Search

The first issue of Georisk was published in March 2007. This journal was launched in recognition of the broader post-millennial challenge facing the geo-engineering community noted in the Editorial...

This paper proposes a method for estimating trends and detecting layer boundaries in depth-dependent soil data based on a least absolute shrinkage statistical operator (lasso). Although the lasso appears to be promising for subsurface modeling because no predetermined basis functions or stratification models are required, it does not provide information on the uncertainty of its estimated solution, i.e., a point estimate. In subsurface modeling, however, characterization of uncertainty is pivotal because soil data can be (spatially) sparse and noisy. A lasso-based method that can quantify its estimation accuracy while preserving its attractive sparsity feature is proposed. The performance of this sparse Bayesian lasso (SBLasso) is demonstrated through numerical tests and an actual case study of its accuracy of trend estimation and layer boundary detection. The degree of accuracy or inaccuracy of estimation provided by the SBLasso clearly corresponds to data quality, such as the number of available data points, noise level, and noise correlation. A method of soil stratification based on SBLasso was also proposed, and the stratification results by SBLasso were compared with those produced by existing methods for validation.

The cone penetration test (CPT) is a method widely used for site characterization as measurements are continuous, fast and economical compared to laboratory testing. However unlike laboratory testing that quantifies soil properties directly, readings obtained from CPT are indirect and needs to be interpreted. Hence many correlations have been proposed to infer a wide range of soil properties from CPT readings. Existing correlations have discovered separate relationships between normalized cone tip resistance (Qt) and effective friction angles (ϕ’), as well as Qt and overconsolidation ratio (OCR). This study employs the press-replace method (PRM), a novel simplified numerical technique, to perform systematic and extensive undrained investigations of CPT in a modified Cam-clay (MCC) soil model. It is the first comprehensive numerical study with an advanced soil model that considers rough cone-soil interactions. The PRM numerical approach overcomes the need to assume the CPT process as analogous to spherical or cylindrical cavity expansions, therefore produces results that are more authentic to real CPT. From the large database of numerical results, an equation is formulated that provides a framework on how Qt is related to both ϕ’ and OCR, rather than separately to ϕ’ or OCR as described in previous correlations. The proposed equation is then simplified so that ϕ’ and OCR can be easily estimated from Qt. Numerical results and predictions from the new equations are then compared to soil behaviour charts, other correlations and field data from different sites.

In this study, a database is compiled for 11 clay parameters covering 50 sites in Shanghai with 4051 data points (labeled as SH-CLAY/11/4051). These sites are distributed in the eight districts of Shanghai covering an area of 145 km2. This is the first sizeable multivariate soil database compiled at the municipal level. General information, including the geology of Shanghai, the borehole locations and depths, the types of parameters, and the basic statistics associated with these parameters are presented first. The quality of the compiled data is then assessed by comparing them with the data in the published global database CLAY/10/7490 with respect to marginal statistics. Results show that data points in SH-CLAY/11/4051 fall within the range of CLAY/10/7490. It is not surprising that local Shanghai data are more clustered than those in CLAY/10/7490. The range of values spanned by each clay parameter is smaller at a regional/municipal scale than at the global scale. A preliminary study to understand what is “unique” in Shanghai clay data is conducted by comparing global correlations between any two parameters with the corresponding Shanghai versions. Some correlations found in SH-CLAY/11/4051 (e.g., normalized effective vertical stress and normalized undrained shear strength) are observed to be similar to those found in CLAY/10/7490. However, there are other distinct correlations produced by SH-CLAY/11/4051, indicating a regional/municipal effect. This study constructs a multivariate probability distribution based on SH-CLAY/11/4051 to capture these distinct correlations between Shanghai clay parameters. One useful application of this multivariate distribution is act as prior for Bayesian updating.

The purpose of this paper is to investigate whether the quasi-site-specific model established based on a municipal or regional database can be more effective in supporting site-specific predictions than one based on a global database. To do this, a Bayesian updating approach that combines site-specific data and a municipal database under realistic MUSIC (Multivariate, Uncertain and Unique, Sparse, and InComplete) attributes for the site data is adopted. This approach assumes that the site data and the municipal database follow the same distribution, which can be partially verified using standard correlation plots. The quasi-site-specific model can then be adopted to make site-specific predictions for the relevant design properties. Because real soil data follow MUSIC attributes, this Bayesian updating is only feasible in the presence of a probability distribution construction method that can handle such data. A real case study for Shanghai shows that a municipal database is more effective in supporting site-specific predictions than a global database. In contrast, another real case study shows that a regional database that covers multiple countries in the Scandinavian region is not necessarily more effective than a global database.

