Horizontal Scale Of Fluctuation Research Articles

Characterisation of the spatial variability of underconsolidated Singapore Marine Clay using random field theory

ABSTRACT This paper characterised the spatial variability of Singapore Kallang Formation Marine Clays using piezocone penetration test (CPT) data from the Marina South area. Both vertical and horizontal scales of fluctuation in undrained shear strength for two distinct clay units, Upper Marine Clay (UMC) and Lower Marine Clay (LMC), are calibrated for autocorrelation and semi-variogram functions. The Marina South area was reclaimed from the sea using fills placed in three phases from 1970s to 1990s and remains underconsolidated up to date. This study identified variations in the degree of consolidation across the area, which has led to variations in the undrained shear strength in this area. However, the variations in the degree of consolidation were found to have no noticeable influence on the vertical and horizontal scales of fluctuation of either UMC or LMC unit. The vertical scales of fluctuation of UMC and LMC units are 0.57 ± 0.3 m and 0.48 ± 0.17 m , respectively, while the horizontal scales of fluctuations are 148 ± 26 m and 138 ± 28 m , respectively. The present study contributes to the database of the statistical and spatial variability of marine clays and can provide useful guidance for assessing reliability of excavations and foundations in Marina South as well as other areas with similar geological settings.

Probabilistic assessment of bearing capacity of skirted foundation under combined loadings with a rigid base

Skirted foundations are critical components in offshore applications where combined loads are common in deep‐water environments. Their ultimate capacity under VH (vertical‐horizontal) combined loading is traditionally determined using VH failure envelopes, which are primarily constructed using numerical methods. These methodologies, however, frequently ignore the spatial variability inherent in seabed soils due to geological formations. This paper investigates the effect of spatial variability of undrained shear strength and embedment ratio impact on the capacity of skirted foundations subjected to VH combined loading. For this, OptumG2 software is used to perform Monte Carlo simulation combined with random finite element limit analysis. This paper investigates the stochastic analysis of bearing capacity and failure envelopes, with a particular emphasis on understanding the effect of spatial correlation on undrained shear strength. The study focuses on the horizontal scale of fluctuation and the soil strength heterogeneity index, shedding insight on previously undiscovered areas. Novel findings highlight how a rigid base affects VH failure envelopes and offer insights into evaluating the vertical bearing capacity of skirted foundations. [Correction added on 8 July 2024, after first online publication: The above statement has been updated in this version.]

Reliability analysis of pile-founded T-walls using the random variable method and random field method

The pile-founded walls (e.g., T-walls) are important civil infrastructure for flood protection. The stability analysis of floodwalls in the face of flooding hazards is a critical problem for the design and maintenance of floodwalls. However, most current studies for the stability assessment of floodwalls are based on deterministic analyses. The uncertainties of soil properties, which cannot be avoidable due to complex geological processes and depositional environment, are not sufficiently studied. To address the uncertainties of soil properties, a two-dimensional pile-founded T-wall system with several clay layers on the top and a sand layer on the bottom is taken as an example for the reliability analysis. Both the random variable method and random field method are adopted for the probabilistic analysis of floodwalls. The effect of the coefficient of variation (COV) and scales of fluctuation of soil properties are investigated by parametric studies. The results show that the probability of failure increases with the COV of soil properties for both methods. For the random field method, the effect of the vertical scale of fluctuation of soil properties is more profound than the horizontal scale of fluctuation. The probability of failure derived from the random variable method is generally larger than that from the random field method at high flood water elevations when the same COV of soil properties is applied. This study can provide useful references for risk-informed decision-making in the stability assessment of floodwalls in the face of flooding hazards.

