Research PaperThe strength reduction method in clay hypoplasticity
Introduction
Various methods can be employed to evaluate the factor of safety (FS) in slope stability analysis. The limit equilibrium (LE) analysis provides an estimation of FS along the predefined failure surfaces of a circular (Bishop, 1955, Petterson, 1955) or polygonal (Morgenstern and Price, 1965, Spencer, 1967, Sarma, 1973) shape. The slope itself is considered as a rigid body with the strength defined by the Mohr-Coulomb parameters and c. Due to the nature of LE, the failure surface corresponding to the lowest FS is determined by the iterative procedure. This, however, does not ensure that the solution represents the slip surface of a minimal factor of safety. The limit analysis (LA) can provide estimations of the upper and lower bounds for the energetic and static equilibrium and, with the aid of the Finite Element Method (FEM), these bounds can be determined even the complex problems (Lyamin and Sloan, 2002a, Lyamin and Sloan, 2002b). A relatively novel approch represented by the Material Point Method (Sulsky et al., 1995) was also introduced into the stability analysis (Zabala and Alonso, 2011).
In finite element method, slope stability is quantified by the Strength Reduction method (SR), which is commonly used in combination with the Mohr-Coulomb model (M–C). This method was initially developed by Zienkiewicz et al. (1975) and has since been extensively studied (Tschuchnigg et al., 2015, Yang et al., 2012), including a solution dependency on both the FEM mesh refinement and elasto-plastic flow rule (Tschuchnigg et al., 2015). Further modifications of the SR can be also found in Tschuchnigg et al., 2015, Dyson and Tolooiyan, 2018. The advantage of connecting staged construction modelling with stability analysis without the need to use additional tools while preserving the stress and variables history is significant.
Even though the M–C model is capable of predicting the stress state at failure, the prediction of the strain response is rather unsatisfactory. Advanced elasto-plastic constitutive soil model theories, such as bounding surface plasticity (Dafalias, 1986), and other modelling frameworks including hypoplasticity (Wu and Kolymbas, 1990) and barodesy (Kolymbas, 2012) were proposed for both sand (Von Wolffersdorff, 1996, Kolymbas et al., 2009) and clay (Mašín, 2013, Medicus and Fellin, 2017) soils. These were further enhanced for various model characteristics to capture intergranular strain stiffness (Niemunis and Herle, 1997), anisotropy (Mašín, 2014) or viscosity (Jerman and Mašín, 2020) in order to overcome the difficulties. These models can account for the nonlinear strain response as well as, state and loading history dependency of soils. The strength reduction method for these advanced constitutive models is, however, usually not avaliable. Consequently, if the advanced constitutive model is used during staged construction and a stability analysis is required, the M–C model has to be employed and a recalibration of the parameters is necessary.
Recently, an example of the strength reduction method for the clay barodesy model (Medicus and Fellin, 2017) was proposed in Schneider-Muntau et al. (2018), giving an option for dealing with the stability analysis while using advanced non-elasto-plastic constitutive models. This paper is dedicated to the strength reduction method applied to the hypoplastic model for clay (Mašín, 2013). In the following, the usual sign convention of continuum mechanics (compression negative) is adopted throughout. In line with Terzaghi’s principle of effective stress, all stresses are effective stresses.
Section snippets
Strength reduction using the Mohr-Coulomb model
The stress state of the M–C model is limited in the principal stress space by the yield surface F of a hexagonal pyramid shape. Since the M–C is an elastic – perfectly plastic model, once the loading path reaches the stress state on the yield surface the model exhibits a plastic strain response. In the integration point, the reduction of the strength parameters from their initial values (internal friction angle) and (cohesion) to their reduced counterparts and directly reduces the
Hypoplastic clay model
The hypoplastic clay model (H–C) (Mašín, 2013) is one of the advanced constitutive models currently available in geotechnical software. It is capable of predicting nonlinear soil behaviour and captures soil hardening as well as post peak softening. The model behaviour is defined by five parameters closely resembling the parameters of the Cam-Clay model (Roscoe and Burland, 1968). The parameters and N are defined in space, see Fig. 2a. The parameter N represents the value of
Strength reduction in clay hypoplasticity
As already mentioned, advanced constitutive models, including hypoplastic models, satisfactorily predict the soil behaviour and characteristics, such as soil hardening and softening during monotonic loading. The strength reduction developed in this article is split into two parts: The method of critical friction angle reduction is described first and the stress return to the ASBS is then elaborated.
