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Random Fields Analysis and Synthesis
Abstract
Existing and new methodologies of random field theory are discussed in
terms of their application to diverse areas in science and technology
where a deterministic treatment is inefficient and conventional
statistics are insufficient. The extent and characteristics of the
random field approach are outlined, the classical theory of
multidimensional random processes is reviewed, and basic probability
concepts and methods in the random field context are introduced.
Second-order analysis of homogeneous random fields in both the
space-time domain and the wave number frequency domain is considered.
Spectral moments and related measures of disorder are discussed, as are
level excursions and extremes of Gaussian and related random fields. A
new analytical framework based on local averages of one, two, and
n-dimensional processes is developed, and its ramifications in important
areas of estimation, prediction, and control are discussed.
... To address this issue, the random failure mechanism method (RFMM) [39,40] encompasses several attractive features pertaining to its practical implementation. By resorting to the kinematic method of limit analysis [41,42] and the spatial averaging technique [43], random bearing capacities can be evaluated in a numerically tractable fashion while explicitly accounting for the three-dimensional variability of soil properties. Even though the RFMM allows considering the effect of soil soundings on random bearing capacities [44][45][46], the practical implementation of the method has been mostly limited to cases involving a single borehole. ...
... To account for the spatial variability of , which is represented as a stationary lognormal random field (see Section 2.1), the vector̄, is assumed to follow a lognormal distribution with a certain correlation structure [39,40,43,51]. Specifically, and employing concepts from Vanmarcke's spatial averaging [43], the vector̄, has mean value and covariance matrix ∈ R × . ...
... To account for the spatial variability of , which is represented as a stationary lognormal random field (see Section 2.1), the vector̄, is assumed to follow a lognormal distribution with a certain correlation structure [39,40,43,51]. Specifically, and employing concepts from Vanmarcke's spatial averaging [43], the vector̄, has mean value and covariance matrix ∈ R × . The latter is determined in terms of the chosen covariance function and the geometry of the failure mechanisms associated with = [53]. ...
This contribution proposes a framework to identify optimal borehole configurations for the design of shallow foundation systems under undrained soil conditions. To this end, the minimization of a performance measure defined in terms of the bearing capacity standard deviations is considered. The random failure mechanism method is adopted for random bearing capacity evaluation, thereby enabling explicit treatment of soil spatial variability with tractable numerical efforts. A sampling-based optimization scheme is implemented to account for the non-smooth nature of the resulting objective function. The proposed framework provides non-trivial sensitivity information of the chosen performance measure as a byproduct of the solution process. Further, the method allows assessing the effect of increasing the number of soil soundings into bearing capacity standard deviations. Three cases involving different foundation layouts are studied to illustrate the capabilities of the approach. Numerical results suggest that the herein proposed framework can be potentially adopted as a supportive tool to determine optimal soil sounding strategies for the design of a practical class of civil engineering systems.
... The focus of the papers is on the design of open-ended piles subjected to uplift (tensile) loading and installed in a single sandy soil layer, noting that this is a common geotechnical problem (e.g., Lehane et al. 2005;Cai et al. 2021). In this paper, the random field theory (Vanmarcke 1984) was used to generate sets of soil strengths data for "seabeds", then the UWA-05 method combined with an advanced Monte Carlo simulation (MCS)-subset simulation (SS)-approach was employed to achieve the aforementioned objectives. The statistical properties of the pile capacity and the p f achieved using different CPT layouts and PNs in different soil conditions were calculated. ...
... This simplification has been made to isolate the already complex statistics of soil variability in this study, although it is noted that real seabeds often have multiple soil layers (e.g., Wang et al. 2019;Zhao et al. 2020). Furthermore, the work used random field calculations (Vanmarcke 1984) to generate sets of synthetic soil strength data for "seabeds" rather than relying on information from realbut unverifiable-site conditions (the methods for generating synthetic soil strength data are detailed in sections "Spatial Variability in Cone Tip Resistance of Seabed Soils" and "Generation of Random Fields"). This allowed multiple sets of "seabed" conditions to be generated, each with known statistical properties allowing direct examination of the consequences of using different PNs on design outcomes (e.g., by "revealing" information on soil conditions contained in a synthetic site investigation, and then being able to calculate how the designed pile behaves). ...
... The coefficient of variation [COVð·Þ] is the ratio between the standard deviation and mean (or trend) of a variable, which parameterizes its relative scatter around a central tendency values. Based on random field theory (Vanmarcke 1984), the vertical spatial variation of q c can be characterized by a deterministic depth-wise trend μ qc (i.e., mean value of q c at a given depth) and the corresponding zero-mean residuals (which can be conveniently modeled as a random variable) as shown on Fig. 1. The random residuals about the mean (trend) reflect the inherent variability of q c , where the spatial correlation of the residuals of q c or its logarithm at two spatial points can be expressed as an autocorrelation function ρ of the lag distance (κ) between these two points and the horizontal and vertical scales of fluctuation (θ h and θ v , respectively) (Vanmarcke 1984;Phoon and Kulhawy 1999). ...
Due to the mechanical properties of the seabed exhibiting horizontal and vertical spatial variability, an important challenge in offshore geotechnical design lies in the selection of the relevant soil strength profile for foundation sizing, called hereafter the “design line.” Design values are most often selected depth-wise using existing field data, through knowledge of the volume and distance of the field data from the designed infrastructure and considering the relevant limit state and targeted level of reliability. The cone penetration test (CPT) is well-suited to this purpose given the quasi-continuity and high repeatability of its measurements. This paper proposes a statistical CPT-based method to identify the design line for the design of open-ended piles under drained uplift loading to achieve a target reliability. Application of
the proposed method reveals that the statistical procedure used to select the design line is dependent on the proximity of the nearest CPTs to the piles, and the horizontal and vertical scales of fluctuation and the coefficient of variation of the cone tip resistance. This information is used in a companion paper to inform optimization and selection of site investigation strategies for the design of uplifted piles for offshore wind farms.