Development of two-dimensional ground models by combining geotechnical and geophysical data

Geotechnical and geophysical testing data are conventionally considered as separated information or combined based on deterministic methods in site investigation programs, which causes loss of information and introduces additional uncertainties. This study aims to reduce the uncertainties and costs in inhomogeneous soil profile characterization and geotechnical analysis by quantitatively integrating these data. The intrinsic collocated co-kriging method (ICCK) is utilized to integrate Multi-channel analysis surface wave (MASW) and CPT data. This research shows the potential of using the MASW data to explore the horizontal scale of fluctuation (SOF) of the cone tip resistance (qc) field which is very difficult to get using the conventional methods due to the limited CPTs. The Markov model is used to avoid the tedious modeling of the cross-covariance relationship in the ICCK method. A series of synthetic case studies show that the combined soil profile is in good agreement with the “true” qc field with significantly reduced uncertainties. Based on estimating the uncertainties, the optimal distance between CPTs is suggested to be 1–2 horizontal SOF. This framework is also applied to a real case in the Christchurch area, which involves integration of CPTs and MASW tests. Cross validation and edge detection methods are used to quantitatively compare the integration results, confirming that the proposed framework is fully applicable to field data and can provide realistic and reliable estimations of qc field.

Reliability Analysis of Soil Liquefaction Considering Spatial Variability of Soil Property

The objective of this study was to investigate the liquefaction response of soil using the spatial variability of the shear modulus by considering different values of the coefficient of variation (COV) and the horizontal scale of fluctuation (SOF). For this purpose, a Monte Carlo simulation, combining the digital generation of a non-Gaussian random field with finite difference analyses, was utilized. Parametric studies were performed from the perspectives of the liquefaction area, excess pore water pressure (EPWP), and displacement at the ground surface. We found that a larger COV of the soil shear modulus was correlated with a slower reduction of liquefaction area and a lower EPWP ratio. To explain the influence from the perspective of the spatial distribution characteristics of the shear modulus, a deterministic model test was carried out. Additionally, it was found that the displacement history and the differential settlement at the end of shaking were regularly affected by the COV and the horizontal SOF, especially for large COV and horizontal SOF.

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.

A framework for the design of vertically loaded piles in spatially variable soil

Each pile in an offshore wind farm may be designed solely on the results of a single cone penetration test (CPT) positioned at its center or based on sample statistics calculated from multiple investigated locations in the same geotechnical unit. This study investigates quantitatively the effects of spatial variability of soil on the efficacy of the two approaches to achieve a reliable design. Random field theory is used to generate sets of cone tip resistance profiles with different characteristics to represent different seabed conditions and piles are ‘designed’ using the above two approaches and the partial factor method to withstand a given vertical drained design load distribution. The effects of the inaccuracies in prediction of pile capacity from the CPT data are quantified in terms of the scatter in calculated reliability (or probability of failure). Outputs of the parametric statistical analysis demonstrate that the ability of the two CPT-based approaches to produce a reliable foundation design is dependent on the ratio between the horizontal scale of fluctuation and the pile diameter, and the overall variability (coefficient of variation) of the seabed properties. Modifications to deterministic, partial factor design approaches are suggested to account for these uncertainties to ensure target reliabilities are achieved.

Improved Vanmarcke analytical model for 3D slope reliability analysis

This paper presents an improved algorithm for evaluating the reliability of three-dimensional soil slopes based on the Vanmarcke model (VM). In the development of the improved Vanmarcke model (IVM), the sliding mass is assumed to be a portion of a cylinder bounded by two ellipsoidal end sections. The soil spatial variability of both the cylindrical and ellipsoidal sections is considered. A set of equations is derived to determine critical failure surfaces and compute their reliability indices. Numerical studies are performed to examine the IVM. The results indicate that (1) the IVM predicts a similar critical failure length compared with that computed by the VM. However, a lower reliability index is obtained by the former. (2) The relative difference between the reliability indices computed by the two analytical models can reach 20% for practical situations. The horizontal scale of fluctuation (SOF) of soil properties is the main factor affecting the relative difference. (3) Compared to the random finite element method, the IVM still gives unconservative analysis results, especially at low horizontal SOFs. This is due to the prediction of smaller critical failure lengths by the IVM for small horizontal SOFs and some retained assumptions of the VM when developing the IVM.

