1 Slope Stability for a Cohesive and Frictional Soil

Transcription

1 Slope Stability for a Cohesive and Frictional Soil Slope Stability for a Cohesive and Frictional Soil 1.1 Problem Statement A common problem encountered in engineering soil mechanics is the stability of soil slopes in frictional materials. In this example, three slope conditions are analyzed. First, a slope in sand with zero cohesion is modeled with an initial slope steeper than the angle of repose of the sand. This slope, of course, should collapse; the progression of this collapse is calculated as it develops. Second, a small cohesion is added to the material and the slope is re-examined to determine whether it is stable. Third, the water level in the slope is raised, and the effect on stability is examined. In this example, the soil is homogeneous, and the stability and factor of safety of the slope can readily be determined using an analytic or graphical technique.* However, the power of the FLAC code lies in its ability to model more complex slope geometries in which, for example, several layers of soil with differing material properties and/or constitutive behaviors may exist. This type of problem can be examined with no greater effort than the homogeneous case by simply assigning different material properties and/or models to different zones. This example also demonstrates two approaches to analyze the effect of a phreatic surface in the slope. In one approach, an effective-stress analysis is performed after adding a pore-pressure distribution directly to the zones in the model. In the other approach, a groundwater flow calculation is performed first to establish the phreatic surface; then the effective-stress analysis is performed. 1.2 Modeling Procedure Initial Model State The following sequence of operations is used to initialize the slope and create a restart file from which the boundary conditions and/or material properties may be varied for the three cases of this example. Please refer to SLOPE.DAT in Section 1.3 for a complete listing of the commands for this example. An initial grid of square zones is set up. Note that this initial grid is assigned the dimensions of 20 units in the x-direction by 10 units in the y-direction, unless otherwise redefined by the user. The command to do this is grid * See Section 10 for a comparison of FLAC results to a limit equilibrium analysis.

2 1-2 Example Applications A Mohr-Coulomb constitutive model is assigned to all zones (assumed because no range is given) with the following properties: density 1500 kg/m 3 shear modulus Pa bulk modulus 10 8 Pa friction angle 20 cohesion Pa tensile strength Pa Note that a high cohesion and tensile strength are assigned to prevent slope failure during the initialization of gravitational stresses in the model (see below). The basic grid is deformed into the shape of a slope and the soil base beneath it. This is done using the GENERATE command. Two different quadrilateral regions are created with the following two GENERATE commands: gen 0,0 0,3 20,3 20,0 j = 1,4 gen same 9,10 20,10 same i = 6,21 j = 4,11 The first GENERATE command defines the base of the slope, and the second GENERATE command creates the slope. Note that the zones are aligned with the angle of the slope so that the zones along the slope face are all quadrilateral-shaped. This is recommended because all zones are then composed of two overlaid sets of triangular elements. These zones are well-suited for plasticity analysis (see Section in Theory and Background). It is also possible to create a slope using the GENERATE line command. However, with this command, single triangular zones will be created along the slope face; these zones are not as accurate for plasticity analysis. The area directly to the left of the slope face is excavated by declaring the appropriate zones as null. This is done by creating a region (i.e., the grid is divided into two regions separated by a boundary) that is defined by marking selected gridpoints as boundaries between regions. The following commands mark the boundary of the excavated region and then null the zones within that region: mark i = 1,6 j = 4 mark i = 6 j = 4,11 model null region 1,10 The marked boundaries can be verified by issuing the PRINT mark command. The MODEL null command will delete zones in the region containing zone (1,10). Figure 1.1 shows the resulting FLAC grid.

