Large deformation analysis of cone penetration testing in undrained clay

Transcription

1 Installation Effects in Geotechnical Engineering Hicks et al. (eds) 13 Taylor & Francis Group, London, ISBN Large deformation analysis of cone penetration testing in undrained clay L. Beuth Deltares, Delft, The Netherlands P.A. Vermeer Deltares, Delft, The Netherlands University of Stuttgart, Germany ABSTRACT: Cone penetration testing is a widely-used in-situ test for soil profiling as well as estimating soil properties of strength and stiffness. In this paper, the relationship between the undrained shear strength of clay and the measured cone tip resistance is investigated through numerical analysis. Such analyses serve to refine and establish correlations between cone penetration testing measurements and soil properties; thus enabling more reliable predictions of soil properties. The presented analyses are performed by means of a Material Point Method that has been developed specifically for the analysis of quasi-static geotechnical problems involving large deformations of soil. Both, the load-type dependency of the shear strength of undrained clay as well as the influence of the anisotropic fabric of natural clay on the undrained shear strength are taken into account through a new material model, the Anisotropic Undrained Clay model. Results indicate that the deformation mechanism relevant for cone penetration in undrained normally-consolidated clay differs significantly from predictions based on the Tresca model, but resulting cone factors appear to be useful. 1 INTRODUCTION The undrained shear strength of soil, s u, represents no unique soil parameter, but largely depends on the type of loading. For normally-consolidated clays, the undrained shear strength found for triaxial compression is for instance much larger than the strength found for triaxial extension. Simple shear tests render an undrained shear strength that lies in between the strength values obtained for triaxial compression and extension. This has to be taken into consideration when using this parameter in geotechnical analyses. When deriving an undrained shear strength from cone penetration measurements of tip resistance, q c, it is important to know which undrained shear strengths dominate the failure mechanism found during cone penetration testing. In this paper, results of numerical analyses of cone penetration testing (CPT) in normally-consolidated undrained clay are presented to provide new insight into the mechanical processes that occur during a cone penetration test. An accurate computation of the stress field that evolves in the vicinity of the penetrating cone requires one to take into account the complex large deformation processes imposed by the cone on the surrounding soil. Because the cone is pushed into the ground with a constant rate of penetration of cm/s, inertia and damping effects can be neglected. Thus, the considered problem fullfills the requirements of quasi-static analysis. The method used in this study, the quasi-static Material Point Method (MPM), has been developed specifically for the analysis of such problems. Furthermore, the non-linear stress-strain relationship of undrained clay must be considered. When using the well-known elastic-plastic Tresca material model for such analyses, s u is specified as the cohesion parameter of the model. Such computations assume an undrained shear strength that is independent of the loading path. Generally, an undrained shear strength obtained from simple shear tests is used as a kind of average for such simplifed analyses. Although the widely-used Mohr-Coulomb model does predict a higher undrained shear strength for triaxial compression than for triaxial extension, it overpredicts the magnitudes of both undrained shear strengths, at least for normallyconsolidated clays. In order to reproduce the magnitudes of undrained shear strengths for triaxial 1

2 compression and extension more accurately, a model based on critical state soil mechanics such as the well-known Modified Cam Clay model might be used. This model takes into account the shear-induced volume change of drained clay (Schofield & Wroth 1968). Indeed, for normallyconsolidated clay as considered in this paper, this model correctly predicts lower undrained shear strengths for triaxial compression and extension than the Mohr-Coulomb model. Due to the deposition process of clay, natural clay generally shows a different strength in horizontal directions than in the vertical direction. In order to further increase the accuracy of CPT analyses, this anisotropic strength of clay must also be taken into consideration. Therefore, the Anisotropic Undrained Clay (AUC) model is used in the presented analyses. It implements the theory of critical state soil mechanics and also considers the strength anisotropy of natural clay (Vermeer