Practical Probabilistic Slope Stability Analysis

Module1lFDumEq0.1051.2871.287SMSForcesVarying l0.000.00For plotting tension crack17.475.00WHhcgwPwMwruxcycRxminxmax17.473.272751.73101561.51795998730.24.917.9513.40-5.8917.467Kahc0.33333333331.7320508076#xybotytopgavecfWaradulPTEls 'LxLy017.473.275.00150.000116.581.375.0020.0010.003047.41.13610.732.1049.7228.5048.10.01212.9812.125.63215.700.005.0020.0010.003076.30.99717.271.6380.9636.3696.30.02532.4911.247.26314.72-1.185.0019.5835.260107.40.87921.891.54112.0142.14155.70.03850.9410.308.54413.74-2.145.0019.0027.150124.20.77025.311.37137.1228.84230.50.05074.979.329.61512.76-2.925.0018.6622.520137.80.67328.091.25149.2621.95306.30.06290.908.3410.47611.78-3.565.0018.4320.000149.00.58230.371.17156.3618.25377.00.072102.757.3611.19710.79-4.105.0018.2720.000158.30.49632.251.12160.9017.34438.40.082111.926.3811.7889.81-4.535.0018.1520.000165.80.41533.801.07165.0616.66489.60.090120.145.4012.2698.83-4.874.5018.0120.000167.00.33634.041.04164.0316.15528.50.096123.774.4212.64107.85-5.134.0017.8420.000161.90.25933.001.02158.1615.78553.80.101122.793.4312.95116.87-5.313.5017.6720.660155.60.18431.711.00152.3716.03565.90.103120.932.4513.17125.89-5.423.0017.5121.100148.10.11030.180.99146.5416.19565.90.105118.261.4713.31134.91-5.462.5017.3421.320139.40.03728.410.98140.4416.27554.80.104114.610.4913.39143.93-5.422.0017.1721.320129.6-0.03726.410.98133.8016.27533.70.101109.85-0.4913.39152.94-5.311.5016.9821.100118.6-0.11024.180.99126.3616.19503.70.097103.81-1.4713.31161.96-5.131.0016.7720.660106.5-0.18421.711.00117.8616.03466.30.09196.36-2.4513.17170.98-4.870.5016.5220.00093.2-0.25919.001.02107.9915.78423.40.08387.37-3.4312.95180.00-4.530.0016.2020.00078.7-0.33616.041.0496.7416.15376.30.07477.03-4.4212.6419-0.98-4.100.0016.0020.00067.7-0.41513.801.0789.1216.66325.10.06469.32-5.4012.2620-1.96-3.560.0016.0020.00060.1-0.49612.251.1285.3417.34269.20.05364.22-6.3811.7821-2.94-2.920.0016.0020.00050.9-0.58210.371.1779.7418.25210.10.04057.52-7.3611.1922-3.93-2.140.0016.0022.52039.7-0.6738.091.2573.9221.95146.90.02750.84-8.3410.4723-4.91-1.180.0016.0027.15026.1-0.7705.311.3768.3928.8478.60.01444.70-9.329.6124-5.890.000.0016.0035.2609.3-0.8791.891.5467.1242.14-0.00.00041.75-10.308.54Undrained shear strength profile of soft clayEmbankmentdepth01.535710(m)cmfmgmcu402820202637(kPa)103020(kPa)(o)(kN/m3)gclay16(kN/m3)

&APage &P&C&"Times New Roman,Regular"&8Spencer_forSARA2003framed cells contain equations

