Local Average Subdivision Research Articles

Impact of Soil Properties’ Spatial Correlation Lengths and Inclination on Permanent Slope Displacements Due to Earthquake Excitation

Natural disasters, when and where they occur, often cause serious social and economic consequences, which require an urgent solution to the problem. In particular, Greece, which is characterized by a complex geological structure and intense tectonic stress, has suffered and continues to suffer the consequences of such catastrophic phenomena. Among the various destructive phenomena recorded on the Earth’s surface, two of the most important problems are landslides and land subsidence. The above phenomena may cause, in addition to the serious case of loss of human life, a threat to the social and economic fabric affecting sustainability in general, i.e., the quality of life of an area (destruction of property, filling of reservoirs, blockage of streams and rivers, etc.). In fact, landslides are a phenomenon with enormous social and economic consequences, since apart from the financial burden due to the collapse of a technical project or the interruption of transportation, they are accompanied by the loss of human life. This research examines the stochastic characteristics of a slopes’ stability to investigate the variation range of permanent earthquake movements. More specifically, the influence of inclination as well as the lengths of the spatial correlation of ground are investigated. The method in the present study follows the development of arbitrary fields of soil properties, which follow the Gaussian distribution characterized by autocorrelation lengths lx and ly in the horizontal and vertical directions, respectively, mean value μ, standard deviation σ, and cross-correlation coefficients ρij. The estimation of permanent displacements is performed by the combination of the Local Average Subdivision algorithm and the FLAC software (Fast Lagrangian Analysis of Continua) used in the parametric investigation of this work. The results of this research showed that the spatial correlation of the properties has an important impact on the permanent displacements of slopes caused by strong earthquake excitations.

Random Characteristics of Hydraulic Gradient through Three-Dimensional Multilayer Embankment

The distribution characteristics of hydraulic gradient in embankment are closely related to seepage failure. Seepage failures such as flowing soil and piping will lead to serious damage and even the overall failure of embankment. The hydraulic conductivity has strong spatial variability, which changes the distribution of hydraulic gradient in embankment and increases the difficulty for predicting the embankment seepage instability. In this study, the distribution of soil hydraulic conductivity in a section of Shijiu Lake embankment was obtained by the permeability test. Based on Local Average Subdivision technique, a three-dimensional multilayer random field of embankment hydraulic conductivity was generated. Then, the mean and standard deviation of overflow point height and hydraulic gradient were calculated by the Monte Carlo method, which combined the generated three-dimensional random model and the deterministic analysis method of seepage field. Finally, the coefficient of variation (COV) of hydraulic conductivity (0.1, 0.3, 0.5, 0.7, 1.0, 2.0, and 3.0), the fluctuation scale in vertical direction (3 m) and the fluctuation scale in horizontal plane (3 m, 6 m, 12 m, 24 m, 36 m, and 48 m) were selected respectively for analyzing the random characteristics of embankment overflow point height and hydraulic gradient under the influence of different COV and fluctuation scale of embankment soil hydraulic conductivity.

Influence of Soil Heterogeneity on the Behavior of Frozen Soil Slope under Freeze-Thaw Cycles

Soil slope stability in seasonally frozen regions is a challenging problem for geotechnical engineers. The freeze-thaw process of soil slope caused by the temperature fluctuation increases the difficulty in predicting the slope stability because the soil property is influenced by the freeze-thaw cycle. In addition, the frozen soil, which has ice crystal, ice lens and experienced freeze-thaw process, could present stronger heterogeneity. Previous research has not investigated the combined effect of soil heterogeneity and freeze-thaw cycle. This paper studies the influence of soil heterogeneity on the stability of frozen soil slope under freeze-thaw cycles. The local average subdivision (LAS) is utilized to model the soil heterogeneity. A typical slope geometry has been chosen and analysed as an illustrative example and the strength reduction method is used to calculate the factor of safety (FOS) of slope. It has been found that when the temperature is steady, the FOS of the frozen soil slope is influenced by the spatial variability of the thermal conductivity, but the influence is not significant. When the standard deviation and the SOF of the thermal conductivity increase, the mean of the FOS is equal to the FOS of the homogeneous case and the standard deviation of the FOS also increases. After the frozen soil goes through freeze-thaw process, the FOS of the frozen soil slope decreases due to the reduction in the cohesion and the internal friction angle caused by the freeze-thaw cycles. Furthermore, the decreasing ratio of the FOS becomes more scattered after the 5th freeze-thaw cycle compared to that of the FOS after the 1st freeze-thaw cycle. The larger variability of the FOS could induce inaccuracy in the prediction of the frozen soil slope stability.

