Random Finite Element Research Articles

An approach for modelling spatial variability in permeability of cement-admixed soil

This paper presents a framework for modelling the random variation in permeability in cement-admixed soil based on the binder content variation and thereby relating the coefficient of permeability to the unconfined compressive strength of a cement-admixed clay. The strength–permeability relationship was subsequently implemented in random finite element method (RFEM). The effects of spatial variation in both strength and permeability of cement-admixed clays in RFEM is illustrated using two examples concerning one-dimensional consolidation. Parametric studies considering different coefficient of variation and scale of fluctuation configurations were performed. Results show that spatial variability of the cement-admixed clay considering variable permeability can significantly influence the overall consolidation rate, especially when the soil strength variability is high. However, the overall consolidation rates also depend largely on the prescribed scales of fluctuation; in cases where the variation is horizontally layered, stagnation in pore pressure dissipation may occur due to soft parts yielding.

Uncertainty quantification of the effect of concrete heterogeneity on nonlinear seismic response of gravity dams including record-to-record variability

Although the effects of the concrete heterogeneity on the seismic response of concrete gravity dams have been explored based on random finite element method, the results of different studies vary greatly and there are few quantitative studies on the effects. In addition, the correlation between concrete modulus and strength is usually ignored, which is a departure from the reality. In this study, the effects of the concrete heterogeneity on the nonlinear seismic response of concrete gravity dams under different input ground motions are systematically studied based on random field theory. The dispersion and uncertainty of the seismic responses and the damage pattern are quantified. In addition, the effects of the correlation between concrete modulus and strength are also investigated and quantified based on the cross-correlated random fields simulation method. It is found that the effects of the concrete heterogeneity on the nonlinear dynamic response of concrete dams are strongly dependent on the input seismic motions, which provides an explanation for the conflicts in the previous studies. The results also show that the correlation between concrete modulus and strength has important effects on the dispersion of the seismic response and damage pattern of concrete gravity dams. Under certain ground motions, ignoring the correlation may underestimate the failure probability of the dams.

Effects of inherent spatial variability of rock properties on the thermo-hydro-mechanical responses of a high-level radioactive waste repository

In France, a deep geological disposal for high level radioactive waste (HLW) and intermediate-level long-lived radioactive waste (ILW-LL) called Cigéo is planned to be constructed in a deep Callovo-Oxfordian claystone (COx) formation. The heat released from the HLW packages leads to a temperature increase in the host rock, which causes a pore pressure increase essentially due to the difference between the thermal expansion of the pore water and the solid skeleton. The low permeability of the COx and its relative rigidity inhibit the discharge of the induced pressure build-up. Moreover, thermal loading also induces thermo-mechanical stresses within the formation. The French National Radioactive Waste Management Agency (Andra) participated in the DECOVALEX-2019 project (Task E) to model the Thermo-Hydro-Mechanical (THM) responses of a benchmark exercise inspired of a HLW repository based on the French concept. This paper aims at studying the THM behavior of the HLW repository by taking into account the inherent spatial variability of the properties of the host rock through thermo-poro-elastic numerical simulations. Only the most influential parameters were selected to perform the spatial variability analysis by using Monte Carlo simulations coupled with the Random Finite Element Method. The selection of the most influential parameters was based on two parametric sensitivity analyses consisting in computing the Sobol indices, which determine the contribution of each input parameter to the overall model output variance. The temporal evolution of the Terzaghi effective stress in the far field is defined as indicator of the influence of each parameter on the THM response.

Effect of soil spatial variability on failure mechanisms and undrained capacities of strip foundations under uniaxial loading

This paper mainly investigates the effect of soil spatial variability on failure mechanisms and capacities of strip foundations under uniaxial loading on marine clay deposits with linearly increasing undrained shear strength using random finite element analysis. The soil spatially variable feature is characterised by a non-stationary random field where the ratio of the standard deviation and mean of undrained shear strength is maintained. The effect of soil parameters, including the dimensionless soil heterogeneity index, the coefficient of variation of shear strength and the dimensionless autocorrelation distance, on the uniaxial capacities are discussed. In addition, shapes of failure envelopes for combined loading in the probabilistic analysis are compared to those shapes in deterministic analysis. Results demonstrate that the difference in shapes of failure envelopes between deterministic and probabilistic cases is minimal and can be neglected for practical purposes. Probabilistic uniaxial capacities derived in this study can be employed to scale probabilistic failure envelopes, which represent sizes of probabilistic failure envelopes. The obtained results offer insights into how soil spatial variability can be comprehensively considered in the design of an offshore shallow foundation.

