Three Dimensional Discrete Failures in Long
Heterogeneous Slopes
Yajun LI, Michael A. HICKS and Philip J. VARDON
Department of Geoscience and Engineering, Delft University of Technology, The Netherlands
Abstract. Long slopes with variable material properties have a variety of different failure modes that are affected by the correlation distances of the material properties and the geometry and total length of the slope. This work follows previous research examining the consequences of increasing slope length, in which analyses were limited by the computational capacity. In this work a long slope (500 m) in a clay, characterised by a spatially varying undrained shear strength, was analysed using the random finite element method (RFEM). Random fields, to simulate the spatial distribution of the strength parameters, were generated and mapped onto finite element meshes, that were used in subsequent strength reduction analyses as part of a Monte Carlo simulation. This involved different, but equally likely, random fields to gain a probabilistic description of the distribution of safety factors and failure mechanisms. A hypothetical study of a slope cut in clay, with an intermediate horizontal correlation distance of the strength parameter, was carried out to investigate how the slope failure develops in a heterogeneous soil. In contrast to previous work, this paper emphasises the quantification of discrete failures. Hence a simple procedure has been developed to quantify the number of discrete failures in a long soil slope. It was found that, although multiple failures are possible for very long soil slopes, a single localised failure still has the greatest likelihood, as is more generally observed in practice. Due to the high computational requirements for a probabilistic analysis of such a physically large problem, Grid computing technologies were used to conduct the investigation, meaning that 1 year of computational analysis could be undertaken in a single day.
Keywords. Discrete failure, finite elements, heterogeneity, long slope, reliability
1. Introduction
Highway embankments, river banks and sea dykes usually have a uniform cross section and extend for a long distance in the third dimension. These long soil structures are generally characterised by spatially varying soil properties, i.e. soil heterogeneity. Slope stability failures of these structures may have significant economic and societal consequences. Thus, it is of particular interest for engineers to investigate the influence of soil spatial variability on the stability and failure mechanisms of such long ‘linear’ structures. For example, Hicks and Spencer (2010) showed that the failure mechanisms of a 100 m long heterogeneous clay slope were highly influenced by the horizontal correlation distance relative to the slope geometry and length.
The fact that heterogeneity and failure mechanisms are three dimensional makes 3D analysis necessary in order to simulate ground conditions and quantify failure likelihood and consequence in a more realistic manner. For
example, Hicks et al. (2014) investigated the influence of heterogeneity on the stability of a 100 m long clay slope, and quantified the failure consequence in terms of failure volume and failure length in the longitudinal direction. This paper continues that work and extends the slope length to 500 m, with the aim of quantifying the range of numbers of discrete failures and their relative frequencies in a longer slope. For such a physically large problem the computational requirements are substantial, with regards to both run-time and memory. Fortunately, with the help of Grid computing, the computational analysis can be undertaken in a significantly reduced time. The high performance computing strategy (Li et al., 2015) used in this study will be utilised to look at various aspects of geotechnical structure performance in the Authors’ research group, such as seepage and piping (Liu et al., 2015), large deformation post failure behaviour using the material point method (Wang et al., 2013) and the applicability of simpler analytical models used in practice (Li et al., 2013; Li and Hicks, 2014).
© 2015 The authors and IOS Press.
This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License.
2. Random Finite Element Method
The random finite element method (RFEM) is used to compute the slope response (e.g. factor of safety and displacement) within a Monte Carlo framework (Fenton and Griffiths, 2008). The procedure is as follows:
a) Generate random property fields using the local average subdivision (LAS) method (Fenton and Vanmarcke, 1990), based on the soil property statistics, e.g. mean, standard deviation and spatial correlation structure (type of correlation function, and horizontal and vertical scales of fluctuation, Th and Tv, respectively);
b) Map random field cell values onto the Gauss points within the finite element mesh modelling the given problem (in this case, a slope stability problem); c) Carry out a traditional finite element
slope stability analysis using the strength reduction method, by applying gravitational loading to generate the in situ stresses (Smith et al., 2013);
d) Repeat the above steps for multiple realisations in a Monte Carlo analysis, until the output statistics (i.e. mean and standard deviation of the factor of safety) converge.
For a given set of statistics, a probability distribution of the factor of safety can be obtained. Moreover, the potential failure volume and failure length are also quantified for each realised factor of safety. This will be clarified in greater detail for a numerical example in the following section.
3. Analysis
3.1. Problem Geometry and Mesh Discretisation The slope dimensions are: length in the third dimension, L = 500 m; height, H = 5 m; width from slope toe to the left boundary, W = 10 m; and slope angle, D . The cross-sectional domain size and finite element mesh details are shown in Figure 1. There are altogether 40,000 eight-node brick elements, 191,281 nodes and 573,843 degrees of freedom.
Figure 1. Slope dimensions, finite element mesh details and boundary conditions in a cross-section.