Influence of Soil Heterogeneity on the Contact Problems in Geotechnical Engineering

Contact problems are widely encountered in geotechnical engineering, such as the contact between soils and concrete used in earth and rockfill dams, tunnels and coastal levees. Due to the unknown contact region and contact forces, the contact problems have strong boundary nonlinearity. In addition, soils have been recognized as heterogeneous materials in geotechnical engineering. The existence of the soil heterogeneity increases the nonlinearity of the contact problems. Currently, the contact problems are mostly analysed without considering the soil heterogeneity, which may not reflect the contact behavior well. In order to investigate the influence of soil heterogeneity on the contact problems, in this paper, a simple plane-strain contact problem is analysed as an example. In this example, Young’s modulus is taken to be a spatially variable. The local average subdivision (LAS) is used to model the heterogeneity of Young’s modulus. The penalty method is utilised to determine the contact behavior. By the first use of linking the penalty method with the LAS, the proposed approach can be used to analyse the contact problems considering soil heterogeneity. The results show that the influence of soil heterogeneity on the elastic contact problems is significant. The contact forces of the heterogeneous case present apparent variation compared to the results of the homogeneous case. The distribution of the contact force at a specific point is also normal when Young’s modulus is normally distributed, moreover, the coefficient of variation (COV) and the horizontal scale of fluctuation of Young’s modulus affect the extent of variation of the normal contact forces. The standard deviation of the normal contact force increases with the increase of the COV and decreases with the increase of the horizontal scale of fluctuation of Young’s modulus. From the analyses, to better predict the deformation/stress in the contact problems, heterogeneity needs to be considered.

Effect of spatial variability of soil properties on permanent seismic displacements of slopes with uniform load

The purpose of this study is to investigate the development of permanent displacements of earth slopes, with spatially varying properties and a uniform surface load, subjected to a strong seismic excitation. The uncertainties associated with the determination of the mechanical properties of the slopes and the intensity of the seismic excitation, make the use of stochastic methods very attractive. As opposed to deterministic methods, stochastic methods allow an acceptable risk level based on the specifications of each project. Recent research described the spatial variability of soil parameters using the Random Field Theory. Apart from the prescribed statistical characteristics and cross-correlation of the various soil properties, the generated random field variables exhibit autocorrelation, a trend in which the soil properties of a point appear to be correlated with the soil properties of neighbouring points. Among the various random field algorithms, a particularly effective one is the Local Average Subdivision (LAS) method by Fenton and Vanmarcke [1]. To this end, a large number of random fields of soil properties is generated for a natural slope, having prescribed mean value μ, standard deviation σ, cross-correlation coefficients pij of properties i and j, and spatial autocorrelation lengths lx and ly for the vertical and horizontal directions, respectively. The numerical simulations are achieved using an automated procedure, based on collaboration of Mathematica, the incorporated LAS algorithm, and the finite difference program FLAC. The studied slope is loaded with a uniform load and subjected to a seismic shaking of various intensities using a series of acceleration records obtained from the five different earthquakes. The paper focuses mainly on the development of permanent seismic displacements of the slope at the end of ground shaking. It is demonstrated that the spatial variability of soil properties has a significant effect on the values of permanent displacements, resulting to a wide range of displacement variation.

