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The problem of a reinforced fill over a void has been the subject of much research in the geosynthetics literature. Previous studies have mainly focused on finding closed-form solutions to predict the tensile loads and strains in the reinforcement layer once a void develops below the fill. In this paper, a 2D finite difference (FLAC) model that implements the hyperbolic isochronous load-strain model for the reinforcement by Bathurst and Naftchali (2021) is used to investigate the influence of the rate-dependent properties of polymeric geosynthetic reinforcement materials on reinforcement tensile strains and load, and overall system performance including vertical deformation at the reinforcement elevation and at the fill surface. The paper also investigates the influence of fill soil properties and constitutive model type, foundation condition, void geometry and fill height on system performance. The results of numerical modelling are compared to predictions made using the closed-form solution of Giroud et al. (1990) and in the BSI 8006-1 (2010) design code. The results of numerical modelling demonstrate that the choice of fill height to void width and the stiffness of the rate-dependent geosynthetic reinforcement layer are important to ensure that the maximum reinforcement strain, allowable strength and fill surface settlement criteria are not exceeded. • A numerical 2D FLAC model is used to perform a parametric analysis of the reinforced fill over a void problem. • The numerical approach is validated against a full-scale test reported in the literature. • A simple rate-dependent hyperbolic isochronous load-strain (creep) model is used for the geosynthetic reinforcement. • The ratio of fill height to void width and the reinforcement stiffness are shown to strongly influence numerical outcomes. • The results of numerical modelling are compared to predictions made using two well-known closed-form solutions. [ABSTRACT FROM AUTHOR]

The paper examines the quantitative influence of uncertainty in the estimate of geosynthetic reinforcement stiffness on numerical outcomes using analytical solutions for a) the maximum outward facing deformation in mechanically stabilized earth (MSE) walls, b) maximum reinforcement tensile loads and strain in MSE walls under operational conditions, and c) the mobilized reinforcement stiffness in a geosynthetic layer used to reinforce a fill over a void. The stiffness of the reinforcement is modelled using an isochronous two-parameter hyperbolic load-strain model. A linear relationship between isochronous stiffness and the ultimate tensile strength of the reinforcement is used to estimate reinforcement stiffness when product-specific creep data are not available at time of design. Solution outcomes are presented deterministically and probabilistically. The quantitative link between nominal factor of safety used in deterministic working stress design practice and reliability index is provided. The latter is preferred in modern performance-based design to quantify margins of safety within a probabilistic framework. Finally, the paper highlights the practical benefit of using product-specific isochronous secant stiffness data when available, rather than estimates of isochronous stiffness values based on reinforcement type or pooled data. • Uncertainty in the estimate of geosynthetic stiffness is investigated for three geosynthetic-reinforced soil applications. • The stiffness of the reinforcement is modelled using an isochronous two-parameter hyperbolic load-strain model. • Isochronous secant stiffness estimated using: a) product-specific creep data; b) type of reinforcement; and c) pooled data. • Closed-form solutions linking the nominal factor of safety to the reliability index for each application are provided. • Solution outcomes are presented deterministically and probabilistically. [ABSTRACT FROM AUTHOR]

Thin granular fill layers are routinely used to aid the construction of shallow footings seated over undrained soft clay foundations and to increase their load capacity. The influence of time- and strain-dependent reduction in reinforcement stiffness on the bearing capacity and load-settlement response of a footing seated on a thin reinforced granular fill layer over undrained soft clay foundations is examined in this paper using finite-difference method (FDM) numerical models. The time- and strain-dependent stiffness of the reinforcement described by a two-component hyperbolic isochronous tensile load-strain model is shown to influence the bearing capacity and load-settlement response of the reinforced granular base scenario. The additional benefit of a reinforced granular layer diminishes as the time-dependent stiffness of the geosynthetic reinforcement increases. An analytical solution for the ultimate bearing capacity of strip footings seated on thin unreinforced and reinforced granular layers over undrained clay is proposed in this study. The main practical outcome from this study are tables of bearing capacity factors to be used with the analytical solution. The bearing capacity factors were back-calculated from the numerical analyses and account for the influence of rate-dependent properties of geogrid reinforcement materials and clay foundations with soft to very soft undrained shear strength. • Load-settlement and ultimate bearing capacity of strip footing on reinforced granular layer over undrained clay is studied. • Stiffness of the reinforcement is described by a two-component hyperbolic isochronous tensile load-strain model. • Benefit of reinforced granular layer diminishes as time-dependent stiffness of the geosynthetic reinforcement increases. • Analytical solution for ultimate bearing capacity of strip footing on thin reinforced granular layer over clay is proposed. • Bearing capacity factors are back-calculated from numerical simulations using large-strain finite difference method model. [ABSTRACT FROM AUTHOR]

