MATERIAL MODELS MANUAL
Back to Main Menu TABLE OF CONTENTS 1
Introduction.........................................................................................................1 - 1 1.1 On the use of three different models ................................................................1 - 1 1.2 Warnings........................................................................................................1 - 2 1.3 Contents ........................................................................................................1 - 3
2
Preliminaries on material modelling ..................................................................2 - 1 2.1 General definitions of stress and strain.............................................................2 - 1 2.2 Elastic strains..................................................................................................2 - 3 2.3 Undrained analysis with effective parameters...................................................2 - 5 2.4 Undrained analysis with undrained parameters.................................................2 - 8 2.5 The initial pre-consolidation stress in advanced models ....................................2 - 8 2.6 On the initial stresses .....................................................................................2 -10
3
The Mohr-Coulomb model (perfect-plasticity) .................................................3 - 1 3.1 Elastic perfectly-plastic behaviour ...................................................................3 - 1 3.2 Formulation of the Mohr-Coulomb model.......................................................3 - 2 3.3 Basic parameters of the Mohr-Coulomb model...............................................3 - 4 3.4 Advanced parameters of the Mohr-Coulumb model........................................3 - 8
4
The Hardening-Soil model (isotropic hardening) ..............................................4 - 1 4.1 Hyperbolic relationship for standard drained triaxial tests.................................4 - 2 4.2 Approximation of hyperbola by the Hardening-Soil model...............................4 - 3 4.3 Plastic volumetric strain for triaxial states of stress............................................4 - 5 4.4 Parameters of the Hardening-Soil model.........................................................4 - 6 4.5 On the cap yield surface in the Hardening-Soil model.....................................4 -11
5
Soft-Soil-Creep model (time dependent behaviour)..........................................5 - 1 5.1 Introduction....................................................................................................5 - 1 5.2 Basics of one-dimensional creep.....................................................................5 - 2 5.3 On the variables τc and ε c ...............................................................................5 - 4 5.4 Differential law for 1D-creep ..........................................................................5 - 6 5.5 Three-dimensional-model..............................................................................5 - 8 5.6 Formulation of elastic 3D-strains....................................................................5 -10 5.7 Review of model parameters.........................................................................5 -11 5.8 Validation of the 3D-model...........................................................................5 -14
6
The Soft-Soil model............................................................................................6 - 1 6.1 Isotropic states of stress and strain (σ1' = σ2' = σ3') ........................................6 - 1 6.2 Yield function for triaxial stress state (σ2' = σ3')...............................................6 - 3 6.3 Parameters in the Soft-Soil model.................................................................. 6 – 5
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7
Applications of advanced soil models ................................................................7 - 1 7.1 HS model: response in drained and undrained triaxial tests...............................7 - 1 7.2 Application of the Hardening-Soil model on real soil tests................................7 - 6 7.3 SSC model: response in one-dimensional compression test.............................7 -13 7.4 SSC model: undrained triaxial tests at different loading rates ...........................7 -18 7.5 SS model: response in isotropic compression test...........................................7 -20 7.6 Submerged construction of an excavation with HS model...............................7 -23 7.7 Road embankment construction with the SSC model......................................7 -25
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References..........................................................................................................8 - 1
A
Appendix A - Symbols .......................................................................................A - 1
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MATERIAL MODELS MANUAL
7 APPLICATIONS OF ADVANCED SOIL MODELS
In this chapter, advanced soil models will be used in various applications in order to illustrate the particular characteristics of those models. For applications of the standard Mohr-Coulomb model, the reader is referred to the Tutorial Manual. 7.1 HS MODEL: RESPONSE IN DRAINED AND UNDRAINED TRIAXIAL TESTS In this Section the behaviour of the Hardening-Soil model is illustrated in simulations of drained and undrained triaxial tests using arbitrary sets of model parameters, representing sands of various densities. The parameters for the different data sets are listed in Table 7.1. The main differences involve the stiffness, the friction angle and the dilatancy. Table 7.1 Arbitrary Hardening-Soil parameters for sands of different densities Parameter
Loose
Medium
Dense
Unit
E50ref
(for pref = 100 kPa)
20000
30000
40000
kN/m2
Eurref (for pref = 100 kPa) Eoedref (for pref = 100 kPa) Cohesion c Friction angle ϕ
60000 20000
90000 30000
120000 40000
kN/m2 kN/m2
0.0 30
0.0 35
0.0 40
kN/m2 °
Dilatance angle ψ Poisson's ratio ν ur
0 0.2
5 0.2
10 0.2
° -
Power m K0nc (using Cap) Tensile strength
0.5 0.5 0.0
0.5 0.43 0.0
0.5 0.36 0.0
kN/m2
Failure ratio
0.9
0.9
0.9
-
A triaxial test can simply be modelled by means of an axisymmetric geometry of unit dimensions (1m x 1m; see Fig. 7.1), representing a quarter of the soil sample. These dimensions are not realistic, but they are just selected for simplicity. The modelling does not influence the results, provided that the soil weight is not taken into account. In fact, in this configuration the stresses and strains are uniformly distributed over the geometry. The values of the deformation in x- and y-direction of the top right hand corner correspond to the horizontal and vertical strains respectively. The left hand side and the bottom of the geometry are axes of symmetry.
