第三章33稳定性分析算咧00

1整体圆弧滑动法（瑞典Petterson）2瑞典条分法（瑞典Fellenius）3毕肖普法（Bishop）4Janbu法5Spencer方法6Morgenstern-Price方法7陈祖煜的通用条分法8不平衡推力传递法9Sarma方法粘性土坡的稳定分析计算方法：粘性土坡的稳定分析1整体圆弧滑动法（瑞典圆弧法）

均质土

二维

圆弧滑动面

滑动土体呈刚性转动

在滑动面上处于极限平衡条件假设条件ORORCBA粘性土坡的稳定分析平衡条件（各力对圆心O的力矩平衡）(1)滑动力矩：(3)安全系数：当=0（粘土不排水强度）时，注：（其中是未知函数）(2)抗滑力矩：dW粘性土坡的稳定分析讨论：ORdCBAW1当0时，n是l(x,y)的函数，无法得到Fs的理论解2其中圆心O及半径R是任意假设的，还必须计算若干组（O,R）找到最小安全系数

——最可能滑动面3适用于饱和软粘土，即=0

情况ORdCBAW粘性土坡的稳定分析2条分法的基本原理及分析源起整体圆弧法：n

是l(x,y)的函数思路离散化分条条分法AORC

ibB-2-10123456710粘性土坡的稳定分析安全系数定义AORC

ibB-2-101234567TiNi极限平衡条件粘性土坡的稳定分析PiHiTiNiihi+1WiPi+1Hi+1未知数：条块间力＋作用点位置＝2(n-1)+(n-1)＝

3n-3滑动面上的力＋作用点位置＝3n安全系数F方程数：静力平衡＋力矩平衡＝3n滑动面上极限平衡条件＝n4n6n-2

未知数－方程数＝2n-2hi3瑞典条分法（简单条分法）忽略所有条间作用力：2(n-1)+(n-1)＝

3n-34n-3PiHiTiNiihi+1WiPi+1Hi+1hi假定滑动面上作用点位置：n

未知数:2n+1方程数:4n假定：圆弧滑裂面；不考虑条间力粘性土坡的稳定分析径向力平衡：TiNiiWiAORC

ibB-2-101234567极限平衡条件：整体对圆心的力矩平衡：滑动力矩=抗滑力矩显式表达圆心O，半径R（如图）分条：b=R/10编号：过圆心垂线为0#条中线列表计算li

Wi

i变化圆心O和半径RFs最小ENDTiNiiWiAORC

ibB-2-101234567计算步骤粘性土坡的稳定分析瑞典条分法的讨论TiNiiWiAORC

ibB-2-101234567(1)一些平衡条件不能满足对0#土条T0N0W0>0粘性土坡的稳定分析瑞典条分法的讨论AORC

ibB-2-101234567(2)假设圆弧滑裂面，与实际滑裂面有差别

忽略条间力，使得计算安全系数Fs

偏小假设圆弧滑裂面，使Fs

偏大最终结果是Fs偏小，

越大Fs

越偏小一般情况下，Fs偏小10%左右工程应用中偏于安全例题3.3.1某边坡及滑动面见下图，c=10kPa，φ=23°，采用瑞典条分法，求其稳定系数。瑞典条分法例题3.3.1粘聚力①内摩擦角②重度③滑面坡角④滑面长度⑤条块面积⑥土条#ciφiγiθi(°）liAi123456710 23 20.3 4.6 15.98 70.2310 23 20.3 12.1 10.42 106.8910 23 20.3 18.2 10.86 143.7910 23 20.3 24.5 11.2 164.6210 23 20.3 31 11.64 169.1410 23 20.3 37.4 10.57 115.5210 23 20.3 43.5 14.86 57.82例题3.3.1土条#ciφiγiθi(°）liAi1102320.34.615.9870.232102320.312.110.42106.893102320.318.210.86143.794102320.324.511.2164.625102320.33111.64169.146102320.337.410.57115.527102320.343.514.8657.82Wici*li(wicos(θi)-uili)*tgφiwisinθi1425.669 159.80 602.87 114.282169.867 104.20 900.10 454.622918.937 108.60 1176.43 911.243341.786 112.00 1290.18 1385.163433.542 116.40 1248.78 1767.602345.056 105.70 790.53 1423.711173.746 148.60 361.33 807.63∑ 855.30 6370.21 6864.23粘性土坡的稳定分析4毕肖甫（Bishop）法忽略条间切向力：n-12n-1hi+1PiHiTiNiiWiPi+1Hi+1hi假定滑动面上作用点位置：n

