于滑移线场理论，按边坡坡面变形量评价其稳定性，提出了均质边坡极限曲线法，该法是求有重边坡极限荷载的逆过程，是强度折减法的对偶过程。以特征线法差分方程组(SCM)和试验方程近似公式 (CCM)求得的极限坡面曲线与坡面线相交为变形破坏准则，定义了安全度(DOS)和破坏度(DOF)两个评价指标。该法不必假设和搜索临界滑动面。经典考题和典型算例的验算表明，随着节点的增加SCM计算精度增加，边界步长不变时，三次样条插值求得的变形破坏准则判断值不变，说明SCM算法稳定；典型算例的计算数据和图例表明，边坡角变大时，边坡稳定性降低，极限坡面曲线与坡面由无交点变为有交点，证明了变形破坏准则的正确性；由两个例题计算结果对比可知，安全系数较大时，SCM/CCM计算结果与其具有可比性，相对于原边界条件增加了外荷载，故安全系数变小时，SCM/CCM偏于保守。34个样本计算正确率：安全系数法67.7%，应力状态法73.5%，CCM为79.4%，SCM为70.6%，表明SCM/CCM正确率较高，计算结果可靠。SCM/CCM因素敏感性分析结论与安全系数法完全一致。在露天矿边坡稳定性和最终边坡角的分析与计算中，SCM/CCM结论与原报告相同，当参数变小时，CCM法更有利于实践，表明该法具有一定的工程应用价值。 关键词：极限曲线法；变形破坏准则；安全度；破坏度 Limit curve method of homogeneous slope stability FANG Hong-wei1，LI Chang-hong2，LI Bo1(1 Department of Road and Bridge, Liaoning College of Communication, Shenyang 110122, China；2 School of Civil and Environmental Engineering, University of Science and Technology Beijing, Beijing 100083, China)Abstract: Based on the theory of slip line field, this paper proposes the limit curve method of slope stability according to the deformation situation; the method is the inverse process for computing a heavy slope ultimate load and the dual process of strength reduction method. The paper defines two evaluation indexes: the Degree of Safety (DOS) and the Degree of Failure (DOF) according to the deformation failure criterion of the limit stable slope curve and the slope surface intersection computed by characteristic line difference method (SCM) and the slope limit experimental approximate formula (CCM). The method does not require assuming and searching critical slip surface. Classic exams and typical examples show that with the increase of nodes, the accuracy of SCM increases; when boundary step is constant, the judgment value obtained by three spline interpolation is unchanged, which proves the stability of SCM. Typical examples show that the larger the slope angle becomes, the lower the slope stability is, limit slope curve and slope is from without intersection to intersection, which proves that the correctness of the deformation failure criterion. Comparing the results from the two examples shows that the safety factor is large and SCM/CCM results are comparable; this paper increases external load relative to the original boundary conditions, so the safety factor becomes smaller, SCM/CCM is conservative. To calculate the correct rate, this paper uses 34 samples: safety factor method is 67.7%, the stress state method is 73.5%, CCM is 79.4% and