3 Slope Stability for a Cohesive and Frictional Soil 1-3 JOB TITLE : SLOPE UNDER GRAVITATIONAL LOAD FLAC (Version 6.00) LEGEND 20-Jun-08 9:54 step E+00 <x< 2.111E E+00 <y< 1.611E+01 Grid plot E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.1 Grid plot of initial slope The acceleration of gravity is set to 9.81 m/sec 2 (positive means acting downward). The boundary conditions consist of roller boundaries on the left and right sides of the model, as well as a fixed base. An initial elastic state in which gravitational stresses are equilibrated is desired. This is achieved with the SOLVE command, using default limits. Equilibrium is obtained when the out-of-balance force ratio limit of 10 3 is reached. To examine the progression of the solution, the y-displacement history is requested at a gridpoint at the slope crest. This is done using the HISTORY command. When the SOLVE command has reached its limit, the history may be plotted to verify that the mesh is, indeed, at an equilibrium state (i.e., PLOT history 1). A numerical and graphical representation of the elastic stresses can be obtained by issuing the following commands: print sxx syy plot sxx yellow bound plot syy yellow bound Initially, very large values for the cohesion and tensile strength are assigned to the slope. In justification, let us reexamine the way in which an explicit model works. An initial grid is created first and, in this case, gravity applied to the gridpoints and gravitational stresses allowed to equilibrate. For most problems, it is desirable that this process occur as rapidly as possible. This can be done by requiring the material to behave elastically during the equilibration process. Once stresses have equilibrated, actual material properties are assigned, excavation is made, loads are applied, etc., and the simulation process continued. In the case illustrated here, a plastic constitutive model is assigned initially, with high cohesion and tensile strength, forcing the material to behave elastically. Then the cohesion and tensile strength are reset to the desired values. This

4 1-4 Example Applications eliminates the necessity of redefining the constitutive model and properties after equilibrium, thus saving a few steps of input. The same effect could be obtained by using an elastic model initially for the equilibration process, followed by a change to the Mohr-Coulomb model, followed by a definition of the material friction, cohesion and tensile strength. Either methodology for initializing a gravity-loaded grid is acceptable, and is selected based on the preference of the user. A restart file is created to save the elastic equilibrium state. This is done to save time in case future runs will be made in which material parameters or constitutive models are varied. Performing these studies requires only that the elastic state be restored, therefore eliminating the necessity of recomputing the equilibrium state. The command save sl1.sav will create a restart file on the default drive called SL1.SAV. FLAC could be halted at this point and the program run with the saved state restored at a later time, simply by typing restore sl1.sav Slope Collapse: Dry Conditions For the next stage of the simulation, the material properties are set to the actual soil values and the calculation continued while examining the possible failure process. During this process, plots of the progressive displacement of the slope are made. To avoid any confusion in analyzing the data, only the change in displacement is monitored, not the cumulative displacements from the beginning of the simulation. The calculation procedure in FLAC does not involve displacements, but keeps the cumulative total for each gridpoint as a convenience to the user. Therefore, the displacements may be initialized at any point in the run without affecting the results. This is done by using the command ini xdis=0 ydis=0 From this point on, plots or printouts will show only the change in displacement from the previous state. Next, the material properties of the zones are reset by using the PROPERTY command. The cohesion is set to zero for all zones that are currently composed of soil.* Finally, the calculation mode is set to large-strain to provide a more accurate geometrical representation of the slope failure as it progresses. Because slope collapse will occur due to the angle of repose of the soil being smaller than the slope angle, the SOLVE command is not used (because equilibrium will not be reached). The STEP command is used to step the simulation a small number of calculational steps at a time, stopping to print and plot intermediate stages. Here, the power of the explicit method is evident in * Note that the tensile strength will also be set to zero in FLAC, because the tensile strength is calculated from cohesion/tan (friction).