Module1

&APage &P

SpencerCircularlFDumEq0.1031.2531.253Ka0.33SForcesSMVarying l0.0000.000hc1.7320508076WHhcgwPwMwruxcycxminxmax2751.73101562.32137410110.28.008.00-18.1215.37-2.94417.028xcycR-15.180-1.661-2818.00Lever arms#xybytgavecfWaradulPTEls 'LxLyxiybiyti018.073.275.0015.000.000EquilCheck15.373.275.00116.881.685.0020.0010.003059.70.93110.101.9855.6532.2140.40.01618.029.475.530.0000.00014.751.405.00215.700.005.0020.0010.003098.40.95816.642.06100.2746.8395.50.03132.168.297.160.0000.00014.120.005.00314.72-0.955.0019.6536.220105.60.76721.511.36104.3239.37139.50.04455.057.218.470.0000.00012.66-2.455.00413.74-1.845.0019.1328.870120.00.73824.451.33127.4430.54202.60.05571.666.239.390.0000.00011.19-4.255.00512.76-2.635.0018.7624.080133.20.68227.151.26143.0524.28273.90.06686.005.2510.240.0000.0009.73-5.664.96611.78-3.375.0018.5020.000145.20.64129.601.22155.9919.53351.50.07697.834.2711.000.0000.0008.26-6.794.21710.79-4.015.0018.3020.000156.10.58331.811.18163.5518.76425.90.084107.343.2911.690.0000.0006.80-7.703.4689.81-4.635.0018.1520.000166.00.56233.831.16172.7418.50502.20.091115.172.3012.320.0000.0005.33-8.442.7298.83-5.154.5017.9720.000170.00.48434.641.11172.6017.69566.90.097121.021.3212.890.0000.0003.86-9.021.97107.85-5.524.0017.7720.990167.10.36034.061.05165.1317.56608.60.100123.440.3413.330.0000.0002.40-9.451.22116.87-5.783.5017.6021.940162.30.26133.071.02159.2717.78632.50.103123.74-0.6413.650.0000.0000.93-9.760.47125.89-5.943.0017.4322.580155.80.16831.751.00153.8217.93640.50.103122.80-1.6213.860.0000.000-0.53-9.940.00134.91-6.022.5017.2622.940147.90.07630.140.98148.4018.01633.80.102120.65-2.6013.980.0000.000-2.00-10.000.00143.93-6.002.0017.0923.030138.5-0.01928.230.98142.7618.03613.00.099117.23-3.5814.010.0000.000-3.47-9.940.00152.94-5.881.5016.9122.820127.6-0.12126.010.99136.8018.00578.60.094112.38-4.5713.940.0000.000-4.93-9.760.00161.96-5.651.0016.7122.290115.0-0.23323.441.01130.4217.94531.00.088105.87-5.5513.760.0000.000-6.40-9.450.00170.98-5.280.5016.4821.390100.5-0.36020.481.05123.6117.89470.70.08197.39-6.5313.460.0000.000-7.86-9.020.00180.00-4.730.0016.1920.01083.5-0.50717.011.12116.1517.92398.60.07286.47-7.5113.000.0000.000-9.33-8.440.0019-0.98-4.080.0016.0020.00069.2-0.58714.101.18105.9218.80324.30.06275.79-8.4912.410.0000.000-10.80-7.700.0020-1.96-3.390.0016.0020.00058.6-0.61511.951.2094.0219.16254.50.05166.33-9.4711.730.0000.000-12.26-6.790.0021-2.94-2.610.0016.0020.00047.1-0.6689.601.2582.9319.95187.40.03956.74-10.4511.000.0000.000-13.73-5.660.0022-3.93-1.810.0016.0024.20034.7-0.6847.081.2769.9624.45124.30.02648.16-11.4310.210.0000.000-15.19-4.250.0023-4.91-0.930.0016.0029.04021.5-0.7344.381.3259.8630.6161.40.01340.92-12.429.370.0000.000-16.66-2.450.0024-5.89-0.000.0016.0036.2907.3-0.7571.481.3548.0839.09-0.00.00034.13-13.408.460.0000.000-18.12-0.000.00Undrained shear strength profile of soft clayEmbankmentdepth01.535710(m)cmfmgmcu402820202637(kPa)103020(kPa)(o)(kN/m3)gclay16(kN/m3)