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At these boundaries the displacements directions normal to the boundary are fixed and the tangential displacements are free so that these sides are 'smooth'. The remaining boundaries are fully free to move. On these remaining boundaries separate distributed loads with input values of 1 kN/m2 are entered. The applied values of the vertical and horizontal loads are controlled by the load multipliers ΣMloadA and ΣMloadB respectively. The values of these multipliers correspond to the absolute values of the principal stresses σ1 and σ3. A very course mesh is sufficient for this simple geometry. Initial stresses and steady pore pressures are not taken into account.
Figure 7.1 Simplified configuration of a triaxial test In the first calculation phase the sample is isotropically compressed up to a confining pressure of p' = 100 kN/m2 assuming drained behaviour (ΣMloadA = ΣMloadB = 100). After this phase the displacements are reset to zero. In the second phase the sample is vertically loaded up to failure while the horizontal load is kept constant. The latter phase is done for drained as well as for undrained material behaviour. These calculations are performed for the three different sets of parameters. The results of these tests are presented in the graphs of Fig. 7.2 and 7.3. Fig. 7.2 shows the development of the principal stress difference as a function of the axial strain. The two left curves present the drained tests. These curves show a hyperbolic relationship between the stress and the strain, which is typical for the Hardening-Soil model. Obviously, the failure level is higher when the sand is denser. The HS model does not include softening behaviour, so after reaching failure the stress level does not reduce, at least in the drained tests. In the undrained tests the failure level is, in principle, lower than in the drained tests. However, on the medium and the dense sands the stress level continues to increase after reaching the failure level due to the fact that dilatancy causes a reduction of excess pore pressures and thus an increase of effective stresses.
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The lower right graph shows the development of the excess pore pressures, which confirms the above statement. The lower left graph presents the relationship between the axial strain and the volume strain. This graph clearly shows the influence of dilatancy of the denser sands. In contrast to the MohrCoulomb model, the transition from elastic behaviour to failure is much more gradual when using the Hardening-Soil model. In fact, in the HS model, plastic straining occurs from the onset of loading.
Figure 7.2a
Results of drained triaxial tests using the Hardening-Soil model, Principal stress difference versus axial strain
Figure 7.2b
Results of drained triaxial tests using the Hardening-Soil model, Volumetric strain versus axial strain
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Figure 7.2c
Results of undrained triaxial tests using the Hardening-Soil model, Principal stress difference versus axial strain
Figure 7.2d
Results of undrained triaxial tests using the Hardening-Soil model, Excess pore pressure vs axial strain
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Fig. 7.3 shows the effective stress paths during the whole test for the dense sand. The first phase of the test is all drained. In the second phase there is a clear distinction between the drained test and the undrained test. In the undrained test the effective horizontal stress reduces while the vertical stress increases due to the development of excess pore pressures. The decrease in horizontal effective stress is more than when the Mohr-Coulomb model would have been used. This is because of the plastic compaction (Cap hardening) in the HS model.
Figure 7.3 Stress paths for drained and undrained triaxial tests using the HS model
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7.2 APPLICATION OF THE HARDENING-SOIL MODEL ON REAL SOIL TESTS In this Section the Hardening-Soil model is subjected to simulations of various laboratory tests on sand in order to validate the response with measured test data. Extensive lab tests were performed on loose and dense Hostun sand. On the basis of these tests the model parameters for the Hardening-Soil model were determined, see also Table 7.2. Table 7.2 Hardening-Soil parameters for loose and dense Hostun sand Parameter
Loose sand
Dense sand
Unit
Volumetric weight γ
17
17.5
kN/m3
E50ref (pref = 100 kPa) Eurref (pref = 100 kPa) Eoedref (pref = 100 kPa) Cohesion c Friction angle ϕ
20000 60000 16000 0.0 34
37000 90000 29600 0.0 41
kN/m2 kN/m2 kN/m2 kN/m2 °
Dilatancy angle ψ Poisson's ratio ν ur Power m K0nc
0 0.20 0.65 0.44
14 0.20 0.50 0.34
° -
Tensile strength Failure ratio
0.0 0.9
0.0 0.9
kN/m2 -
Triaxial test The first test that is considered is a standard drained triaxial test on both the loose sand and the dense sand. For the simulation of triaxial tests the same configuration is used as the one presented in Fig. 7.1. In the first phase the sample is isotropically compressed up to a confining pressure of p' = 300 kN/m2. In the second phase the sample is vertically loaded up to failure while the horizontal stress is kept constant. The result of these tests and the comparison with the measured data are presented in Fig. 7.4. The calculational results show a reasonable agreement with the test data. There is a gradual transition from elastic to plastic behaviour. The relation between the deviatoric stress and the axial strain can well be approximated by a hyperbola. The failure level is fully controlled by the friction angle (the cohesion is zero). The test results on dense sand show softening behaviour after the peak load has been reached.