5 Slope Stability for a Cohesive and Frictional Soil 1-5 its ability to follow highly nonlinear problems, which may never converge to an equilibrium state, through progressive failure. The model is now stepped in intervals to evaluate the progressive collapse of the slope. The collapse is revealed when printing and plotting results after each step. The following commands are used: print xv yv xd yd state plot xv z disp bou Figures 1.2 and 1.3 show the state of the slope at 897 steps and 1697 steps. These figures illustrate the progressive collapse and, in particular, indicate the location of the failure (slip) surface. The slope is collapsing in an attempt to reach its angle of repose. At some point, the displacements of the gridpoints become unrealistic because of extreme distortion of the grid. FLAC automatically checks for excessive grid deformation and will stop the calculation process if the condition is detected, displaying an error message. JOB TITLE : SLOPE UNDER GRAVITATIONAL LOAD FLAC (Version 6.00) LEGEND 20-Jun-08 9:54 step E+00 <x< 2.111E E+00 <y< 1.611E+01 Boundary plot E 0 Displacement vectors max vector = 2.518E E -2 X-velocity contours Contour interval= 2.50E-05 Minimum: -2.25E-04 Maximum: 0.00E+00 (zero contour omitted) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.2 Plot of displacement vectors and x-velocity contours at step 897

6 1-6 Example Applications JOB TITLE : SLOPE UNDER GRAVITATIONAL LOAD FLAC (Version 6.00) LEGEND 20-Jun-08 9:54 step E+00 <x< 2.111E E+00 <y< 1.611E+01 Boundary plot 0 5E 0 Displacement vectors max vector = 5.732E E 0 X-velocity contours Contour interval= 2.00E-04 Minimum: -1.00E-03 Maximum: 0.00E+00 (zero contour omitted) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.3 Plot of displacement vectors and x-velocity contours at step 1697 An interesting comparison can be obtained by running another simulation in which a small cohesion and zero tensile strength are assigned to the soil. Because the initial elastic equilibrium state has been saved, the problem can be restored from this state: rest sl1.sav Following the procedure used earlier, enter the following commands: ini xdis=0 ydis=0 prop coh=1e4 tens=0.0 set large solve Under these conditions, the results will show that the slope is stable. (Note the small magnitude of the calculated displacements.)

7 Slope Stability for a Cohesive and Frictional Soil Effective Stress Analysis with WATER table Next, we wish to assess the stability of the slope with a water table present. Continuing with the model at the present state, the water level is raised in the slope to a height of 9montheright side of the model and 5montheleft side (i.e., 2 m above the base of the excavation). It is a simple matter to use the WATER table command to specify the phreatic surface. Note that the correct wet and dry densities must also be supplied. The FISH function wet den in SLOPE.DAT at BRANCH:WATER TABLE prescribes the wet density to zones below the water table, assuming a soil porosity of 0.3. The APPLY press command must also be used to apply the weight of the water in the excavation to the surface of the excavation. Figure 1.4 illustrates the location of the water table, the applied forces representing the weight of the water in the excavation, and the wet and dry densities in the zones. The calculation is continued with the STEP command. Now, after an additional 6000 steps, the slope is observed to fail. The pore pressure distribution and velocity vectors indicating the slope failure in the model are plotted in Figure 1.5. JOB TITLE : SLOPE UNDER GRAVITATIONAL LOAD FLAC (Version 6.00) LEGEND 9-Jul-08 13:54 step E+00 <x< 2.111E E+00 <y< 1.611E+01 Density 1.35E E E E E E E Contour interval= 7.50E+01 Extrap. by averaging Grid plot E 0 Water Table Net Applied Forces max vector = 2.000E Itasca 0 Consulting Group, 5E 4Inc. Minneapolis, Minnesota USA Figure 1.4 Location of water table, applied forces along slope and wet and dry densities