... In the current work, an analytical framework is presented to capture the effects of sample length scale on the FIP maximum value distribution (MVD) in metals and alloys without requiring the condition of convergence for the EVD of FIPs. The framework is an extension of the theory developed in the area of wind engineering [16,17], which has been extended to include non-Gaussian fields in [18,19] through the use of translation field theory [20,21]. The model has been demonstrated conceptually for 1-dimensional structural mechanics problems [22], and in this work it is extended to response metrics of 3-dimensional crystal plasticity finite element (CPFE) simulations. ...
... This section provides a brief overview of the derivation of the probability distribution of the maximum value of a random field over a finite domain for completeness. The classical theory developed for Gaussian random fields dates back to the work of Davenport [16], which is further elaborated by Vanmarcke [17]. The distribution of the maximum value of a random field over a domain refers to the sample-tosample uncertainty in the observed maximum value. ...
... The SVE size is therefore well below the size of the representative volume element (RVE) necessary to describe all higher moments of the distribution of FIPs. While the current approach has not been previously adapted for fatigue applications, the distribution of the maximum value of a Gaussian random field over a finite volume has been studied extensively in the area of wind engineering [16,17], and it has been extended to include non-Gaussian fields in [18,19] through the use of translation field theory [20,21]. ...
... Based on unconditional random fields (URF), considerable effort has been made to investigate the effects of the spatial variability of soil's undrained shear strength (s u ) on the bearing capacity of monopiles (e.g., Fenton and Griffiths 2007;El Haj et al. 2019;Yi et al. 2023) and other foundations (e.g., Griffiths et al. 2002;Li et al. 2015;Liu et al. 2023). URF is realized using only the statistics [e.g., mean, coefficient of variation (COV), autocorrelation function and scale of fluctuation (SOF)] of the site investigation data, and discards the actual data at "sampled" locations (Vanmarcke 1983), which is considered a waste of site investigation effort. Additionally, neglecting the actual data increases the simulation variance of the random field, which subsequently affects the responses, such as the probability of failure of the slope system (Liu et al. 2017). ...
... Vertical and horizontal scales of fluctuation (δ v and δ h ), which describe the correlation distance of soil properties (Vanmarcke 1983), are important feature parameters used in both URFs and CRFs, and affect the random bearing capacity of foundations (Griffiths et al. 2002;Liu et al. 2023). However, the quantification of SOF is not easy, because (1) in situ data are sparse, especially in the horizontal direction (Zhang et al. 2022), and (2) the quantification of SOF has a size effect and its value depends on the observed scale (Zou 2018). ...
The spatial variability of soil is usually modeled as a nonstationary unconditional random field (URF), which utilizes only the statistics of the soil field, and conditional random field (CRF), which utilizes both the statistics and the actual data at "sampled" locations. The simulated undrained shear strength of soil and random bearing capacity of monopiles using URF and CRF are compared, and the results show that CRF can more accurately reflect the real soil strength profile and significantly reduce the coefficient of variation of random bearing capacity of monopiles. However, of course, the priority of CRF over URF is based on the quality of the sampled data. The effects of various site investigation parameters (vertical investigation interval, investigation depth, and horizontal investigation distance) on the bearing capacity of monopiles are investigated. It can be concluded that, to ensure the quality of sampled data for establishing CRF, the vertical investigation interval and horizontal investigation distance should be less than 0.75 times the vertical and horizontal scales of fluctuation, respectively, and the investigation depth should be greater than 1.2 times the embedment depth of monopiles. Because the estimation of the horizontal scale of fluctuation is rarely performed due to the sparse in situ data in the horizontal direction, the worst-case scale of fluctuation for monopiles is also investigated. The worst-case horizontal scale of fluctuation mainly depends on the monopile diameter and is 5-7 times the diameter of monopiles. Therefore, to be on the safe side, it is suggested that the horizontal scale of fluctuation be 5-7 times the diameter of monopiles in the design and evaluation of monopile foundations.
... One of the practical tools to express the soil spatial variability is random field (RF) theory, which seeks to model complex patterns of variation and interdependence in cases where deterministic treatment is inefficient and conventional statistics are insufficient [6,7]. The first geotechnical application combined the random field theory with the finite element method on a footing settlement analysis using Taylor series expansion, entitled the stochastic finite element method (SFEM) [8]. ...
... Once the random variable fits a probability distribution, the field is then determined by the correlation function, which depends on the variation distance of the random parameter in the field called correlation length . Markov correlation function in Eq.1 was adopted for 2D Gaussian random field case [7]: ...
The present study aimed to create a series of hazard curves against maximum total settlement and angular rotation of strip footings for probabilistic shallow foundation design on clays. Random field finite element method (RFEM) was adopted with elasto-plastic clay-like soil behavior, deformation modulus (E d) and shear strength parameters (c and ) were employed as random field inputs. Parameters were defined and assigned to the analysis models with varying correlation lengths ( h , v). Models have been iteratively solved one thousand times, and output distributions of maximum settlement and angular rotations were recorded. Probability density functions (PDF) were fitted to the outputs, and probability of failure (P f) for footing deformation limits was subsequently estimated. Proposed hazard curves for two anisotropy and three variability categories were developed employing the estimated P f s. The method proposed has been validated using an independent database of in-situ results, and a worked example was provided to illustrate the implementation of the process. The key contribution of the research is to form hazard curves for shallow foundations considering elasto-plastic soil behavior with the impact of all influencing parameters, respecting the limit values for foundation deformation in the design codes. The proposed technique offers a probabilistic evaluation of strip footings with spatial variation of clayey soils and a valid method for the reliability-based design of foundations in the serviceability limit state.
... Other researchers have focused on the probabilistic analysis of different geotechnical structures (Fenton & Griffiths, 2003;Lacasse, 2001;Phoon & Kulhawy, 1999;Popescu, Deodatis, & Nobahar, 2005;VanMarcke, 1983). In these studies, heterogeneous soil is considered, characterized by its spatial properties such as the mean, coefficient of variation, and correlation length (Fenton & Vanmarcke, 1990). ...