The importance of accurate prediction of limiting cavity depths during offshore spudcan foundations installation has been variously highlighted in the literature. Nonetheless, most of the previous research is deterministic in nature and confined to homogenous soils. Since offshore clayey soils can be highly spatially variable, there is a practical need to take proper account of the spatial variability in the prediction of limiting cavity depths. In a bid to remedy this situation, large deformation finite element calculations combined with three-dimensional random fields were repeatedly conducted in this study within a Monte-Carlo framework. The continuous penetration of a spudcan initiated from surface was explicitly modeled until a full-localized flow-around mechanism was observed. Spatial variability was found to clearly affect the soil back-flow and thereby the limiting cavity depth, the latter of which takes a range of values that can be approximately modeled as a log-normal distribution. Characteristic limiting cavity depths at various probability levels were ascertained. An algebraic expression was proposed to explicitly predict the characteristic limiting cavity depths in random soils from the fractile. Particular attention was paid to the lower and upper 5% characteristic values, which are likely to be useful for reliability-based design.

Eurocode 7 recommends that when statistical methods are used, the characteristic value of a geotechnical parameter can be selected as the 0.05 quantile. However, Eurocode 7 does not state how to select characteristic values when there are multiple correlated input geotechnical parameters, e.g., cohesion and friction angle. One possible interpretation is that the 0.05 quantile has to be applied to each geotechnical parameter. Another possible interpretation is that the 0.05 quantile should be imposed on the overall factor safety (FS) or the performance function G = FS − 1. This study shows that the first interpretation may lead to overly conservative designs when there are multiple correlated geotechnical parameters. The second interpretation does not have this issue, but it is challenging to implement the second interpretation in practice because it requires the ability to compute the cumulative density function (CDF) of G and also control this CDF. The challenge is practical rather than theoretical; otherwise a fundamental shift to direct probability-based design would address this issue. This paper examines how multivariate characteristic values can be selected in a sensible way and aligned to the non-probabilistic basis in Eurocode 7. One promising proposal is to apply a simplified version of the quantile-value method (QVM) in conjunction with the concept of the “effective random dimension” (ERD). The resulting design verification format is identical to the current one in Eurocode 7. The key difference is that the quantile defining the characteristic value is not fixed at 0.05, but it is adjusted through calibration by the code drafting committee (not practitioner) using the ERD-QVM method.

Geotechnical design codes have been migrating toward reliability-based design (RBD) concepts. ISO 2394 identified the characterization of model uncertainty as one of the critical elements i...

Direct Monte Carlo simulation (MCS) could be prohibitively expensive for slope reliability updating problem with relatively low-probability level. This paper aims to propose an efficient slope reliability updating method based on the direct MCS. First, the direct MCS is briefly introduced. Thereafter, a weighted approach based on the direct MCS is proposed for slope reliability updating. Furthermore, the implementation procedure and the flow chart of the proposed method are presented. Two illustrative examples are investigated to demonstrate the capacity and validity of the proposed method. The results indicate that the proposed method can properly utilize the information contained in failure samples generated by performing direct MCS based on the baseline distribution, which is much more efficient than repeatedly performing direct MCS as the probability distributions of soil properties are updated. The proposed method can solve the slope reliability updating problems involving multiple uncertain parameters and implicit performance functions. The accuracy of the proposed method depends on the failure samples generated from the baseline distribution. When the failure samples cannot completely cover the failure region underlying the reliability updating problem, the proposed method should be used with caution. The coefficient of variation (COV) of probability of failure from the proposed approach is derived and used as a measure of the accuracy in updated probability of failure. In geotechnical engineering practices, the updated COV of an uncertain parameter is often smaller than its baseline COV. In this case, the proposed method can reasonably update the probability of failure.

This paper aims to propose a bootstrap method for characterizing the uncertainty in probabilistic models and its effect on geotechnical reliability. First, the copula theory is briefly introduced. Second, both the uncertainties in parameters and type of the best-fit marginal distributions and copulas are characterized by the bootstrap method. Finally, four load-test datasets of load-settlement curves of piles are used to illustrate the proposed method. The serviceability limit state reliability analysis of piles is presented to illustrate the practical application of the proposed method. The results indicate that the bootstrap method can effectively characterize the uncertainty in probabilistic models derived from a small sample. Through bootstrapping, the uncertainties in both the parameters and type of the specified probabilistic models are simultaneously incorporated into geotechnical reliability analyses. The probability of failure of piles is represented by a confidence interval at a specified confidence level instead of a single fixed probability, which enables the engineers to make a more informed decision.