8 1-8 Example Applications JOB TITLE : SLOPE UNDER GRAVITATIONAL LOAD FLAC (Version 6.00) LEGEND 9-Jul-08 14:07 step E+00 <x< 2.111E E+00 <y< 1.611E+01 Boundary plot 0 5E 0 Velocity vectors max vector = 2.686E E -5 Pore pressure contours Contour interval= 1.00E+04 B: 1.000E+04 I: 8.000E+04 B C D E F G H I Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure Pore pressure distribution and velocity vectors in failed slope (using WATER table) Effective Stress Analysis with CONFIG gw Alternatively, the groundwater flow option in FLAC can be used to find the phreatic surface and establish the pore pressure distribution before the mechanical response is investigated. The model is run in groundwater flow mode by using the CONFIG gw command. This command must be given before the GRID command, so we create a new model state in SLOPE.DAT, at BRANCH:SLOPE WITH GW in Section 1.3 for this case. We turn off the mechanical calculation (SET mech off) in order to establish the initial pore pressure distribution. We apply pore pressure boundary conditions to raise the water level to 5 m at theleft boundary and 9 m at theright. The slope is initially dry (INI sat 0). We also set the bulk modulus of the water to a low value ( ) because our objective is to reach the steady-flow state as quickly as possible. The groundwater time scale is wrong in this case, but we are not interested in the transient time response. The steady-flow state is determined by using the SOLVE ratio command. When the groundwater flow ratio falls below the set value of 0.01, steady-state flow is achieved. This can be checked by using the FISH function QRATIO.FIS to assess the flow state. (See Section 3 in the FISH volume for a description of this function.) The steady-flow state is indicated by the plot of flow vectors and phreatic-surface contour in Figure 1.6. (The FISH function PS.FIS is used to locate the phreatic surface see Section 3 in the FISH volume.) The bumpy phreatic-surface line is due to the coarse discretization.

9 Slope Stability for a Cohesive and Frictional Soil 1-9 JOB TITLE : SLOPE UNDER GRAVITATIONAL LOAD FLAC (Version 6.00) LEGEND 20-Jun-08 9:54 step 2054 Flow Time E E+00 <x< 2.111E E+00 <y< 1.611E+01 Boundary plot 0 5E 0 Flow vectors max vector = 5.672E E -6 phreatic surface Contour interval= 5.00E-01 Minimum: 0.00E+00 Maximum: 5.00E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 1.6 Steady-state flow through slope Mechanical equilibrium is then established including the pore pressure. This is accomplished with the following commands: set flow off mech on apply press 2e4 var 0-2e4 from 1,4 to 6,6 water bulk = 0.0 These commands turn off the flow calculation, turn on the mechanical calculation, apply the weight of the water to the excavation surface, and set the bulk modulus of the water to zero. The last command prevents pore pressures from generating as a result of mechanical deformation. This is done so that the results can be compared to the previous case using the water table. The SOLVE command is then used to find the equilibrium state. Finally, the cohesion is reduced to , the tensile strength to zero, and the calculation is continued with the STEP command. The slope fails, as shown in Figure 1.7. The result is the same as that which occurs using WATER table. The approach using the groundwater flow mode can take longer to reach a solution because of the extra calculation needed to establish the pore pressure distribution. However, this method is helpful when the pore pressure distribution or phreatic surface location is unknown. Also, this approach avoids the necessity of assigning wet density values to zones beneath the phreatic surface, which can become difficult for complex geometries.

10 1-10 Example Applications JOB TITLE : SLOPE UNDER GRAVITATIONAL LOAD FLAC (Version 6.00) LEGEND 20-Jun-08 9:54 step 3826 Flow Time E E+00 <x< 2.111E E+00 <y< 1.611E+01 Boundary plot 0 5E 0 Velocity vectors max vector = 1.531E E -4 Pore pressure contours Contour interval= 1.00E+04 B: 1.000E+04 I: 8.000E+04 B C D E F G H I Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure Pore pressure distribution and velocity vectors in failed slope (using CONFIG gw)