SCM is 70.6%, which indicates that SCM/CCM correct rate is higher. The conclusions of SCM/CCM factor sensitivity analysis and safety coefficient method are completely consistent. By analyzing and computing the slope stability and the ultimate slope angle of open pit mining, the report about SCM/CCM is the same as the original; when the parameter variable is smaller, CCM is more conducive to practice, which indicates that the method has a certain value in engineering applications. Key words: Limit Curve Method; deformation failure criterion; Degree of Safety; Degree of Failure1 引 言 边坡稳定性分析主要是条分法和有限元法，包括计算安全系数和搜索临界滑裂面两个方面1。条分法局作者简介：方宏伟(1980-)，男，博士，讲师，从事岩土工程的教学与研究工作。E-mail: fanghongwei911126.com。限性主要是需假定滑裂面的位置2，有限元法分为滑裂面应力分析法和强度折减法两类，前者为有限元与极限平衡理论的结合，仍需假定和搜索临界滑裂面，后者克服了以上不足，但是对边坡临界破坏的失效判据学术界尚无统一的意见3，而且不同程序计算的结果并不一致，因此有限元法还不能够替代极限平衡法4。为解决上述问题，学者们一方面努力寻找准确可靠的搜索方法5，另一方面不断提出新观点，如朱大勇6认为直接求解最小安全系数是解决问题的突破口，并提出了边坡临界滑动场法。以上都是从力的角度进行研究的，陈震7认为可以按边坡变形发展过程定量估定稳定性，提出可用破坏坡度与实际坡度之比作为安全系数，并指出联系变形发展情况还可能有其它表达形式。滑移线法在边坡中的应用主要是计算无重边坡极限荷载8，近年来已有学者认为滑移线具有更重要的意义，并且提出与有限元法结合确定滑裂面9，张天宝10认为该理论对土石坝合理边坡形状设计具有启发意义，并与高广岩11通过有限元论证堆石坝合理边坡形状是凹形曲面的结论相一致。基于以上研究，以边坡变形量评价其稳定性为出发点，本文提出应用滑移线场理论计算得到的极限坡面曲线分析边坡稳定性的新方法，称极限曲线法。计算极限坡面曲线有两种方法：B.B. Sokolovskii研究得出的特征线法差分方程组 7 和A.M.Cehkob根据试验得到的极限稳定边坡曲线方程近似公式 10 ，为节省篇幅，本文不列出相关公式，读者可查阅索引文献，对应的极限曲线法也分为两种：前者称S曲线法(S curve method，SCM) ，后者称C曲线法(C curve method，CCM) ，采用Mat lab编程计算，作者可提供源程序。通过经典考题与典型算例及34个边坡样本2 的验算考察方法合理性和结果可靠性，进行因素敏感性分析，并应用于露天矿边坡稳定性和最终边坡角的分析与确定。 2 极限曲线法简介 2.1基本概念与公式滑移线法已计算得到了无容重边坡极限荷载，对于有容重边坡则要求坡面为凹形曲面才能求得解析解8，本文采用该计算逆过程，则对有容重边坡，在极限荷载作用下，坡面形状应为凹形曲面，二维坐标下为凹形曲线，需要说明的是，即使边坡所受荷载不是极限荷载，也可以求得极限坡面曲线，这个曲线是将现有荷载定义为极限荷载后边坡极限平衡状态下的坡面形状。将地基线以上边坡体放入第一象限，以坡脚为坐标原点，设坡高为H, F1(x)为极限坡面曲线拟合二次函数，当F1(x)与正x轴的交点x110时，将坡面线与极限坡面曲线之间的面积S1和坡面线与正x轴所成面积S2之比定义为安全度DOS(Degree of Safety)，见图1，设坡脚到坡顶横坐标x22，S2 = x22 ·H/2，极限坡面曲线与正x轴所成面积S3=，则S1= S2-S3，DOS= S1/S2，DOS越大稳定性越好，值域为(0,1)；以极限坡面曲线与坡面线相交为变形破坏准则，其交点横坐标x1与x22之比的负值定义为破坏度DOF(Degree of Failure)，见图2，此时x110，DOF=-x1/x22，DOF越小稳定性越差，值域为(-1,0)。 由以上分析可知，x11为变形破坏准则判断值。强度折减法构筑一个与真实边坡相同轮廓的“虚拟”边坡，强度指标缩减，缩减的系数即安全系数。极限曲线法正好是这个方法的对偶过程，即强度指标不变，但坡面缩减变形,按其变形量评价稳定性，因此不必假设和搜索临界滑动面。 图1 DOS计算示意图 图2 DOF计算示意图 Fig. 1 Calculation schematic diagram of DOS Fig. 2 Calculation schematic diagram of DOF 2.2算法函数与流程 极限曲线法函数： F1(x) = f1 (, c, , , H, N1, N2, x) (1)DOS/DOF= f2 (F1 (x), F0 (x) (2)式中，容重(kN/m3)，c 粘聚力(kPa)，内摩擦角(°)，坡度(°)，N1为主动区边界步长数，N2为过渡区边界步长数，x为SCM主动区边界步长(m)，当CCM时为y=H/N1，N1为H剖分数，无N2，f1为SCM/CCM，F0(x) 为边坡坡面函数(本文为过零点的一次函数)，f2为上节求DOS/DOF的方法。