In this study, we show that existing simplified reliability-based design methods, including the partial factor method and the quantile-based method (quantile value method; QVM), are not robust over design scenarios with variable degree of redundancy. This variable degree of redundancy is quite common in geotechnical design. For instance, a pile group has more redundancy than a single pile. Less obviously, a pile embedded in multiple soil layers has more redundancy than one embedded in a single soil layer. Implementing a fixed set of partial factors or a fixed quantile will not produce designs with uniform reliability indices. This paper shows that the degree of redundancy can be effectively quantified by the concept of an “effective random dimension” (ERD), and it proposes a practical method of estimating ERD for reliability-based design. It is shown by numerical examples that by incorporating ERD, the ERD-QVM outperforms the original QVM in achieving a more uniform reliability level across different design scenarios.

The objective of this paper is to investigate the effect of copulas for constructing the bivariate distribution of shear strength parameters on system reliability of geotechnical structures. First, the bivariate distribution of shear strength parameters is constructed using copulas. Second, the implementation procedure of system reliability analysis using direct Monte Carlo simulation (MCS) is developed. Finally, the system reliability of a retaining wall and a rock wedge slope is presented to explore the effect of copula selection on geotechnical system reliability. The results indicate that the system reliability of geotechnical structures under incomplete probability information could not be determined uniquely because the bivariate distribution of cohesion and friction angle with given marginal distributions and correlation coefficient could not be determined uniquely. The copulas for modeling dependence structure between cohesion and friction angle have a significant influence on the system reliability of geotechnical structures. Such an influence includes two phases separately. The first phase is that the dependence structure between shear strength parameters characterized by copulas affects the reliability of single failure mode, depending on the marginal distributions, dependence structure between shear strength parameters, and reliability level of each failure mode. The second phase is that the reliability of each failure mode influences on system reliability, only depending on reliability level of each failure mode and correlations among various failure modes. It is important to distinguish between the effect of copula selection on reliability of each failure mode and that on geotechnical system reliability.

This paper thoroughly investigates the bearing capacity of strip footings on sand under combined loading by using a lower-bound limit analysis in conjunction with finite elements and second-order cone programming (SOCP). Two analyses were performed: one using a constant friction angle and the other using a variable friction angle. The analyses with a constant friction angle were used to calibrate the existing results, including the classical solutions commonly used in foundation design practice and other numerical or experimental solutions. The analyses with a variable friction angle allow us to investigate the effect of footing width B on the bearing capacity of strip footings. An iteration computational procedure is employed to account for the dependency of the friction angle on the stress level. According to the numerical results, it is found that the bearing capacity factor Nγ for eccentrically or obliquely loaded strip footings linearly decreases with the increase of B on a log–log scale, where the range 0.3–5 m of footing width was considered in this paper. In addition, it is found that the footing width has a negligible effect on the shape and size of the normalized failure envelopes.

Reliability-based design (RBD) can play a useful complementary role to overcome some limitations in the Eurocode 7 (EC7) design approach, for example in situations with parameters not covered in EC7, different parametric sensitivities across different problems, cross-correlated or spatially correlated parameters, design aiming at a target reliability or failure probability, or when uncertainty in unit weight of soil is modeled. The complementary role played by RBD under these circumstances is illustrated and discussed for a shallow foundation, a reinforced rock slope, a Norwegian clay slope with spatial variability, a laterally loaded pile requiring implicit numerical analysis, and an anchored sheet pile wall. A pragmatic RBD approach involving first-order reliability method (FORM) only and a more rigorous RBD approach involving both first-order and second-order reliability method (SORM) are offered. Both approaches are implementable using either spreadsheet-based FORM and SORM procedures, or using various commercially available FORM/SORM packages.