3.2. Soil Properties
The clay has a unit weight of 20 kN/m3, an elastic modulus of 100 MPa, a friction angle of zero and a Poisson’s ratio of 0.3. The statistics of undrained shear strength (assuming a normal distribution) are listed in Table 1, and are within the range normally encountered in practice. The Tresca failure criterion is used to model the clay behaviour in an undrained condition.
Table 1. Statistics of soil undrained shear strength cu
Mean (kPa) Standard
deviation (kPa)
Scale of fluctuation (m) 40.0 8.0 Th= 12.0,Tv = 1.0
3.3. Boundary Conditions
The finite element boundary conditions are: a fixed base, rollers at the back boundary preventing displacements perpendicular to the back face, and rollers on the two end faces allowing only settlement and preventing movement in the other two directions (i.e. out-of-slope-face and longitudinal directions) (see Spencer (2007), Hicks and Spencer (2010), and Hicks et al. (2014) for further details). These are also shown in Figure 1.
Note that, for a very long slope (e.g. 500 m), the boundary conditions at the two ends (either smooth, rough, or the intermediate case utilised above) have a marginal effect on the calculated factor of safety.
3.4. Computing Strategy
There are two types of solvers in a finite element analysis; one is the direct solver, and the other is the ‘element by element’ iterative solver.
Spencer (2007) showed that the direct solver was approximately ten times faster than the iterative solver in his nonlinear plasticity iterative factor of safety analyses. Therefore, the direct solver is used in this investigation. However, as the problem becomes large in terms of the total number of degrees of freedom or total numbers of equations (e.g. 509,760 equations are involved in the current investigation), it requires more computer memory to store the global stiffness matrix, and also more time to factorise it and run the plasticity iterations. For a 500 m long slope, such as that investigated in the current paper, around 6 GB of memory is needed; therefore a standard desktop computer, which usually has >8 GB of standard memory, is adequate. However, hundreds of Monte Carlo realisations are needed to simulate the range of spatial variability of the ground conditions. On average, it takes 3 hours to run a single realisation for a problem of this size. If, for example, 2000 realisations need be run on a desktop computer, it would take 250 days to finish. It is therefore not a feasible task for a desktop computer.
As each realisation or job can be run independently on a single serial computer, Grid computing, which is generally the allocation of jobs over a network of distributed computers, is ideally suitable for Monte Carlo simulation.
The Authors have access to the Dutch grid infrastructure and have been authenticated as a member in a virtual organization where 3 computing clusters are available (4400 cores for the gina cluster, 900 cores for the rug cluster and 3800 cores for the nikhef cluster). For high memory jobs (as is the case in the current investigation), it is the gina cluster that is used. The computing capacity is 4400/2 = 2200 jobs (it has 3-8 GB of memory per core and therefore 2 free cores are needed for a single 6 GB job).
With these available resources, it is possible to carry out the investigation for a 500 m long slope in a reasonable time. Usually it takes 15-18 hours (i.e. 5-6 times that of a single realisation) to finish 2000 jobs/realisaitons, including the waiting time in the queue (Li et al., 2015). 3.5. Analysis and Results
Figure 2 shows the probability density function of the realised factor of safety from 1000 Monte
Carlo realisations. A fitted normal distribution with mean E[FS] and standard deviation V[FS] is also shown. It can be seen that a normal distribution fits the simulated results reasonably well. It is also interesting to note that all the realised factors of safety are less than 2.46, which is the deterministic factor of safety based only on the mean strength.
Figure 2. Probability density function (PDF) of the realised factor of safety.
Apart from the distribution of the realised factor of safety, the failure volume and failure length, associated with taking the slope to failure in each realisation using the strength reduction method, are quantified by a simple procedure using the out-of-slope-face displacement. That is, the slide volume is approximated as the volume of all elements with an average nodal displacement greater than some threshold value (in this case, 26% of the maximum nodal displacement for the mesh), which is calibrated against a more rigorous ridge-finding technique for a 2D slice taken through the centre of a homogeneous slope (Hicks et al., 2014). The failure length is estimated to be the integrated length along the row of elements directly above the slope toe in the longitudinal direction, for which the displacement is greater than the threshold displacement (Hicks et al., 2014).
Note that the failure length is approximated to an accuracy of 1.0 m, which is the element size in the horizontal direction. This failure length, in some cases, is the accumulated length of multiple discrete failures along the slope. It may not be necessary to differentiate between multiple failures for a relatively short (e.g.
50-100 m long) slope, due to the relatively lower probability of encountering multiple failures. However, for a slope length of, e.g., 500 m, it is informative to quantify the number of discrete failures and their respective lengths. Hence, in this paper, it is considered to be a separate failure if the row of failed elements directly above the slope toe is not continuously linked.