... However, the spatial varying random soil is generated using the local average theory developed by VanMarcke (1983), which is then combined with the elastic-plastic finite element algorithm using the Monte Carlo and the Newton-Raphson methods. This procedure is implemented in a Matlab program. ...
The nonlinear analysis of a beam resting on nonlinear random tensionless soil was studied with the aim of quantifying the influence of the spatial variability of the tension soil characteristics on the behavior of the beam and illustrating the importance of the geometric nonlinear analysis of a beam. The soil–structure interaction mechanism is taken into account where the soil is modeled as nonlinear. Due to large deflections and moderate rotations of the beam, the Von-Kàrman type nonlinearity based on the finite element formulation of nonlinear beam response is adopted and the interfacial shearing resistance along the beam is taken into account. To assess the probabilistic nonlinear behavior of the beam resting on nonlinear random soil, a Monte Carlo approach was used and the response of the beam was statistically analyzed for different beam and soil parameters. The equation of motion is solved using the Newton-Raphson iterative method which is implemented by using a Matlab software. The results indicate that the probabilistic approach of the soil and the geometric nonlinearity of the beam serve a major role in the evaluations of the beam response (of the real response of the buried beam).
... In a statistically stationary soil layer, the coefficient of variation (COV, equal to the standard deviation divided by the mean) and the horizontal and vertical scales of fluctuation (θ h and θ v , respectively) of soil properties do not change spatially. The scale of fluctuation is an indication of the distance within which a soil property is correlated along a specific spatial direction (Vanmarcke 1984). The framework was applied for piles with a ratio of length L over outer diameter D equal to 10, a ratio of D over wall thickness t equal to 50, subject to an environmental load described by a Gumbel distribution with an arbitrary mean and a constant coefficient of variation of 0.3, and designed to a target probability of failure, p f ¼ 10 −4 . ...
... Synthetic sites conditions with different seabed conditions were simulated using random field theory (Vanmarcke 1984). This approach allows: (1) conceptually, to explore how geotechnical site conditions (specifically, the horizontal scale of fluctuation) affect the efficacy of different SI layouts; and (2) operationally, to implement Monte Carlo simulation (MCS) analysis to check the achieved p f of the designed piles (i.e., verify the integrated design method) and investigate the statistical robustness of such findings. ...
Interpreting seabed properties for future offshore wind farm development appears challenging given the requirement to investigate very large areas. Current approaches, where significant numbers of geotechnical boreholes and cone penetration tests (CPTs) are conducted-often at the location of each foundation or anchor-may prove prohibitive given the scale of modern wind farms (typically over 100 turbines). This paper presents a framework for the refinement of the design of piles under axial tension [for example, to anchor floating offshore wind turbines (OWTs)] in seabeds where the spatial variability of soil properties exhibits isotropy or anisotropy in the horizontal and vertical directions. The framework relies on the approach to the rational selection of design lines representing the soil resistance for achieving a target probability of failure (p f) presented in the companion paper. The framework is validated via application to the design of piles for OWTs in both artificially generated (synthetic) and real seabeds using standard deterministic design methods, and then comparing the achieved p f values to the target. The framework is also implemented jointly with a cost model to investigate the overall project cost for different CPT layouts for an example floating wind farm layout anchored in synthetic seabeds. The spatial variability levels of cone tip resistance for these seabeds are assumed to be the same in the vertical direction but different in the horizontal direction. The optimum CPT layouts that achieved the minimum total project cost are shown to depend on the per-CPT cost and the horizontal spatial variability of cone tip resistance, so that a generally applicable "best" CPT layout cannot be identified. However, CPT layouts that include clustered CPTs and/or have small spacings between CPTs and piles (compared to the underlying actual horizontal spatial variability scale of soil properties) resulted both in a lower total project cost and require fewer total CPTs, suggesting significant potential in this approach.
... One of the practical tools to express the soil spatial variability is random field (RF) theory, which seeks to model complex patterns of variation and interdependence in cases where deterministic treatment is inefficient and conventional statistics are insufficient [6,7]. The first geotechnical application combined the random field theory with the finite element method on a footing settlement analysis using Taylor series expansion, entitled the stochastic finite element method (SFEM) [8]. ...
... Once the random variable fits a probability distribution, the field is then determined by the correlation function, which depends on the variation distance of the random parameter in the field called correlation length . Markov correlation function in Eq.1 was adopted for 2D Gaussian random field case [7]: ...
The present study aimed to create a series of hazard curves against maximum total settlement and angular rotation of strip footings for probabilistic shallow foundation design on clays. Random field finite element method (RFEM) was adopted with elasto-plastic clay-like soil behavior, deformation modulus (Ed) and shear strength parameters (c and ) were employed as random field inputs. Parameters were defined and assigned to the analysis models with varying correlation lengths (h, v). Models have been iteratively solved one thousand times, and output distributions of maximum settlement and angular rotations were recorded. Probability density functions (PDF) were fitted to the outputs, and probability of failure (Pf) for footing deformation limits was subsequently estimated. Proposed hazard curves for two anisotropy and three variability categories were developed employing the estimated Pfs. The method proposed has been validated using an independent database of in-situ results, and a worked example was provided to illustrate the implementation of the process. The key contribution of the research is to form hazard curves for shallow foundations considering elasto-plastic soil behavior with the impact of all influencing parameters, respecting the limit values for foundation deformation in the design codes. The proposed technique offers a probabilistic evaluation of strip footings with spatial variation of clayey soils and a valid method for the reliability-based design of foundations in the serviceability limit state.
... is the variation function and it affects the correlation properties of the field. According to Vanmarcke [10], the product of 1 ...
... It allows in turn the reliability index determination, whose two different algorithms-the FORM and Bhattacharyya relative entropy based-return the same patterns and values while increasing external load during its incrementation procedure. Further implementations towards elasto-plasticity with random field analysis [18][19][20] or other probability distributions 13 do not demand tremendous mathematical and numerical efforts. ...