Slope reliability under incomplete probability information is a challenging problem. In this study, three copula-based approaches are proposed to evaluate slope reliability under incomplete probability information. The Nataf distribution and copula models for characterizing the bivariate distribution of shear strength parameters are briefly introduced. Then, both global and local dispersion factors are defined to characterize the dispersion in probability of slope failure. Two illustrative examples are presented to demonstrate the validity of the proposed approaches. The results indicate that the probabilities of slope failure associated with different copulas differ considerably. The commonly used Nataf distribution or Gaussian copula produces only one of the various possible solutions of probability of slope failure. The probability of slope failure under incomplete probability information exhibits large dispersion. Both global and local dispersion factors increase with decreasing probability of slope failure, especially for small coefficients of variation and strongly negative correlations underlying shear strength parameters. The proposed three copula-based approaches can effectively reduce the dispersion in probability of slope failure and significantly improve the estimate of probability of slope failure. In comparison with the Nataf distribution, the copula-based approaches result in a more reasonable estimate of slope reliability.

This paper addresses the statistical uncertainties associated with the estimation of a depth-dependent trend function and spatial variation about the trend function using limited site-speci...

This paper presents a critical evaluation of the model factor M = qu,m/qu,c for Eurocode 7 calculating the bearing capacity of circular footings on dense sand, where qu,m = measured capacity and qu,c = Eurocode 7 calculated capacity. Regression analysis is required to remove the dependency of M on the input parameters. Because the input parameters cannot be varied systematically in load tests, previous studies showed that finite-element limit analysis (FELA) can be used as an alternative to load tests for regression. This is further verified from the model factor MFELA = qu,m/qu,FELA with a mean of 1 and a coefficient of variation (cov) of 0.1, where qu,FELA = FELA predicted capacity. A correction factor (Ms = qu,FELA/qu,c) is next defined, which can be decomposed as a product of a systematic part f and a residual part η (i.e., Ms = fη), which is modeled as a lognormal random variable with mean = 1 and cov = 0.11. Finally, a new model factor (M′ = qu,m/q′u,c = qu,m/fqu,c) is defined. The model sta...

The model uncertainties for the static analysis (determination of the settlement at a given load and the ultimate bearing capacity) of square foundations on sand under axial compression are characterized using 67 field load tests. The model uncertainty for calculating the ultimate bearing capacity is represented by a ratio (model factor) of measured capacity to calculated capacity. The capacity model factor is found to be significantly affected by the foundation width. This statistical dependency is expressed as an exponential function. The residual part of the model factor is characterized as a lognormal random variable. A hyperbolic equation with two curve-fitting parameters is used to estimate the load-settlement curves normalized by measured capacity. Using the same database, the best-fit copula for the hyperbolic parameters is Gaussian copula.

The multiple modes of Rayleigh waves (R-waves) or leaky Lamb waves (L-waves) may be present in the activated and/or the back-scattered wave fields. The presence of these waves in the wave fields is related to the source and the cavity depths. Two types of layered media are used to investigate the influences of the source and the cavity depths. One is the layered half space where the half space is stiffer than all the overlying layers; the other is the pavement system where there is significant velocity contrast between layers and the shear velocity of layers decreases with the layer depth. The discrete displacement expressions developed for spherical point sources are used to study the effects of the source depth on the excitability and the propagation behavior of R-waves or leaky L-waves. In layered half spaces, the multiple modes of R-waves can be activated by the sources within a certain depth. The effective phase velocity of these activated modes is derived. In pavement systems, the leaky L-waves can be activated by the sources at any depth. The discrete displacement expressions of the scattered waves are derived and used to evaluate the effects of the cavity depth on the presence of R-waves or the leaky L-waves in the back-scattered wave field.

This paper studies the reliability of infinite slopes in the presence of spatially variable shear strength parameters that increase linearly with depth. The mean trend of the shear strength parameters increasing with depth is highlighted. The spatial variability in the undrained shear strength and the friction angle is modeled using random field theory. Infinite slope examples are presented to investigate the effect of spatial variability on the depth of critical slip line and the probability of failure. The results indicate that the mean trend of the shear strength parameters has a significant influence on clay slope reliability. The probability of failure will be overestimated if a linearly increasing trend underlying the shear strength parameters is ignored. The possibility of critical slip lines occurring at the bottom of the slope decreases considerably when the mean trend of undrained shear strength is considered. The linearly increasing mean trend of the friction angle has a considerable effect on the distribution of the critical failure depths of sandy slopes. The most likely critical slip line only lies at the bottom of the sandy slope under the special case of a constant mean trend.