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Softening is not included in the Hardening-Soil model, so in contrast to the test data the deviatoric stress remains at the same level. It can also be seen from the test data that the dilatancy reduces during the softening traject. In the Hardening-Soil model the dilatancy continues to infinity, unless the dilatancy cut-off option has been used.
Figure 7.4a
Results of drained triaxial tests on loose Hostun sand, principal stress ratio versus axial strain
Figure 7.4b
Results of drained triaxial tests on loose Hostun sand, volumetric strain versus axial strain
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Figure 7.5a
Results of drained triaxial tests on dense Hostun sand, principal stress ratio versus axial strain
Figure 7.5b
Results of drained triaxial tests on dense Hostun sand, volumetric strain versus axial strain
Oedometer test The second test is an oedometer test on both the loose sand and the dense sand. For the simulation of the oedometer test a simplified axisymmetric geometry is considered with dimensions of 1m x 1m (see Fig. 7.6). At the left hand side, the right hand side and the bottom of the geometry the displacements normal to the boundary are fixed whilst the tangential displacements are free so that these boundaries are 'smooth'. The displacements of the upper boundary are free and a distributed load with an input value of 1 kN/m2 is applied here.
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Figure 7.6 Simplified configuration of an oedometer test The same parameters for the Hardening-Soil model are selected as in the triaxial tests (loose and dense sand; see Table 7.1). The type of behaviour is Drained. A course mesh may be generated for this simple geometry. Initial stresses and steady pore pressures are not taken into account. From a stress free state the loose sample is subsequently loaded to loads of 25 kPa, 50 kPa, 100 kPa and 200 kPa with intermediate unloadings. The dense sample is loaded to loads of 50 kPa, 100 kPa, 200 kPa and 400 kPa with intermediate unloadings. The results of these tests are presented in Figs. 7.7 and 7.8.
Figure 7.7
Results of oedometer test on loose Hostun sand, axial stress versus axial strain
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Figure 7.8
Results of oedometer test on dense Hostun sand, axial stress versus axial strain
The above results show a reasonable agreement with the test data. A small offset of 0.15% has been applied to the calculational results of the loose sample in order to account for the relative soft response at the beginning of the test. The results show that it is well possible to back-calculate both triaxial tests and oedometer tests. No doubt, distinction should be made between loose and dense soil, but it seems that for a soil with a certain density the stiffness behaviour under different stress paths can be well captured with a single set of model parameters. A further illustration of this aspect will be considered in the next section. Pressiometer test The final test is a pressiometer test on the dense sand with the parameters as listed in Table 7.1. The pressiometer is placed in a circular calibration chamber with a diameter of 1.2 m and a height of 0.75 m. The pressiometer itself has a diameter of 44 m and a height of 160 mm. A large overburden pressure of 500 kPa is applied at the surface to simulate the stress state at larger depths. Only half of the problem is included in an axisymmetric geometry model. In addition to the overburden pressure (load A), the expansion of the pressiometer is simulated by imposing a horizontal distributed load (load B). Therefore the initial standard boundary conditions have to be changed near the pressiometer in order to allow for free horizontal displacements. Besides, a vertical interface is placed along the shaft of the pressiometer borehole and a horizontal interface just above the pressiometer to allow for a discontinuity in horizontal displacements. Both interfaces are rigid (Rinter = 1.0).
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Extra geometry lines are created around the pressiometer in order to locally generate a finer mesh. The geometry model is presented in Fig. 7.9.