et al. 1; Beuth 1). It is largely based on the S-CLAY1 model developed by (Wheeler et al. 3) but considers neither density and rotational hardening nor softening. To the authors knowledge, usage of a quasistatic MPM in combination with the AUC model for CPT analysis exceeds the accuracy of numerical studies reported so far in literature, such as (Van den Berg 1994) and (Lu et al. 4). In the following section, the used numerical method will be introduced briefly. A detailed description of it would exceed the scope of this paper. For further information, the reader is therefore referred to (Beuth et al. 7; Vermeer et al. 9; Beuth et al. 11) and (Beuth 1). The constitutive modelling of undrained clay is treated in Section 3. Subsequently, the geometry and discretisation of the performed numerical analyses are presented in Section 4. In Section 5, results obtained for the analysis with the AUC model are compared to results for the Tresca model. The paper ends with an outlook on possible future work. QUASI-STATIC MATERIAL POINT METHOD The quasi-static MPM can be considered as an extension of the classical Updated Lagrangian Finite Element Method (UL-FEM). With the UL- FEM, a solid body is discretised by finite elements that follow the deformations of the solid body. In case of large deformations of the solid body, the finite element grid might eventually experience severe distortions which lead to numerical inaccuracies and can even render the calculation impossible. The Material Point Method discretises a solid body by means of a cloud of material points that move through a fixed finite element grid. Thereby, the material points capture the arbitrary large deformations of the solid body without the occurrence of severe deformations of the finite elements. Material and state parameters of the solid body as well as applied loads are stored in material points whereas the mesh does not store any permanent information. With the MPM, the underlying finite element grid is used as with the UL-FEM to solve the system of equilibrium equations for an applied load increment on the basis of those finite elements that contain material points at the considered loading step. Once displacement increments are mapped from nodes to material points, once strain increments are computed at the locations of material points, the mesh is usually reset into its original state. It might be changed arbitrarily. Obviously, the finite element grid used with the MPM must cover not only the solid in its initial configuration as with the UL-FEM but the entire region of space into which the solid is expected to move. It should be emphasised, that in contrast to mesh-less Lagrangian methods such as the Discrete Element Method, the material points represent subregions of a solid body and not individual particles such as sand grains. The mass and volume of subregions is memorised with material points, but changes in the shape are not traced. With one exception (Guilkey & Weiss 3), existing implementations of the MPM are dynamic codes that employ an explicit time integration scheme (Sulsky et al. 1994; Wi eckowski et al. 1999; Coetzee et al. 5). Using these codes for the analysis of quasi-static problems is computationally inefficient as explicit integration requires very small time steps. The quasi-static MPM makes use of an implicit integration scheme and thus circumvents the limitation on step size of dynamic codes. In recent years, the quasi-static MPM has been validated with numerous geotechnical benchmark problems. Furthermore, it has been extended by a contact formulation for modelling reduced frictional or adhesive contact between structure and soil based on interface elements (Vermeer et al. 9). 3 CONSTITUTIVE MODELLING 3.1 Modelling of undrained elasticity The total mean stress rate of the undrained soil is split into the effective mean stress rate, p, and the change of excess pore pressures, p w, through consideration of strain compatibility between the two materials

3 K p w = water εvol and p K n = ε (1) vol where K water is the bulk modulus of water, n is the porosity and K is the bulk modulus of the soil skeleton. The term K water /n can be written as K n water = 3( vu v ) 1 v 1 v ( )( + ) u K () where v u is an undrained Poisson ratio and v the effective Poisson ratio of the soil skeleton. Incompressibility of the soil implies v u being close to.5. In this paper, it is taken to be.49 for the computation with the Tresca model and.495 for the computation with the AUC model. The effective Poisson ratio is assumed to be v =.5 and the Young s modulus is E = 6 MPa. This yields K = 4 MPa and K water /n = 11 MPa for v u =.49 and 35 MPa for v u =.495. It should be noted that elastic behaviour (inside the yield surface) is assumed to be isotropic according to Hooke s law. 3. Modelling of undrained plasticity The AUC model requires 4 