Consolidation of a 1D column

Transcription

1 1 Introduction Consolidation of a 1D column This example simulates a one-dimensional consolidation test. The sides are constrained as in an odometer test. Analytical solutions are available for a case like this, making it possible to verify the SIGMA/W formulation and implementation. 2 Feature highlights GeoStudio feature highlights include: Using the fully coupled stress / pore-pressure formulation to simulate a consolidation test Using a step-boundary function to apply a surface load only for the first time step Using a watertable definition to establish the initial pore-pressure conditions 3 Configuration and set-up Figure 1 configuration and set-up. Initially, the pore-pressure is hydrostatic with depth, with zero pressure at the top surface. This makes the initial pressure at the base equal to 10 kpa (γ w = 10 kn/m 3 ). Applied pressure = kpa using a step function Elevation - m Distance - m Figure 1 One-dimensional consolidation setup SIGMA/W Example File: Consolidation 1D.doc (pdf) (gsz) Page 1 of 5

2 4 Material properties The pore-pressures in this analysis will remain positive at all times, and therefore we can deal properties only for satuated conditions. Volumetric water content and hydraulic conductivity functions have been specified for the sake of completeness, but they do not enter into the calculations. Only the saturated hydraulic conductivity (K sat ) is used. The coefficient of volume compressibility m v is required, but is computed from the E modulus so it does not need to be specified as part of the volumetric water content function. When Poisson s ratio ν is 0.334, m v = 1/E. With E = 00 kpa, m v is equal to 5 x 10-4 / kpa. K sat is 1 x 10-6 m/sec. 5 Boundary function In this analysis, we want to apply a surface load and then allow the resulting excess pore-pressure to dissipate. Since SIGMA/W is based on an incremental formulation, we cannot apply a surface load and keep it there during the dissipation stage. The surface load has to be applied only during the first time step, and then it has to be numerically removed for the remaining time steps. This can be accomplished with a stepfunction such as shown in Figure 2. Loading function Normal Boundary Stress (kpa) Time (sec) Figure 2 Surface load boundary function The time step duration is 1 sec. SIGMA/W obtains the function values at time n and (n-1). In this case they are and zero. The difference is 0 kpa, which is the surface load applied for the first integration time step. For all other time steps the applied load increment is zero. For example, the function value at t = 1 is 10 and at t = 2 is also, and therefore the applied load increment is zero as intended. SIGMA/W Example File: Consolidation 1D.doc (pdf) (gsz) Page 2 of 5

3 6 Pore-pressure response Figure 3 shows the pore-pressure response to the applied kpa and the subsequent dissipation. The form and shape is exactly as what is typically in text books. The increase everywhere is kpa except right at the top, where there has been a little dissipation during the first second. At the bottom, the initial pressure was 10 kpa and, after applying the load, the pore-pressure is 110 kpa. 1.0 PWP dissipation 0.8 Y (m) Pore-Water Pressure (kpa) Figure 3 Pore-pressure response and dissipation Most textbooks on soil mechanics present graphical solutions to the consolidation equation in the form of dimensionless charts such as Figure 4. The curves in this figure are known as isochrones. Time factors T, can be computed as T Cv t = where 2 H C v ksat = γ m w v Once T has been computed, the percentage of pore-pressure dissipation at the bottom of the problem, for example, can be estimated. SIGMA/W Example File: Consolidation 1D.doc (pdf) (gsz) Page 3 of 5

4 Figure 4 After Lambe & Whitman, Soil Mechanics, p. 8 Figure 5 shows the pore-pressure response at the bottom of the column for times between 250 and 3500 seconds. These results are compared in Figure 6 with estimates from the isochrones in Figure 4. The chart values are somewhat higher at the same dissipation time as those computed by SIGMA/W. However, the forms of the curves are in reasonable agreement. The shift between the curves is likely due to the early pore-pressure dissipation near the top at the early times, which is not captured in the analytical solution and associated charts. Once the pore-pressure starts to diminish at the base, the rate of dissipation is nearly identical. It is not possible to say which curve is more correct. It is likely that neither would exactly correspond to laboratory measurements. The good agreement in the form and shape of the curves in Figure 6 are, however, sufficient to indicate that SIGMA/W is correctly formulated and implemented within the bounds of a numerical solution to the consolidation equation. SIGMA/W Example File: Consolidation 1D.doc (pdf) (gsz) Page 4 of 5

5 PWP at base 110 Pore-Water Pressure (kpa) Time (sec) Figure 5 Pore-pressures at the column base Pore-pressure at base 1 Pore-pressure - kpa SIGMA Chart Time - sec Figure 6 Comparison of pore-pressures with time at the column base 7 Conclusion This example illustrates that SIGMA/W can be used for saturated consolidation analyses and that it produces reasonable results. SIGMA/W Example File: Consolidation 1D.doc (pdf) (gsz) Page 5 of 5