Units: m, kN/m3, kN/m2, kN, or other consistent set of units.Array formulasslope angleSoft clayEmbankmentlx0xnCenter of Rotationxnx0cu (kPa)Depth40282020263701.535710SPENCER methodVarying side-force angle lFormulas: hc = 2*cm/(lmKa), Pw = 0.5*gw*hc^2, Mw = Pw*(yc-H+0.6667*hc)xmin = -(abs(R^2-yc^2))+xc, xmax = (abs(R^2-(yc-H+hc)^2))+xcx(0) = xmax, x(2) = xc+(abs(R^2-yc^2)), x(1) = (B16+B18)/2 x(3) = B18-($B$18-xmin)/22, drag downwards to autofill. ybot(0 )= -(abs(R^2-(B16-xc)^2))+yc, autofill. (Columns ytop, gave and c invoke the user-defined functions shown in Fig. 2.)f(1) = IF(0.5*(C16+C17)>0, $Q$45,0), autofill (also for the formulas below.)W(1) = (B16-B17)*0.5*((D16-C16)+(D17-C17))*E17, arad(1) =ATAN((C16-C17)/(B16-B17))u(1) =ru*E17*0.5*(D16-C16+D17-C17), l(1) =(B16-B17)/COS(I17)Formulas for P and T as shown in text.E(0) = Pw, E(1) = N16+L17*SIN(I17)-M17*COS(I17)l(0)=0, l(1)=IF($R$5,$N$2*SIN(($B$16-B17)/($B$16-$B$40)*PI()),$N$2)s = (P/l - u), Lx(1) = 0.5*(B16+B17)-xc, Ly(1) =yc-0.5*(C16+C17), DumEq = F*1SM =SUM((M17:M40*SIN(I17:I40)+L17:L40*COS(I17:I40)-H17:H40)*Q17:Q40+ (M17:M40*COS(I17:I40)-L17:L40*SIN(I17:I40))*R17:R40)-F11, (Ctrl & Shift, then Enter) SForces =SUM(M17:M40*COS(I17:I40)-L17:L40*SIN(I17:I40))-E11, (Ctrl & Shift, then Enter)EiliEili-1Ei-1Ei-1TiPiWiailybotytopWRead the Low (2003) paper "Practical Probabilistic Slope Stability Analysis", downloadable from http://alum.mit.edu/www/bklow.For hands-on:(1) Initialize l (Cell N2) to zero and F (Cell O2) to 1.0.(2) Initialize (xc, yc, R) to arbitrary initial circle (6, 8, 12).(3) Invoke Solver (Menu bar: Tools/Solver);click Solve. Then click OK to accept the solution found.- This file contains VBA codes, as shown in Tools\Macro\Visual Basic Editor\ Modules\Module 1, and explained in the paper.- Need to set Tools\Macro\Security to medium, then close and re-open file with macros enabled. VBA codes will function only when Macro Security is set to medium.- If never use Excel's built-in Solver before, will need to activate it once via Tools\Add-Ins\Solver Add-in.&C&"Times New Roman,Regular"&8from Spencer_forSARA2003framed cells contain equationsUnits: m, kN/m3, kN/m2, kN, or other consistent set of units.Array formulasslope angleCenter of RotationEiliEili-1Ei-1Ei-1TiPiWiail

SpencerCircular000000000000000000000000000000000000000000000000000000000000000000000000000000000

