Research Paper
Finite-element upper-bound analysis of seismic slope stability considering pseudo-dynamic approach

https://doi.org/10.1016/j.compgeo.2020.103530Get rights and content

Abstract

The present study presents a novel procedure to assess the seismic slope stability, using the finite-element upper-bound method combined with the pseudo-dynamic approach. A pioneering work is performed to incorporate the pseudo-dynamic accelerations into the finite-element upper-bound analysis which combines the advantages of upper bound and finite element methods. The finite element method is principally used to discretize the domain of soil mass into finite elements, aiming to construct a kinematically admissible failure mechanism based on which the external and internal rates of work can be expressed. Based on the upper bound theorem, the upper bound formulations are derived from the virtual work rate equation, in the form of slope safety factor and limit surcharge acting on the slope crest. The problem of seeking the optimal upper bound solution is transformed to solve a linear programming problem within numerous kinematically admissible velocity fields, and this is achieved by using the interior-point algorithm implemented into the MATLAB. Pseudo-dynamic solutions are calculated and compared with the pseudo-static, limit equilibrium and FLAC results. The proposed procedure is versatile in considering the homogeneity and non-homogeneity of soil strength properties in seismic slope stability analysis. A benchmark problem with 0.5 m thick weak interlayer is discussed.

Introduction

Finite-element limit-analysis method has been widely applied to geotechnical stability analyses, since it combines the merits of finite element and limit analysis methods. It consists of two bifurcations: finite-element upper-bound and finite-element lower-bound, aiming to estimate the stability of geotechnical structures in the form of upper and lower bounds. The authentic collapse load is accordingly limited to a range of lower and upper bound solutions. In a sense, the actual failure load is likely to be sought if the upper bound is infinitely approaching to the lower bound, by constructing a kinematically admissible velocity field and a statically allowable stress field. In geotechnical engineering, slopes widely exist and its stability/safety must be ensured, or casualties and engineering loss are to be induced once slope failure occurs. Note that slopes are vulnerable to extreme loading conditions such as earthquake disturbance, and in this case catastrophic hazard is likely to occur, such as landslides and/or debris flows.

The finite-element limit-analysis method is superior to the conventional limit analysis approach which was systematically discussed in Chen [1] for dealing with classical geotechnical problems including slope stability, footing bearing capacity and earth pressure calculations. A pioneering work was made by Lysmer [2] where the finite element method was incorporated to the limit analysis to assess the ultimate bearing capacity of geotechnical structures and the safety factor of slopes. The numerical limit analysis was then improved for plane strain problems, which further facilitates the development of the finite-element limit analysis [3], [4]. A large-scale linear programming model for a lower bound analysis was developed by discretizing the area of interest to linear triangular elements [5]. The shared side between elements is regarded as the interface in the model, and the yield function is linearized within each element. Based on the characteristics of sparse linear programming problems, a steepest edge active set algorithm was proposed by Sloan [6], with the problems solved in high efficiency. After that, Sloan [7], [8] further proposed the finite-element upper-bound method without/with consideration of the velocity discontinuous surface. One can find that Sloan’s work on the finite-element limit-analysis method has been strengthening its status as an effective approach for the assessment of geotechnical stability problems. Many scholars applied this method to deal with actual geotechnical problems in tunnels, slopes and strip footings, considering different geomaterials [9], [10], [11], [12], [13], [14], [15].

In theory, rigorous upper and lower bound solutions can be obtained from the finite-element limit-analysis method, and hence it has been widely applied to slope stability assessment. Yu et al. [16] used the above mentioned method to evaluate the stability of a soil slope under undrained condition, and the research finding shows that an approximate linear relationship is observed between the stability coefficient of slopes and dimensionless parameter λcp. This method was also used to investigate the influence of pore water pressure on slope stability by considering the pore ware pressure as an external force [17], [18]. The seismic slope stability was discussed with a pseudo-static approach, in combination with the finite-element limit-analysis method [19]. This method was also adopted to investigate the stability of a bi-layered purely cohesive soil slope [20]. Lim et al. [21] used this method to study the slope stability considering a slope with a soft band, a post-quake slope and a rock slope. The upper bound formulation of the ultimate bearing capacity for a stone retaining wall slope was derived based on a mixed numerical discretization procedure of finite element method and rigid finite element method [22].

When geotechnical structures are subjected to an earthquake, a reliable prediction of geotechnical stability is highly dependent upon the approach used to represent the earthquake input. The acceleration time-history is commonly used in a numerical analysis where commercial software has certain merits in dealing with complicated issues. The use of the pseudo-static approach is more straightforward by quantifying the earthquake as constant horizontal and/or vertical accelerations [23], [24]. However, this approach is incapable of considering the dynamic earthquake effect varying with time and space within slopes. In an effort to solve this drawback, the pseudo-dynamic approach was proposed by Steedman and Zeng [25], for a retaining wall analysis, considering shear waves propagating upwards. Then this approach was used to evaluate the seismic stability of a retaining wall, with the horizontal and vertical accelerations expressed in the form of sinusoidal waves [26], [27], [28]. Note that the limit equilibrium analysis was adopted only and hence rigorous upper or lower bounds cannot be obtained. In view of this, a discretization-based kinematic analysis was proposed to investigate the seismic stability of a non-uniform soil slope and a rigid retaining wall, combined with the pseudo-dynamic approach [29], [30], [31].

The present study aims to investigate the seismic stability of a soil slope, using the finite-element upper-bound analysis combined with the pseudo-dynamic approach. Such a combined procedure provides an avenue to derive the upper bound solutions of limit bearing capacity that a slope can support at limit state and the slope safety factor, besides the critical failure mechanism.