The generalized iterative stochastic perturbation approach to the stress‐based Finite Element Method has been proposed in this work. This approach is completed using the complementary energy principle, Taylor expansion of the general order applicable to all random functions and parameters as well as nodal polynomial response bases determined with the use of the Least Squares Method. The main aim of this elaboration is the usage of such a probabilistic approach to determine Bhattacharyya relative entropy for some nonlinear engineering stress analysis with uncertainty. Mathematical apparatus with its numerical implementation has been used to study elastoplastic torsion of some prismatic bar with Gaussian material uncertainty and the corresponding reliability measures. This problem has been solved using the Constant Stress Triangular (CST) plane finite elements, the modified Newton–Raphson algorithm, whereas the first four probabilistic characteristics resulting from the iterative generalized stochastic perturbation method have been contrasted with these obtained with the crude Monte‐Carlo sampling and the semi‐analytical probabilistic approach.
... Lumb [10] firstly put forward the concept of "spatial variability" of geotechnical parameters, pointing out that due to the differences in material composition during sedimentation and the role of various uncertain external forces in the later stage, the geotechnical parameters show variability characteristics. Vanmarcke [11,12] developed a random field model describing the spatial variability of soils by considering the soil properties parameters as a random field. Random field modeling of geotechnical parameters is to characterize the spatial variability of the entire soil profile by using a limited number of sample point observations, i.e., reflecting the process of the concrete to the general. ...
The excavation of a super-large foundation pit is a greatly risky project in coastal water-rich silt strata, and it is of great significance to establish a numerical model to analyze and predict the stability of the foundation pit retaining its structure for safe construction. Based on a geological investigation report and key node monitoring data, the model parameters have been modified. Subsequently, the random field theory and numerical analysis were employed to proceed with deterministic analysis and uncertainty analysis, respectively. Using the uniform distribution method, Gaussian distribution method and the covariance matrix method to generate a random field model, finite difference software was applied to analyze the impact of spatial variability of cohesion and the internal friction angle on structural deformation. The study shows that the overall distribution of axial force is small on both sides and large in the center, and the axial force is larger near the shaped region. Due to the principle of “lever”, there is a tendency for the horizontal displacement of surrounding piles to partially rebound from the pit when the bearing platform pit is excavated. The spatial variability of the internal friction angle and cohesion has an important influence on the numerical value of the enclosure structure and surface deformation, and the variation pattern is basically unchanged.
... The international society for soil mechanics and geotechnical engineering (ISSMGE-TC304, 2021) conducted an extensive review of inherent uncertainty in geotechnical engineering, noting that the properties of two soil samples tend to be more similar when the spatial distance between them is smaller. Such a spatial correlation structure with significant anisotropy can be described by an advanced formalism based on random field theory (RFT) (Vanmarcke, 2010). Given this issue, a probabilistic safety assessment (PSA) of shield tunnels that considers soil property uncertainties is preferable (Li et al., 2021;Shi et al., 2023;Zhang et al., 2022a). ...
Probabilistic analysis based on random field (RF) has been widely adopted in the safety assessment of shield tunnels. However, its practical applicability has been limited by the intricacy involved with integrating geotechnical data and tunneling information. This paper addresses the following research question: How can the RF-based probabilistic safety assessment be carried out efficiently? In addressing this research question, we suggested an RF-based tunneling information modeling (TIM) framework to realize the probabilistic safety assessment of shield tunnels. In the proposed framework, the modeling of tunnel structure and geological conditions is initially introduced. The numerical safety assessment model is then created via an automated procedure using the RF-based TIM. A case study is conducted to verify the suggested framework, and results demonstrate that the framework can offer an automated design-to-analysis solution to improving the safety assessment of shield tunnels by comprehensively considering the uncertainties of geological conditions.
... This spatial variability is a major contributor to soil uncertainty [32]. The spatial variability of soil properties can be explained by the correlation structure within the theoretical framework of the random field model [33]. The autocorrelation distance is used to indicate the spatial extent over which the soil properties display a strong correlation. ...
The maintenance of water-retaining structures involves evaluating their performance against current and future operating water levels. Fragility curves are commonly used for this purpose, as they indicate the conditional probability of failure for various load conditions and accurately characterize a structure’s performance. Monte Carlo simulation (MCS) can be used to determine the fragility curve of water-retaining structures by calculating the probability of failure as the water level changes. However, performing repetitive MCS involves extensive calculations, thus making it inefficient for practical applications. Therefore, it is essential to develop efficient methods that require a minimum number of MCS runs to estimate the fragility curve. This study proposed two methods to estimate the fragility curves of water-retaining structures, thereby allowing for the assessment of failure probabilities related to important quantities such as the steady-state seepage rate, exit gradient, and uplift force, which make them suitable for practical applications. The fragility curves obtained using the proposed methods are valuable for risk assessment, design, and decision-making purposes, as they offer information to evaluate the performance of the water-retaining structure under various water level conditions.
... For further details concerning stochastic processes, their transformations (linear filtering, derivatives, etc.), random vibration and stochastic dynamics, we refer the reader to Bendat and Piersol, 1980;Crandall and Mark, 1973;Doob, 1953;Elishakoff, 1983;Guikhman and Skorokhod, 1979;Jenkins and Watt, 1968;Kree andSoize, 1983 andLin, 1967;Priestley, 1981;Roberts and Spanos, 1990;Soize, 1988Soize, , 1993aSoize, , 1994Soong, 1973;Vanmarcke, 1983. 6. Random Case: Nonstationary Stochastic Excitation ...
... Therefore, this study model ψ with random field theory. Random field theory is commonly used to model the inherent spatial variability in homogeneous soils (Vanmarcke, 1983;Zhao et al., 2022a). Due to the spatial correlation of ψ test data, ψ was considered to be modeled as random field rather than a set of independent random variables. ...