未知数:4n-1方程数:4n假定：圆弧滑裂面；条间力切向力=0粘性土坡的稳定分析Pihi+1TiNiiWiPi+1hiAORC

ibB-2-101234567∑Fz=0极限平衡条件方程组求解，得到：粘性土坡的稳定分析hi+1PiTiNiiWiPi+1hiAORC

ibB-2-101234567整体力矩平衡：Ni

过圆心；Pi互相抵消licosi=bi隐式表达AORC

ibB-2-101234567圆心O，半径R设Fs=1.0计算mqi变化圆心O和半径R

Fs

最小END

计算No计算步骤10.3粘性土坡的稳定分析PiTiNiihi+1WiPi+1hiAORC

ibB-2-101234567毕肖甫法的讨论(2)大多数情况下是精确的(1)假设圆弧滑裂面例题3.3.2某边坡及滑动面见下图，c=10kPa，φ=23°，采用Bishop条分法，求其稳定系数。Wici*bi(wicos(θi)-uili)*tgφiwisinθi1425.67 159.29 604.82 114.282169.87 101.89 920.53 454.622918.94 103.17 1238.31 911.243341.79 101.93 1417.70 1385.163433.54 99.79 1456.63 1767.602345.06 83.99 994.85 1423.711173.75 107.83 497.94 807.63

土条#ciφiγiθi(°）liAi1102320.34.615.9870.232102320.312.110.42106.893102320.318.210.86143.794102320.324.511.2164.625102320.33111.64169.146102320.337.410.57115.527102320.343.514.8657.82∑ 6864.231.0291.0621.0761.0771.0651.0391.003742.514962.5361246.9731410.9241461.8191038.139604.0517466.956F11.053mθi=（①+②/Fs)(③+④)/mθiF21.0881.0281.0601.0721.0721.0581.0310.994743.261964.9901251.6551417.9581470.9881046.017609.4417504.309F31.0931.0281.0591.0711.0711.0571.0300.993743.373965.3601252.3631419.0241472.3781047.213610.2617509.973F41.0941.0281.0591.0711.0711.0571.0300.992743.390965.4161252.4701419.1851472.5881047.394610.3857510.828F51.094土条#cos(θi)①sinθitanφi②10.9970.03420.9780.08930.9500.13240.9100.17650.8570.21860.7950.25870.7260.292土条#cibi③wi*tanφi④1159.29604.822101.89920.533103.171238.314101.931417.70599.791456.63683.99994.857107.83497.94Fs=1.094∑5简布（Janbu）法条块间力的作用点位置：n-12n-1PiHiTiNiihi+1WiPi+1Hi+1hi假定滑动面上作用点位置：n

未知数:4n-1方程数:4nabhi推力线假定：假定各土条间推力作用点连线为光滑连续曲线“推力作用线”即假定了条块间力的作用点位置粘性土坡的稳定分析粘性土坡的稳定分析PiHiTiNiihi+1WiPi+1Hi+1hi极限平衡条件∑Fz=0∑Fx=02345678910111P1=0PU=0

NiTi

Pi方程组求解P1=P1P2=P1+P2

=P1

+P2

Pj=Pi

(i=1,j)Pn=Pi=0

(i=1,n)与

Hi

有关，但Hi

可以通过每个土条的力矩平衡由hi

得到粘性土坡的稳定分析AORC

ibB-2-101234567圆心O，半径R设ΔHi=0变化圆心O和半径R

Fs

最小END

计算No计算步骤计算

计算

计算粘性土坡的稳定分析2345678910111P0=0Pn=0简布法的讨论(1)任意形状滑裂面，不一定是圆弧(2)计算较复杂(3)条块较多时，不收敛例题3.3.3某边坡及滑动面见下图，c=10kPa，φ=23°，采用Janbu条分法，求其稳定系数。12土条#ciφiγiθi(°）liAiuiWiΔhihiΔXi1102320.34.615.9870.2301425.674.070.0015.912102320.312.110.42106.8902169.873.522.8010.183102320.318.210.86143.7902918.944.354.1410.314102320.324.511.2164.6203341.795.175.0910.185102320.33111.64169.1403433.546.035.619.966102320.337.410.57115.5202345.064.285.628.387102320.343.514.8657.8201173.756.973.4810.54ΔHi(wi+ΔHi)sinθi0000000114.28454.62911.241385.161767.601423.71807.63∑6864.233土条#