&APage &P

SpencerNoncircularlFDumEqKahc0.0981.0001.0000.321.7815148835meanStDevvnCorrelation matrix, "crmat"SForcesSMVarying lDistNamevimismNsN(v-mN)/sNcmfmgmcu1cu2cu3cu4cu5cu60.0000.000lognormalcm10.4721071455101.509.8731.5620.3841-0.30.5000000lognormalfm30.9485048337303.0029.8313.0870.362-0.310.5000000lognormalgm20.7576088869201.0019.9601.0370.7690.50.51000000lognormalcu134.0729809574406.0039.1585.083-1.00100010.60653065970.36787944120.18887560280.09697196790.03567399330lognormalcu222.7355057338284.2027.2183.391-1.3220000.606530659710.60653065970.31140322390.15987974610.05881647161.5WHhcgwPwMwruxcyclognormalcu315.9068070156203.0019.3722.373-1.4610000.36787944120.606530659710.5134171190.26359713810.096971967932751.781015.868976399966.45508332120.28.008.00lognormalcu415.8520032655203.0019.3602.365-1.4840000.18887560280.31140322390.51341711910.5134171190.18887560285lognormalcu522.4631074225263.9025.4983.351-0.9060000.09697196790.15987974610.26359713810.51341711910.36787944127lognormalcu634.8174096798375.5536.5475.194-0.3330000.03567399330.05881647160.09697196790.18887560280.3678794412110Lever arms01.535710dv#xybytgavecfWaradulPTEls 'LxLybPrFail017.223.225.0015.870.000EquilCheckArray formula: Ctrl+Shift, then Enter1.9620.0249116.191.665.0020.7610.4730.948504833754.60.99010.641.8745.4334.8934.70.01513.658.715.560.0000.000215.170.005.0020.7610.4730.948504833788.91.01617.321.9583.9050.4879.40.02925.767.687.170.0000.000314.28-0.865.0020.3830.81098.80.76922.141.2496.8938.26119.30.04155.876.728.430.0000.000413.38-1.695.0019.7924.440110.80.74424.841.21117.4629.64177.00.05272.015.839.270.0000.000512.49-2.435.0019.3720.190122.00.69727.341.16131.6823.50243.50.06285.824.9410.060.0000.000DeterministicCriticalNormals, reliability-based critical611.60-3.115.0019.0616.950132.20.64829.621.12142.2818.98314.30.07197.454.0410.770.0000.000xybytxybyt710.71-3.725.0018.8315.900141.40.60031.681.08149.6817.19384.60.079106.743.1511.410.0000.00018.073.275.0017.25433.21445.000089.81-4.305.0018.6415.880149.90.57833.591.07157.4516.92456.50.086114.142.2612.010.0000.00016.881.685.0016.23951.64805.0000LogNormals, reliability-based critical98.92-4.794.5518.4415.860153.40.50534.371.02157.0516.18518.30.091119.651.3712.550.0000.00015.700.005.0015.22460.00005.000017.22046405883.21849098235108.03-5.144.0918.2115.850150.90.37033.820.96149.8115.18558.40.095122.660.4712.970.0000.00014.72-0.955.0014.3227-0.87585.000016.1940518791.65516702865117.14-5.373.6418.0216.700146.60.25632.860.92144.1115.41580.00.097123.34-0.4213.260.0000.00013.74-1.845.0013.4208-1.70925.000015.167639699305126.24-5.533.1817.8317.350141.00.17431.600.91139.2115.72588.70.098122.04-1.3113.450.0000.00012.76-2.635.0012.5188-2.46035.000014.2752101133-0.86367877515135.35-5.622.7317.6517.750134.30.09730.100.90134.3215.92585.80.097119.70-2.2013.570.0000.00011.78-3.375.0011.6169-3.14085.000013.3827805274-1.68508980515144.46-5.632.2717.4617.920126.60.01628.380.89129.2715.99571.80.094116.45-3.1013.630.0000.00010.79-4.015.0010.7150-3.75265.000012.4903509414-2.43183810645153.57-5.571.8217.2717.840117.9-0.06926.410.89123.8815.96547.40.090112.07-3.9913.600.0000.0009.81-4.635.009.8131-4.33595.000011.5979213555-3.10804718435162.67-5.431.3617.0717.500108.0-0.16224.190.90118.1115.82512.80.084106.43-4.8813.500.0000.0008.83-5.154.508.9111-4.82704.540410.7054917696-3.71860954565171.78-5.180.9116.8416.85096.7-0.26821.680.93111.9715.60468.10.07799.32-5.7713.300.0000.0007.85-5.524.008.0092-5.16994.08099.8130621836-4.30123170795180.89-4.780.4516.5715.85083.7-0.42418.760.98107.5815.52409.70.06991.11-6.6612.980.0000.0006.87-5.783.507.1073-5.39903.62138.9206325977-4.79450450284.545284861519-0.00-4.170.0016.2315.87068.1-0.60015.251.08104.1617.15336.70.05981.09-7.5612.470.0000.0005.89-5.943.006.2054-5.55243.16188.0282030117-5.14094993344.09056972320-0.90-3.410.0016.0015.89054.1-0.70312.131.1796.2218.59260.30.04870.11-8.4511.790.0000.0004.91-6.022.505.3034-5.63432.70227.1357734258-5.37490997533.635854584621-1.79-2.590.0016.0015.91042.8-0.7449.601.2183.5819.29189.50.03759.31-9.3411.000.0000.0003.93-6.002.004.4015-5.64522.24276.2433438398-5.53169082953.181139446122-2.68-1.790.0016.0019.60031.3-0.7327.001.2068.2923.51126.40.02549.94-10.2310.190.0000.0002.94-5.881.503.4996-5.58031.78315.3509142539-5.61819903542.726424307623-3.57-0.910.0016.0023.87019.3-0.7774.321.2559.7029.8763.30.01343.39-11.139.350.0000.0001.96-5.651.002.5976-5.43121.32364.4584846679-5.63228189882.271709169124-4.47-0.000.0016.0030.6306.5-0.7961.461.2850.3039.06-0.00.00037.99-12.028.460.0000.0000.98-5.280.501.6957-5.18210.86403.566055082-5.5708169931.8169940306-0.00-4.730.000.7938-4.77210.40452.6736254961-5.42532923931.3622788922-0.98-4.080.00-0.1081-4.15790.00001.7811959101-5.1805674790.9075637537Undrained shear strength profile of soft clayEmbankment-1.96-3.390.00-1.0101-3.40790.00000.8887663242-4.77773458110.4528486152depth01.535710(m)cmfmgm-2.94-2.610.00-1.9120-2.59210.0000-0.0036632618-4.16749505250cu34.072980957422.735505733815.906807015615.852003265522.463107422534.8174096798(kPa)10.472107145530.948504833720.7576088869-3.93-1.810.00-2.8139-1.79300.0000-0.8960928477-3.41077946280(kPa)(o)(kN/m3)-4.91-0.930.00-3.7158-0.91360.0000-1.7885224337-2.58930911510gclay-5.89-0.000.00-4.6178-0.00000.0000-2.6809520196-1.7880316888016-3.5733816056-0.91089117720(kN/m3)-4.4658111915-00

