SUMMARY, CONCLUSIONS AND FUTURE SCOPE
CHAPTER 2 BACKGROUND STUDY AND LITERATURE REVIEW
2.3. UNCERTAINTY DUE TO SOIL HETEROGENEITY
2.3.1. Inherent Spatial Variability of Shear Strength Parameters of Soil
2.3.1.2. Random Finite Element Method (RFEM)
variable. This approach succeeds to gain much popularity among researchers and often used in probabilistic slope stability study as this method completely accounts for spatial correlation and local averaging (Griffiths and Fenton, 2007). Further, the method does not make any presumptions regarding the location and shape of the critical slip surface, which is automatically determined using FEM that can search for actual weakest path through the soil domain.
If the mean and covariance of a random field vary with the location, the characterising joint pdf is highly inconvenient to use in mathematical practice as well as to estimate from real field data.
Therefore, simplifications of stationarity or statistically homogeneity of random field is essential.
A statistically homogeneous or stationary random field indicates that the joint pdf characterising the random field is spatially invariant; i.e., the mean, variance and all the higher order moments are constant at any location within the random field. The correlation between any two random variables entirely depends on their separation distance, not on their absolute locations. In order to characterize a random field under the simplifying assumptions of stationary random field, the mean, variance and the pattern of soil spatial variability should be known. The pattern of spatial variability in soil can be characterized using autocorrelation function, variance function or spectral density function.
The most commonly used algorithms available in literature to generate multi-dimensional random fields in geotechnical engineering are the Moving average (MA) method, Discrete Fourier Transformation (DFT) method, Covariance Matrix Decomposition (Sarma et al., 2014), Turning Bands Method (TBM), Fast Fourier Transformation (FFT) method and Local Average Subdivision (LAS) method (Hicks and Spencer, 2010). The accuracy of probabilistic study highly depends on
the aptness of the algorithm under consideration for generation of random field realisations. Fenton and Griffiths (2007) compared different random field generator with respect to their accuracy, efficiency, ease of use and implementation.The FFT, TBM, and LAS methods were found to be much more efficient than the first three methods. It has been highlighted that every method has some advantages and disadvantages, and the selection of random field generator algorithm entirely depends on the particular problem under consideration; further details are furnished in Fenton and Griffiths (2007).
Over the years, several researchers have considered the theory of random fields in order to incorporate the spatial variability of soil properties in geotechnical engineering practise (Vanmarcke, 1977; Vanmarcke, 1983; Griffiths and Fenton, 2000; Griffiths and Fenton, 2004).
However, in most of these studies, the spatial variation in soil shear strength was characterised as stationary random field, i.e., the mean of the shear strength parameters is invariant with depth of soil slope. However, it is well understood that the soil properties are non-stationary i.e., soil property progressively changes with depth from the surface (Phoon and Kulhawy, 1999). Several in-situ test data showed that soil properties of a homogeneous soil layer exhibit variable trends with depth (Elkateb et al., 2003; Hicks and Samy, 2002; Kulatilake and Um, 2003). Srivastava and Babu (2009) reported that for a data exhibiting no trend with depth, the reliability index (β) values underestimate the reliability of slope failure as compared to those obtained with linear trend in data. Further, Li et al. (2014b) stated that ignoring this increasing trend in shear strength parameters with depth results in overestimation of failure probability. Moreover, the chances of critical slip surface developing at the bottom of the slope decreases significantly when the increasing trend of mean shear strength parameters is considered. In this regard, to incorporate the progressive
increment of the various geotechnical parameters with depth, Griffiths et al. (2015) introduced the incorporation of non-stationary random fields in probabilistic slope stability studies. Further Huang et al. (2021) investigated the effect of rotated transverse anisotropy, occurring due to various geological processes, combined with non-stationary random field on reliability of the stability of an undrained soil slope.The study considered two different cases of non-stationary random field: the trend of soil strength increasing with depth and the trend increasing along the direction perpendicular to bedding. The study revealed that the reliability of the slope stability highly depends on the directions of the trend.
Further, it is worth mentioning that although the random field theory is well established in geotechnical slope stability literature, for small or medium sized projects, it is understandable that the data collected from the site is usually sparse. Therefore, random field parameters estimated from such sparse data may contain significant uncertainty resulting in inaccurate estimation of random field samples for slope stability analysis. To overcome such limitations, Wang et al.
(2018a) recently proposed a random field generator based on Bayesian Compressive Sampling (BCS) and Karhunen-Loeve (KL) expansion, which can be suitable for generating random field samples based on sparse measurements obtained from a given site.
As mentioned earlier, lognormal pdf is often chosen in RFEM analysis to avoid negative values for soil properties considered as random variables. However, while using lognormal pdf the correlation length of the random soil property Y relates to X = lnY instead of Y itself. Few research (Fenton and Griffiths, 2004; Pieczy´nska-Kozłowska et al., 2015) stated that the differences between the correlation length in the lognormal random field Y and the underlying Gaussian
random field X should not crucially influence the results of RFEM estimations. Pula and Griffiths (2021) investigated the theoretical relationship between spatial correlation lengths in transformed lognormal and hyperbolic tangent random fields, and the underlying Gaussian random fields from which they are derived. The study shows that for CoV less than 0.3, the untransformed and transformed spatial correlation lengths are essentially the same. An application to a bearing capacity problem shows that the transformations result in more conservative design values as compared with the untransformed results, however, the differences are very modest.