The state parameter (ψ) is based on a framework of critical state soil mechanics, which reflects the influences of soil compactness and stress level and has significant advantages for liquefaction analysis. In engineering practice, a simplified procedure based on the cone penetration test (CPT) to assess the liquefaction potential in the field involves model and parameter uncertainties. Furthermore, the use of limited CPT measurements involves project-specific test uncertainties, and the spatial variability of soil properties has a remarkable effect on soil liquefaction. To tackle these challenges, this study develops a ψ-based liquefaction probability framework to integrate the prior information of project-specific CPT, and uses limited tests to characterize the 2D spatial distribution of liquefaction potential with proper consideration of various uncertainties and soil spatial variability. The framework is developed based on simulation of Gaussian stationary random fields and Markov Chain Monte Carlo simulation. The proposed method is demonstrated using real CPT data from the 2011 Tohoku-Oki earthquake in Japan. The results indicate that the ψ-based evaluation performs well and can reasonably assess the liquefaction phenomenon of heterogeneous soils. This study incorporating the spatial variability of ψ into the liquefaction probability framework provides valuable information for geotechnical engineering design.
... /0000-0001-6341-5316. Email: tianning19@mails.ucas.ac.cn the parameters as constants. A large number of studies have shown that there are uncertainties in soil parameters affected by depositional history and weathering conditions (Vanmarcke 1984;Phoon and Kulhawy 1999;Cho 2007;Griffiths and Fenton 2004;Gong et al. 2021;Huang et al. 2017;Fenton and Vanmarcke 1990). This uncertainty, namely spatial variability, is not only embodied in the randomness of the soil parameters, but also in the overall spatial correlation . ...
Existing tunnels are inevitably disturbed by adjacent engineering projects, with ground surface surcharge accounting for the largest proportion. Meanwhile, the influence of spatial variability of soil parameters on tunnel deformation cannot be ignored, especially for the cross-correlation when a multivariate random field is modeled. Unfortunately, studies about the coupled effect of multivariate crosscorrelated random fields and surcharge on longitudinal deformation of tunnels in three-dimensional (3D) space are quite scarce. Thus, this study firstly investigated the coupled impact of the multivariate cross-correlation of soil parameters and ground surface surcharge on the maximum longitudinal deformation of an existing tunnel in 3D space by using the random FEM (RFEM) combining the parallel technology integrated within ABAQUS. The Young’s modulus, Poisson’s ratio, and friction angle were discretized and highlighted by isotropic crosscorrelated random fields. The results showed that the surcharge has quite a significant impact on the deformation of an operating tunnel especially in spatially variable soil. The calculation result obtained by random variable theory (RVT) was smaller than that of random field theory (RFT) because the former ignored the structural characteristics of soil parameters. Conventional single-parameter random fields (SRF) and multivariate independent random fields (IRF) may overestimate the tunnel deformation compared with multivariate cross-correlation random fields (CRF). Further parameter studies found that a larger cross-correlation coefficient between soil parameters will result in a smaller tunnel deformation in the CRF model.
... Associated uncertainty in the observational component can thus mainly be related to inability to bound the variability and randomness of the intrinsic properties of the system being modelled. Generally most aspects of observational uncertainty can be calculated and measured with various mathematical, quantitative methods i.e., Random Field approaches (Vanmarcke 1984, Vessia & Russo, 2018, Geostatistical methods such as Kriging methods (Vessia et al., 2020a, b), and Stochastic simulations (Di Curzio & Vessia, 2022). Although this paper does not delve into any detail on these quantitative methods, their usefulness where applicable is highlighted. ...
In existing load models, the crowd bouncing load is often simplified as a single-point excitation; moreover, these models lack data support from crowd bouncing experiments. Inspired by the random field models widely adopted in seismic ground motion fields, a random field model for crowd bouncing loads was established in this research. The bouncing frequency, time lag, and amplitude of the coherence function were modeled to quantify the crowd synchronization; an auto-power spectral density (PSD) model from the author’s previous study was adopted for an individual bouncing load. The values of these parameters were obtained based on data from a crowd bouncing experiment involving 48 test subjects on the first day and 42 test subjects on the second day, in which the trajectories of reflective markers fixed at the clavicle of every test subjects were simultaneously recorded using three-dimensional motion capture system. Based on the PSD matrix of the crowd bouncing loads as simulated by the proposed random field model, the structural acceleration can be analyzed using random vibration analysis in the frequency domain. The established random field model and spectral analysis framework can be adopted to evaluate the vibrating performances of lightweight and high-strength structures. Moreover, the established load model is also the basis of structural vibration control.
Liquefaction is a phenomenon which tends to cause damage to buildings and structures founded on liquefiable soils. However, existing probabilistic liquefaction-induced lateral spread hazard analysis (PLLSHA) is based on empirical lateral spread (LS), in which the spatial variability of soil parameters is ignored in liquefaction hazard evaluations. This paper thus proposes a PLLSHA method which can consider the spatial variability of soil parameters. The response spectrum matching method was used to select 15 ground motions from NGA-West2 database under the earthquake scenario with magnitude of 7.0 and rupture distance of 10 km. Based on the Cholesky midpoint method, random fields for the relative densities of the liquefiable layer with different mean values, coefficients of variation, horizontal and vertical correlation lengths were generated and then embedded into the sloping grounds containing liquefiable layers. A set of random variables for relative densities of the sloping ground is also generated for comparative purpose. The finite element software OpenSees is used to establish the two-dimensional sloping ground model and conduct several dynamic response analyses based on the ground motions selected. The mean LS on surface ground are monitored and recorded as indicators of liquefaction consequence for analysis. An assumed site source with variable magnitudes and 10 km from sloping ground is used for conducting the PLLSHA. Result shows that the spatial variability of relative density of soils has a great influence on the liquefaction hazard results. Specifically, the liquefaction-induced LS hazard would be greater when the mean value of the relative densities or vertical correlation lengths are smaller, as well as when the coefficient of variation or horizontal correlation lengths are larger. When a random variable model is utilized, the liquefaction-induced LS hazard is significantly overestimated. This study would provide a theoretical reference for lateral spread hazard analysis of liquefiable soil layers exhibiting strong spatial variability.