Three‐Dimensional Stochastic Seepage Field Analysis of Multimedia Embankment

The soil hydraulic conductivity of an embankment has strong spatial variability due to the spatiotemporal variation, both natural and artificial. The strong randomness of the hydraulic conductivity can be expressed by the coefficient of variation (COV) and the fluctuation scale θ. Moreover, different coefficients of variation and fluctuation scales correspond to different random field structures. To study the characteristics of the three‐dimensional stochastic seepage field in an embankment under different COVs and fluctuation scales, we generate a three‐dimensional random field of the hydraulic conductivity of multimedia embankment based on the local average subdivision technique. In particular, a calculation method for a three‐dimensional random seepage field based on the Monte Carlo method combined with a three‐dimensional multimedia random field and a deterministic analysis is proposed. The results showed that after three thousand realizations and considering the randomness of the hydraulic conductivity, the position of the free surface of each section in the embankment differed. The mean value of the total head decreased when the COV increased. Furthermore, when the COV was small, the change in the total head with anisotropy ratio was not evident, while the COV was large. The mean value of the total head increased with the anisotropy ratio. When the anisotropy ratio increased, the mean value of the standard deviation of the total head increased first and then decreased.

Influence of random soil strength properties on the earthquake vulnerability of slopes with embedded oil and natural gas pipelines

Numerous pipeline systems carrying oil and natural gas, passing through geologically unstable areas worldwide are vulnerable to seismically induced slope instability, large ground displacements and settlements. Such ground movements constitute significant obstacles towards achieving certain standards for the pipeline concerning both safety and structural integrity, as well as conformity to safety rules regarding environmental safety in case of wall rupture and loss of containment. The two-dimensional distribution as well as the intensity of the subsequent ground movement could significantly depend on the two-dimensional mutability of soil properties in an area alongside the pipeline system. This paper aims at rationally accounting for the effects of the two-dimensional mutability of soil properties on the seismic performance of interred oil and natural gas conduits passing through natural slopes. More specifically, the effects of two-dimensional mutability of soil properties, slope inclination and earthquake excitation characteristics on the development of permanent displacements are investigated using random fields created by the Local Average Subdivision method by Fenton and Vanmarcke (J Eng Mech 116(8):1733–1749, 1990). These random fields of soil properties are characterized by specific mean, variance, cross-correlation coefficients and autocorrelation lengths and used for performing a series of seismic analyses of various slopes by utilizing the FLAC software to calculate finite difference through a Monte Carlo type simulation methodology. The outcomes of an all-encompassing parametric inquiry allow the development of vulnerability curves, based on the permanent horizontal or resultant displacement for several seismic intensity levels. Findings clearly show that the influence two-dimensional mutability has on soil properties on the ground movement imposed on the embedded pipeline are very important and should be accounted for in the evaluation of its seismic safety.

Simulation of random fields of soil properties by the local average subdivision method and engineering applications

The seismic performance of oil and natural gas pipelines founded or embedded in earth slopes encompasses great uncertainty related both to the earthquake shaking characteristics and to the natural heterogeneity of geomaterials. Regarding the latter, the parameters of shear strength, stiffness, density, etc., may vary indeed from point to point even within the same soil layer as a result of the natural formation process. Apart from their cross-correlation, such random variables exhibit autocorrelation, in which the soil properties at a given point appear to be spatially correlated with the properties of neighbouring points. Therefore, there is a need to use stochastic methods in the safety evaluation of such systems. Aiming at the reliability assessment of such soil-pipeline systems under seismic shaking, this paper introduces an automated methodology for generating random fields using the Local Average Subdivision (LAS) method by Fenton and Vanmarcke (J Eng Mech 116(8):1733–1749, 1990). Subsequently, it performs rigorous nonlinear dynamic analysis of a given slope using the finite difference method. The automated procedure is used in a Monte-Carlo simulation scheme for computing the probability of exceeding different levels of anticipated permanent slope movement for different levels of shaking intensity. The results demonstrate that the effect of the spatial variability of the soil properties on the permanent displacements of natural slopes is important, leading to a range of variation of about ± 60%.

A method for generating virtual soil profiles with complex, multi-layer stratigraphy

ABSTRACTThis paper presents a framework for generating multi-layer, unconditional soil profiles with complex stratigraphy, which simulates the effects of natural erosion and sedimentation processes. The stratigraphy can have varying degrees of randomness and can include features such as lenses, as well as sloped and undulating layers. The method generates the soil comprising the layers using local average subdivision (LAS), and a random noise component that is added to the layer boundaries. The layers are created by generating coordinates of key points in the simulated ground profile, which are then interpolated with a customised, 2D, linear interpolation algorithm. The resulting simulations facilitate more accurate probabilistic modelling of geotechnical engineering systems because they provide more realistic geologies, such as those usually encountered in the ground. Fortran code implementing this framework is included as supplementary material.

