Reliability-Based Stability Analysis of a Baltic Cliff by the Combined Response Surface Method

Since the interest in reliability-based slope stability analysis began over 50 years ago, different methods have been proposed to assess slope reliability, including the correction factor methods, First-Order Second-Moment (FOSM), Second-Order Second-Moment (SOSM), First-Order Reliability Method (FORM), Second-Order Reliability Method (SORM) and Monte Carlo (MC) method (Wu and Kraft 1970; Cornell 1971; Alonso 1976; Tang et al. 1976; Vanmarcke 1977; Low et al. 1997; Chowdhury and Grivas 1982; Chowdhury et al. 1987; Tobutt 1982; Christian et al. 1994; El-Ramly et al. 2002; Babu and Mukesh 2004).

A milestone in probabilistic analysis of slopes was the application of the Finite Element Method (FEM) in geotechnical problems. Various FEM-based probabilistic methods have been proposed. Developments in both hardware and software have made the MC simulation dominant. The FEM-based MC application is usually called the Random Finite Element Method (RFEM), which fully incorporate spatial correlation and averaging (Griffiths and Fenton 2004). This method has been applied to slope reliability (Griffiths et al. 2009; Cho 2007; Huang et al. 2010, 2013; Wang et al. 2011; Li et al. 2014; Farah et al. 2015; Liu et al. 2015). In the case of complicated slopes, the application of RFEM involves extensive computational effort. In order to improve the computational efficiency of slope reliability assessment, research was conducted to simplify the slope stability performance function and to reduce the sample for Monte Carlo simulations.

A method addressing slope stability problems is the Response Surface Method (RSM) (see Wong 1985; Cho 2010; Huang et al. 2010; Li et al. 2011, 2013; Tan et al. 2013, 2016; Ji and Low 2012; Zhang et al. 2011, 2013; Jiang et al. 2014; Li et al. 2016; Zhou and Huang 2018). RSM is usually applied to relatively simple cases; it is difficult to find a literature example of its application to multi-layered slopes with more than a few uncertain soil parameters.

The Point Estimate Method (PEM) proposed by Rosenblueth (1975) and its modified procedures have also attracted interest in many engineering fields. Various authors apply this method in probabilistic slope design, e.g. Gibson (2011), Wang and Huang (2012), Ahmadabadi and Poisel (2015).

Solving the same analytical problem by different probabilistic approaches may often produce diverse results. The differences are even clearer in non-linear cases. On the other hand, confirming the results by at least two methods is required to obtain a better insight into the problem. An MC simulation is usually a reference here. Moreover, random variable and random field generation by the MC method is relatively easy and widely used in great many engineering applications. Thus it seems reasonable to combine traditional MC sampling with other algorithms, such as PEM or RSM, to accelerate computations and verification of computational results.

The paper aims at a directed approach to RSM, the so-called combined Response Surface Method (CRSM), which is based on simple random methods (MC, PEM and RSM) and combines their major advantages. Note that CRSM is not universal; it is dedicated to cases of high or moderate failure probability. Thus, it is particularly suitable for analyses of landslides and soils with a complex structure.

The applicability of CRSM is discussed here with regard to a stability study of a Baltic cliff in Jastrzębia Góra, Poland. The cliff has a complex multi-layered geological structure with over a dozen uncertain soil parameters. Complex investigations of the critical section of the cliff (approximately 750 m long) conducted over the years have been addressed in numerous works and reports, e.g. Tejchman et al. (1995a), Zabuski and Korzec (2017). Some works, e.g. Tejchman et al. (1995b), have been presented at conferences, and some include a probabilistic stability analysis of the cliff, e.g. Tejchman et al. (1996). All these publications provide a background for the in-depth analysis presented here. It is also worth noting that a damaged building is situated on the crest of the cliff (in the landslide-prone zone), and numerous attempts have recently been made to repair it and restore it to use. The paper complements a feasibility study of this project, and therefore the computations regard both the present situation (the case of high failure probability) and the hypothetical post-drainage situation (the case of moderate failure probability).

The proposed computational CRSM algorithm draws on three popular reliability assessment methods: Monte Carlo simulation (MC), Point Estimate Method (PEM) and Response Surface Method (RSM).

