1 Introduction Modified Cam-clay triaxial test simulations This example simulates a series of triaxial tests which can be used to verify that Modified Cam-Clay constitutive model is functioning properly. The simulations include: Consolidating the sample to an initial isotropic stress state A drained strain-controlled test An undrained test with pore-pressure measurements A load-unload-reload test Consolidation with a back-pressure The verification includes comparisons with hand-calculated values and discussions relative to the Cam-Clay theoretical framework. 2 Feature highlights GeoStudio feature highlights include: Using the axisymmetric option to simulate a triaxial test Displacement-type boundary conditions to simulate a strain-controlled test Using the MCC model with a Load/Deformation analysis with no pore-pressure changes due to the loading A fully coupled analysis with specified initial pore-pressures A fully coupled analysis with zero-flow boundary conditions to simulate an undrained test 3 Included files Full details of this example and the GeoStudio files are included as: MCC-triaxial tests.gsz MCC-triaxial tests.pdf The specifics of each analysis are available in the GeoStudio data file. 4 General Methodology The simulated shearing phases are preceded by the simulation of the consolidation phase of a triaxial test. Consolidation is isotropic with the confining pressure equal to 1 kpa or 15 kpa. The isotropic stress state is simulated by applying a normal stress on the top and on the right side of the sample equal to 1 kpa or 15 kpa. The consolidation stage is set as the Parent ; that is, the initial condition, for the subsequent simulations involving shearing. SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 1 of 13
metres (x.1) 6 5 4 3 2 1-1 -1 1 2 3 4 metres (x.1) Figure 1 Triaxial test configuration for establishing initial stress state The shearing phase of the analysis is simulated as a strain-rate controlled test. The definition of the strain-rate involves defining the number of time steps and the displacement that occurs over each step. Although the time steps are being defined, it is more appropriate to think of the time steps as load steps. Absolute time has no meaning in the context of these analyses. The number of load steps defined in the shear stage simulations is generally 5 or 25 and the incremental y-displacement of the top of the specimen is defined as -.2 m (per load step), where the negative sign indicates downward displacement. Consequently, 5 and 25 load steps multiplied by a y-displacement of -.2 m per load step results in a total vertical displacement of.1 m and.5 m, respectively. Symmetry is assumed about the vertical and horizontal centre-lines; consequently, only ¼ of the specimen is simulated. The dimensions of the simulation portion of the specimen are.25 m by.5 m, which is half of the width and height of a conventionally sized triaxial specimen. Total vertical y-displacements of.1 m and.5 m produce axial strains of.2 (or 2%) and.1 (1%), respectively. Notice that Linear-Elastic parameters are used when setting up confining stresses; non-linear models are not required for this and the value of E is not relevant 5 Initial yield surface with OCR 1.25 The Cam-Clay properties are in the data file. The material has a Ø of 25.84 degrees and an OCR of 1.25. A Ø of 25.84 is equivalent to an Μ value of 1.. SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 2 of 13
The first step in using the Cam-clay models is to establish the yield surface created sometime in the past under some particular stress state condition. In the field this would be some past insitu condition. This is referred to as the past or initial yield surface. SIGMA/W uses the initial vertical stress specified together with a specified Ø value and a specified OCR (over-consolidation ratio) value to establish the past or starting yield surface. The initial confining stress is 1 kpa. The past vertical effective stress then is, σ y = 1 x OCR = 1 x 1.25 = 125 kpa K o = 1- sin Ø = 1.44 =.56 (formula in SIGMA/W code) σ x = σ z = 125 x.56 = 7.52 σ mean = (125 + 7.52 + 7.52) /3 = 88.68 The shear stress q at the past mean stress is, 1 2 2 2 q ( y x) ( z y ) ( x z ) = 54.48 kpa 2 This is one point on the yield surface created by the past stresses. Since the sample is slightly over-consolidated, the past stress state was higher than the current stress state. Currently, the sample is isotropically consolidated to 1 kpa (σ x = σ y = σ z). The past maximum-mean stress (pre-consolidation pressure) is, 1 2 2 2 1 2 2 2 p c 2 q p 2 54.48 1 88.68 122.15 p 1 88.68 p x is at the center of the yield surface where q is at its maximum value on the initial yield surface; p x = 122.15 / 2 = 61.8 kpa. Now that p c is known, we can draw the initial yield surface for an assumed range of p values between. and p c using the equation, q p p p 2 2 2 c Figure 2 shows the initial or past yield surface. The three means stresses computed above are marked on the diagram. The maximum q value occurs where the yield surface passes through the CSL (critical state line), as it properly should. The q past and p past values form the starting point for establishing the initial yield surface. SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 3 of 13