Effect of sampling plan and trend removal on residual uncertainty

ABSTRACTThe ground is one of the most highly variable of all engineering materials. As a result, geotechnical designs depend upon a site investigation to estimate the ability of the ground to perform acceptably. For example, when a shallow foundation is being proportioned to avoid a bearing capacity failure under a certain applied load, the frictional and cohesive properties of the ground under the foundation must first be estimated through a site investigation. Questions which arise are: How does the quality and intensity of the site investigation affect the design? Is more investigation cost effective? If the site is sampled at one location and the foundation placed at a different location, how does this mismatch affect the target design and the reliability of the final foundation? By modelling the ground as a spatially variable material, questions such as the above can be investigated through Monte Carlo simulation and sometimes theoretical probabilistic models. Using such tools, this paper looks specifically at how the intensity (frequency and spatial distribution) of a site sampling plan, and how the samples are used, affects the understanding of the ground properties under a foundation. Interestingly, it is found that removing the sample mean outperforms removing the best linear unbiased estimate (BLUE) when the actual field correlation length is small but the BLUE correlation length is assumed equal to the field size. Recommendations are made regarding number of samples and the type of trend to best characterise the field.Abbreviations: BLUE: best linear unbiased estimate; MCS: Monte Carlo simulation; LAS: local average subdivision

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This work focuses on the probabilistic analysis of slope stability and rigid shallow footing problems. For this purpose, mathematical models based on the Finite Element Method (FEM), Montecarlo (MC) method and Local Average Subdivision (LAS) procedure are studied. The LAS procedure is used to generate random fields, which properly represent the associated uncertainties in the properties of the materials. The FEM focuses on the numerical response of the problem in terms of displacements and stresses. The plasticity of the soil can be included via a visco-plastic algorithm beside a Mohr-Coulomb law. The LAS and MEF procedures are implemented in the framework of a MC analysis, where each MC execution requires several simulations of the problem at hand. This permits to quantify the failure probability of the system and report the most probable settlement to occur in the case of shallow foundations. After many executions of the numerical model, it is suggested that at least 4000 and 500 simulations are needed for the slope stability and shallow foundation problems, respectively, in order to obtain stable and reliable values. The obtained results show that the failure probability of the slope is relatively low and equal to 0.18, while the expected settlement of a shallow rigid foundation is around 1.96 cm.

Soil Probabilistic Slope Stability Analysis Using Stochastic Finite Difference Method

The paper contrasts results obtained by the partially factored limit state design method and a more advanced Random Finite Difference Method (RFDM) in a benchmark problem of slope stability analysis with variable undrained shear strength. Local Average Subdivision method was used to simulate the non-Gaussian random variables. The key difference between the methods is that RFDM takes into account spatial variability of soil parameters allowing slope failure to occur naturally along the path of least resistance. The probabilistic method leads to predictions of the "probability of slope failure" as opposed to the more traditional "factor of safety" measure of slope safety in the limit state design method; however, they give significant different results depending on the level of the variability. Analyses conducted using Monte Carlo Simulation show that the same partial factor can have very different levels of risk depending on the degree of uncertainty of the mean value of the soil shear strength. Calibration studies show the partial factor necessary to achieve target probability values.

Random analysis of bearing capacity of square footing using the LAS procedure

AbstractIn the present paper, a three-dimensional problem of bearing capacity of square footing on random soil medium is analyzed. The random fields of strength parameters c and φ are generated using LAS procedure (Local Average Subdivision, Fenton and Vanmarcke 1990). The procedure used is re-implemented by the authors in Mathematica environment in order to combine it with commercial program. Since the procedure is still tested the random filed has been assumed as one-dimensional: the strength properties of soil are random in vertical direction only.Individual realizations of bearing capacity boundary-problem with strength parameters of medium defined the above procedure are solved using FLAC3D Software. The analysis is performed for two qualitatively different cases, namely for the purely cohesive and cohesive-frictional soils. For the latter case the friction angle and cohesion have been assumed as independent random variables. For these two cases the random square footing bearing capacity results have been obtained for the range of fluctuation scales from 0.5 m to 10 m. Each time 1000 Monte Carlo realizations have been performed. The obtained results allow not only the mean and variance but also the probability density function to be estimated. An example of application of this function for reliability calculation has been presented in the final part of the paper.