Figure 7.9 Geometry model for pressiometer test After the generation of initial stresses, the vertical overburden load (load A) is applied using the standard boundary fixities. The lateral stress around the pressiometer appears to be 180 kPa. Subsequently, the horizontal fixity near the pressiometer is eliminated in the input. In order to restore equilibrium, an initial pressiometer pressure of 180 kPa (load B) is applied. In the next calculation the pressure (load B) is further increased in an Updated Mesh analysis. The results of this calculation are presented in Fig. 7.10 and 7.11.
Figure 7.10
Stress distribution in deformed geometry around the pressiometer at a pressure of 2350 kPa
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Fig. 7.10 shows a detail of the deformations and the stress distribution when the pressure in the pressiometer is 2350 kPa. The high passive stresses appear very locally near the pressiometer. Just above the pressiometer the vertical stress is very low due to arching effects. At a larger distance from the pressiometer a normal K0-like stress state exists.
Figure 7.11 Comparison of numerical results and pressiometer test data Fig. 7.11 shows a comparison of the numerical results with those of the real pressiometer test. In the graph the pressiometer pressure is presented as a function of the relative volume change. The latter quantity cannot directly be obtained from PLAXIS and was calculated from the original radius R0 and the lateral expansion ux of the pressiometer:
∆V V0
=
( R0 + u x )2 - R02 R 02
Up to a pressure of 1600 kPa the results match quite well. Above 1600 kPa there is a sudden decrease in stiffness in the real test data which cannot be explained. Nevertheless, the original set of parameters for the dense sand that were derived from triaxial testing also seem to match the pressiometer data quite well. Conclusion The above results have indicated that it is quite well possible to match the results of different tests following completely different stress paths with the same set of input parameters.
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This cannot be obtained with simple models like Mohr-Coulomb without changing input parameters. Hence, the parameters in the Hardening-Soil model are consistent and more or less independent from the particular stress path that is followed. This makes the HS model a powerful and accurate model that can be used in many applications.
7.3 SSC MODEL: RESPONSE IN ONE-DIMENSIONAL COMPRESSION TEST In this section the behaviour of the Soft-Soil-Creep model is illustrated on the basis of a onedimensional compression test on clay. In a first analysis the test is performed assuming drained behaviour to show the logarithmic relationships between strain and stress and the logarithmic timesettlement behaviour on the long term (secondary compression). In a second analysis the test is simulated more realistically by including undrained behaviour and consolidation. Since the consolidation process highly depends on the drainage length, it is important in this example to use realistic geometry dimensions. In this case the sample height is 0.01 m. The used axisymmetric configuration is presented in Fig. 7.12. The Soft-Soil-Creep model parameters used in this example are arbitrarily selected, but they are realistic for normally consolidated clay (see Table 7.3). The vertical preconsolidation stress is installed at 50 kPa (POP = 50 kPa).
Figure 7.12 Realistic configuration for one-dimensional compression test In the first analysis subsequent plastic loading steps are applied using drained behaviour. The load is doubled in every step using a time increment of 1 day. In this analysis it is not necessary to perform intermediate consolidation steps (because of the drained behaviour). After the final loading step an additional creep period of 100 days is introduced.
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The calculation scheme is listed in Table 7.4. All calculations are performed with a reduced tolerance of 1%. Table 7.3 Soft-Soil-Creep model parameters for one-dimensional compression test Parameter
Symbol
Value
Unit
Unit weight
γwet
19
kN/m3
Permeability Modified compression index Modified swelling index Secondary compression index Poisson's ratio
kx, ky λ* κ* µ* ν ur
0.0001 0.10 0.02 0.005 0.15
m/day -
Cohesion Friction angle Dilatancy angle Coeffient of lateral stress
c ϕ ψ K0nc
1.0 30 0.0
kN/m2 ° °
0.5
-
Table 7.4 Calculation scheme for first analysis Phase
Calculation type
ΣMloadA
ΣMtime
1
Plastic
10
1
2 3 4
Plastic Plastic Plastic
20 40 80
2 3 4
5 6
Plastic Plastic
160 320
5 6
7 8
Plastic Plastic
640 640
7 107
In the second analysis the loading steps are instantaneously applied using undrained behaviour. After each loading step a consolidation of 1 day is introduced to let the excess pore pressures fully dissipate. After the final loading step an additional creep period of 100 days is again introduced. The calculation scheme for this analysis is listed in Table 7.5. All calculations are performed with a reduced tolerance of 1%. Fig. 7.13 shows the load-settlement curve of both analyses. The results of the undrained test after consolidation exactly correspond with the results of the drained test.