The Effect of the Selection of Three-Dimensional Random Numerical Soil Models on Strip Foundation Settlements

This paper delivers a probabilistic attempt to prove that the selection of a random three-dimensional finite element (FE) model of a subsoil affects the computed settlements. Parametric analysis of a random soil block is conducted, assuming a variable subsoil Young’s modulus in particular finite elements. The modulus is represented by a random field or different-sized sets of random variables; in both cases, the same truncated Gaussian model is assumed. Mean values and standard deviations of random soil settlement are estimated by a Monte Carlo simulation procedure. With regard to the adopted FE model, the estimated settlement mean values do not vary significantly, but standard deviations do strongly. Similarities also appear in the diagrams of random field correlation length versus settlement standard deviation and the diagrams displaying a total number of model random variables versus settlement standard deviation. Thus, relevant single random variable models represent the random field approach well with regard to settlement parameter estimation. This remark is verified upon a settlement analysis of a three-dimensional FE model of a hypothetical strip foundation. Following the preliminary model observations, various probabilistic geotechnical analyses may be supported, e.g., continuous footing design, slope stability analysis, and foundation reliability assessment.

Reliability Analysis of Embedded Strip Footings in Rotated Anisotropic Random Fields

This study explores the influence of rotated anisotropy of soil property on the bearing capacity of embedded strip footings using the random finite element method. An extensive parametric study was carried out considering the different scales of fluctuations, embedded depths, and rotation angles of the random fields. Results show that for various footing embedded depths considered in this study, the major scale of fluctuation has a large influence on the probability of bearing capacity failure in all cases. Furthermore, the mean bearing capacity remains almost constant regardless of the change of rotation angle, while the probability of bearing capacity failure decreases with the increase of rotation angle. The significant role of rotation angle in probabilistic analysis highlights the need to understand and consider the soil layer orientation in geotechnical foundation design to achieve safety and reliability.

Random finite element analysis of backward erosion piping

Backward erosion piping (BEP) is a type of internal erosion that threatens the integrity of dams and levees. BEP has been shown to be highly sensitive to spatial variations in soil properties; however, there are presently no assessment methods that permit incorporating spatial variation in soil properties into BEP analysis. The Random Finite Element Method (RFEM) is a numerical approach for incorporating spatially variable properties into finite element analysis. In this study, an RFEM approach to simulating BEP was developed to assess pipe progression through variable soils. The soil hydraulic conductivity and critical hydraulic gradient for pipe progression were treated as log-normal random variables. Analyses were conducted for a range of hydraulic conductivity and critical gradient random fields with varied spatial correlation lengths, distribution parameters, and field correlations. Results indicate that the probability of failure increases with increasing spatial correlation length. Additionally, increased variance in soil permeability was shown to increase the probability of failure for large correlation lengths and decrease the probability of failure for short correlation lengths. Regardless of correlation length, increasing values of the mean critical hydraulic gradient led to decreased failure probabilities, and increasing values of critical hydraulic gradient variance led to increased failure probabilities.

Modeling Seepage Flow and Spatial Variability of Soil Thermal Conductivity during Artificial Ground Freezing for Tunnel Excavation

