Random Finite Element Analysis Research Articles

Prediction of wall deflection induced by braced excavation in spatially variable soils via convolutional neural network

Recently, the random field finite element method (RF-FEM) has attracted significantly increasing attention in the field of geotechnical engineering, especially for the purpose of analyzing the response of geotechnical systems due to the inherent variability of physical and mechanical properties. However, the method requires repeated finite element calculations based on a mass of sampling processes, making the computing effort expensive. The surrogate model is one of the techniques commonly adopted to alleviate the computational burden. In addition, some architectures of deep learning surrogate models are so unique that it is difficult to transfer to similar cases and to be familiar and reproducible by readers. In this study, we propose a convolutional neural network (CNN) surrogate model based on classical architecture, VGG6, to perform random field finite element analyses (RF-FEM). We pre-process the tabular data generated by the random filed method into an image-like format as input data. The VGG6 is used as a surrogate model to replace the original RF-FEM simulations for all subsequent calculations. The applications of the proposed method to assess wall deflection of braced excavation in clays with randomly varying cohesion cu and the friction angle φ are illustrated and compared in different cases. The excellent agreement between the VGG6 outputs and the FEM predictions demonstrated the promising potential of using VGG6s as a surrogate model for reliability analysis in spatially variable soils. Moreover, the model saves a lot of computing time and computing power and fully proves the generalization performance of the model under various cases.

Homogenizing spatially variable Young modulus using pseudo incremental energy method

Homogenization is the process of representing a spatially variable property by an equivalent (or effective) homogenous property. The second and fifth authors proposed the pseudo incremental energy (PIE) method to represent a spatially variable Young’s modulus by an effective Young’s modulus for a rigid footing problem, the overall Young’s modulus actually “felt” by the footing. The effective Young’s modulus is based on a non-uniform geometric average with weights provided by PIE. It performs better than the conventional uniform geometric average at a cost of computing PIE weights from one deterministic finite element analysis. The current paper extends the previous work on PIE in four respects. First, a new PIE procedure is developed to address a known issue in the previous PIE. Second, the applicability of the new PIE is verified for a wide variety of two-dimensional (2D) and three-dimensional (3D) geotechnical problems, including 2D retaining wall, 2D & 3D axially loaded pile, 2D & 3D laterally loaded pile, and 2D & 3D base heave. Third, the new PIE is extended to problems with layered soils, where each layer is modeled by a separate random field. Finally, the new PIE is implemented to a real case study of a 3D rigid footing. The new PIE method can approximately match the results from a random finite element analysis (RFEA) at the realization level (not ensemble statistics level) at an acceptable cost of one deterministic finite element analysis. It is much cheaper than RFEA that requires hundreds of thousands of deterministic analyses. It is slightly more costly than the original PIE.

Random finite element analysis on uplift bearing capacity and failure mechanisms of square plate anchors in spatially variable clay

Square plate anchors are increasingly used to provide uplift bearing capacity for both onshore and offshore infrastructures. Past research on the behavior of plate anchors mostly assumed that the soil strength is uniform or linearly increasing with depth without considering the geological uncertainty due to soil spatial variability. This study therefore conducted a series of three-dimensional random finite element analyses on the uplift bearing capacity and failure mechanisms of square plate anchors considering the combined effects of anchor buried depth and soil spatial variability. It was found that two typical types of failure mechanisms, namely, a localized rotational scoop failure mechanism and a global shear failure mechanism would form in random soils with relatively weak and strong soil strengths, respectively. A larger buried depth of at least 3B (B = width of plate anchor) is required to ensure a localized failure mode in random soils. Moreover, the mean uplift bearing capacity in random soils is generally lower than the corresponding deterministic value. For this reason, the deterministic uplift bearing capacity is generally overestimated. Finally, this study provided a quantitative approach to predict the probability of failure for plate anchors considering the combined effects of anchor buried depth and soil spatial variability, which may benefit the estimation of the probability of failure for square plate anchors as part of the conventional factor of safety design approach.

Response of vertically-loaded pile in spatially-variable cement-treated soil.