The direct MC approach estimates the probability of failure \(P_{f}\) in the entire domain of \({\mathbf{x}}\) by means of the expressionapplying the so-called indicator function \(I_{0/1}\)where \(LS({\mathbf{x}})\) is the limit state function, and \(NS\) denotes the number of samples used in the approximation.

Cornell’s reliability index \(\beta\) (Cornell 1971) is estimated in the formin which \(\varPhi ( \cdot )\) denotes the standard Gaussian cumulative distribution function.

The main idea behind the PEM is to properly discretize the continuous random variable, see Rosenblueth (1975). In the case of n-dimensional random vector \({\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} }} = \left\{ {x_{i} } \right\},\,\,i = 1,\,2,\,3,\, \ldots ,\,n\), the structural response function is given aswhere values of the response function \(y({\mathbf{x}})\) are computed at relevant combinations of PEM discretization points.

For example, two combinations related to the variable \(x_{i}\) defined by the mean value \(\mu_{i}\) and standard deviation \(\sigma_{i}\) take the formIn turn, the mean value \(\mu_{y}\) and the standard deviation \(\sigma_{y}\) of the response function \(y({\mathbf{x}})\) are defined asHence, the reliability index \(\beta\) readsand the probability of failure \(P_{f}\) is obtained by inversion of Eq. (3).

The aim of RSM is to determine a function to approximate the actual structural response induced by random variation of input parameters. The response surface \(\hat{y}({\mathbf{x}})\) maps a set of discrete values of actual response \(y({\mathbf{x}})\) corresponding to a finite number of discrete realizations of the random vector \({\mathbf{x}}\):where \(x_{1} ,\,\,x_{2} ,\,\,x_{3} , \ldots \,\,,\,\,x_{n}\) are realizations of basic random variables, and \(\varepsilon\) is the assessment error of the actual structural response.

If the response is approximated in a small subspace only or its detected skewness is low, the first-order response surface (RS) model may be applied. On the other hand, cases of spacewise response mapping, large curvature of the actual response or an inaccurate first-order RS model approximation require higher-order RS models, e.g. the second-order model, defined asIn order to assess surface slope factors \(b_{( \cdot )}\) given in Eq. (10), standard approximation techniques are commonly used, e.g. the least square method or the ANOVA tabular variance reduction technique, see Montgomery (1997). On the basis of the surface slope factors determined in RSM-Win (Winkelmann and Górski 2014), the reliability index \(\beta\) is estimated as the shortest distance from the system origin to the so-called design point \({\mathbf{x}}^{ * }\) on the failure surface in the reduced (standardized) space (Hasofer and Lind 1974).

In general, MC results are usually used as reference values, especially in non-linear and multidimensional random variable cases. However, this method requires multi-sample computations and may thus become extremely inefficient. Computational effort, even in multidimensional cases, is significantly reduced by PEM. Unfortunately, due to the small number of samples used in PEM, its accuracy is frequently unsatisfactory, therefore it is recommended to verify its results by other methods. On the other hand, PEM samples may be successfully introduced in the sensitivity assessment of structural response to variations in input parameters.

The idea behind CRSM is to combine the advantages of the three abovementioned methods: MC, PEM and RSM. Samples generated by the MC method and PEM may be directly used in surface approximation by RSM. CRSM enables parallel comparison of the MC, PEM and RSM results. Their convergence is checked at every approximation step, so that CRSM computations can be terminated as soon as the required convergence level has been achieved. Thus, a CRSM solution is obtained in a relatively short time.

MC computations in the course of CRSM are not independent, since this would involve a high computational expenditure due to a large sample space. MC samples are generated primarily to be used in the approximation of further, higher-accuracy RSM functions. However, MC-generated samples may be distant from the limit state boundary, so surface approximation by RSM is not precise in this region. Hence CRSM is dedicated to cases of moderate failure probability. The introduction of MC samples imposes limitations on the methodology, but this approach is highly accessible, requiring neither a vast theoretical background on probability, nor specialized, advanced software. Such requirements are inherent to the target sampling methods that iteratively search for the limit state boundary. CRSM seems simple and straightforward to use in engineering applications, but its simplicity does not exclude non-linear, multi-variable unknown limit state functions.

CRSM is applied here to analyse the stability of a Baltic cliff in a region where progressive erosion threatens a building situated on the crest of the cliff.