Artificial ground freezing (AGF) technology has been commonly applied in tunnel construction. Its primary goal is to create a frozen wall around the tunnel profile as a hydraulic barrier and temporary support, but it is inevitably affected by two natural factors. Firstly, seepage flows provide large and continuous heat energy to prevent the soil from freezing. Secondly, as a key soil parameter in heat transfer, the soil thermal conductivity shows inherent spatial variability, binging uncertainties in freezing effects and efficiency. However, few studies have explored the influence of spatially varied soil thermal conductivity on AGF. In this study, a coupled hydro-thermal numerical model was developed to examine the effects of seepage on the formation of frozen wall. The soil thermal conductivity is simulated as a lognormal random field and analyzed by groups of Monte-Carlo simulations. The results confirmed the adverse effect of groundwater flow on the formation of frozen wall, including the uneven development of frozen body towards the downstream side and the higher risk of water leakage on the upstream face of the tunnel. Based on random finite element analysis, this study quantitively tabulated the required additional freezing time above the deterministic scenario. Two levels of the additional freezing time are provided, namely the average level and conservative level, which aim to facilitate practitioners in making a rule-of-thumb estimation in the design of comparable situations. The findings can offer practitioners a rule of thumb for estimating the additional freezing times needed in artificial ground freezing, accounting for the seepage flow and spatial variation in soil thermal conductivity.

Effect of the spatial variability of soil parameters on the deformation behavior of excavated slopes

The prediction of slope stability is difficult due to the spatial variability of soil parameters and measurement errors from core samples. This study proposed a Bayesian updating framework that based on the random finite element model (RFEM) method. The proposed approach was applied in a field excavation slope project. The field data of a soil slope with anti-slide bored piles under four-stage excavations were compared to elucidate the spatial variability of the cohesion, friction angle and stiffness in different soil layers. Then, a Bayesian updating model with random finite element analysis was developed by considering the COV and SOF of the soil parameters. The proposed analysis approach utilized the prior information from the site investigation and field measurement results. The posterior statistical characteristic values of the soil parameters were obtained from equivalent samples calculated by Markov Chain Monte Carlo simulations. The results indicated that the proposed method can reduce the effect of uncertainty related to the soil properties. Finally, the effects of soil parameters and excavation conditions on the slope stability were established, enabling an evaluation of safety based on field measurement results.

Probabilistic characteristics analysis for the time-dependent deformation of clay soils due to spatial variability

Consolidation and creep are two types of time-dependent deformation of clay soil. However, the soil deformation presents probabilistic behaviours when the spatial variability is considered. The Monte–Carlo method is used to sample anisotropic random fields that are mapped into two-dimensional random finite element models (RFEM) to analyse the influences of spatial variability on the time-dependent deformation of clay soil. The anisotropic random fields of soil parameters are generated by a spectral representation method (SRM) to describe the spatial variability of clay soil. The consolidation and the creep problems are modelled by Biot’s consolidation theory and the creep model proposed by Yin et al. (2010) respectively. The spatial variability of Young modulus and permeability coefficient in probabilistic consolidation analysis and the secondary consolidation coefficient in probabilistic creep analysis are emphasized by the RFEM simulation. The results show that the scale of fluctuation (SOF) in vertical direction and the coefficient of variation (COV) of Young modulus and permeability have significant influences on the probabilistic characteristics of consolidation behaviour. Besides, the bunched region of low permeability soil produces a blockage effect, which seriously affects the consolidation characteristics. It is found that the uncertainty of creep deformation will accumulate over time until it remains constant. The variability of creep deformation increases with the increase in the vertical SOF and COV of the secondary consolidation coefficient. The research also provides insightful clues for the time-varying reliability design related to the time-dependent deformation of clay soil.

Transformations of spatial correlation lengths in random fields

The paper describes the theoretical relationship between spatial correlation lengths in lognormal and hyperbolic tangent (“tanh”) random fields, and the underlying Gaussian random fields from which they are derived following transformation. The inevitable change in the spatial correlation length following transformation has been noted in several studies, but has not, to the authors’ knowledge, been rigorously investigated before. The paper presents derivations that show the dependencies for normal/log-normal and normal/tanh transformations together with three different correlation functions. The one-dimensional derivations are extrapolated to two- and three-dimensional models, with particular emphasis on vertical and horizontal correlation lengths. The paper concludes with an example probabilistic bearing capacity analysis using the Random Finite Element Method (RFEM). For the example considered, the untransformed solutions were slightly unconservative, but the differences were generally quite modest.