Probabilistic Bearing Capacity of a Pavement Resting on Fibre Reinforced Embankment Considering Soil Spatial Variability

This paper intends to examine the influence of spatial variability of soil properties on the probabilistic bearing capacity of a pavement located on the crest of a fibre reinforced embankment. An anisotropic random field, in combination with the finite difference method, is used to carry out the probabilistic analyses. The cohesion and internal friction angle of the soil are assumed to be lognormally distributed. The Monte Carlo simulations are carried out to obtain the mean and coefficient of variation of the pavement bearing capacity. The mean bearing capacity of the pavement is found to decrease with the increase in horizontal scale of fluctuation for a constant vertical scale of fluctuation; whereas, the coefficient of variation of the bearing capacity increases with the increase in horizontal scale of fluctuation. However, both the mean and coefficient of variation of bearing capacity of the pavement are observed to be increasing with the increase in vertical scale of fluctuation for a constant horizontal scale of fluctuation. Apart from the different scales of fluctuation, the effects of out of the plane length of the embankment and randomness in soil properties on the probabilistic bearing capacity are also investigated in the present study.

Probabilistic analysis of the diaphragm wall using the hardening soil-small (HSs) model

The main goal of this article is to perform a probabilistic analysis on a cantilever diaphragm wall constructed in sandy subsoil characterized by spatially variable strength and stiffness parameters. Three quantitates are considered, namely, the deflection of the wall, the settlements behind the wall and the maximum bending moment along the wall. To enable high accuracy of the evaluation of these quantities, a hardening soil model with a small strain overlay is used. To account for soil spatial variability, the friction angle of the sand is modelled using an anisotropic random field with specified vertical and horizontal scales of fluctuation (SOFs). All computations of the considered boundary value problem are carried out by means of the random finite element method (RFEM). The influence of both the vertical and horizontal SOFs on the probabilistic distributions of all the considered quantities is investigated. Reliability analyses are carried out based on the obtained distributions. Additionally, the effect of the number of realizations in the simulation process is analysed. It is shown that in the case of a diaphragm wall, the value of the horizontal SOF can substantially affect the probability of both excessive deformations as well as high bending moment values. It is also shown that additional modelling of soil stiffness moduli by a random field may induce small but noticeable changes in the probability of exceeding the conditions of the serviceability limit state.

Reliability analysis of sheet pile wall in spatially variable soil including CPTu test results

The paper deals with reliability analysis of cantilever sheet pile wall located in non-cohesive soil with random properties. Spatial variability of friction angle has been described using random fields theory. The influence of both vertical as well as horizontal scale of fluctuation on the mechanical response of sheet pile wall is investigated. Deflection of wall top as well as maximum bending moment in the sheet pile wall are tested. The point distribution of soil friction and its vertical fluctuation scale is estimated using quasi-continuous results of cone penetrometer tests (CPTu). The boundary value problem is solved using finite difference code FLAC. The Fourier series method (FSM) allowing for non-uniform meshes is used to generate random fields for individual realizations. By utilizing Monte Carlo Simulation (MCS) technique the probability distributions of the results for different values of vertical and horizontal scales of fluctuation are obtained and used for reliability analysis. The results of analysis show that in case of cantilever sheet pile wall it is very important to properly estimate value of vertical fluctuation scale for the reliability analysis. It is also illustrated that in the considered problem horizontal scale of fluctuation significantly influences probability of failure.

Probabilistic assessment of tunnel convergence considering spatial variability in rock mass properties using interpolated autocorrelation and response surface method

This study aims at the probabilistic assessment of tunnel convergence considering the spatial variability in rock mass properties. The method of interpolated autocorrelation combined with finite difference analysis is adopted to model the spatial variability of rock mass properties. An iterative procedure using the first-order reliability method (FORM) and response surface method (RSM) is employed to compute the reliability index and its corresponding design point. The results indicate that the spatial variability considerably affects the computed reliability index. The probability of failure could be noticeably overestimated in the case where the spatial variability is neglected. The vertical scale of fluctuation has a much higher effect on the probabilistic result with respect to the tunnel convergence than the horizontal scale of fluctuation. And the influence of different spacing of control points on the computational accuracy is investigated.