11 Slope Stability for a Cohesive and Frictional Soil Data File SLOPE.DAT ;Project Record Tree export ;*** BRANCH: SLOPE WITHOUT GW **** new ;... STATE: SL1... config ats grid 20,10 ;Mohr-Coulomb model m m ; soil properties -- note large cohesion to force initial elastic ; behavior for determining initial stress state. This will prevent ; slope failure when initializing the gravity stresses prop s=.3e8 b=1e8 d=1500 fri=20 coh=1e10 ten=1e10 ; warp grid to form a slope : gen 0,0 0,3 20,3 20,0 j 1,4 gen same 9,10 20,10 same i 6 21 j 4 11 mark i=1,6 j=4 mark i=6 j=4,11 model null region 1,10 ; displacement boundary conditions fix x i=1 fix x i=21 fix x y j=1 ; apply gravity set grav=9.81 ; displacement history of slope his ydis i=10 j=10 ; solve for initial gravity stresses solve ; save initial state save sl1.sav ;*** BRANCH: DRY **** ;... STATE: SL2... ; reset displacement components to zero ini xdis=0 ydis=0 ; set cohesion to 0 prop coh=0 ; use large strain logic set large step 200 save sl2.sav

12 1-12 Example Applications ;... STATE: SL3... step 800 save sl3.sav ;*** BRANCH: WATER TABLE **** restore sl1.sav ;... STATE: SL4... ini xdis=0 ydis=0 prop coh=1e4 tens 0.0 set large solve ; ini xdis=0.0 ydis=0.0 ; install phreatic surface in slope water table 1 den 1000 table 1 (0,5) (6.11,5) (20,9) def wet den loop i (1,izones) loop j (1,jzones) if model(i,j)>1 then xa=(x(i,j)+x(i+1,j)+x(i+1,j+1)+x(i,j+1)) xc=0.25*xa ya=(y(i,j)+y(i+1,j)+y(i+1,j+1)+y(i,j+1)) yc=0.25*ya if yc<table(1,xc) then density(i,j) = 1800 end if end if end loop end loop end wet den apply press 2e4 var 0-2e4 from 1,4 to 6,6 save sl4.sav ;... STATE: SL5... step 6000 sclin save sl5.sav ;*** BRANCH: SLOPE WITH GW **** new ;... STATE: SLGW1...

13 Slope Stability for a Cohesive and Frictional Soil 1-13 config gw ats ex 1 grid 20,10 ;Mohr-Coulomb model m m ; soil properties -- note large cohesion to force initial elastic ; behavior for determining initial stress state. This will prevent ; slope failure when initializing the gravity stresses prop s=.3e8 b=1e8 d=1500 fri=20 coh=1e10 ten=1e10 ; warp grid to form a slope : gen 0,0 0,3 20,3 20,0 j 1,4 gen same 9,10 20,10 same i 6 21 j 4 11 mark i=1,6 j=4 mark i=6 j=4,11 model null region 1,10 prop perm 1e-10 por.3 water den 1000 bulk 1e4 ; displacement boundary conditions fix x i=1 fix x i=21 fix x y j=1 ; pore pressure boundary conditions apply pp 9e4 var 0-9e4 i 21 j 1 10 apply pp 5e4 var 0-3e4 i 1 j 1 4 ini pp 2e4 var 0-2e4 mark i 1 6 j 4 6 fix pp mark ; apply gravity set grav=9.81 call qratio.fis hist gwtime hist qratio hist inflow hist outflow set mech off solve save slgw1.sav ;... STATE: SLGW3... set flow off mech on app press 2e4 var 0-2e4 from 1 4 to 6 6 water bulk 0.0 ; displacement history of slope hist reset his ydis i=10 j=10 solve save slgw3.sav

14 1-14 Example Applications ;... STATE: SLGW4... ini xdis 0.0 ydis 0.0 prop coh 1e4 ten 0.0 set large step 1000 sclin save slgw4.sav ;*** plot commands **** ;plot name: grid plot hold grid ;plot name: Displacement vectors plot hold bound displacement xvel zero ;plot name: Water Table plot hold density fill inv grid water apply lmagenta ;plot name: Pore pressure distribution plot hold bound velocity pp ;plot name: Steady-state flow plot hold bound flow saturation alias phreatic surface min 0.0 max 0.5 & int 0.5 lmagenta