In part I, spectral moments and kurtosis were established as parameters in analytic models of correlation and spectral density functions for dynamic reverberation fields. In this part II, several practical limitations affecting the accuracy of estimating these parameters from measured stir sweep data are investigated. For sampled fields, the contributions of finite differencing and aliasing are evaluated. Finite differencing results in a negative bias that depends, to leading order, quadratically on the product of the sampling time interval and the stir bandwidth. Numerical estimates of moments extracted directly from sampled stir sweeps show good agreement with values obtained by an autocovariance method. The effects of data decimation and noise-to-stir ratios of RMS amplitudes are determined and experimentally verified. In addition, the dependencies on the noise-to-stir-bandwidth ratio, EMI, and unstirred energy are characterized.
This paper addresses Bayesian-based data-driven site characterization methods for estimating soil parameters used in top-hole casing design. Different models are applied to datasets of piezocone tests (CPTu) conducted in Brazilian fields and their performance is compared to some characterization techniques currently employed by the oil and gas industry. Regression models are crucial for soil characterization for top-hole casing design. Data-driven methods consist of a powerful tool for this purpose, allowing handling uncertainties originating from soil variability. This paper addresses machine learning models devoted to sparse data, namely Geotechnical lasso (Glasso) and Gaussian Process Regression (GPR) from a Bayesian perspective considering prior knowledge of site information. This approach provides statistical information on soil parameters like undrained shear strength, supporting structural analysis of wellhead systems, and conductor/surface casing strings. Datasets were collected from in-situ CPTu tests conducted in the Campos basins, in eastern Brazil. The first case study addressed primary CPTu data, including cone tip resistance, friction sleeve, and total pore pressure, to characterize undrained shear resistance as a random variable. In this context, the parameter was modeled using Phoon´s modified Bartlett test to calculate sample sizes that ensure stationarity. This approach presented relevant results in assessing the probability of failure for conductor and surface casing design based on the operator´s internal design criteria. As these applications, more robust techniques were used to improve data characterization. Glasso and GPR models are used to model parameter tendencies and evaluation of the random data. These methods differ in prior probability density functions adopted - Laplace and Gaussian, respectively - and they are compared with regression techniques widely used in design practice. All the models were trained to estimate undrained shear resistance. Preliminary results confirm that the technique improves the characterization of soil strata and undrained shear strength, with a beneficial effect on the analysis of offshore top-hole structural design cases. Some metrics such as Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) indicate that GPR outperforms other methods. This is an innovative methodology applied in real-case scenarios with data ceded from the partner operator. The formulation evaluates uncertainties associated with the spatial heterogeneity of the material, continuously improving robustness with each new project data. This enables a better understanding of soil behavior in specific oilfields and can assist the decision-making process in well design, improving operational safety. Furthermore, the results of the statistical modeling support reliability-based analysis to deliver probability-based indicators for well integrity design.
This article is the second part of a previous article devoted to the deterministic aspects. Here, we present a comprehensive study on the development and application of a novel stochastic second-gradient continuum model for particle-based materials. An application is presented concerning colloidal crystals. Since we are dealing with particle-based materials, factors such as the topology of contacts, particle sizes, shapes, and geometric structure are not considered. The mechanical properties of the introduced second-gradient continuum are modeled as random fields to account for uncertainties. The stochastic computational model is based on a mixed finite element (FE), and the Monte Carlo (MC) numerical simulation method is used as a stochastic solver. Finally, the resulting stochastic second-gradient model is applied to analyze colloidal crystals, which have wide-ranging applications. The simulations show the effects of second-order gradient on the mechanical response of a colloidal crystal under axial load, for which there could be significant fluctuations in the displacements.
Auto- and cross-spectral density functions (SDFs) for dynamic random fields and power are derived. These are based on first- and second-order Padé approximants of correlation functions expanded in terms of spectral moments. The second-order approximant permits a characterization of stir noise observable at high stir frequencies in the autospectral density. A relationship between stir imperfection and spectral kurtosis is established. For the latter, lower bounds are established. A novel alternative measure of correlation time for mean-square differentiable fields is introduced as the lag at the first point of inflection in the autocorrelation function. A hierarchy of Padé deviation coefficients is constructed that quantify imperfections of correlations and spectra with increasing accuracy and range of lags. Analytical models of the spectral densities are derived and their asymptotic behavior is analyzed. The theoretical spectral density for the electric field (or
$\mathrm{S}_{21}$
) as an input quantity is compared with that for power (or
$|\mathrm{S}_{21}|^{2}$
) as the measurand. For the latter, its inverted-S shape conforms to experimentally obtained stir-spectral power densities. The effect of additive noise on the stir autocorrelation function and SDF is quantified.
Why should moral philosophers, moral psychologists, and machine ethicists care about computational complexity? Debates on whether artificial intelligence (AI) can or should be used to solve problems in ethical domains have mainly been driven by what AI can or cannot do in terms of human capacities. In this paper, we tackle the problem from the other end by exploring what kind of moral machines are possible based on what computational systems can or cannot do. To do so, we analyze normative ethics through the lens of computational complexity. First, we introduce computational complexity for the uninitiated reader and discuss how the complexity of ethical problems can be framed within Marr’s three levels of analysis. We then study a range of ethical problems based on consequentialism, deontology, and virtue ethics, with the aim of elucidating the complexity associated with the problems themselves (e.g., due to combinatorics, uncertainty, strategic dynamics), the computational methods employed (e.g., probability, logic, learning), and the available resources (e.g., time, knowledge, learning). The results indicate that most problems the normative frameworks pose lead to tractability issues in every category analyzed. Our investigation also provides several insights about the computational nature of normative ethics, including the differences between rule- and outcome-based moral strategies, and the implementation-variance with regard to moral resources. We then discuss the consequences complexity results have for the prospect of moral machines in virtue of the trade-off between optimality and efficiency. Finally, we elucidate how computational complexity can be used to inform both philosophical and cognitive-psychological research on human morality by advancing the moral tractability thesis.