Despite the extensive application prospects of piles in cement-treated soil, few studies have explored the ultimate bearing capacity especially in consideration of the spatial variability of cement-treated soil. This study examines the performance of driven piles which were installed inside the cement-treated ground, considering the inherent spatial variability of the cemented soil and the positioning error during piles installation through finite element analyses. The deterministic and random finite element analysis results have shown that the shaft resistance mainly provided the ultimate bearing resistance of piles in cement-treated soil. The spatial variability reduced the global performance of pile installed through a cement-treated soil. The ultimate bearing resistance of the pile inserted in cement-treated soil was controlled by drained condition. Drained ultimate bearing resistance should be used to determine the design working compression load of pile in cement-treated soil.

Three-Dimensional Seepage Investigation of Riverside Tunnel Construction Considering Heterogeneous Permeability

Seepage is a major concern during the construction of a riverside tunnel, because the seepage may cause both hydraulic and mechanical problems. At present, there are some well-accepted methods combining the hydraulic-mechanical coupling effect. This paper investigated a tunnel near a riverside under seepage forces using hydraulic-mechanical coupling analysis. Two cases were considered: an unlined tunnel without waterproofing, and a tunnel with a cement-treated soil surround acting as water barrier. The spatial variability of hydraulic conductivity was taken into account using random finite-element analysis. Results indicated that the analysis is able to account reasonably for the variation range of soil parameters. The arithmetic mean of hydraulic conductivity could serve as a rule of thumb estimation for a macroscale representative hydraulic conductivity. The results are likely to yield a scientific estimation of the seepage flow as well as a better understanding of the potential tunnel collapse mechanism.

A large deformation finite element analysis of uplift behaviour for helical anchor in spatially variable clay

Helical anchors are emerging as an innovative foundation design option for offshore applications to resist uplift loading. Numerous studies have been conducted to investigate the uplift behaviour of helical anchor under different helical bearing plate configurations in clay, using field or model tests and numerical analyses. However, almost all of them are limited to deterministic analyses, ignoring the influence of inherent spatial variability of soil. This work conducts a 3D large deformation random finite element analysis to examine the uplift behaviour of helical anchor in spatially variable clay. Both the stochastic and deterministic trends are reflected in the 3D non-stationary random field adopted. It is found that the spatial variability of soil strength significantly influences the failure mechanism of helical anchor. Parametric studies indicate that the coefficient of variation and vertical scales of fluctuation of soil strength can much influence on the uplift capacity, while the effect of horizontal scales of fluctuation is trivial. The goodness-of-fit test of probability distribution suggests that random uplift capacities follow log-normal distribution, whereby the relationship between factor of safety and probability of failure is established. The findings of this paper might facilitate the safety design of helical anchor.

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.

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.

Novel approach to efficient slope reliability analysis in spatially variable soils

The random field finite element method (RF-FEM) provides a robust tool for carrying out slope reliability analysis that incorporates the spatial variability of soil properties. However, it has a major drawback of being computationally very time-consuming. To address this common criticism, the current study proposes a novel metamodel-based method for efficient slope reliability analysis in spatially variable soils. The proposed method involves the use of Convolutional Neural Networks (CNNs) as metamodels of the random field finite element model. With proper training using a small but sufficient number of random field samples, the CNN can potentially replace the computationally demanding random field finite element analyses for Monte-Carlo simulations. This paper examines the capability of CNNs to learn high-level features that contain information about the random variabilities in both spatial distribution and intensity, and the accuracy of the subsequent predictions of the RF-FEM results. Application of the proposed method to slope reliability analysis in spatially variable soils is illustrated and compared against other metamodel-based approaches, using a case study involving a multi-layered soil system with randomly varying cohesion c and the friction angle ϕ. The results show that (i) the proposed CNN approach predicts a probability of slope failure that is within 5% of the corresponding value obtained using direct RF-FEM Monte-Carlo simulations, but at a small fraction of the computational cost, and (ii) the proposed method also compares favourably against other metamodel-based methods in terms of computational efficiency and accuracy.