input parameters: the elastic parameters, E and v, the friction angle for the critical state, φ cs, and the overconsolidation ratio, OCR, defined as the preconsolidation stress, σ p, divided by the vertical effective stress. In this paper, only normally-consolidated clay with OCR = 1 is considered, φ cs is taken to be 3. The yield function of the AUC model is defined as 3 α α s p s s p s q : q f = p + p p M α p T ( ) (3) where s is the deviatoric stress tensor, and s is the initial in-situ one. Furthermore, it yields 6sinφcs 3 T M and q s : s (4) 3 sinφ = = ( ) cs For special stress loading paths with s proportional to s, implying q proportional to q, the yield function can be written in the simpler form ( q α p ) f = p + p p (5) M α p ( ) Figure 1. Yield locus of the AUC model for triaxial compression and extension. giving the well-known Modified Cam Clay yield condition for α =, i.e. for isotropic consolidation with K = 1. In this particular case, the yield condition is represented by an isotropic ellipse in p -q-space. For non-isotropic consolidation, the yield locus is rotated away from the p -axis, as illustrated in Figure 1. In the special case of 1-dimensional consolidation, the rotation parameter, α, follows from η + 3η M α = 3 η 3 1 K with = (6) 1+ K where K is the coefficient of earth pressure at rest. In this paper, normally-consolidated clay is considered with K = 1 - sin φ cs. Only associated plasticity is considered, so that a separate specification of a plastic potential function is not needed. For OCR = 1, as considered in the following, the isotropic preconsolidation stress, p po, can be initialised for the initial stress state from the condition f =. 4 GEOMETRY AND DISCRETISATION A slice of the axisymmetric cone penetration problem is considered as shown in Figure. The mesh extends outwards by 14 cone diameters D, D being for the standardised cone penetrometer 3.57 cm. In the initial configuration, the cone tip is located 4.9 cm below the soil surface which initially coincides with the cone base, see Figure 3 (b). The cone is slightly rounded in order to circumvent numerical problems induced by a discontinuous edge at the base of the cone. Apart from this modification, the dimensions of the penetrometer correspond to those of a standard penetrometer: the apex angle is 6 and the horizontal base area is 1 cm. 3

4 Figure. (a) Material point discretisation for the initial; (b) Final configuration at a penetration depth of 18 D. Figure 3. (a) FE discretisation with local mesh refinement along the cone tip; (b) Geometry of the cone tip. The height of the discretised space has been chosen so as to accomodate a penetration depth of up to 18 D measured from the base of the cone. The lower mesh boundary is located 18 D below the tip of the cone. Considering a rigid penetrometer, there is no need to discretise the penetrometer itself which simplifies the application of boundary conditions along the penetrometer surface. Rather than incrementally moving the penetrometer downwards into the ground, the soil is pushed upwards against the fixed penetrometer surface. The upward displacement of the material point discretisation is realised by prescribed displacements applied on the bottom boundary of the mesh. Material points are continuously fed along the lower boundary into the mesh from a cloud of material points that is initially located outside the discretised region (feeder). Figure shows the material point discretisation including the feeder in its initial and final configuration. Along the sides of the mesh, displacements are fixed normal to the vertical planes (roller boundaries). Only smooth contact between soil and penetrometer is considered by prescribing roller boundaries fixed normal to the penetrometer surface. Reduced adhesive or frictional contact along the penetrometer surface might be considered by placing interface elements along the fixed boundary of the penetrometer but this is not considered here. Soil weight is not considered with the presented computations since the gradient of vertical stress is not relevant at the greater depths of cone penetration that are reached in the performed analyses. Because the computation with the AUC model requires initial stresses for initialisation of the extent and orientation of the yield surface in principal stress space, a surface traction of 1 kpa is applied. K is set to.5. Thus, in an initial loading stage, a vertical stress of 1 kpa and horizontal stresses of 5 kpa are realised. For the AUC model, the isotropic preconsolidation pressure is then 7.1 kpa. The computations were performed with 4-noded tetrahedral elements that were extended by a strain-smoothing algorithm to prevent locking. A mesh with 15,678 degrees of freedom has been used. Mesh re-finement has been applied in the region around the penetrometer cone as shown in Figure 3 (a) where large stress gradients occur. Furthermore, a mesh re-finement is required on the narrow edge along the longitudinal axis of the discretised slice. Initially, the soil surface coincides with element boundaries adjacent to the cone base, see Figure 3 (b). An equal number of 1 material points is initially placed inside each finite element. In total, 86,41 material points are used for the performed analyses. 4