In the random field model’s consideration of the spatial variability of soil, soil properties at different locations play different roles in the reliability analysis of the foundation. Investigating the importance distribution of the random field through reliability sensitivity analysis (RSA) is beneficial for understanding how the random field affects the reliability of the foundation. However, many existing RSA methods for the random field model are deficient in terms of efficiency, accuracy, and applicability under complex engineering conditions. Consequently, this study proposes an efficient RSA method for the random field model based on the Karhunen–Loève (KL) expansion method and the first-order reliability method (FORM) to identify the important random field domain in foundation engineering. In the proposed method, the mean reliability sensitivity index (MRSI) is extended to a random field model of continuous form to characterize the importance distribution of the random field. The MRSI is analytically derived based on the results of the KL expansion method and the FORM without additional limit state function (LSF) calculations. Subsequently, the important random field domain, in which the variation of the mean of the soil property contributes significantly to the reliability index, is identified based on the MRSI. Last, two foundation engineering examples that consider the cross-correlated random fields of cohesion and friction angle, including strip footing on single-layer soil and pile in multiple-layer soil, were used to verify the proposed method. The results showed that an important random field domain with a small area dominates the variation of the reliability index of a foundation, and important random field domain area increases with autocorrelation distance (ACD). This innovative identification method holds great engineering significance, because it allows geotechnical practitioners to gain a comprehensive understanding of the failure modes and foundation treatment areas of foundations in spatially varying soil.
Reliability-based design is increasingly being applied to geotechnical problems because it allows the robust consideration of various sources of uncertainty, such as the inherent variability of soil properties. Some soil properties, however, are mutually dependent, and ignoring this cross-correlation may lead to biased estimates of the probability of unsatisfactory performance. Hence, in this study, the Gaussian copula was used to evaluate how the applied cross-correlation or independence between the com-pressibility properties affects the uncertainty in the stress-strain response of two marine soft clays. Two settlement calculation methods were considered: the compression index and Janbu tangential stiffness methods. The correlation coefficients were defined from the site-specific oedometer data at two extensively studied clay sites, and from a database. The simulated oedometer curves were compared to the observed variability in the site-specific data. The settlements in the clay sublayers were then computed, and different cases were compared by means of box-plots. It is concluded that the Janbu method leads to a significant overestimation of uncertainty in settlement if the cross-correlation between the compressibility parameters is ignored. On contrary, the compression index method seems less sensitive to the assumed correlation structure, and as such, the parameters can be treated as independent in most cases.
Toe excavation of the hills is a common practice in the highlands of India for construction or extension of roadway projects. Landslides in such cut slopes occurs very often and have dire consequences on human lives, properties and communication network in the hilly regions. Rise in natural ground water level due to torrential rain during monsoon season and seismic activity are two most crucial factors responsible for most of these landslides in cut slopes. The customary stability assessment techniques for such cut slopes is based on deterministic method, where the stability is assessed with reference to a single safety measure commonly known as factor of safety. Nonetheless, due to uncertainty related to various geotechnical parameters, the standard deterministic approach may end up resulting in mismatched design solutions. This paper reports the seismic behaviour of cut slopes in presence of water table for different seismic zones in India utilising a probabilistic approach. The study also exhibits the influence of correlation coefficient, spatial variation of shear strength parameters and the coefficient of variation on the seismic response of the partially saturated cut slope. Furthermore, the study reports the effect of incorporation of uncertainty in water table location and pseudo-static earthquake forces on seismic response of the partially saturated cut slope. A nonlinear time-history analysis is also carried out to estimate the more realistic seismic behaviour of the partially saturated cut slope during earthquake.
The uncertainties related to the inherent properties of pavement subgrade subjected to the dynamic effect of vehicular and earthquake loading were studied extensively using probabilistic analysis. The dynamic effect of the loading on pavement was modeled using lower bound shakedown analysis. Pavement subgrade was considered to be purely cohesive and was assumed to fail through the Mohr–Coulomb yield criterion. The soil spatial variability was simulated using random fields, which were represented in the form of finite-element random variables using the Karhunen–Loéve expansion method. The effect of parameter uncertainty was investigated using stochastic results such as the mean, coefficient of variation, and failure probability of the structure using the Monte Carlo simulation technique. The effect of dynamic loading of vehicles was studied by varying the period of vehicular movement. The dynamic shakedown was analyzed using two different methods. Seismic waves were generated using modified pseudodynamic approach. The combined effect of moving vehicles and earthquake was studied to determine the worst-case scenario. It was found that the generation of seismic waves affects the direction and frequency of the moving vehicle, which reduces the effect of shakedown on pavements due to vehicular load. The worst-case scenario was found to be the pavements with period ratio of 0.06 for a deterministic analysis, whereas it was 0.48 for a probabilistic analysis.
This paper presents a new geotechnical database for the soils of Quito, Ecuador. The geotechnical database is then used to investigate the best fit probability distributions for the key geotechnical parameters contained in the database. Using the Akaike information criteria for best fit selection, SPT (N), plasticity index, Vs30, peak friction angle (direct shear), and apparent cohesion (triaxial) are best represented by a Weibull distribution. The peak friction angle (triaxial) is best fitted with a truncated normal distribution. The database is also used to develop transformation models to allow for the estimation of more complex geotechnical parameters from intrinsic ones. This analysis shows that the transformation model between Vs30 and SPT (N) has high coefficients of determination and is statistically significant. Finally, the systematic collection of information in the database is used to investigate the assumption, based on engineering judgement by local practitioners, that soil derived from volcanic deposits and volcano-lacustrine sediments in the northern part of Quito has different geotechnical properties with respect to the southern zone of the city beyond the value of shear wave velocity whose difference is embedded in the soil classification map of the seismic code.
A profile of geotechnical properties is often needed for geotechnical design and analysis. However, site-specific data might be characterized as MUSIC-X (i.e., Multivariate, Uncertain and Unique, Sparse, Incomplete, and potentially Corrupted with “X” denoting the spatial/temporal variability), posing a significant challenge in accurately interpreting geotechnical property profiles. Different sources, or types, of data are commonly available from a specific site investigation program, and they are usually cross-correlated, and thus can provide complementary information. This leads to an important question in geotechnical site investigation: how to integrate multiple sources of sparse data for enhancing the profiling of different geotechnical properties. To address this issue, this study proposes a novel method, called fusion Bayesian compressive sampling (Fusion-BCS), for integrating sparse and non-co-located geotechnical data. In the proposed method, the auto- and cross-correlation structures of different sources of data are exploited in a data-driven manner through a joint sparse representation. Then, profiles of different geotechnical properties are jointly reconstructed from all measurements under a framework of compressive sampling/sensing. The proposed method is illustrated using simulated and real geotechnical data. The results indicate that accuracy of the interpreted geotechnical property profiles may be significantly improved by integrating multiple sources of site investigation data.