A large-deformation random finite-element study: failure mechanism and bearing capacity of spudcan in a spatially varying clayey seabed

Although the random heterogeneity of a clayey seabed has been long recognised, it is relatively recently that the influence of spatial variability on the spudcan foundation has been explicitly explored by random finite-element analyses. Owing to technical and computational challenges, these are still limited to the small-strain Lagrangian finite-element calculation with the ‘wished-in-place’ assumption, which is a departure from the reality. Attempts were successfully made in this research to combine the large-deformation finite-element calculation with random field theory in a Monte Carlo simulation framework. The continuous penetration of a spudcan in a clayey seabed was modelled through the coupled Eulerian–Lagrangian finite-element calculation with simultaneous consideration of the strain-softening effect and linearly increasing strength with depth. The results suggest that the randomness of soil strength can obviously alter the flow pattern and failure mechanism of soil, and in turn affects the bearing capacity of the spudcan. The log-normal probability distribution function was found to fit the distribution of the bearing capacity factor well. A correlation between the failure probability and the safety factor was established, from which the level of reliability corresponding to different safety factors can be clearly evaluated. Such a correlation may facilitate the further development of reliability-based design.

Analysis of cement-treated soil slab for deep excavation support – a rational approach

In this study an effective stress model for cement-treated soil is employed to carry out a systematic study on cement-treated soil slab used to control the wall displacement in excavations with thick, soft clay layers. Deterministic analyses were first conducted to assess the drainage state of the treated soil slab. These indicate that, under timescales which are representative of those in actual construction, cement-treated soil slab conditions may range from nearly undrained to drained. Random finite-element analyses also show that the global undrained and drained behaviour of the treated soil slab differ significantly. Undrained loading leads to a rapid rise in stress with strain to a peak strength with subsequent softening. Drained behaviour, on the other hand, leads to a more compliant response, which suggests that the main issue in the drained condition is likely to be large displacement, rather than complete collapse. A rational design framework for cement-treated soil slab is proposed, which explicitly considers the distribution in global behaviours of the treated soil slab. This differs from current limit state approaches which are largely based on statistical properties of the input data, such as the unconfined compressive strength of sampled cores.

Analysis of tunnelling through spatially-variable improved surrounding – A simplified approach

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

On characteristic values and the reliability-based assessment of dykes

ABSTRACTA case study involving the assessment and re-design of an existing dyke, founded on a layered soil, has compared deterministic analysis based on 5-percentile property values and a reliability-based random finite element analysis consistent with the requirements of Eurocode 7. The results show that a consideration of the spatial nature of soil variability generally leads to higher computed factors of safety and, for those dyke sections requiring remedial action, to more economic designs. Back-figured characteristic values are shown to be considerably higher than the 5-percentile soil properties; hence, a reduction in over-conservatism is achieved.

A three-dimensional algorithm for estimating water-tightness of cement-treated ground with geometric imperfections

Jet-grouted cut-off walls (JGCOWs) and jet-grouted earth-plugs (JGEPs) are commonly used as artificial water barriers to impede ground water from seeping downstream of dams or into deep excavation pits. However, due to positioning errors and variability of column diameters, seepage passages may occasionally penetrate through the water barrier, thus undermining its water-tightness. A three-dimensional discretized algorithm (TDA) is proposed for quantitatively estimating the effect of the above geometrical imperfections on the water-tightness. With given random geometries of the treated ground, this algorithm is able to examine whether a water barrier is penetrated by seepage passages. It further quantitatively estimates the flow rate through the water barrier with the given random geometry. When compared with the results from 3-D random finite element analysis, it was found that the TDA results show strong correlations with FEM results, but at a remarkably reduced computational cost. The TDA generally gives a slightly more conservative estimation of flow rate than the FEM counterpart due to the lengthened seepage distance due to random corrugations in the penetrating untreated zones.