• The model was made purposely simple to focus attention on the qualitative influence of stiffness on reinforcement loads. • Using numerical models to infer quantitative performance of actual MSE walls must be undertaken with caution and skill. • Small changes in numerical model details can give widely different estimates of reinforcement loads. • The veracity of any numerical model should be based on comparisons with more than one physical wall of realistic height. • The accuracy of design methods must be assessed using measured loads from full-scale walls, not numerical models. [ABSTRACT FROM AUTHOR]

Many analytical and numerical analysis and design methods for geosynthetic-reinforced soil structures require a single-value (constant) estimate of reinforcement stiffness. However, geosynthetic reinforcement products are rate-dependent polymeric materials meaning that they exhibit time and strain-dependent behaviour under load. Hence, the appropriate selection of a constant (elastic) stiffness value requires careful consideration. A simple hyperbolic stiffness model is shown to be a useful approximation to the constant-load isochronous creep-strain behaviour of these materials at low load levels applicable to operational (serviceability) conditions of geosynthetic-reinforced soil structures. A large database of 606 creep tests on 89 different geosynthetic reinforcement products falling within seven different product categories was collected. From these data, isochronous stiffness values were determined for different combinations of duration of loading and strain level. Data from products falling within the same category were collected together to provide approximations linking the isochronous load-strain (creep) stiffness to the ultimate tensile strength of the material. These approximations are useful for analytical and numerical modelling particularly when parametric studies are undertaken to identify the sensitivity of model outcomes to reinforcement stiffness. Finally, three different geosynthetic-reinforced soil application examples are provided to demonstrate the important role of tensile stiffness on analysis and design outcomes. • A simple hyperbolic model is developed for the isochronous load-strain (creep) behaviour of geosynthetic reinforcement materials. • The model is used to estimate reinforcement stiffness at low load levels consistent with operational conditions of reinforced soil systems. • Model parameters are fitted from a large database of creep tests on geosynthetic reinforcement materials from different product categories. • Approximations linking the isochronous load-strain (creep) stiffness to the ultimate tensile strength of the material are provided. • Three different geosynthetic-reinforced soil examples demonstrate the important role of tensile stiffness on analysis and design outcomes. [ABSTRACT FROM AUTHOR]

The mean (point) estimate of reliability index (β) for simple limit state performance functions is most often based on load and resistance model bias statistics taken from bias data of limited size. The paper shows how the confidence in the point estimate of β can be computed using the bootstrap method and a closed-form solution for the special case of lognormally distributed nominal and bias random variables. Bias data collected for the pullout limit state for steel strip mechanically stabilized earth walls are used to demonstrate the approach but the method is general. Mean and confidence intervals on β are computed using the entire available bias data sets and smaller sample sizes located in the tails of the original populations, or smaller sample sizes that result in upper and lower extreme point estimates of β. The closed-form solution for β varies linearly with the logarithm of the nominal factor of safety for the example pullout limit state. Depending on the load and resistance bias sample sizes and where they are taken from in the original larger population, results in a wide range of nominal factors of safety to satisfy a target β at time of design. [ABSTRACT FROM AUTHOR]

The paper demonstrates deterministic and reliability-based assessment of strength limit states (tensile resistance and pullout) and the service limit state for soil failure for mechanically stabilized earth (MSE) walls constructed with polyester (PET) strap reinforcement. The general approach considers the accuracy of the load and resistance models that appear in each limit state equation plus uncertainty in the estimate of nominal load and resistance values at time of design. Reliability index is computed using a closed-form solution that is easily implemented in a spreadsheet. Three PET strap MSE wall case studies are used to demonstrate the reliability-based assessment approach and to compare margins of safety using different load and resistance model combinations. In some walls using the Coherent Gravity Method to compute loads, the recommended nominal factors of safety for tensile strength and pullout limit states were not satisfied. However, reliability analyses showed that the walls satisfy recommended minimum target reliability index values for the limit states investigated, usually by large amounts. The most critical limit state is the soil failure limit state which is used in the Simplified Stiffness Method to keep the reinforced soil zone at working stress conditions assumed for geosynthetic MSE walls under operational conditions. • Describes deterministic and reliability-based analyses of internal limit states for PET strap walls. • Reliability index is computed using a closed-form solution that is easily implemented in a spreadsheet. • Three PET strap MSE wall case studies are used to demonstrate the reliability-based assessment approach. • Most critical limit state is the soil failure limit state which is used in the Simplified Stiffness Method. • General approach is applicable to MSE walls with extensible and inextensible reinforcement materials. [ABSTRACT FROM AUTHOR]