5 5 RESULTS FEM simulations of triaxial compression and extension tests were performed with the AUC model for anisotropically consolidated soil under K =.5 up to σ p = 1 kpa. Similarly, a direct shear test was computed for K -consolidation up to σ p = 1 kpa in order to obtain the respective values of undrained shear strength relevant for the performed CPT analyses. The resulting undrained shear strengths are for triaxial compresssion s u,comp = 3 kpa, for triaxial extension s u,ext = 1. kpa and for simple shear s u,ss =.15 kpa (see also Figure 1). For the computation with the Tresca model, an undrained shear strength of 4 kpa is specified. An overview of the undrained shear strengths relevant for the two computations is provided in Table 1. In Figure 4 the obtained relative tip stresses σ c /s u,ss for the two computations are plotted over the relative cone penetration. Once the influence of the soil surface on the movement of soil particles around the penetrating structure subsides, a steady-state deformation process evolves, as can be seen in Figure 4. The tip stress σ c for the fully developed steady-state penetration process corresponds to the tip resistance q c measured during cone penetration testing. The relative tip resistance q c /s u corresponds to the cone factor N c which is commonly used to correlate the undrained shear strength of soil to measurements of cone tip resistance. Equating s u to s u,ss results in approximately the same cone Table 1. Undrained shear strengths [kpa]. s u,comp s u,ext s u,ss Tresca AUC Figure 4. Relative tip stress plotted over the relative cone displacement for the Tresca and AUC model. factor for the computations with the Tresca and AUC model. For the Tresca model, a cone factor of 1.5 is obtained, for the AUC model a cone factor of The Tresca cone factor of 1.5 agrees well with results of previous numerical studies based on this model. Indeed, (Van den Berg 1994) reports for a smooth cone and a rigidity index I r = G/ s u = 6 a cone factor of approximately 13. (Lu et al. 4), who additionally take into consideration the initial stress state, report a matching cone factor of 1.5. Total stress analyses were considered in these studies whereas in the computations of this paper distinction is made between effective stresses and excess pore pressures as described in Subsection 3. Supplementary numerical analyses show that the choice of analyses has little impact on the obtained cone factors, at least for undrained Poisson ratios above a value of.49 (Beuth 1). Figure 5 (a) shows the loading types found within the soil body at a penetration depth of 1 D; on the left half for the AUC model, on the mirrored right half for the Tresca model. Points colored dark grey represent regions of triaxial compression statesof-stress. Light grey points represent regions that cannot be associated with a specific loading type. For the Tresca model, triaxial compression statesof-stress dominate in the vicinity of the cone. The AUC model predicts triaxial compression statesof-stress to occur only in a small wedge below the advancing cone. Figure 5 (b) shows the effective vertical stresses for the AUC model and the Tresca model at a penetration depth of 1 D. Compressive stresses are negative. Low vertical effective stresses occur next to the cone for both material models forming wingshaped stress bulbs. This can be explained by the reduction of vertical compaction once soil leaves the shadow of the penetrating cone. Within the shadow of the cone, soil is pushed to the sides and downward, outside it, soil moves to the sides in an upward direction as noted by Lu et al. Some of the vertical compression previously applied on soil particles is consequently released. For the AUC model, as a consequence, mean effective stresses decrease within the region of the wing-shaped stress bulb. The stress paths obtained for the AUC model correspond to soil yielding in a critical state: stress states in this region follow the critical state line. They do not correspond to triaxial compression nor triaxial extension which corresponds to the results shown in Figure 5 (a). Soil yields in the vicinity of the cone with effective mean stresses decreasing down to.5 kpa and deviatoric stresses q as low as 3 kpa which explains the lower tip resistance found with the AUC model. 5