Evaluation of spatial soil variability in the Pearl River Estuary using CPTU data

For a project in the Pearl River Estuary a very extensive site characterisation was carried out. The aim of this paper is to use this extensive investigation campaign to understand the spatial variability of the soil in both horizontal and vertical directions based on cone penetration tests with pore pressure measurement (CPTU). The investigated area stretches over 6762m length and 50m width covering an area of 338100m2 with CPTU soundings every 25m. Based on boreholes and CPTU soundings the stratigraphy is known and thus the correlation characteristics can be estimated specifically for the individual soil types. Soil types found in the Pearl River Estuary are: marine clay and sand, continental clay, marine alluvial clay and sand, fluvial alluvial clay and sand. For all these soil types the vertical and horizontal scales of fluctuation are assessed. This is an important parameter for probabilistic analysis of geotechnical problems such as settlements, differential settlements, bearing capacity and slope stability.

Undrained strength for a 3D spatially variable clay column subjected to compression or shear

The mobilized shear strength (τfm) of a soil is the most important design parameter governing ultimate limit states. The authors have proposed some models and equations to characterize several useful aspects of the mean, variance, and probability density function of τfm for a two dimensional (2D) spatially variable clay specimen. This paper demonstrates that the models and equations developed for 2D can be extended to three-dimensional (3D) cases. An important class of 3D cases are examined in detail: two horizontal scales of fluctuation (SOF) are equal, and both are larger than the vertical SOF. A key observation for 3D cases is that the averaging effect over the potential slip planes is significantly stronger than that for 2D cases. The upshot is that the phenomenon of the critical SOF (e.g., SOF producing the lowest mean value of τfm) is less clear for 3D cases.

Effective Young's modulus for a spatially variable soil mass subjected to a simple stress state

ABSTRACTThe effect of spatial variability on the effective Young's modulus of a soil mass is investigated. The soil mass is a two-dimensional plane strain square domain subjected to a simple stress state, with Young's modulus being modelled as a stationary lognormal random field. The effective Young's modulus is simulated by random field finite-element analysis. It is found that under the condition of horizontal scale of fluctuation (SOF) = vertical SOF, the effective Young's modulus can be satisfactorily approximated as the geometric average over the square domain. However, the conclusion changes dramatically when the spatial variability is highly anisotropic, e.g. horizontal scale vertical SOF. In this case, the effective Young's modulus can be approximated as the arithmetic average or harmonic average, depending on the direction of loading. A unified spatial averaging model is further proposed in this study. It is shown that the effective Young's modulus can be satisfactorily approximated by this unified model without the need to switch among arithmetic, geometric and harmonic averages.

Influence of heterogeneity on undrained clay slope stability

This paper concentrates on the fact that natural materials are variable, and that representation of this variability appears crucial to getting a realistic understanding of certain geotechnical problems. In this case, finite elements have been used to assess the influence of heterogeneity on the stability of a clay slope, characterized by a spatially varying undrained shear strength c u . For low to intermediate values of the coefficient of variation (e.g. 0.1–0.3), as are often observed in practice, pointwize variability can be approximated by a normal distribution. Data on spatial variability are more scarce, although vertical scales of fluctuation are usually much smaller than horizontal scales of fluctuation, and will often be small relative to the height of a slope. The results show that, for a given factor of safety, reliability is greatest for c u constant with depth. For c u increasing linearly with depth, reliability decreases due to the greater range of possible rupture surfaces: hence, in conventional (deterministic) design, higher factors of safety (or lower characteristic values) would be needed. In this study, horizontal anisotropy (of the heterogeneity) causes modest changes in reliability. However, problems are identified in which anisotropy is likely to have a significant influence.