The scattered distributions of the failure volume and failure length are plotted in Figure 3 (a-b), as a percentage of the total mesh volume and total slope length, respectively. The number of discrete failures in a realisation and the overall frequency distribution are shown in Figure 3 (c-d). It can be seen that, for a horizontal scale of fluctuation of 12 m, there is a wide range of possible slide volumes and slide lengths. Note also the relatively small scale (<0.3% of the problem size) of the slide volumes and slide lengths for a 500 m long slope, compared to those for a 100 m long slope as reported in Hicks et al. (2014); i.e. the % slide volume and length reduce as the slope length increases, as well as being functions of the horizontal scale of fluctuation. What is also apparent in Figure 3 (a-b) is the left tail of the scatter plot, which implies that a smaller failure volume or length is often associated with a smaller realised factor of safety. In Figure 3 (a-b), the failure volumes and failure lengths associated with a single failure are highlighted with circles, whereas the crosses indicate the total volume and length of multiple failures in individual realisations. Clearly, a single failure usually results in a small failure volume and small failure length. Note that single failures are often associated with smaller realised factors of safety. Moreover, it can be seen from Figure 3 (c-d) that, although multiple failures (up to 10 failures) are possible for a 500 m long slope, there is a greater likelihood of a single failure in any realisation. The probability of two or more failures is less (Figure 3d).
3.6. Failure Mechanisms
It is difficult to visualise a 3D failure mechanism in 2D, especially when the third dimension size is significantly larger than the cross-section size. Nevertheless, typical failure mechanisms (for slopes with 1-4 failures) are illustrated in Figure
4, in terms of contours of the out-of-slope-face displacement.
Figure 3. Failure volume (a), failure length (b), and number of discrete failures (c) versus realised factor of safety, and frequency distribution of number of discrete failures (d).
Note that although multiple failures are possible, their extents are generally different, with some failures having larger displacements than others. In reality, those failures involving larger displacements tend to fail first, which, in the case of a sea dyke failure, will usually release the hydraulic loading before some other failures of smaller extent develop fully themselves, resulting in fewer failures.
Figure 4. Illustration of failure mechanisms in terms of contours of out-of-slope-face displacement: (a) single failure, (b) two failures, (c) three failures, (d) four failures.
4. Conclusion
A 500 m long slope, characterised by an undrained shear strength profile, has been analysed using the random finite element method. Failure consequences including sliding volume, length and number of discrete failures are quantified using a simple procedure based on the displacement in the out-of-slope-face direction. It was found that multiple failures are possible for a very long slope, although single failures develop with the highest frequency. This seems to agree with reality, as single failures and double failures are the most frequent scenarios observed in practice.
Acknowledgements
This work was carried out on the Dutch National e-infrastructure with the support of SURF Foundation. Special thanks are given to SURFsara advisor Anatoli Danezi for her kind support in developing the computing strategy. The work is funded by the China Scholarship Council (CSC) and by the Section of Geo-Engineering, Delft University of Technology.
References
Fenton, G.A., Griffiths, D.V. (2008). Risk assessment in geotechnical engineering, John Wiley & Sons, New York.
Fenton, G.A., Vanmarcke, E.H. (1990). Simulation of random fields via local average subdivision, Journal of Engineering Mechanics ASCE 116 (8), 1733–1749. Hicks, M.A., Spencer, W.A. (2010). Influence of
heterogeneity on the reliability and failure of a long 3D slope, Computers and Geotechnics 37 (7-8), 948–955. Hicks, M.A., Nuttall, J.D., Chen, J. (2014). Influence of
heterogeneity on 3D slope reliability and failure consequence, Computers and Geotechnics 61, 198–208. Li, Y., Hicks, M.A. (2014). Comparative study of
embankment reliability in three dimensions, Proceedings of the 8th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE), M.A. Hicks et al. (eds.), 467–472, Delft, The Netherlands.
Li, Y., Hicks, M.A., Nuttall, J.D. (2013). Probabilistic analysis of a benchmark problem for slope stability in 3D, Proceedings of the 3rd International Symposium on Computational Geomechanics, S. Pietruszczak and G. N. Pande (eds.), 641–648, Krakow, Poland.
Li, Y., Hicks, M.A., Vardon, P.J. (2015). High performance computing strategies for nonlinear finite element (a)
(b)
(c)
analysis of long heterogeneous soil slopes. Proceedings of the 23rd UK Conference of the Association for Computational Mechanics in Engineering, A.J. Gil & R. Sevilla (eds.), 427–430, Swansea, UK.
Liu, K., Hicks, M.A., Vardon, P.J., Jommi, C. (2015). Probabilistic analysis of velocity distribution under earth embankments for piping investigation. Proceedings of the 5th International Symposium on Geotechnical Safety and Risk (ISGSR), Rotterdam, submitted for publication.
Smith, I.M., Griffiths, D.V., Margetts, L. (2013). Programming the finite element method (5th edition), John Wiley & Sons, New York.
Spencer, W.A. (2007). Parallel stochastic and finite element modelling of clay slope stability in 3D, PhD thesis, University of Manchester, UK.
Wang, B., Vardon, P.J., Hicks, M.A. (2013). Implementation of a quasi-static material point method for geotechnical applications, Proceedings of the 3rd International Symposium on Computational Geomechanics, S. Pietruszczak & G.N. Pande (eds.), 305–313, Krakow, Poland.