采用准确率最高的三次样条差值12计算x11，该法可以回避插值问题的Runge现象，又是连续光滑的。为保证F1(x)与x轴有交点，要求F1(x)纵坐标最小值ymin<-1。对SCM，从理论上讲，应为x110时x10，而x11<0时x1>0，但是由于F1 (x)随着取点变化而发生多项式摆动13，会产生x11<0时x1<0的情况，此时DOS/DOF=0，当坡面无荷载时，满足计算条件的坡顶最小荷载7为Pmin= c·cot·(1+sin)/(1-sin)，此时无过渡区，则N2=0，而CCM不存在上述两个问题，计算流程见图3。边界条件SCM/CCM极限坡面曲线拟合二次函数F1(x)坡面一次函数F0 (x)三次样条插值x11<0 x10 是否DOS 是DOF (SCM)否DOS=DOF=0(SCM)DOF (CCM) 图3 算法流程图Fig.3 Algorithm flow chart3 考题和算例验算与样本分析王国体建立了边坡稳定性应力状态法2，引用了其它文献的大量例题，为便于比较，本文例题皆取之于该文献，不同之处是用文献1的经典考题b代替c。 3.1经典考题与典型算例的验算考题a的参数为 =20kN/m3，c=3kPa，=19.6°，=27°，H=10m；考题b的参数为 =20kN/m3，c=32kPa，=10°，=27°，H=10m；检验有限元强度折减法的典型算例参数为 =20kN/m3，c=42kPa，=17°，为变化范围为30°，50°，均分五个区间计算，H=20m。由于SCM采用特征线差分法，而拟线性双曲型微分方程组没有唯一解14，相应的数值方法也具有类似的特点，即SCM计算的DOS/DOF会随着边界条件不同而发生变化。为考察该法稳定性，采用不同的x和N1及节点数k计算分析，结果见表1和2，随着x的减小和N1与k的增加，计算精度随之增加，计算结果表明SCM算法稳定。计算范围影响分析见表3和4，当x不变，N1增加时k变多，对函数拟合有影响，DOS/DOF会发生变化，但x11不变，说明解的变化完全是由于拟合振荡产生的，对边坡稳定性的判断不会产生影响。由于计算机性能的限制，N1不可能取得很大，本文计算N1999。CCM算法没有稳定性问题。 不同坡度SCM(x=0.02)计算图例4、5、6、7、8，CCM(y=0.001)计算图例9、10、11、12、13，随着边坡角增大，安全系数逐渐变小，边坡稳定性变差，DOS/DOF也在变小，x11由大于0变为小于0，即极限坡面曲线与x轴的交点逐渐向左偏移，当x110时极限坡面曲线与坡面相交，且安全系数越小，x11越小，极限坡面曲线与坡面相交点x1越大，由此证明了变形破坏准则的正确性。 DOS/DOF与安全系数对比见表5，安全系数较大时，SCM/CCM与其具有可比性，当安全系数变小时，SCM/CCM偏于保守，且CCM保守程度大于SCM，原因应该是相对于坡顶无荷载的初始条件，SCM计算时在坡顶附加了最小荷载值，而CCM将坡顶以下部分边坡体自重作为外荷载。 表1 经典考题SCM稳定性分析 Table 1 Stability analysis of SCM参数xN1kyminx11DOSDOS考题a1881-2.330214.5630.83430.14590.515256-3.781511.29870.68840.30280.1452116-1.025.09670.38560.09470.05806561-1.24593.26490.29090.02480.0410010201-1.99612.78010.26610.03730.0315022801-6.1782.22650.22880.02750.0220040401-4.47631.57450.20130.03550.01350123201-2.83450.76610.16580.00620.009400160801-3.84850.67240.15960.00210.008420177241-2.21590.57570.15750.00480.007480231361-2.44280.47570.15270.00460.006550303601-2.21270.3720.14810.00490.005650423801-2.06260.26430.1432考题b115256-4.79514.08060.81140.06160.525676-2.567411.29960.74980.10210.111012321-1.28198.51210.64770.01840.0522048841-1.30168.1230.62930.00470.0428078961-1.49548.04370.62460.00490.03380145161-1.69067.9640.61970.00190.02550303601-1.31477.88380.6178 表 2 典型算例SCM稳定性分析(不同坡度)Table 2 Stability analysis of SCM (Different slope)x10.50.10.050.040.030.02N12035150300400500750k44112962280190601160801251001564001ymin-8.7823-5.9757-3.0265-3.262-5.2398-3.3622-3.413630°0.76860.70220.61550.60030.59320.59390.590735°0.71930.63880.53370.51520.50660.50750.503640°0.66370.56720.44120.41910.40870.40980.405145°0.59920.48420.33410.30770.29530050°0.52230.3853-0.0836-0.119-0.1381-0.135-0.1434 