Use of local average subdivision to characterize marine mineral reserves—A conceptual framework

Use of local average subdivision to characterize marine mineral reserves—A conceptual framework

Three-dimensional finite element analysis of spatially variable PVD improved ground

A stochastic approach that investigates the effects of soil spatial variability on stabilisation of soft clay via prefabricated vertical drains (PVDs) is presented and discussed. The approach integrates the local average subdivision of random field theory with the Monte Carlo finite element (FE) technique. A special feature of the current study is the investigation of impact of spatial variability of soil permeability and volume compressibility in the smear zone as compared to that of the undisturbed zone, in conjunction with uncoupled three-dimensional FE analysis. A sensitivity analysis is also performed to identify the random variable that has the major contribution to the uncertainty of the degree of consolidation achieved via PVDs. The results of this study indicate that the spatial variability of soil properties has a significant impact on soil consolidation by PVDs; however, the spatial variability of soil properties in the smear zone has a dominating impact on soil consolidation by PVDs over that of the undisturbed zone. It is also found that soil volume compressibility has insignificant contribution to the degree of consolidation estimated by uncoupled stochastic analysis.

Bearing Capacity of Spatially Random Cohesive Soil Using Numerical Limit Analyses

This paper describes a probabilistic study of the two-dimensional bearing capacity of a vertically loaded strip footing on spatially random, cohesive soil using numerical limit analyses (NLA-CD). The analyses use a Cholesky decomposition (CD) technique with midpoint discretization to represent the spatial variation in undrained shear strength within finite-element meshes for both upper- and lower-bound analyses and assumes an isotropic correlation length. Monte Carlo simulations are then used to interpret the bearing capacity for selected ranges of the coefficient of variation in undrained shear strength and the ratio of correlation length to footing width. The results are compared directly with data from a very similar study in which bearing capacity realizations were computed using a method of local average subdivision (LAS) in a conventional displacement-based finite-element method (FEM-LAS). These comparisons show the same qualitative features but suggest that the published FEM calculations tend to ov...

A method to assess risk reduction when utilizing geosynthetic clay liners (GCLs) with compacted soil liners

This paper presents an analytical solution developed to estimate probabilities of “failure” or advective flux “exceedance” for the case of a spatially variable geosynthetic clay liner (GCL) situated over a spatially variable compacted soil liner (CSL). The risk of utilizing such a liner system is assessed relative to a regulatory compacted clay-based soil liner. The analytical solution developed is validated over a range of parameters against random field simulation using the Local Average Subdivision method, and the analytical solution is shown to be in good agreement with simulation. The analytical solution is then used to examine the “probability of exceedance” for a spatially variable GCL and CSL combined liner system. It is shown that the use of a GCL can potentially result in a low probability of exceedance when used with a spatially varying, high hydraulic conductivity CSL. The risk of exceedance generally decreases as the hydraulic conductivity of the CSL decreases. An example problem is presented to demonstrate the capabilities of the analytical solution.

Probabilistic Analysis of Coupled Soil Consolidation

Coupled Biot consolidation theory was combined with the random finite-element method to investigate the consolidation behavior of soil deposits with spatially variable properties in one-dimensional (1D) and two-dimensional (2D) spaces. The coefficient of volume compressibility ( mv ) and the soil permeability (k) are assumed to be lognormally distributed random variables. The random fields of mv and k are generated by the local average subdivision method which fully takes account of spatial correlation, local averaging, and cross correlations. The generated random variables are mapped onto a finite-element mesh and Monte Carlo finite-element simulations follow. The results of parametric studies are presented, which describe the effect of the standard deviation, spatial correlation length, and cross correlation coefficient on output statistics relating to the overall “equivalent” coefficient of consolidation. It is shown that the average degree of consolidation defined by excess pore pressure and settlemen...