The Jastrzebia Góra cliff (Fig. 1) is a several-kilometre-long section of the Polish Baltic cliff coast, which extends over 100 km. The cross-section of the cliff, descending steeply into the sea, rises 31 m above sea level (Fig. 2). The complex structure of this cliff stretch is dominated by fluvioglacial sediments prone to active landslides. In this case, the initial layers of the cliff, composed mainly of clays, induce the sliding of the other beds.

Investigations of coastline movement started at the end of the 19th century. They were initially carried out using historical topographic maps, then with the use of aerial photography and finally, in recent years, by terrestrial laser scanning (TLS), capable of the most accurate monitoring of movement. According to long-term observations, the average landward movement of the coastline between 1875 and 1976 was 0.3 m/year, causing the cliff coastline to move 80 m landwards. Later, in the years 1977–1990, the movement accelerated substantially to 0.94 m/year. Since 1991, the average velocity of displacements has decreased to 0.5 m/year. This slowdown has been achieved by erecting a gabion barrier at the base of the cliff, at the sea level (Fig. 2). In spite of that, several small landslides have developed. In 2006, one of the landslides caused the collapse of a seaward wall of the building (Fig. 1). Therefore, remedial works were conducted to replace the colluvium with geogrid-reinforced gravel and erect a shelf-like structure between the building and the cliff’s edge (Fig. 2).

An extensive research programme was conducted (Tejchman et al. 1995b) in the most active section of the cliff. Based on deep borings in the crest of the cliff, up to a depth of 30 m, three cross-sections were identified in a 750 m range. All boreholes were also used to install piezometers and inclinometers. Physical and strength parameters of particular soils composing the cliff layers were determined on the basis of laboratory tests of undisturbed soil samples (NNS) taken from these sections. The majority of laboratory tests were carried out with a simple shear apparatus, although some additional complex tri-axial tests were also performed. The primary objective was to estimate the strength parameters of soils lying in the vicinity of potential slip surfaces. The in situ investigations and numerical computations revealed the dominant role of clays in the stability analysis. The results are collected in Fig. 2 and Table 1, which are complemented by other data presented in this paper. Two formations have been distinguished on this basis. The first one, up to an altitude of 15 m a.s.l., is composed of relatively strong impermeable soils, i.e. clays, loams and silts. The second, upper formation is a complex of sand-loam soils (Zabuski and Korzec 2017).

An urban project has recently been initiated to restore the damaged building (Fig. 1). Thus, it was necessary to verify the cliff’s stability in the vicinity of the building. Due to the lack of data on the considered cross-section, the arrangement of soil layers, water conditions and geomechanical parameters of particular soils were assumed on the basis of the three nearest cross-sections available, included the closest located 70 m from the building (Tejchman et al. 1995b).

Landslide processes in the Jastrzębia Góra cliff are also triggered by unfavourable hydrogeological conditions, mostly by temporary increases in the groundwater level. Measurements of the ground water table (GWT) by all piezometers proved that the GWT was located between the permeable granular soil layer and the impermeable clay layer (Tejchman et al. 1996). Thus the same GWT location, i.e. at the contact between layers L4 and L3, was assumed in the section analysed, see Fig. 2. The continuous GWT is relatively high and appears on the cliff face in the form of seepage springs.

Summing up, it is reasonable to extrapolate the data available for the nearest cross-sections to the section analysed in the paper. Furthermore, a number of redundant inquiries may be avoided this way.

Slope stability was investigated with the Itasca FLAC (FLAC 4.0, 2000) commercial software in a two-dimensional explicit model based on the Finite Difference Method integrated with the strength reduction technique. The safety of the cliff is assessed on the basis of the safety factor \(F\), which is calculated using the strength reduction technique fundamentals (Dawson et al. 1999). It is typically used to calibrate the factor during a stepwise reduction in the material shear strength in order to reach a state of limit equilibrium of the slope. The method is applied with the Mohr–Coulomb failure criterion. A simulation sequence is performed using trial values of the \(F\) factor to modify cohesion \(c\), friction angle \(\phi\) and tensile strength \(\sigma_{t}\). An incremental reduction (or increase) in strength is performed until a failure state is reached, and on this basis the selected technique finds strength values to match the final safety factor \(F\). It is then normalized appropriately, and the safe situation means that \(F > 1\).