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The influence of the preconsolidation stress can clearly be seen, although the transition between reloading and primary loading is not as sharp as when using the Soft-Soil model. In fact, the results presented here are more realistic. The transition is indeed around 50 kPa. Table 7.5 Calculation scheme for second analysis Phase
Calculation type
ΣMloadA
ΣMtime
1
Plastic
10
0
2 3 4 5 6
Consolidation Plastic Consolidation Plastic Consolidation
10 20 20 40 40
1 1 2 2 3
7 8 9 10 11
Plastic Consolidation Plastic Consolidation Plastic
80 80 160 160 320
3 4 4 5 5
12
Consolidation
320
6
13 14 15
Plastic Consolidation Consolidation
640 640 640
6 7 107
From the inclination of the primary loading line one can back-calculate the modified compression index λ* = ∆ε 1 / ln((σ1+∆σ1)/σ1) ≈ 0.10. Note that 1 mm settlement corresponds with ε 1 = 10%. For an axial strain of 30% one would normally use an Updated Mesh analysis. However, in PLAXIS an Updated Mesh analysis cannot be combined with a Consolidation analysis. If the Soft-Soil-Creep model would have been used in an Updated Mesh analysis with axial strains over 15% one would observe a stiffening effect as indicated by the dashed line in Fig. 7.13. Fig. 7.14 shows the time-settlement curve of the drained and the undrained analyses. From the last part of the curve one can back-calculate the secondary compression index µ* = ∆ε 1 / ln(∆t/t 0) ≈ 0.005 (with t0 = 1 day).
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Figure 7.13
Load-settlement curve of oedometer test with Soft-Soil-Creep model. A) Transient loading with doubling of loading within one day. B) Instantaneous loading with doubling of load at the beginning of a new day. C) As 'A' using Updated Mesh calculation
Figure 7.14
Time-settlement curve of oedometer test with Soft-Soil-Creep model. A) Transient loading with doubling of loading within one day. B) Instantaneous loading with doubling of load at the beginning of a new day
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Another interesting phenomenon is the development of the lateral stress. During primary loading, the lateral stress should be given by the value of K0NC, appropriate for normally consolidated soil. During unloading, the lateral stress decreases much less than the vertical stress, so that the ratio σ′xx / σ′yy increases. In order to show these effects the calculation is continued after the final step with a drained unloading phase down to a vertical stress of 80 kPa. Fig. 7.15 shows the stress state for two different calculation phases, both at a vertical stress level of 80 kPa. The left hand plot represents the stress state after primary loading In this case, the horizontal stress is found to be approximately 40 kPa. Hence, the stress ratio corresponds to the input value K0NC = 0.5. The right hand plot represents the final situation after unloading down to 80 kPa. In this case the horizontal stress is found to be approximately 220 kPa. The decrease of the horizontal stress (∆σ′xx = 100 kPa) during unloading is much less than the decrease of the vertical stress (∆σ′yy = 560 kPa), which results in a situation in which σ′xx is larger than σ′yy. During sudden unloading in a one-dimensional compression test, the behaviour is purely elastic. Hence, the ratio of the horizontal and vertical stress increments is determined by Poisson's ratio ν ur:
∆ σ ’ xx ν ur = ∆ σ ’ yy 1 - ν ur
(7.2)
It is easy to verify that the results correspond with Poisson's ratio ν ur = 0.15 as listed in Table 7.3.
Figure 7.15
Stress states at a vertical stress level of -80 kPa. Left, after primary loading σxx ≈ -40 kPa. Right, after unloading from -640 kPa σxx ≈ -220 kPa
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7.4 SSC MODEL: UNDRAINED TRIAXIAL TESTS AT DIFFERENT LOADING RATES In this section the Soft-Soil-Creep model is validated for its response in undrained triaxial tests at different rates of strain. The model parameters used in the current application are derived from the data of the Haney Clay as given in Chapter 5. The parameters for the Soft-Soil-Creep model were selected on the basis of the original test data and are listed in Table 7.6. Table 7.6 Soft-Soil-Creep model parameters for Haney clay Parameter
Symbol
Value
Unit
Modified compression index
λ*
0.105
-
Modified swelling index Secondary compression index Poisson's ratio Cohesion Friction angle
κ* µ* ν ur c ϕ
0.016 0.004 0.15 0.0 32
kN/m2 °
Dilatancy angle Coeffient of lateral stress Permeability
ψ K0nc Kx , k y
0.0 0.61
° -
0.0001
m/day
Figure 7.16
Modelling of triaxial test on Haney clay. Left, Initial configuration. Right, configuration for phase 9 - 11
The modelling of the triaxial test is similar as described in 7.1. Instead of a unit dimension, the real test dimensions are used (17.5 x 17.5 mm2). During the isotropic loading phases an external horizontal and vertical load (both System A) are applied.