Section snippets

Finite-element upper-bound method

In the realm of the upper bound analysis, the core work is to express the internal and external rates of work in a kinematically admissible velocity field. When the external rates of work obtained from the strain rate field are equal to or greater than the internal dissipated energy, the geotechnical structure is under critical limit state or fails. Therefore, the upper bound theorem states that the authentic failure load is no greater than the load computed from the power-based balance

Problem statement

In this study, a soil slope model is discussed, as presented in Fig. 2 where the slope height is h. In a numerical analysis, the height of the model, H, is supposed to be larger than h to avoid the geometric boundary condition. Apart from soil self-weight, traction forces such as surface surcharge acting on the slope crest surface exist, and the surcharge force is characterized by λF(t)q¯n where λF(t) is the dynamic overload coefficient and q¯n is the prescribed surcharge. For an upper bound

Comparison

In order to verify the robustness of the proposed approach, comparison is carried out considering different approaches. Since the factor of safety calculated in this study is based on the strength reduction technique which is similar to that used in the limit equilibrium, the limit analysis and limit equilibrium solutions (calculated from Bishop and Morgenstern-Price method) are directly compared, as summarized in Table 1. Meanwhile, the comparison between pseudo-static and pseudo-dynamic

Limit surcharge load

After having obtained the upper bound formulation of the overload coefficient, the limit surcharge load acting on the slope crest surface can be sought with the procedure presented above. In order to better understand the effect of dynamic earthquake and soil strength parameters on limit overload, a parametric study is carried out herein. The numerical results are calculated, with the basic input parameters including soil properties: γ = 18 kN/m3, φ = 15° and c = 20 ~ 35 kPa, seismic load: kh

‘NC’ soil slope

In the above calculations, a homogeneous soil slope is considered with constant shear strength parameters. However, there are some situations where non-uniform soil properties widely exist in site, such as in normally consolidated or deposited soils. In this case, the shear strength profile is supposed to be adopted in the finite-element upper-bound analysis, for the sake of providing a reliable solution, although it is hardly able to find similar research with considerations to non-uniform

Conclusion

This study presents a pseudo-dynamic procedure with finite-element upper-bound analysis to assess the seismic stability of a slope in a uniform and non-uniform soil stratum. Such an analysis retains the merits of upper bound theorem and finite element method, and hence can be applied to resolve complicated stability problems, such as under earthquake effect. The finite element principle is used to discretize the soil mass into finite elements so as to construct a kinematically admissible

CRediT authorship contribution statement

Jianfeng Zhou: Methodology, Software, Data curation, Visualization, Validation. Changbing Qin: Conceptualization, Writing - original draft, Writing - review & editing, Supervision, Investigation.

Acknowledgements

This research was supported by the Natural Science Foundation of Fujian Province, China (Grant No.: 2019J05088) and Scientific Research Funds of Huaqiao University, China (Grant No.: 18BS112), which are greatly appreciated.

References (32)

  • S.W. Sloan

    A steepest edge active set algorithm for solving sparse linear programming problems

    Int J Numer Anal Met Geomech

    (1988)
  • S.W. Sloan

    Upper bound limit analysis using finite elements and linear programming

    Int J Numer Anal Met Geomech

    (1989)
  • J.S. Shiau et al.

    Bearing capacity of a sand layer on clay by finite element limit analysis

    Can Geotech J

    (2003)
  • D.W. Wilson et al.

    Undrained stability of dual square tunnels

    Acta Geotechnica

    (2015)
  • X.S. Shi et al.

    A homogenization equation for the small strain stiffness of gap-graded granular materials

    Comput Geotech

    (2019)
  • Y. Xiao et al.

    Finite Element Limit Analysis of the Bearing Capacity of Strip Footing on a Rock Mass with Voids

    Int J Geomech

    (2018)
  • Cited by (24)

    • Strain-dependent slope stability for earthquake loading

      2022, Computers and Geotechnics
      Citation Excerpt :

      For the latter part, the analysis of the slope stability and the investigation of the factor of safety, methods such as LEM (Leshchinsky and San, 1994; Baker et al., 2006; Hleibieh and Herle, 2019a; Hazari et al., 2020), LA (Ausilio et al., 2000; Loukidis et al., 2003), SRFEA (Baker et al., 2006) or FELA (Loukidis et al., 2003; Zhou and Qin, 2020; Li et al., 2021) have been applied, considering additional forces to account for inertial effects. To approximate the seismic effects, pseudo-static (Leshchinsky and San, 1994; Loukidis et al., 2003; Baker et al., 2006; Bray and Travasarou, 2009; Hleibieh and Herle, 2019a; Macedo and Candia, 2020; Li et al., 2021) or pseudo-dynamic (Zhou and Qin, 2020; Hazari et al., 2020) methods can be utilized. In contrast to the pseudo-static approach, in which constant horizontal and vertical seismic coefficients are applied, the pseudo-dynamic approach enables to account for spatial and temporal variation of the seismic coefficients and amplification factors considering the soil types and layering between the source of the earthquake signal and the slope.

    • Slope topographic impacts on the nonlinear seismic analysis of soil-foundation-structure interaction for similar MRF buildings

      2022, Soil Dynamics and Earthquake Engineering
      Citation Excerpt :

      For the sake of convenience, many works on the TSSI and SSSI simulate the structures as a simplified reduced-order model with a limited number of stories [35–39] or as a solid block [8,16] or in the 2D planar modeling approach [40–44]. For the TSSI systems, most of such studies evaluated the model with simple assumptions of the pseudo-static approach [43,45–47], and the earthquake loads are considered to be a constant force. This approach may not meticulously simulate the wave propagations and coupled effects of dynamic TSSI.

    View all citing articles on Scopus
    View full text