In models based on Winkler springs for tunnel lining design, designers always face the difficulty of selecting appropriate values for the radial subgrade modulus (kr). The widely used solution kr for a circular tunnel in elastic ground proposed by Wood (1975) was found to be applicable only when the tunnel radial deformation is oval-shaped. On the basis of the Wood׳s solution, this note presents a general solution for kr when the radial deformation of the tunnel is described by a Fourier series. This modified Wood׳s solution of kr using compatible stress functions is validated by a numerical example. The modified solution for the example shows good consistency with the original Wood׳s solution when the tunnel becomes an oval shape with deformations. The example indicates that the magnitude of kr is significantly affected by the distribution shape of the tunnel radial deformation. The value of kr is no longer a constant value around the tunnel when the tunnel deforms into a general shape described by a Fourier series. It is quite different from the value of kr for a distribution shape described by a single Fourier term, i.e. one involving a single frequency. The application of a general solution for kr is illustrated by a design case using a bedded beam model.

This study proposes a set of simple equations for the mean value and variance of the mobilized shear strength for spatially variable soil masses subjected to uniform stress states. These equations are fairly effective in explaining the complicated behaviors for the mobilized shear strengths, regardless of stress states (e.g., compression or shear), spatial variability patterns (e.g., isotropic or anisotropic), and inherent mean and variance of the random field. Two mechanisms that affect the behaviors of the mobilized shear strength are identified: (1) the averaging effect along the potential slip curves, and (2) the emergent feature of a critical slip curve. The emergence is associated with the slip curve with the minimum averaged strength. In any realization of the random field, it is not possible to know a priori the location of the minimum average; hence, it would not coincide with a prescribed average. It is shown that the well-known phenomenon of critical scale of fluctuation is the result o...

In this paper, the formulation of a lower-bound limit analysis for axisymmetric problems by means of finite elements leads to an optimization problem with a large number of variables and constraints. For the Mohr-Coulomb criterion, it is shown that these axisymmetric problems can be solved by second-order cone programming (SOCP). First, a brief introduction to SOCP is given and how axisymmetric lower-bound limit analysis can be formulated in this way is described. Through the use of an efficient toolbox (MOSEK or SDPT3), large-scale SOCP problems can be solved in minutes on a desktop computer. The method is then applied to estimate the collapse load of circular footings and uplift capacity of single or multiplate circular anchors. By comparing the present analysis with the results reported in the literature, it is shown that the results obtained from the proposed method are accurate and computationally more efficient than the numerical lower-bound limit analysis incorporated with linear programming.

Recent studies have shown that the sample path smoothness in soil spatial variability can have a significant effect on the failure probability of geotechnical problems. The purpose of the current study is to propose a procedure that can identify the sample path smoothness based on site investigation data. It is shown that two factors determine whether or not the sample path smoothness can be identified: the type of auto-correlation function (ACF) model and the parameter estimation method. In order to identify the sample path smoothness, a non-classical two-parameter ACF model, such as the powered exponential (PE) model and Whittle-Matern (WM) model, must be adopted together with the maximum likelihood (ML) method. The method of moments (MM) is incapable of identifying the sample path smoothness regardless of the ACF model type, classical or otherwise, although it is effective in identifying the scale of fluctuation (SOF). Between the two non-classical ACF models, the WM model is more flexible because it covers a wider range of sample path smoothness than the PE model. Neither the PE model nor the WM model is able to model the “hole effect” (non-monotonic auto-correlation). The development of a sufficiently flexible non-classical model that can simultaneously identify SOF, sample path smoothness, and hole effect remains an open research question.

This study explores the mechanism of the undrained response of tunnelling through spatially variable improved soil. It is found that spatial variability produces a more realistic failure mechanism compared to that produced by the conventional homogeneous assumption. The overall failure of the improved ring through which the tunnel passes occurs when four or more localized plastic hinges are fully formed. The radial line average strength traversing the cement treated soil surrounding the tunnel changes along the circumferential direction (from crown to invert) because of spatial variability. The locations of the local yield zones are found to correspond to minimums of radial line average passing through the improved ring and the mobilized strength is approximately equal to the 4th minimum of the radial line average in 70% of the cases. A closed-form probabilistic model based on this “4th weakest path model” is proposed to characterize this relatively complex failure mechanism. Random finite element analyses are only required to calibrate this probabilistic model of the mobilized strength. The practical outcome of this paper is to provide a rational method of estimating the characteristic strength that is consistent with both the statistics of the point strength and the mechanics of failure in a spatially variable improved ring.

ISO2394:2015 contains a new informative Annex D on “Reliability of Geotechnical Structures”. The emphasis in Annex D is to identify and characterize critical elements of the geotechnical reliability-based design process, while respecting the diversity of geotechnical engineering practice. This paper highlights the main features of Annex D and gaps for future work.