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The isotropic loading phases consist of instantaneous undrained plastic calculations and consolidation analyses. The two external boundaries (top en right hand side) are draining whereas the other boundaries are closed. Before the first axial straining phase the vertical load is replaced by a prescribed displacement in vertical direction and free horizontal displacements at the top. At the beginning of axial loading the displacements are reset to zero. The axial loading is applied by activating the prescribed displacements at different velocities. In this way a total of 12% axial strain (2.1 mm) is applied in 8.865 days (0.00094%/minute), 0.0556 days (0.15%/minute) and 0.00758 day (1.10%/minute) respectively. These three axial loading phases all start from the end of the isotropic loading phase (8). The full calculation scheme is listed in Table 7.7. Table 7.7 Loading scheme of triaxial tests at different velocities Phase
Calculation
ΣMloadA
ΣMdisp
ΣMtime [day]
1 2 3 4 5
Plastic Consolidation Plastic Consolidation Plastic
65 65 130 130 260
0 0 0 0 0
0.00 0.01 0.01 0.02 0.02
6 7 8
Consolidation Plastic Consolidation
260 520 520
0 0 0
0.03 0.03 0.04
Replace vert. load by prescribed vert. displ. of 1 m and free hor. Directions 9 10 (start from 8)
Plastic Plastic
520 520
0.0021 0.0021
8.905 (+8.865) 0.0956 (+0.0556)
11 (start from 8)
Plastic
520
0.0021
0.04758 (+0.00758)
The results of the triaxial tests are presented in Fig. 7.17 and 7.18. Fig. 7.17 shows a stress-strain curve of the axial loading phase, from which it can be seen that the shear strength highly depends on the rate of straining. For lower strain rates the shear strength is higher than for higher strain rates. Fig. 7.18 shows the p-q stress paths during axial loading. For higher strain rates there is a smaller reduction of the mean effective stress, which allows for a larger ultimate deviatoric stress. It should be noted that the stress state is not homogeneous at all, because of the inhomogeneous (excess) pore pressure distribution. This is due to the fact that points close to draining boundaries consolidate faster than points at a larger distance.
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Figure 7.17
Average deviatoric stress versus axial strain for different rates of straining
Figure 7.18
p-q stress paths for different rates of straining for a point at position (0.01 , 0.01)
7.5 SS MODEL: RESPONSE IN ISOTROPIC COMPRESSION TEST In this Section it will be demonstrated that the Soft-Soil model obeys a logarithmic relationship between the volumetric strain and the mean stress in isotropic compression. For this example the same axisymmetric geometry with unit dimensions is used as the one presented in Fig. 7.1. In fact, the vertical load (A) and the horizontal load (B) are simultaneously applied to the same level so that a fully isotropic stress state occurs.
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The parameters of the Soft-Soil model are chosen arbitrarily, but the values are realistic for normally consolidated clay. The parameters are given in Table 7.8. Table 7.8 Model parameters for isotropic compression test Modified compression index
λ*
0.10
Modified swelling index Poisson's ratio Friction angle Cohesion Normally consolidated K0
κ* ν ur ϕ c K0NC
0.02 0.15 30° 1.0 kPa 0.5
From a stress-free state, the model is isotropically loaded to a mean stress of p′ = 100 kPa, after which the displacements are reset to zero. This is the starting point of the test. The material is now in a 'normally consolidated' state, i.e. the preconsolidation stress is equivalent with the current stress state. From this point, the isotropic pressure is increased to p′ = 1000 kPa. This loading path is denoted as 'primary loading'. After this, the sample is isotropically 'unloaded' to p′ = 100 kPa. Finally, the sample is loaded to p′ = 10000 kPa. In this loading path, the maximum preload of 1000 kPa is exceeded. Hence, this final loading path consists of two parts: the portion of the loading path for which p′ < 1000 kPa is referred to as 'reloading' and the portion of the loading path for p′ > 1000 kPa consists of further primary loading. The different calculation phases are specified in Table 7.9. Table 7.9 Calculation phases for isotropic compression test on clay Stage
Initial stress
Final stress
0 1
Initial situation Primary loading
p0 = 100 kPa
p0 = 100 kPa p1 = 1000 kPa
2 3 4
Unloading Reloading Primary loading
p1 = 1000 kPa p2 = 100 kPa p3 = 1000 kPa
p2 = 100 kPa p3 = 1000 kPa p4 = 10000 kPa
Results of the calculation are presented in Fig. 7.19. This plot gives the relation between the vertical strain ε yy and the vertical stress σ′yy. The latter quantity is plotted on a logarithmic scale. The plot shows two straight lines, which indicates that there is indeed a logarithmic relation for loading and unloading. The vertical strain is a third of the volumetric strain, ε v, and the vertical stress is equal to the mean stress, p′. The volumetric strains that follow from the calculation are given in Table 7.10.