Testing in single-element model
The proposed procedure was tested in the simulation of the element test. An example of undrained triaxial simulation is illustrated in Fig. 9, which is divided into two parts: Firstly, the strength reduction is conducted simultaneously with the triaxial loading; the reduction was executed from the critical state friction angle down to . Subsequently, the strength reduction was cancelled while the triaxial loading continued to imposE further deformation. The reduction stopped at an
umat implementation
As mentioned above, the strength reduction method was implemented in the umat format of ABAQUS and executed in the PLAXIS finite element code through an interface. The reduction of the critical state friction angle is mapped to auxiliary quantity t (time) as seen in Fig. 10. Since not only a time increment but also the exact value of the time at any given step i is provided by the FEM software, the value of the reduced friction angle at the step i can be directly evaluated as
Study of the strength reduction method
The reliability of the strength reduction method was further tested in the PLAXIS 2D software. Three FEM models of different geometries were created with slope inclinations and , see Fig. 11. In addition, a finer FEM mesh was adopted in the areas associated with the development of the slip surface. To fully test the developed method, models of embankments of loose soil as well as excavation in overconsolidated soil were tested.
Different densities of FEM mesh were used
Conclusions
This study demonstrated the quantitative and qualitative predictions of a newly developed strength reduction method for the hypoplastic clay model (Mašín, 2013). The method adopts a strength parameter reduction, combined with an analytical formulation of the stress projection back to the ASBS in the octahedral plane of stress at a constant Lode angle. It was demonstrated that this stress reduction is a necessary component of the strength reduction method since the lack of the stress return
CRediT authorship contribution statement
Tomáš Kadlíček: Conceptualization, Methodology, Software, Validation, Formal analysis, Data curation, Writing - original draft. David Mašín: Conceptualization, Methodology, Writing - review & editing, Supervision, Project administration, Funding acquisition.
Declaration of Competing Interest
The authors declared that there is no conflict of interest.
Acknowledgement
Financial support from a research grants LTACH19028 of the Czech Ministry of Education, Youth and Sports and grant 17-21903S by the Czech Science Foundation is greatly appreciated. The second author acknowledges institutional support by the Center for Geosphere Dynamics (UNCE/SCI/006).
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2022, Computers and GeotechnicsCitation Excerpt :This makes the evaluation of the factor of safety based on the approach proposed by Nitzsche and Herle (2020) more accurate but also significantly slower compared to other stress-based approaches, for instance the ones by Zou et al. (1995) or Farias and Naylor (1998). Besides the concept presented by Nitzsche and Herle (2020), stability analyses taking into consideration advanced constitutive models have also been conducted via SRFEA, for instance using clay hypoplasticity (Kadlíček and Mašín, 2020) or barodesy (Schneider-Muntau et al., 2018; Tschuchnigg et al., 2019). However, these studies (Nitzsche and Herle, 2020; Kadlíček and Mašín, 2020; Schneider-Muntau et al., 2018; Tschuchnigg et al., 2019) have not been extended and applied to slopes subjected to earthquake loading.
A consistent calibration process for the Matsuoka–Nakai friction angle under direct simple shear conditions for clay hypoplasticity
2022, Computers and GeotechnicsCitation Excerpt :We further evaluate stress paths in principal stress space, investigate different overconsolidation ratios and evaluate and interpret possible soil states in the framework of Critical State Soil Mechanics. The outcomes of this studies can therefore also be relevant for plane-strain applications of clay hypoplasticity as carried out by e.g. Nitzsche and Herle (2014), Stutz et al. (2017), Kadlíček and Mašín (2020) and Jerman and Mašín (2021). For a summary of the symbols and notation that we have adopted, please refer to Appendix A.