A general closed-form solution to compute the reliability index of a simple limit state function with two load terms and one resistance term is derived. The formulation considers contributions to margins of safety expressed in probabilistic terms due to the choice of load and resistance models, bias values, dependencies between nominal values and bias, uncertainty in estimates of nominal values for uncorrelated load and resistance terms at time of design, and average margin of safety expressed as the operational factor of safety. A sensitivity analysis and example application demonstrate the quantitative influence of the contributing random variables on reliability index. [ABSTRACT FROM AUTHOR]

Reliability analysis of bearing capacity of a strip footing at the crest of a simple slope with cohesive soil was carried out using the random finite element method (RFEM). Analyses showed that the coefficient of variation and the spatial correlation length of soil cohesion can have a large influence on footing bearing capacity, particularly for slopes with large height to footing width ratios. The paper demonstrates cases where a footing satisfies a deterministic design factor of safety of 3 but the probability of design failure is unacceptably high. Isotropic and anisotropic spatial variability of the soil strength was also considered. [ABSTRACT FROM AUTHOR]

The paper reports the details of numerical models used to predict the performance of two 3.6 m-high well-instrumented wrapped-face walls. The walls were nominally identical except that the reinforcement material in one wall was a steel welded wire mesh and in the other a biaxial polypropylene geogrid. The backfill soil was modelled using both linear and nonlinear elastic-plastic constitutive models. A general hyperbolic (nonlinear) axial load-strain-time model was used for the reinforcement. The numerical results show good agreement with measured performance features for the welded wire mesh wrapped-face wall. Agreement between numerical predictions of facing displacements and maximum reinforcement loads was less accurate for the very flexible geogrid wrapped-face wall. The discrepancies are believed to be related to the unusually flexible wrapped face used in the geogrid wall construction. Numerically predicted and measured maximum reinforcement loads are compared to loads using the AASHTO reinforcement strength-based design approach (Simplified Method) and the Simplified Stiffness Method which is an empirical reinforcement stiffness-based method. The paper provides physical test data that can be used to benchmark other numerical models, highlights lessons learned during the development of the models, and identifies reasonable expectations for numerical model accuracy for models of similar complexity used to simulate the performance of mechanically stabilized earth (MSE) wall structures under operational conditions. [ABSTRACT FROM AUTHOR]

Soil-cement deep mixing (DM) columns combined with geosynthetic basal reinforcement are an accepted technique in geotechnical engineering to construct road and railway embankments over soft foundations. Both full-width and unit cell models have been used to numerically simulate the performance of geosynthetic-reinforced and column-supported (GRCS) embankments. However, the typical unit cell model with horizontally fixed side boundaries cannot simulate the lateral spreading of the embankment fill and foundation soil. As a result, the calculated reinforcement tensile loads using typical unit cell models are much less than those from matching full-width models. The paper first examines GRCS embankments using a full-width model with small- and large-strain modes in FLAC and then compares the calculated results from the full-width model with those using a typical unit cell model, a recently proposed modified unit cell model, and a closed-form solution. The paper also examines the influence of the soft foundation soil modulus, reinforcement tensile stiffness, and DM column modulus on the reinforcement tensile loads. Numerical analyses show that the reinforcement tensile loads from the modified unit cell model are in good agreement with those from the full-width model for zones under the embankment crest for all cases and conditions examined in the paper. Both the full-width model and modified unit cell model perform better than the typical unit cell model for the prediction of the reinforcement tensile load when compared to the closed-form solution. However, while the modified unit cell developed by the writers is shown to be more accurate than the typical unit cell when predictions are compared to results using full-width numerical simulations, the benefit of using this approach to reduce computation times may be limited in practice. [ABSTRACT FROM AUTHOR]

Probabilistic slope stability analyses of simple geosynthetic reinforced soil slopes were carried out using the shear strength reduction method in combination with the finite element method (FEM). An existing open-source FEM code was modified to include bar elements to model horizontal layers of geosynthetic reinforcement. Analysis results using the modified code (mFEM) demonstrate that large reductions in probability of failure can be realized by adding geosynthetic reinforcement layers to constructed slopes. The modified code was also used to investigate the effect of variability of soil friction angle on probabilistic outcomes for constructed unreinforced and reinforced purely frictional soil slopes. [ABSTRACT FROM AUTHOR]