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Figure 7.19 Results of isotropic compression test Table 7.10 Volumetric strains from various calculation phases Phase
Initial strain
Final strain
0 1
ε v0 = 0.000
2
ε v1 = - 0.235
ε = 0.000 ε v1 = - 0.235 ε v2 = - 0.188
3 4
ε v2 = - 0.188 ε v3 = - 0.235
ε v3 = - 0.235 ε v4 = - 0.471
0 v
From these strains and corresponding stresses, the parameters λ* and κ* can be back-calculated using Eqs. (5.1) and (5.2). Phase 1 λ* = -
0.235 ε 1v - ε 0v = 0.102 = ln ( p 1 / p 0 ) ln ( 1000 / 100 )
Phase 2 κ * = -
0.188 - 0.235 ε 2v - ε 1v = 0.020 = 2 1 ln ( p / p ) ln ( 100 / 1000 )
Phase 3 κ * = -
0.235 - 0.188 ε 3v - ε 2v = = 0.020 3 2 ln ( p / p ) ln ( 1000 / 100 )
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Phase 4 λ* = -
0.471 - 0.235 ε 4v - ε 3v = = 0.102 4 3 ln ( p / p ) ln ( 10000 / 1000 )
The back-calculated values correspond to the input values as given in Table 7.8. Note that the Soft-Soil model does not include time effects like secondary compression. The latter type of behaviour is included in the Soft-Soil-Creep model. An example of an application with this model is described in section 7.7.
7.6 SUBMERGED CONSTRUCTION OF AN EXCAVATION WITH HS MODEL In this example a particular advantage of the Hardening-Soil model is demonstrated, namely the distinction between loading and unloading stiffness. This feature becomes particularly important in quasi unloading problems like excavations and tunnels. For example, in Lesson 2 of the Tutorial Manual (submerged excavation) the heave at the bottom of the excavation is unrealistically large. In order to show that the results are better when using the Hardening-Soil model, this example is used here again. For convenience, the situation of Lesson 2 is redisplayed in Fig. 7.20. The geometry model is similar as the one used in Lesson 2, but instead of the Mohr-Coulomb model the two soil layers are modelled here with the Hardening-Soil model. The applied model parameters are listed in Table 7.11.
Figure 7.20. Geometry model of submerged excavation
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PLAXIS
Table 7.11 HS model parameters for the two layers in the excavation project Parameter
Clay layer
Sand layer
Unit
Volum. weight γdry/γwet
16 / 18
17 / 20
kN/m3
E50ref
(pref = 100 kPa)
8000
30000
kN/m2
Eurref
(pref = 100 kPa)
24000
90000
kN/m2
Eoedref (pref = 100 kPa)
4000
30000
kN/m2
Cohesion c
5.0
1.0
kN/m2
Friction angle ϕ
25
32
°
Dilatancy angleψ
0
2
°
Poisson's ratio ν ur
0.2
0.2
-
Power m
0.8
0.5
-
K0
0.5
0.47
-
Tensile strength
0.0
0.0
kN/m2
Failure ratio
0.9
0.9
-
nc
After the generation of the initial pore pressures and effective stresses, the excavation is executed in two phases. The results of the excavation are shown in Fig. 7.21.
Figure 7.21 Deformed mesh after excavation
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MATERIAL MODELS MANUAL
The deformed mesh clearly shows that there is a limited heave of the bottom of the excavation. Most of the deformation is caused by the horizontal movement of the diaphragm wall which pushes the soil up. The vertical heave at the bottom further away from the wall is very low compared to the results as presented in the Tutorial Manual for the Mohr-Coulomb model. The difference can be explained by the fact in contrast to the Mohr-Coulomb model the Hardening-Soil model distinguishes between loading and an unloading stiffness. The unloading stiffness is selected here as three times higher than the loading stiffness, which is the default setting in PLAXIS.