Shear stress - q 8 7 6 q max = 61.8 q past = 54.48 5 4 3 CSL 1 1 2 1 1 2 3 4 5 6 7 8 9 1 11 12 13 p x = 61.8 p past = 88.68 Mean stress - p' p c = 122.15 Figure 2 Initial yield surface for the past stresses at OCR = 1.25 (produced in EXCEL) 6 Analysis 1 Drained-load deformation A drained test can be done with a Load/Deformation analysis, which does not involve any changes in pore-pressures due to the loading (straining). During drained loading, the yield surface continues to increase in size as Figure 3. The total stress path (which is equal to the effective stress path in this case) on a q-p plot for a triaxial test will have a slope of 1h:3v. This being the case, the stress path intersects the CSL at 15 kpa. SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 4 of 13
Deviatoric Stress (q) (kpa) Shear stress - q 16 14 12 3 1 CSL 1 8 6 Effective stress path & Total stress path 4 2 2 4 6 8 1 12 14 16 18 2 Mean stress - p' Figure 3 Total stress path and final yield surface under drained loading The final deviatoric stress (q) will be 15 kpa. The final vertical (y) stress will be the confining stress plus the deviatoric stress; that is, 1 + 15 = 25 kpa. Moreover, this being a drained test, the sample will undergo some volumetric strain. The next three figures from SIGMA/W confirm this behavior. 15 Stress path 1 5 1 11 12 13 14 15 Mean Effective Stress (p') (kpa) Figure 4 Stress path under drained loading SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 5 of 13
Volumetric Strain Y-Total Stress (kpa) 25 Y stress - strain 2 15 1 Y-Strain Figure 5 Vertical stress versus vertical strain.1 volumetric:axial strain.8.6.4.2.5.1.15.2.25.3.35.4.45 Y-Strain Figure 6 Volumetric strain versus load step number 7 Analysis 2 Drained with fixed pwp This analysis is a repeat of the previous drained test simulation, but now using the fully coupled formulation with pore-pressure specified as a constant equal to zero pore-pressure. The specified porepressure becomes a specified hydraulic boundary condition. Notice in Figure 7 how the hydraulic boundary condition is specified at all nodes and outside edges. SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 6 of 13
Deviatoric Stress (q) (kpa) metres (x.1) 6 5 4 3 2 1-1 -1 1 2 3 4 metres (x.1) Figure 7 Hydraulic boundary conditions for coupled drained simulation 15 q versus strain 1 5 Y-Strain Figure 8 Deviatoric stress versus y-strain for coupled drained test The results from the Coupled analysis are identical to the previous Load/Deformation analysis. This verifies that two different formulations give matching results. This example also illustrates that drained conditions in a Coupled analysis can be simulated by fixing the pore-pressures with a specified boundary condition. 8 Analysis 3 Undrained OCR 1.25 The previous coupled analysis is now repeated, but the hydraulic boundary conditions are specified as no flow. Actually, leaving the hydraulic boundary conditions undefined has the effect of zero flow across the SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 7 of 13
Shear stress - q perimeter. By not allowing flow to exit from the sample, this analysis simulates a consolidated-undrained test with pore-pressure measurements. Based on theoretical consideration for the MCC model, the effective stress path should be vertical until it meets the past maximum yield surface as in Figure 9. Once the effective stress path meets the past yield surface, the path bends to the left and continues to rise slightly until it hits the CSL. The total stress path again is straight line rising at a 1h:3v slope. The different between the total and the effective stress paths is the excess pore-pressure. The final total mean stress is 122 kpa. The effective mean stress where the effective stress path hits the CSL is 66 kpa. The final pore-pressure therefore is 122 66 = 56 kpa. 9 8 7 u = 56 kpa 6 5 3 4 3 Effective stress path 1 2 1 Total stress path 1 2 3 4 5 6 7 8 9 1 11 12 13 Mean stress - p' Figure 9 Stress paths for undrained loading Two SIGMA/W output graphs below confirm these values. The SIGMA/W q-p plot ends at p = q = 66 kpa (Figure 1). The SIGMA/W pore-pressure versus y-strain plot indicates the maximum pore-pressure is 56 kpa (Figure 11). SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 8 of 13
Pore-Water Pressure (kpa) Deviatoric Stress (q) (kpa) 7 Stress path 6 5 4 3 2 1 65 7 75 8 85 9 95 1 15 Mean Effective Stress (p') (kpa) Figure 1 Effective stress under undrained loading 6 pore-pressure:axial strain 5 4 3 2 1.2.4.6.8.1.12 Y-Strain Figure 11 Excess pore pressure in undrained test The volumetric strain for this test is zero, as it properly should be. The sample, however, undergoes some plastic strain, which results in some strain-hardening and the yield surface consequently expands such that it passes thorough the point where the effective stress path meets the CSL. 9 Analysis 4 Load-unload-reload The MCC model treats the soil as elastic when the stress state is under the past maximum yield surface. In an undrained test, the effective p :q stress path is vertical inside the yield locus whether the loading path is one of unloading or loading. Figure 12 reveals that this is indeed the case: the stress path resumes its non-linear behavior once the yield locus is crosses upon re-loading. SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 9 of 13