Simulations performed to assess the safety factor \(F\) of the cliff (Fig. 2) in its natural (undrained) state, using strength parameters presented in Table 1, give the safety factor \(F_{undr} = 1.06055\). The slip, expressed by the maximum shear strain increments (SSI), encompasses the entire slope (Zabuski and Korzec 2017). Hence, the safety margin is extremely low. The potential landslide zone is shown in Fig. 3. The building at the crest of the cliff is threatened by the slip, as it is situated within the critical zone.

After drainage, which is one of the possible stabilization methods, the safety factor of the cliff increases significantly to \(F_{dr} = 1.29492\).

To properly take into account the variation in strength parameters, the deterministic cliff analysis is extended by a probabilistic approach. It is aimed at determining both the safety margin \(F\) and slope reliability.

The basic random variables of the problem are related to the parameters of the assumed soil layers forming the slope, as shown in Fig. 2. Eight different soil layers can be distinguished in the slope: six cohesive and two non-cohesive layers. For cohesive soils, both soil cohesion \(c\) and the internal friction angle \(\phi\) are assumed random, while for non-cohesive soils the internal friction angle \(\phi\) is the only variable. Thus, a total of 14 random parameters are adopted here. The volumetric weight of each layer is assumed deterministic.

On the basis of several hundred tests classified into several groups (Tejchman et al. 1995a), the coefficients of variation of cohesion \(\nu_{c}\) and coefficients of variation of the internal friction angle \(\nu_{\phi }\) were defined for the soil of the highest impact on stability, i.e. clay: \(\nu_{c}\) was assumed to range from 0.20 to 0.27, and \(\nu_{\phi }\) was assumed to fall within the interval from 0.05 to 0.11. In the case of silts, these coefficients were \(\nu_{c} = 0.30\) and \(\nu_{\phi } = 0.075\) (Tejchman et al. 1996). Eventually, the values of \(\nu_{c} = 0.30\) and \(\nu_{\phi } = 0.10\) were adopted in the subsequent probabilistic analysis. Identical coefficients of variation were applied to assess the strength parameters of all geotechnical layers of the cliff, consistently with remarks found in the literature, see e.g. (Christian et al. 1994). All random parameters are assumed Gaussian. Other random variable types are applicable here as well, e.g. log-normal ones or more advanced (Brejda and Moorman 2000; Fenton and Griffiths 2003; Tobutt 1982), but the insufficient database makes the Gaussian choice reasonable. The statistical characteristics of the random variables related to particular soil layers of the slope are presented in Table 1. The high diversity in soil cohesion \(c_{i}\) and the internal friction angle \(\phi_{i}\) is intended to produce a high dispersion in cliff response results. Moreover, a uniform variation of an entire layer was assumed here, whereas a real slope may display a random field-like character, see e.g. Griffiths et al. (2009), Metya and Bhattacharya (2011), Liu et al. (2014), Zhu et al. (2019). Neglecting the two-dimensional parameter variation leads to an overestimation of safety factor variance. Thus the simplifications concerning strata variability are justified from the engineering viewpoint.

This paper proposes a combined response surface approximation approach (CRSM). It is based on the sum of specific PEM samples and random MC samples. CRSM may be successfully employed in cases of a moderate or high failure probability, i.e. \(P_{f} > 0.05\), representing various tasks in geotechnics.

The main feature of CRSM is multiple use of the same samples:

In both cases presented here (i.e. an undrained cliff and a drained cliff), the parallel MC-RSM algorithm works well, each time resulting in a reasonable number of software runs. The time savings due to the possibility of sample re-usage and parallel computing is the greatest advantage of this novel approach. The estimated reliability indices \(\beta\) and probabilities of failure \(P_{f}\) result in randomization of a deterministically computed slope stability factor \(F\). In the undrained cliff, the factor was estimated at close to 1.0, but the CRSM results indicate its potential high variability, and hence a high instability of the cliff. Thus, the probabilistic computations confirm the preliminary deterministic analysis and the subsequent conclusion that the repair and operation of the building at the cliff top will be possible only after reinforcement of the cliff.

The CRSM computations prove that the proposed drainage-based cliff stabilization is effective. The probability of stability loss decreases from 44 to 6%. However, the idea of overall drainage is merely theoretical. On the other hand, the preliminary sensitivity analysis performed by CRSM identifies strata with the highest contribution to slope stability, so an alternative solution, such as a partial drainage of the key strata, may be proposed. This case needs to be re-analysed, so the variability parameters of particular soil layers should be verified to evaluate the repair works performed.