Bayesian inference of spatially varying parameters in soil constitutive models by using deformation observation data

AbstractParameters of soil constitutive models are usually identified through laboratory tests. The spatial variability of these parameters is generally not considered due to the limitation of the test scale. This study proposes a data‐driven approach to infer the spatially varying parameter of the modified Cam‐clay model from limited field observations and subsequently improves soil settlement predictions. The observation data and numerical results of random finite element method are assimilated in an inverse modeling process based on the iterative Ensemble Kalman filtering (iEnKF). Different unknown variables and number of observations are used to study their effects on parameter estimations and settlement predictions. The effectiveness of the proposed approach is illustrated through a synthetic partial‐loading test. The results show that the site‐specific spatial variability can be estimated reasonably, and predictions of settlement can be improved by using the inferred parameter field. When the variables to be inferred change from all 60 variables to the selected 17 important variables, the average error of the estimated fields increases, but the variance decreases. A reduction in the observation spacing and an increase in the number of observations lead to a slightly smaller error of the mean and considerably reduced uncertainties of soil parameters. Although the inferred results of parameter field show different accuracies, the corresponding calculated settlements are generally similar and satisfactory.

Predictive Computational Model for Damage Behavior of Metal-Matrix Composites Emphasizing the Effect of Particle Size and Volume Fraction.

In this paper, an integrated numerical model is proposed to investigate the effects of particulate size and volume fraction on the deformation, damage, and failure behaviors of particulate-reinforced metal matrix composites (PRMMCs). In the framework of a random microstructure-based finite element modelling, the plastic deformation and ductile cracking of the matrix are, respectively, modelled using Johnson–Cook constitutive relation and Johnson–Cook ductile fracture model. The matrix-particle interface decohesion is simulated by employing the surface-based-cohesive zone method, while the particulate fracture is manipulated by the elastic–brittle cracking model, in which the damage evolution criterion depends on the fracture energy cracking criterion. A 2D nonlinear finite element model was developed using ABAQUS/Explicit commercial program for modelling and analyzing damage mechanisms of silicon carbide reinforced aluminum matrix composites. The predicted results have shown a good agreement with the experimental data in the forms of true stress–strain curves and failure shape. Unlike the existing models, the influence of the volume fraction and size of SiC particles on the deformation, damage mechanism, failure consequences, and stress–strain curve of A359/SiC particulate composites is investigated accounting for the different possible modes of failure simultaneously.

Effect of pile inclination on the lateral deformation behaviour of pipe piles in calcareous sand

The lateral deformation behaviour of inclined piles in calcareous sand is investigated with model testing and numerical analysis. A model pile is instrumented with quasi-distributed strain sensors and installed in calcareous sand with various inclination angles. The lateral deformation, bending moment and shear force of the pile are analysed by investigating the effect of the inclination angle and embedded depth. The experimental results of piles with inclination angles of −30°, 0° and 30° are analysed. A random finite element method (RFEM) is established to investigate the effect of the pile inclination. A single exponential function is used to characterize the spatial autocorrelation of soil parameters. A random field is obtained by the Cholesky method. The RFEM is verified with experimental results and used to further analyse the lateral deformation behaviour. The results indicate that both of the orthogonal maximum influence distances from the pile centre have a linear relationship with the pile inclination. A linear relationship between the maximum lateral deformation on the pile top and the pile inclination is proposed. The outcome of this study may be helpful for understanding the lateral deformation behaviour of piles in calcareous sand and designing inclined piles.

Reliability analysis and risk assessment of deep excavations using random-set finite element method and event tree technique

In the recent past years, the importance of reliability analysis and risk assessment in solving Geotechnical Engineering problems is underlined in literature. This study is aimed to show the feasibility of Random Set Finite Element Method (RS-FEM) and Event Tree Analysis (ETA) in assessing the reliability and risk to fatality of urban excavations. In order to fulfill this aim, a case study (22-meter excavation) which is stabilized with soil anchors is considered in North of Tehran, Iran. Prior to risk assessment, the failure probability of the deep excavation is estimated using RS-FEM, which is a relatively novel method in analyzing the reliability of deep excavations. Findings of the reliability analysis suggest that the upper and lower bounds of failure probabilities (Pf) for the considered case study are 0.0002 and 0.00001, respectively. Subsequently, using ETA, different possible scenarios, which may lead to fatality, are defined and proper probabilities are assigned to each scenario. Finally, for risk quantification purposes, the outcome probability of each scenario is multiplied by a considered Value of Statistical Life (VSL) as well as an aversion factor. Overall, findings recommend that the presented methodology (i.e. RS-FEM coupled with ETA) can be employed for Quantitative Risk Assessment (QRA) in deep urban excavations. Nevertheless, further research is recommended for other case studies as practicing the aforementioned method can constitute common sense on QRA of deep excavations.