表3 经典考题SCM计算范围的影响分析Table 3 Calculation influence range of SCM考题ax=0.005x11=0.2643N1700750800850900950k491401564001641601724201811801904401ymin-4.1686-6.5277-9.156-12.0690-15.2814-18.8072DOS0.140.13360.1260.11840.11140.1052DOS0.00640.00760.00760.00700.0062考题bx=0.02x11=7.8838N1600650700750800850k361201423801491401564001641601724201ymin-2.2744-3.2615-4.2800-5.3334-6.4243-7.5552DOS0.61150.60200.59150.58130.57190.5639DOS0.00950.01050.01020.00940.008 表4 典型算例SCM计算范围的影响(x=0.02) Table 4 Calculation influence range of SCM (x=0.02)N1800850900950x1130°0.58660.58050.57350.566314.429835°0.49870.49130.48280.4748.351840°0.39920.39040.38020.36973.623945°0000-0.211250°-0.1581-0.1838-0.2158-0.2498-3.4292k641601724201811801904401ymin-5.3575-7.3856-9.5031-11.7148 图4 30º的SCM计算图(典型算例) 图5 35º的SCM计算图(典型算例)Fig.4 SCM diagram of 30º (Typical example) Fig. 5 SCM diagram of 35º (Typical example) 图6 40º的SCM计算图(典型算例) 图7 45º的SCM计算图(典型算例)Fig.6 SCM diagram of 40º (Typical example) Fig. 7 SCM diagram of 45º (Typical example) 图8 50º的SCM计算图(典型算例) 图 9 30º的CCM计算图(典型算例)Fig.8 SCM diagram of 50º (Typical example) Fig. 9 CCM diagram of 30º (Typical example) 图10 35º的CCM计算图(典型算例) 图11 40º的CCM计算图(典型算例)Fig.10 CCM diagram of 35º (Typical example) Fig. 11 CCM diagram of 40º (Typical example) 图12 45º的CCM计算图(典型算例) 图13 50º的CCM计算图(典型算例)Fig.12 CCM diagram of 45º (Typical example) Fig. 13 CCM diagram of 50º (Typical example)表5 DOS/DOF与安全系数法的对比Table 5 Results comparison算法SCMCCM应力状态法有限元折减法Spencer裁判答案考题a0.1432-0.67660.926,1.1620.99,1考题b0.61780.56991.65,1.7030°0.59070.5391.91.56,1.931.5535°0.50360.44041.661.42,1.771.4140°0.4051-0.09391.461.31,1.651.3645°0-0.1931.291.21,1.541.350°-0.1434-0.29271.151.12,1.441.123.2 样本计算与正确率分析取文献2中(去掉c=0和=0的实例)共34个边坡样本并按原序列重新排序，计算结果见表6。分析可知，安全系数法计算正确率67.7%，应力状态法73.5%，CCM为79.4%，SCM为70.6%，SCM判断错误的样本安全系数法也都判断错误，CCM判断错误最少。表 6 样本分析与结果对比Table 6 Samples analysis and result comparison算法SCMCCM安全应力实际系数法状态法状态xN1kDOS/DOFyN1DOS/DOF10.06900811801-0.31290.001115000-0.58410.890.87滑坡20.0399910000000.52040.0011200000.59121.371.3稳定30.029509044010.77180.001305000.75192.141.63稳定40.029008118010.35600.0011000000.21161.241.06稳定50.029008118010.4770.0011000000.38251.421.14稳定60.029008118010.21430.00188000-0.48561.060.95滑坡70.029509044010.10690.001470000-0.69291.031.06稳定80.059509044010.75390.0012000000.76712.291.43稳定90.0049509044010.69250.001100000.69191.91.28稳定100.0129909820810.25310