Deviatoric Stress (q) (kpa) 7 p':q stress path 6 5 4 3 2 1 65 7 75 8 85 9 95 1 15 Mean Effective Stress (p') (kpa) Figure 12 Effective loading unloading stress path 1 Initial yield surface with OCR 5. In this analysis, the sample is subjected to a cell pressure of 15 kpa and the back pressure is set to 5 kpa. The cell pressure is simulated with normal stress boundary conditions equal to 15 kpa. The back pressure is specified as a material activation PWP equal to 5 kpa. This makes the effective confining consolidation stress equal to 1 kpa. The next step is to simulate a consolidated undrained test with pore-pressure measurements for an overconsolidated soil. In this test, we will set the confining stress to 15 kpa with a back pressure of 5 kpa. The effective confining stress is again 1 kpa. This is discussed further in the next section. With and initial effective confining stress at 1 kpa, the past vertical effective stress is, σ y = 1 x OCR = 1 x 5. = 5 kpa K o = 1- sin Ø = 1.44 =.56 (formula in SIGMA/W code) σ x = σ z = 5 x.56 = 282.7 σ mean = (5 + 282.7 + 282.7) /3 = 354.71 The shear stress q at the past mean stress is, 1 2 2 2 q ( y x) ( z y ) ( x z ) = 217.93 2 This is one point on the yield surface created by the past stresses. Since the sample is slightly over-consolidated, the past stress state was higher than the current stress state. Currently, the sample is isotropically consolidated to 1 kpa (σ x = σ y = σ z). The past maximum-mean stress (pre-consolidation pressure) is, 1 2 2 2 1 2 2 2 p c 2 q p 2 217.9 1 354.7 488.6 p 1 354.7 SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 1 of 13
q - kpa p x is at the center of the yield surface where q is at its maximum value on the initial yield surface; p x = 488.6 / 2 = 244.3 kpa. Now that p c is known, we can draw the initial yield surface for an assumed range of p values between. and p c using the equation, q p p p 2 2 2 c Figure 13 shows the initial or past yield surface. The three p stresses computed above are marked on the diagram. The maximum q value occurs where the yield surface passes through the CSL (critical state line) as it properly should. The q past and p past values form the starting point for establishing the initial yield surface. 35 3 25 q past = 217.93 2 15 1 CSL 5 5 1 15 2 25 3 35 4 45 5 p' - kpa p x = 244.3 p past = 354.71 p c = 488.61 Figure 13 Initial yield surface for OCR = 5 Now the effective stress path starts at 1 kpa and rises vertically until it hits the initial yield surface. The soil behaves in a elastically up to this point, as illustrated in Figure 14. After meeting the yield surface, the effective stress path bends to the right and rises slightly until it intersects the total stress path. This is the point at which the excess pore-pressure is zero. After this point the excess pore-pressure diminishes until the effective stress path intersects the CSL. Beyond this point the soil behaves in a plastic manner with no further change in load or pore-pressure. SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 11 of 13
Deviatoric Stress (q) (kpa) q - kpa 35 3 25 u =. u = -39 CSL 2 15 1 5 Effective stress path Total stress path 5 1 15 2 25 3 35 4 45 5 p' - kpa Figure 14 Effective stress for over-consolidated case (produced in EXCEL) Of significance in this case is the fact that the effective stress path remains below the initial yield surface. This is in response to dilation that occurs once the stress path meets the initial yield surface. 11 Analysis Undrained OCR 5. The following graphs from SIGMA/W confirm this behavior. The effective stress path is vertical until meets the initial yield surface. Then it bends over to the right and continues to the right until the path meets the CSL. 25 Stress path 2 15 1 5 8 1 12 14 16 18 2 22 Mean Effective Stress (p') (kpa) Figure 15 Effective stress path with OCR = 5 SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 12 of 13
Pore-Water Pressure (kpa) 12 pore-pressure:axial strain 1 8 6 4 2.5.1.15.2.25 Y-Strain Figure 16 Pore-pressure variations with OCR = 5 The pore-pressure starts at 5 kpa, which is the starting back pressure. The pore-pressure then rises until the effectives stress path meets the initial yield surface. After that, the pore pressure diminishes due to the tendency for dilation until it approaches the CSL. The ending pore-pressure is around 1 KPa. Without the initial back pressure, the ending pore-pressure would be around -4 kpa. In SIGMA/W the MCC model is actually formulated only for saturated conditions and in the SIGMA/W formulation, this means the pore-pressure must be positive. This is the reason for the back pressure. The pore-pressure in this case can fall below the initial value and yet remain positive, so the SIGMA/W results can be compared with the hand-calculated values in Figure 14. SIGMA/W Example File: MCC - triaxial tests.docx (pdf) (gsz) Page 13 of 13