Soft clayEmbankmentxnx0ybytytinitialcriticalcu (kPa)depth000000000000SPENCER: Search for noncircular slip surfaceInitialCirclelx0xnVarying side-force angle lRead the Low (2003) paper "Practical Probabilistic Slope Stability Analysis", downloadable from http://alum.mit.edu/www/bklow.For hands-on:(1) Click the button "InitialCircle"(2) Invoke Solver (Menu bar: Tools/Solver); click Solve. Then click OK to accept the solution found. May run Solver one more time.&C&"Times New Roman,Regular"&8from Spencer_forSARA2003framed cells contain equationsUnits: m, kN/m3, kN/m2, kN, or other consistent set of units.Array formulasslope angleCenter of RotationEiliEili-1Ei-1Ei-1TiPiWiail

SpencerNoncircular000000000000000000000000000000000000000000000000000000000000000000000000000000000

&APage &P

Soft clayEmbankmentxnx0ybytytinitialcriticalcu (kPa)depth000000000000SPENCER: Search for noncircular slip surfacelx0xnVarying side-force angle l00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000=SQRT(MMULT(TRANSPOSE(vn),MMULT(MINVERSE(crmat),vn)))Equation 9 autofilledProbability of failureEqvN(, 1)EqvN(, 2)deterministic critical=normsdist(-b)Read the Low (2003) paper "Practical Probabilistic Slope Stability Analysis", downloadable from http://alum.mit.edu/www/bklow.

For hands-on: Read the section in paper related to Figure 6.

MBD001F3B7C.unknown

MBD0002AFFA.unknown

MBD001A40AB.unknown