Characteristic strength of soils underlying foundations considering effect of spatial variability

In Eurocode EC7, a “characteristic” strength is used as a cautious estimate of the local average strength that governs the bearing capacity of the foundation. The objective of this paper is to examine the correlation between the local average strength and the bearing capacity of a stiff caisson foundation resting on spatially variable ground using random finite element analyses. The results show that using the local average strength over some assumed or postulated failure zones tends to overestimate the mean bearing capacity of the ground. This can be attributed to two possible reasons. Firstly, the postulated failure zone is unlikely to be fully reflective of the real failure zone in spatially variable ground. Secondly, the bearing capacity is more affected by the strength of the weak zones than that of the strong zones. Both of these factors lead to a lowering of the bearing capacity. A more indicative way of determining a characteristic strength that will give a better indication of the bearing capacity is also recommended, together with a strength reduction factor that accounts for the effect of spatial variability.

Restraining requirements for lateral elastic-plastic buckling of columns accounting for random imperfections

This paper dealt with the restraint stiffness and strength requirements for axially loaded columns with lateral restraints, accounting for inelastic flexural buckling and random imperfections. Firstly, the buckling behaviour of restrained columns with classic deterministic imperfections was discussed to verify the effect of section dimensions, material properties and imperfection shapes. Then by using probabilistic distribution of imperfections along the column, extensive numerical results were gained through random imperfection finite element analyses, followed by restraint stiffness and restraint force proposed statistically corresponding to full restraining for columns. Results showed that, the full restraining stiffness of perfect columns proposed by Winter is sufficient for columns with imperfections and inelastic buckling. The traditional approach with predetermined artificial imperfections is deficient to evaluate the restraint force properly and rationally, due to ignoring the effect of random imperfections.

Effective Young’s modulus of a spatially variable soil mass under a footing

This study investigates the possibility of representing the effective Young’s modulus (Eeff) for a footing problem supported on a spatially variable medium - the Young’s modulus actually “felt” by the footing - using a spatial average. The Eeff is simulated by a homogenization procedure that matches the responses between a random finite element analysis (RFEA) and a homogeneous finite element analysis. Emphasis is placed on whether the spatial average can well represent the numerical value of Eeff in each spatially varying realization, not just the statistics of Eeff within an ensemble (a weaker requirement). It is found that the conventional spatial averaging model that treats all soil regions equally important cannot satisfactorily represent Eeff. Extensive numerical results show that the concept of “mobilization” is essential: highly mobilized soil regions close to the footing should be given larger weights than non-mobilized remote regions. Moreover, the non-uniform weights can be prescribed prior to RFEA, that is, they do not depend on the specific response corresponding to a specific random field realization. The “prescribed mobilization” for the spatially variable Young’s modulus can be contrasted with the “emergent” mobilized shear strength in a spatially variable medium that results from the emergent nature of the critical failure path – it cannot be predicted prior to random finite element analysis. A key contribution of this paper is the development of a simple method based on the “pseudo incremental energy” to estimate the non-uniform weights for the spatial averaging using a single run of a homogeneous finite element analysis.

Effects of material and drilling uncertainties on artificial ground freezing of cement-admixed soils

The artificial ground freezing method can be used jointly with the deep cement mixing method during break-in and break-out processes of shield machines in a tunnel shaft. The frozen ground can fully cut off groundwater seepage, thus ensuring a watertight working platform. Cement-admixed soils can restrict frost heave and thaw-induced settlement because of the decreased permeability. Both methods can also enhance mechanical strength of the soil to enable construction to proceed. Two main sources of heterogeneity are likely to influence the freezing effect: spatial variability in in situ water content in natural soil and spatial variability in binder concentration in cement-admixed soils. Furthermore, positioning error when installing freeze pipes can also affect freezing efficiency. This study simulates in situ water content and binder concentration as Gaussian random fields, whereby variations in the thermophysical properties are estimated. Positioning error is also assessed by prescribing an incline angle in freeze pipes. The influences of those two sources of spatial variability as well as positioning error are examined with random finite-element analyses and statistical characteristics are estimated based on the results. Results are tabulated to offer practitioners a rule of thumb for estimating additional efforts needed in artificial ground freezing, accounting for variations in the thermophysical properties and positioning errors in installing freeze pipes.