An ideal solution for disposal of large volumes of unwanted dredged clays is to stabilize and use them as fill materials for land reclamations. This kind of stabilized dredged fill (SDF) requires a lower cement amount (Cm) compared with traditional cement-treated soils. Strength behavior might be different for mixes within the inactive zone (lower Cm) and those within the active zone (higher Cm). The SDF is fully/partially within the inactive zone. A new set of unconfined compressive strength (qu) data as well as qu data compiled from literature covering both lower and higher Cm are analyzed. First, main parameters governing qu of cement-stabilized clays are identified: (1) cement amount (Cm), and (2) water content (W). Then, qu behaviors in the inactive zone and active zone are compared. Results indicate that a nonlinear normalized qu-Cm trend is commonly observed in the inactive zone whereas a linear one is observed in the active zone. In both zones, the normalized qu-W curve (or normalized qm-C...

Various empirical equations have been proposed to estimate the unconfined compressive strength (qu) of cement-treated soils based on the proportions of water, cement, and clay or based on measured early strength. Most equations provide only a point estimate of qu and no measure of uncertainties associated with the estimation. As the design’s safety level is governed by the uncertainties involved, it is essential to quantify uncertainties associated with the estimation. The coefficient of variation (COV) is taken as a measure of uncertainties. It is straightforward to estimate the mean and COV of qu based on a single parameter. Similar estimation based on multiple parameters, which may be intercorrelated, is not as straightforward. This paper aims to integrate multiple parameters (water-to-cement ratio, cement content, and 7-day strength) that may contribute to the estimation of 91-day strength using a multivariate normal distribution. The multivariate normal model can be used to derive the mean, C...

The adequacy of two approximate methods based on incomplete information, namely method P and method S, for constructing multivariate distributions with given marginal distributions and covariance has not been studied systematically. This article aims to study the errors of the method P and method S. First, the method P and method S as well as the exact method are presented. Second, the performance of the two approximate methods is evaluated based on their abilities to match exact solutions for system probabilities of failure. Finally, an illustrative example of a parallel system is investigated to demonstrate the errors associated with the two methods. The results indicate that the errors in system probabilities of failure for the two methods highly depend on the level of system probability of failure, the performance function underlying the system, and the degree of correlation. Such errors increase greatly with decreasing system probabilities of failure. When the target system probability of failure is ...

Simplified reliability-based design (RBD) formats in the form of Load and Resistance Factor Design (LRFD) and Multiple Load and Resistance Factor Design (MRFD) are applied to 42 drilled shafts installed in layered soil profiles underlying the city of Taipei. Clay, sand, gravel and rock layers or some partial combination thereof are encountered within the depth of penetration. It is reasonable to require a foundation reliability-based design code to achieve a reasonably consistent target reliability index over this range of design scenarios, as they are representative of the ground conditions underlying Taipei. The simplified RBD formats are calibrated using two approaches. They are the design value method (DVM) and the more recently proposed quantile value method (QVM). In general, for layered soil profiles, conventional formats containing resistance and load factors are unable to achieve the prescribed target reliability index with the same consistency as that reported for homogeneous soil profiles, regardless of the reliability calibration approach. For the drilled shaft examples considered in this study, it was found that the direct application of quantiles in the simplified RBD format is more reasonable for layered soil profiles.

This paper studies the probability distribution for the mobilised shear strength of a spatially variable soil mass that is subjected to a uniform stress state. Based on the mechanisms identified in two previous studies conducted by the authors, this study further proposes a probability distribution model for the mobilised shear strength that is based on the extreme value of normal random variables. It is concluded that the probability distribution of the mobilised shear strength of a spatially variable soil mass is affected by the line averaging effect along the potential slip plane and the number of independent potential slip planes. These two factors depend on the stress state and the orientation of the potential slip planes. With this model, the mobilised shear strength of a spatially variable soil mass can be simulated without the need of conducting random-field finite-element analyses. In addition, the strength characteristic value that is the 5% quantile in the Eurocodes can be easily derived from t...

In random field finite element analysis (FEA), continuous random fields must be discretized, and the discretized solution is different from the continuous one. In this paper, it is found that the tolerable maximum element size to achieve acceptable accuracy for soil shear strength is surprisingly small. The discrepancy is governed by the line averaging effect along potential slip curves - the discrepancy is small if the line averaging is strong. Discretization based on element-level averaging (ELA) is always unconservative, giving mobilized shear strength always larger than the actual value, while discretization based on the midpoint method can be sometimes conservative.