An extended Prandtl solution for analytical modelling of the bearing capacity of a shallow foundation on a spatially variable undrained clay

Classical bearing capacity theory was developed mainly based on spatially uniform soil properties, which cannot account for the influence of inherent soil variability. If the soil strength is heterogeneous, then using the average strength may overestimate the bearing capacity of foundations, because the failure mechanism may preferentially mobilise the weaker soils. In this study the aim is to establish a theoretical model using upper-bound solutions applied to the bearing capacity analysis of shallow foundations on undrained clay considering spatial variability. The model is derived on the principle of least energy dissipation using a four-parameter variation on Prandtl's mechanism. The theoretical model developed is verified by the random finite-element (FE) method in spatially varying soil conditions. The results show that the model can accurately capture the effect of spatially varying strength on the shallow foundation failure mechanism. The difference of bearing capacity factor between the proposed model and the FE model is within 5%, which demonstrates that the four-parameter model has an accuracy that is comparable to FE analysis with many hundreds of degrees of freedom. Another advantage of the theoretical model is that the possible non-convergence in FE analysis can be avoided, and hence, the calculation efficiency is significantly enhanced. The model is therefore suitable for rapid quantification of bearing capacity in spatially varying soils.

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.

Undrained stability analysis of strip footings lying on circular voids with spatially random soil

Random adaptive finite element limit analysis (RAFELA) is introduced to investigate the stability of the footing-voids system with random soils. The emphasis of this study is, for the footing-void system, on quantifying the influence of spatial variability on its bearing capacity and the failure probability. Of particular interest is that random field theory is utilized to characterize the spatial variability of soil, and at the same time, failure probability is calculated by imposing Monte-Carlo simulations on independent cases. Strict close bearing capacity could be obtained within the scope embraced by upper bound (UB) results and lower bound (LB) results. By imposing the proposed modelling, comprehensive parametric studies were carried out, especially for the soil spatial correlation parameters and geometry parameters. The results indicated that the failure probability would increase with the farther distance between the void and the footing, the greater coefficient of variation (COV) and the smaller vertical correlation length. Detailed design tables were presented to assess potential failure probability under different conditions. Typical failure patterns are also summarized, revealing a positive correlation between the failure probability and the diffusion degree of failure curves.

Analyses of the Impacts of Climate Change and Forest Fire on Cold Region Slopes Stability by Random Finite Element Method

Increasing number of landslides occurred in the cold regions over the past decades due to rising temperature or forest fires associated with climate change. The instability of thawing slopes caused serious damages to transportation infrastructure, residential properties, and losses of human lives. These types of landslides, however, are difficult to analyze by the traditional limit equilibrium methods due to the coupled thermos-hydro-mechanical multiphysics processes involved. This paper describes a novel microstructure-based random finite element model (RFEM) to simulate the stability of permafrost slope subjected to climate change, including the increasing extent of thawing by warm atmosphere and thermal load due to forest fire. The properties of frozen soil are captured with random finite element model incorporating soil-phase coding. The thermomechanical responses of a permafrost slope are simulated to obtain the temperature, displacement, and stress fields. From these, the local factors of safety are obtained, which predict failure slumps along the slope that are consistent with the field-observed failure behaviors in permafrost slopes. The effects of climate change on the permafrost slope stability are analyzed for the years of 1956, 2017, and 2045. The results demonstrated appreciable amount of effects of climate change on the extent of slope failure zones. Forest fire led to melting of frozen soil and also affects the slope stability primarily in the shallow depth.

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.