Reply to the discussion by R. Popescu on "Bearing capacity prediction of spatially random c – ϕ soils"

Fenton and Griffiths The discussor brings up a number of good points regarding the difficulties in characterizing spatial variability of soils. The main points of concern raised by the discussor are the high coefficients of variation (COVs) considered by the authors and the methodology used to simulate the random soil property fields. It is hoped that the following discussion will shed some light on these concerns. 369 Regarding the range in COV considered by the authors, which was 0.1–5.0, the authors believe that it is still unknown what value(s) of COV should be used in geotechnical characterization. The appropriate COV depends on several things, such as the intensity of site investigation; the level of deterministic site characterization (e.g., higher order trend, layer-wise descriptions); and how the soil variance affects the response quantity, or engineering property, of interest (i.e., is the property itself a measure of some form of local average, or is it highly dependent on microscale “defects”?). The issue of site investigation intensity is intimately connected to the degree of deterministic site characterization, for example, if only a single value global average property is employed in the design of a footing, then the site investigation results could yield a large COV, particularly if the site is large and the investigation points are widely separated. The COV would then be interpreted as one’s “uncertainty” about the value of the property at the footing if no test results were available near the footing. If a particular test result were available near the footing, then that result would be preferentially used to design the footing, and the corresponding COV would be reduced. In such a case, the site characterization moves away from a single value global average to a more detailed deterministic description that incorporates observation versus footing locations. The COV used for design depends on how the investigation data are used and on where the investigation points are relative to the footing. For one site with considerable data near the footing, the COV to be used might be quite small, whereas for another site with limited data and (or) data well removed from the footing location, the appropriate COV might be quite large. In this sense, the comprehensive results reported by authors such as Phoon and Kulhawy (1999) are really just a start at the characterization of soil variability. They are basically reporting COVs estimated from a particular dataset, reflecting the residual variability about the locally estimated mean (or mean trend). These results tell us little about how to handle uncertainty about the soil properties at some distance from where the soil was actually sampled. There is much that is unknown about this problem, and considerable research that needs to be done before definitive levels of COV can be stated for any given situation. For this reason, the authors chose to perform their analysis over a wide range in COV values. This is not viewed as a recommendation that designers should be considering COVs as high as 5.0, but rather allows the results to be used in the event that a designer determines such a high COV is appropriate. Alternatively, if a lower COV seems appropriate, these results are also included in the paper. It is not clear to the authors why the discussor is introducing the idea of the finite element model size into the choice of COV. The important issue is the estimate of the “point” COV (where point is usually some local average over a small volume) from a set of data collected in the field. How the data are collected, how the statistical analysis is carried out, and how the results are to be used will affect the value of the estimated point COV. Once that value has been determined, it is appropriate to use it in whatever numerical model one chooses. The quality of the numerical model in representing reality is another issue, but the authors have strived to produce a model that reflects the material behaviour as well as possible given current computational resources. In particular, the Local Average Subdivision method employed by the authors correctly reflects the transformation from the true point statistics to the element averages that is consistent with the continuum finite element model. The second issue raised by the discussor has to do with the ability of the authors’ simulation technique to adequately represent the prescribed random fields. The discussor is concerned that the nonlinear transformation, when going from the underlying Gaussian random field to the target soil property, affects the final correlation structure. This transforma-

Conditional LAS stochastic simulation of regionalized variables in random fields

Ore reserves forecasts are required to aid in investment decisions, mine design and valuation, short and long term production plans and proper and efficient mill design. In random multivariable fields with limited data and high levels of uncertainty, the kriged block estimates produce a smoothing effect resulting in underestimating high values and overestimating low values. The modified conditional simulation (MCS) methodology solves these problems by simulating the random field to preserve its mean and the variance structure. The simulation model is conditioned to reproduce the data at known sample points to minimize the variability between the simulated data and the true field data. In this study, the authors develop the MCS methodology to simulate ore reserve grades using the best linear unbiased estimation (BLUE) and the local average subdivision (LAS) techniques. The MCS methodology is applied to simulate block grades in a section of the Sabi Gold Project in Zimbabwe. The results are compared with the kriged estimates for this section. Analysis of the results shows that the MCS methodology reproduces the known sample grades with minimum estimation error of 0.001 while the estimation error associated with the kriged estimates is 1.104, a 100% efficiency of the MCS method over the kriging technique.