1234567土条#

1234567cos(θi)①0.9970.9780.9500.9100.8570.7950.726Sinθitanφi②0.0340.0890.1320.1760.2180.2580.292Cibi③159.29101.89103.17101.9399.7983.99107.83(wi+ΔHi)*tanφi④604.82920.531238.311417.701456.63994.85497.94∑mθi1=①+②/Fs1.0081.0070.9940.9690.9300.8800.823(③+④)/mθi757.9511014.8761349.3551568.7841673.3741225.308736.1088325.758F13.000=6864.23F2

1.2131.0251.0511.0591.0551.0371.0070.966

745.599972.7271266.4951440.3691500.3531071.387626.8987623.829F31.1111.0271.0581.0691.0681.0541.0270.988743.726966.5231254.5871422.3731476.7541050.981612.8467527.790F41.0971.0281.0591.0711.0701.0561.0290.992743.443965.5911252.8051419.6891473.2471047.961610.7747513.510F51.095

4土条#cibi③(wi+ΔHi)*tanφi④1159.29604.822101.89920.533103.171238.314101.931417.70599.791456.63683.99994.857107.83497.94（wi+ΔHi)*tanθi⑤⑥114.6480.892464.9360.882959.1770.8981522.0650.9382061.7991.0081791.7011.1161112.9851.269土条#1234567Δpi=⑥*（③+④）-⑤566.704437.105245.320-97.269-492.405-587.363-344.409pi0.00566.701003.811249.131151.86659.4572.09Hi0.000316.405522.420585.095419.540-57.193-66.147ΔHi316.405206.01562.675-165.555-476.733-8.95466.147Fs=1.0955土条#1234567Wi1425.672169.872918.943341.793433.542345.061173.75ΔHi316.405206.01562.675-165.555-476.733-8.95466.147(wi+ΔHi)sinθi139.64497.78930.811316.541522.171418.28853.146678.35(wi+ΔHi)*tanφi④=6878.35739.051007.931264.901347.471254.38991.06526.01土条#cos(θi)①sinθitanφi②10.9970.03420.9780.08930.9500.13240.9100.17650.8570.21860.7950.25870.7260.292土条#cibi③(wi+ΔHi)*tanφi④1159.29739.052101.891007.933103.171264.904101.931347.47599.791254.38683.99991.067107.83526.01∑F11.095mθi1=①+②/Fs

1.0281.0271.0271.0271.0591.0571.0571.0561.0711.0681.0671.0671.0711.0661.0661.0661.0571.0511.0511.0511.0301.0231.0231.0230.9920.9850.9840.984(③+④)/mθi

873.994744.020744.094744.1051047.980967.493967.740967.7761277.3631256.4431256.9161256.9851353.6881425.1701425.8831425.9881281.3491480.4111481.3451481.4821043.8181054.1351054.9401055.059638.741615.011615.564615.6467516.9337542.6837546.4827547.041F2F3F4F51.1261.1291.1301.130土条#cibi③(wi+ΔHi)*tanφi④1159.29739.052101.891007.933103.171264.904101.931347.47599.791254.38683.99991.067107.83526.01土条#Δpi=⑥*（③+④）-⑤PiHiΔHi1636.5350.0000.000341.8172441.614636.535341.817199.5443214.2611078.149541.36152.2424-124.1581292.410593.603-137.4915-445.0561168.252456.112-498.6476-614.047723.197-42.535-14.2397-390.151109.149-56.77456.7746Fs=1.130（wi+ΔHi)*tanθi⑤⑥140.0930.8645206509.0790.85662979.7720.872781446.6600.912451775.5260.982501784.8601.089081175.7081.239377土条#cos(θi)①10.99720.97830.95040.91050.85760.79570.726土条#cibi③1159.292101.893103.174101.93599.79683.997107.83sinθitanφi②0.0340.0890.1320.1760.2180.2580.292(wi+ΔHi)*tanφi④749.831005.191260.481359.371245.08988.81522.03∑F11.130mθi1=①+②/Fs