7.7 ROAD EMBANKMENT CONSTRUCTION WITH THE SSC MODEL This example demonstrates some of the particular features of the Soft-Soil-Creep model in a practical application. One of these features is the reduction of the mean effective stress during undrained loading due to compaction of the soil. This feature becomes particularly important in embankment construction projects, since it highly influences the stability of the embankment during construction. For example, in Lesson 5 of the Tutorial Manual (construction of a road embankment) the safety factor is low during construction. When using the Soft-Soil-Creep model for the clay and peat layer with similar effective strength properties as for the Mohr-Coulomb model, the safety factor will be even lower. In order to show these effects this example is used here again. For convenience, the situation of Lesson 5 is redisplayed in Fig. 7.22. The geometry model is similar as the one used in Lesson 5, but for the clay layer the Soft-Soil-Creep model is used. The applied model parameters for this layer are listed in Table 7.12. The peat layer and the sand embankment are still modelled with the Mohr-Coulomb model using the same parameters as given in Lesson 5 of the Tutorial Manual.
Figure 7.22 Geometry model of road embankment project
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Table 7.12 Soft-Soil-Creep model parameters for undrained clay layer Parameter Clay layer 15 Dry soil weight γdry Wet soil weight γwet Horizontal permeability k x Vertical permeability k y Modified compression index λ* Modified swelling index κ* Secondary compression index µ* Poisson's ratio ν ur Cohesion c Friction angle ϕ Dilatancy angle ψ Coeffient of lateral stress K0NC
18 1•10-4 1•10-4 0.035 0.007 0.002 0.15 2 4 0 0.59
Unit kN/m3 kN/m3 m/day m/day kN/m2 ° ° -
The clay layer is lightly overconsolidated. In the generation of initial stresses, an overconsolidation ratio (OCR) of 1.3 is used for this layer. In this case a total of four calculation phases are executed. The first phase is the construction of the first part of the embankment. This is done in 1 day, so in addition to the staged construction setting in the plastic calculation, the parameter ΣMtime is set to 1.0. Time is relevant since the Soft-SoilCreep model is used here. The second phase is a consolidation analysis to an ultimate time of 200 days. For this analysis the manual setting has to be selected in which the desired maximum parameter is set to 50 and the first time increment to 1 day. This is necessary to improve the numerical performance of the calculation process because in the beginning of the consolidation analysis the situation is near failure. The third phase is the construction of the second part of the embankment, again with a time increment of 1 day (ΣMtime = 201). In this calculation the number of additional steps should be set to 30 in order to avoid unnecessary calculation steps. In contrast to the results described in the Tutorial Manual, the embankment collapses in the current calculation phase. The final phase is a safety analysis of the first construction stage. This is a plastic calculation, load advancement number of steps, starting from phase 1 and the loading input is phi-c-reduction. A summary of all calculation phases is given in Table 7.13. At the end of the first construction phase there is indeed a reduction of the mean effective stress, due to irreversible compaction (volume creep strain).
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MATERIAL MODELS MANUAL
In the beginning of the consolidation analysis the construction is near failure. Although the excess pore pressures can dissipate at the upper and lower boundaries, there is a further increase of excess pore pressures in the middle of the clay layer due to secondary compression. This part of the calculation is very difficult and requires a manual setting of the control parameters. The construction of the second part of the embankment cannot be completed because of soil failure. This is also due to the decreasing mean effective stress as mentioned above. Fig. 7.23 shows a plot of the failure mechanism that occurs in this phase. The phi-c-reduction analysis gives a safety factor of 1.10 for the first part of the embankment construction. This is indeed less than the value of 1.17 in the Tutorial Manual, which was based on a Mohr-Coulomb modelling of the clay layer. Since failure occurs during the construction of the second part of the embankment, it is not possible to do a safety analysis for this phase (Safety factor < 1). Table 7.13 Overview of calculation phases Phase Start Type 1 0 Plastic, staged construction 2 1 Consolidation, ultimate time 3 2 Plastic, staged construction 4 1 Plastic, phi-c-reduction
Figure 7.23
Loading 1st embankment part
ΣMtime 1 day
Desired maximum 50 First time incr. = 1 day 2nd embankment part
200 days
Msf = 0.1
1 day
201 days
Failure mechanism during construction of second part of the embankment
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Conclusion The reduction of mean effective stress during undrained loading is a known phenomenon in soil soils. This phenomenon has a negative influence on the strength and stability of the soil structure. When using simple soil models, like the Mohr-Coulomb model, this effect is not taken into account, which leads to an overprediction of the stability when using effective strength properties. In such cases it is better to use undrained strength properties in the Mohr-Coulomb model (c = cu and ϕ = 0). The Soft-Soil-Creep does include the effect of reducing mean effective stress during undrained loading. This model gives a more realistic prediction of soft-soil behaviour, including time-dependent behaviour (secondary compression and consolidation). Disadvantage of this model are the fact that no undrained strength properties can be specified (only c' and ϕ') and that the numerical procedure becomes more complicated (less robust) when soil failure is approached.
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