6 Figure 5. (a) Regions of triaxial compression states-of-stress colored dark gray; (b) Comparison of effective vertical stresses σ yy [kpa]. 6 CONCLUSIONS For a normally-consolidated clay, the relationship between cone tip resistance and the strength of undrained clay has been investigated. Penetration of a smooth cone has been simulated. Large soil deformations are taken into consideration through use of a quasi-static MPM. The load-type dependency of the undrained shear strength and the strength anisotropy of natural clay are considered through use of the new AUC model. The cone factor obtained for a computation with the Tresca model agrees well with results from the AUC model. Hence, the Tresca model seems well suited for determining cone factors for normallyconsolidated undrained clays, provided the cone factors are related to an undrained shear strength for simple shear. Apart from the Tresca model, the Von Mises yield criterion might be used for such CPT analyses. However, when fitting the Von Mises to the Tresca yield surface so that the yield stress coincides for triaxial compression and extension, the Von Mises model will give larger cone factors. For a proper cone factor, the Von Mises criterion should be matched to the Tresca yield criterion in the case of simple shear. In order to obtain this, the Von Mises tensile strength, σ tension, should be equal to 3 s u,tresca. It would be important to also investigate cone penetration in overconsolidated clays since the overcon-solidation ratio has a considerable influence on the undrained shear strength of clay. It would seem that the AUC model is not only suited for normally-consolidated but also for overconsolidated clays. This work would allow one to identify, for given soil parameters of strength and stiffness, the relationship between cone tip resistance and undrained shear strength as a function of OCR as suggested by Vermeer (1). Formulating such a relation would simplify the analyses of geotechnical problems involving undrained clay and increase their reliability. ACKNOWLEDGEMENTS The research leading to these results has received funding from the European Community s Seventh Framework Programme FP7/7 13 under grant agreement n PIAG-GA (Geo- Install). The authors would like to express their gratitude to Deltares, especially to Peter van den Berg and Hans Teunissen, for their support of the numerical study on which this publication is based and of the development of the AUC model. The authors would like thank Plaxis B.V. and Deltares for their funding of the development of the quasi-static MPM. Here, the authors would especially like to thank Paul Bonnier from Plaxis B.V for kindly providing his expert advise in this work and Issam Jassim from the University of Stuttgart for his contributions to the research efforts leading to this publication. REFERENCES Beuth, L. 1. Formulation and application of a quasistatic material point method. Ph. D. thesis, University of Stuttgart, Holzgartenstr. 16, 7174 Stuttgart. 6

7 Beuth, L., Benz, T., Vermeer, P., Coetzee, C., Bonnier, P., & Van Den Berg, P. 7. Formulation and validation of a quasi-static Material Point Method. In Proceedings of the 1th International Symposium on Numerical Methods in Geomechanics, Volume 1, pp Taylor & Francis Group. Beuth, L., Więckowski, Z., & Vermeer, P. 11. Solution of quasi-static large-strain problems by the material point method. International Journal for Numerical and Analytical Methods in Geomechanics 35(13), Coetzee, C., Vermeer, P., & Basson, A. 5. The modelling of anchors using the material point method. International Journal for Numerical and Analytical Methods in Geomechanics 9(9), Guilkey, J., & Weiss, J. 3. Implicit time integration for the material point method: Quantitative and algorithmic comparisons with the finite element method. International Journal for Numerical Methods in Engineering 57(9), Lu, Q., Randolph, M., Hu, Y., & Bugarski, I. 4. A numerical study of cone penetration in clay. Geotechnique 54(4), Schofield, A., & Wroth, P Critical state soil mechanics. McGraw Hill New York. Sulsky, D., Chen, Z., & Schreyer, H A particle method for history dependent materials. Computer Methods in Applied Mechanics and Engineering 118(1 ), Van den Berg, P Analysis of soil penetration. Ph. D. thesis, Delft University of Technology, The Netherlands. Vermeer, P., Jassim, I., & Hamad, F. 1. Need and performance of a new undrained clay model. Vermeer, P., Yuan, Y., Beuth, L., & Bonnier, P. 9. Application of interface elements with the Material Point Method. In Proceedings of the 18th International Conference on Computer Methods in Mechanics, Volume 18, pp Polish Academy of Sciences. Wheeler, S., Näätänen, A., Karstunen, M., & Lojander, M. 3. An anisotropic elastoplastic model for soft clays. Canadian Geotechnical Journal 4(), Więckowski, Z., Youn, S., & Yeon, J A particle in cell solution to the silo discharging problem. International Journal for Numerical Methods in Engineering 45(9),