1.0271.0271.0561.0561.0671.0671.0661.0651.0511.0501.0231.0220.9840.983(③+④)/mθi885.323885.3851047.9111048.0981277.7631278.0991371.2691371.7491280.1411280.7061049.1681049.729640.136640.5397551.7107554.304F2F31.1331.133Fs=1.133粘性土坡的稳定分析6不平衡推力传递系数法假定：条块间力的作用方向：n-1假定滑动面上作用点位置：n2n-1

未知数:4n-1方程数:4n∑FY=0∑Fx=0极限平衡条件方程组求解，得到：当Pn=0时，所对应的Fs为坡体的稳定系数具体求解时需要要试算法：1首先假定一个Fs，可以求得传递系数Ψi,

2求出不平衡下滑力P1（从边坡顶部第一块算起），3然后依次求出P2……Pn，如果Pn=0，则Fs即为所求之稳定性系数。4如果Pn不等于，根据Pn为负或正，增减原定Fs，重复1-3一般我们可先取3个Fs值同时计算，将Fs—Pn绘制成曲线，从图上找出Pn为“零”对应的Fs值，即所求之值粘性土坡的稳定分析为了简化，对每个条块采用下式计算不平衡下滑力不平衡下滑力=下滑力×Fs-抗滑力超载法显式FS作为强度安

全储备系数强度储备法超载法

FS作为超载安全储备系数稳定系数求解需要迭代,又称隐式法对超载法解又称显式法《岩土工程勘察规范》GB50021-2001无论隐式解与显式解法，传递系数法都存在一个缺陷：即对折线形滑面有严格的要求，如果两滑面间的夹角(即转折点处的两倾角的差值)过大，就会出现不可忽视的误差。因而当转折点处的两倾角的差值超过10°时，需要对滑面进行处理，以消除尖角效应。一般可采用对突变的倾角作圆弧连接，然后在弧上插点，来减少倾角的变化值，使其小于10°处理后，误差可以达到工程要求。不平衡推力法的讨论粘性土坡的稳定分析例题3.3.4某边坡及滑动面见下图，c=10kPa，φ=23°，采用不平衡推力传递系数法，求其稳定系数。1土条#坡面滑面

XYXY012345671234567（95.47,44.39）95.47 44.3984.93 44.3976.54 44.3966.58 38.3456.40 32.1446.09 25.8735.91 19.6820.00 10.0095.47 44.3984.93 33.9476.54 27.5266.58 21.5156.40 16.8646.09 13.4635.91 11.2720.00 10.00重度粘聚力内摩擦角γiciφi

20.3102320.3102320.3102320.3102320.3102320.3102320.31023土条#

01234567条块面积 滑面坡角 滑动面长 条块重A αi(°） li Wi

55.09 44.74 14.84 1118.34114.54 37.46 10.56 2325.15167.83 31.10 11.63 3406.90163.46 24.56 11.19 3318.17142.72 18.23 10.85 2897.30105.95 12.14 10.42 2150.8466.84 4.58 15.96 1356.81①Wisinαi786.931413.561758.871378.27906.16452.05108.21②cili148.39105.59116.31111.94108.54104.15159.59Wicosai*tgφi③337.11783.191237.811280.501167.47892.08573.77④ψi1.00000.94070.94910.94750.94950.95150.9382Pi(kN/m)324.55872.421297.251281.25907.43366.64-246.26Fs1.05Pn-246.261.101.0000 345.570.9431 931.470.9511 1413.810.9496 1454.920.9515 1130.500.9534 624.220.940628.6528.651.151.00000.94520.95300.95150.95330.95520.9428364.75985.471520.521614.201335.47861.39282.61282.61a=5288.700b=-5795.903Fs=1.096隐式方法显式方法土条#坡面滑面条块面积土体重度粘聚力内摩擦角滑面坡角

XYXYAγiciφiαi(°）liWi095.47244.39595.47244.395

184.92844.39584.92833.94555.0920.3102344.7414.841118.34276.54444.39576.54427.521114.5420.3102337.4610.562325.15366.58338.33666.58321.513167.8320.3102331.1011.633406.90456.40032.14256.40016.860163.4620.3102324.5611.193318.17546.09125.87146.09113.464142.7220.3102318.2310.852897.30635.90819.67735.90811.273105.9520.3102312.1410.422150.84720.00010.00020.00010.00066.8420.310234.5815.961356.81