This study investigates the effect of multivariate cross-correlated random fields on the failure behaviour of passive trapdoors in c-ϕ soil utilising random finite element limit analysis (RFELA). Failure mechanisms of a trapdoor in c-ϕ soil depend on an activity of the soil unit weight (γ), thus triggering downward and a restriction of soil shear strength (c, ϕ). The cross-correlated coefficients between any two soil parameters were used to generate the spatial variability of correlation, single and multiple dependent random fields. The failure mechanism and probability of design failure were explored with these random fields for a shallow to a deep passive trapdoor in c-ϕ soil. The results indicated that considering the greater cross-correlated coefficients produced a lower probability of design failure. Ignoring multivariate cross-correlated random fields can result in either an underestimation or an overestimation of the probability of design failure, depending on the specific scenarios. This study also provided a required factor of safety to satisfy the probability of design failure for each random field and normalised depth of the trapdoor. Additionally, the multivariate adaptive regression spline model was applied to predict passive load, thereby reducing computational time in practical applications.
Due to the complex geologic, environmental, and physical–chemical processes, geotechnical properties vary spatially even in the same geological unit, referred to as spatial variability.
The working stress design has long been used in civil engineering (e.g., Barker et al. in Manuals for the design of bridge foundations: shallow foundations, driven piles, retaining walls and abutments, drilled shafts, estimating tolerable movements, and load factor design specifications and commentary. Transportation Research Board, Washington, DC, 1991), in which a FOS used to consider the effect of all uncertainties on the safety of structures.
This study explores the impact of including the vertical spatial variability in effective stress friction angle of clay on the probabilistic analysis of deep excavations. The proposed methodology is demonstrated and verified by conducting random finite element modeling (RFEM) of an instrumented deep excavation project situated in Ankara, Turkey. The excavation has a depth of 20 meters and is supported by six levels of pre-stressed ground anchors. To simulate the vertical spatial variability of effective stress friction angle in the clay, Monte Carlo simulation method and the random field theory are employed. The simulated parameters are then inserted into the finite element model via Python programming language to analyze the probabilistic distribution of lateral deflections and bending moments in the drilled shaft wall. The results obtained from the Monte Carlo simulations reveal that the incorporation and selected value of spatial variability significantly impacts the resulting lateral movements, bending moments, and the probability of failure of the system.
The basal heave stability of supported excavations is an essential problem in geotechnical engineering. This paper considers the probabilistic analysis of basal heave stability of supported excavations with spatially random soils by employing the random adaptive finite element limit analysis and Monte Carlo simulations to simulate all possible outcomes under parametric uncertainty. The effect of soil strength variability is investigated for various parameters, including the width and depth of the excavation ratio, strength gradient factor, and vertical correlation length. Probabilistic basal stability results have also been employed to determine the probability of design failure for a practical range of deterministic factors of safety. Considering probabilistic failure analysis, the more complete failure patterns caused by the various vertical correlation length would decrease the probability of design failure. There are different tendencies between the probability of design failure at the same safety factor with various vertical correlation lengths. These results can be of great interest to engineering practitioners in the design process of excavation problems.
Coupled fluid-solid phase continuum problems associated with large deformation as geotechnics experts encounter in slope stability problems have been extensively reviewed. This has been done with a view to exploring the most efficient numerical approach to solving them for the sake of our environment. Decades ago, analytical solutions known as the limit equilibrium methods (LEM), e.g. Navier-Stokes and the likes were celebrated with the level of mathematical solutions they offered. In order to overcome the limitations of the LEM in handling more complex slope failure problems, numerical solutions were born, which solved these problems by method of mesh discretization. However, mesh discretization suffered distortions as these mesh-based numerical solutions like finite element method (FEM), finite difference method (FDM), material point method (MPM), discrete element method (DEM) and boundary element method (BEM) were deployed to solve large deformation problems encountered in slope failures like steep watersheds, road embankments, landslides, debris flow, etc. Future developments were made in this line and the birth of the meshfree approach to solving these largely proportional geophysical flows known as the smoothed particle hydrodynamics (SPH) took place. In this extensive review, previous works have been studied and explored the limitations of the LEM and the mesh-based numerical solutions. Also, the superiority and the use of the SPH in efficiently solving large deformations environmental geotechnics problems especially those related to slope failure have been proposed despite the environmental conditions of the watershed. However, it has been remarked that the SPH interface is yet to integrate the intergranular force and the slope angle into its framework but maintains its superiority over the other methods.
The paper presents a proprietary procedure for the analysis of normal stress distributions in post-tensioned cross-sections. It has a significant advantage over conventional commonly used approaches based solely on the envelope analysis as it provides stress levels in all components of the cross-section. The procedure was used in a series of probabilistic analyses with the adoption of random fields. These fields represented uncertainties in strain-stress relationship in concrete. The analysis covered several types of cross-sections and several types of random fields. Key observations from the conducted simulations are as follows: (I) the widest ranges of the probable maximum stresses (i.e. the lowest indexes of reliability) were obtained for sections with relatively low heights of the compressive zone. (II) The highest probabilistic sensitivity to the type of random field used was found in tall sections with a relatively large compressive zone. (III) The greatest sensitivity to batch uncertainties was evident in all cross-sections when using squared exponential random fields. (IV) The greatest relative sensitivity to the batch uncertainties in the form of the random field compliant with the guidelines of the Joint Comity of Structural Safety (JCSS) was evident in the analyses of the tallest cross-section corresponding to the incrementally launched bridges.
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