.00150000-0.18321.071.03滑坡110.035990982081-0.25150.001200500-0.61460.90.98滑坡120.035990982081-0.74610.001292000-0.8480.80.92滑坡130.029909820810.1060.001480000-0.69941.031.06稳定140.04990982081-0.30470.001305000-0.71040.890.98滑坡150.035990982081-0.01340.001213000-0.56170.931滑坡160.019909820810.60640.001305000.48071.751.14稳定170.0069909820810.50000.001200000.3831.431.09稳定180.0129909820810.27990.00150000-0.12081.091.02滑坡190.0059008118010.75480.00152300.83891.111.75滑坡200.019909820810.60700.001305000.48071.881.14稳定210.0259008118010.76760.001305000.75192.051.63稳定220.029008118010.36170.0011000000.21821.781.06稳定230.029008118010.4770.0011000000.38251.991.14稳定240.0159008118010.3630.001400000.17521.251.01滑坡250.029008118010.21430.00188000-0.48561.020.95滑坡260.0359008118010.52090.0011200000.59121.31.3稳定270.059008118010.49440.0012000000.52421.21.28稳定280.039509044010.68890.0011150000.72821.111.39滑坡290.0089008118010.77240.001106700.76811.41.68稳定300.019008118010.3560.00112190-0.39711.351.22滑坡310.0079008118010.3140.00112800-0.34761.031.01滑坡320.029008118010.80440.001457200.78141.281.32滑坡330.0159008118010.31040.00110670-0.37091.631.38稳定340.01900811801-0.20280.00121000-0.61731.090.9滑坡4因素敏感性分析按文献2的作法，分析SCM/CCM在不同坡度下各因素变化对DOS/DOF的敏感性，增加了坡高H一项，计算结果分别见表7-1、7-2、7-3、7-4和8-1、8-2、8-3、8-4。安全系数法中各因素对边坡稳定性影响趋势是15：单一参数变化而其它参数固定时，容重、边坡角、坡高H增加时，边坡稳定性降低，而粘聚力c和摩擦角增加时，边坡稳定性增大。由各个表格数据可知SCM/CCM因素敏感性结论与安全系数法完全一致。 表 7-1 SCM容重敏感性计算(不同坡度)Table 7-1 SCM density sensitivity calculation (different slope)SCMx=0.02N1=750k=564001(kN/m3)181920212230°0.62560.60790.59070.57380.557235°0.54590.52450.50360.48310.463040°0.45580.43020.40510.38050.356545°0.35150.32090-0.0001-0.029650°-0.0561-0.0974-0.1434-0.1926-0.2433表 7-2 SCM粘聚力敏感性计算(不同坡度)Table 7-2 SCM cohesive force sensitivity calculation (different slope) SCMx=0.02N1=750k=564001c(kPa)1520253035404530°00.28970.38880.46500.52520.57380.613835°-0.4648-0.11700.25870.35110.42410.48310.531640°-0.6441-0.4598-0.2004-0.00510.30990.38050.438745°-0.7252-0.6294-0.4724-0.2706-0.1014-0.00010.331050°-0.7705-0.7148-0.6293-0.5021-0.3425-0.1926-0.0830表 7-3 SCM摩擦角敏感性计算(不同坡度)Table 7-3 SCM friction angle sensitivity calculation (different slope) SCMx=0.05N1=450k=203401(°)101214161830°0.33260.42080.48190.53160.574835°-0.123600.37160.43190.484340°-0.3240-0.166300.31920.382045°-0.4963-0.3956-0.2665-0.09990.263550°-0.6117-0.5492-0.4722-0.3757-0.2526表 7-4 SCM坡高敏感性计算(不同坡度)Table 7-4 SCM Slope high sensitivity calculation (different slope