Geosynthetic-reinforced and column-supported (GRCS) embankments have proven to be an effective construction technique for fills on soft foundations. The paper introduces a modified unit cell approach to model GRCS embankments supported by deep mixed column walls. The modified unit cells include linear elastic springs at one or both vertical boundaries to simulate lateral displacements of the embankment fill and foundation soil. The finite difference program FLAC is used to compare numerical outcomes using the modified unit cells with those using the typical unit cell arrangement with lateral rigid side boundaries. Numerical results demonstrate good agreement between simulations using small-strain and large-strain modes in some cases and large differences in other cases. Lateral displacements of the embankment fill and foundation soil using the modified unit cells are shown to have large influence on reinforcement loads. Finally the paper demonstrates that calculated reinforcement loads are sensitive to choice of small-strain or large-strain mode when using program FLAC. [ABSTRACT FROM AUTHOR]

• Buried steel grids used in mechanically stabilized earth (MSE) walls can corrode. • Probability of failure is computed for limit states of galvanized steel grids. • Three-stage corrosion model is coupled with reinforcement load and resistance models. • Probability of steel grid tensile and pullout failure with time is computed. • Pullout failure of coated steel members due to corrosion is shown to be unlikely. Design for internal limit states of steel grid reinforcement in mechanically stabilized earth (MSE) walls must provide for loss of steel thickness due to corrosion. The paper combines key features of zinc and steel corrosion models used in North American practice with corrosion rate statistics reported in the literature for the zinc coating on buried galvanized steel strips and for steel rebar. Examples of loss of strength and pullout capacity of steel grid elements constructed with different member diameters and up to 120 years after construction are presented in probabilistic terms. Results of reliability-based analysis for the tensile strength and pullout limit state at design lives of 75 and 120 years are demonstrated for a typical wall under operational conditions using tensile load and resistance models in combination with non-aggressive soils recommended in North American design codes. The results of calculations show that for the wall example the influence of different zinc cover thickness on probability of tensile failure of longitudinal members is detectable at the end of design life. However, the typical minimum specified zinc thickness of 0.086 mm recommended in North American practice is likely sufficient to keep probabilities of failure to 1% or less (i.e., reliability index of 2.33 or greater). The influence of a range of initial zinc thickness reported in the literature on probability of pullout failure of the same steel grid elements was found to be negligible. [ABSTRACT FROM AUTHOR]

Model checking, evaluation and comparison are critical steps in Bayesian data analysis but not routinely considered in statistical analysis of geotechnical data. This paper presents an overview of model checking and comparison concepts and techniques employed in modern Bayesian data analysis that are useful for analysis of geotechnical engineering data. These methods are applied to statistical analysis of load and pullout model bias data for steel strip mechanically stabilized earth walls. It is shown how model checking can result in statistical models with better fits. The practical implications of using improved statistical models are discussed in the context of reliability-based design and load and resistance factor design calibration of the internal stability pullout limit state. Finally, statistical model uncertainty and Bayesian model averaging are discussed and explored using an example of reliability-based design for the same limit state. [ABSTRACT FROM AUTHOR]

This paper describes a strategy to compute deterministic and probabilistic estimates of maximum outward deformation of mechanically stabilized earth (MSE) modular block wall structures of particular type, materials and boundary conditions. The reference walls in this study were a group of MSE modular block walls constructed under carefully controlled laboratory conditions. A numerical FLAC model was first demonstrated to give reasonable estimates of measured maximum wall deformations. The model was then used to carry out a sensitivity analysis using a modest number of simulations with different values of key input parameters. Six key control parameters were selected based on the results of the sensitivity analysis and a review of the literature on wall performance. A response surface of full quadratic form was used to predict maximum normalized wall deformations as a function of the six key parameters. The results of a second set of 729 numerical simulations corresponding to the maximum number of possible permutations of three different values for each of the six control parameters were used to find the coefficient terms for the response surface equation using a conventional least squares method. The response surface method (RSM) equation was then used to carry out a large number of Monte Carlo simulations using random sampling of distributions for soil friction angle, surcharge pressure, block–block interface shear stiffness and reinforcement stiffness input parameters. The resulting statistics were used to construct a cumulative distribution function (CDF) for maximum normalized wall deformations. The CDF is then used to link reliability performance-based design to a reasonable range of serviceability criteria for wall deformations generated under operational conditions. [ABSTRACT FROM AUTHOR]

The choice of structure element to simulate soil reinforcement and soil–structure interaction details for numerical modelling of mechanically stabilized earth (MSE) walls can have a significant influence on numerical outcomes. Program FLAC (finite difference method) offers three different options (beam, cable and strip element) to model the reinforcement and program PLAXIS (finite element method) has two (beam and geogrid element). Both programs use different models and properties to simulate the mechanical behaviour of the interface between dissimilar materials. The paper describes the details of the linear elastic Mohr–Coulomb interface model available in the two software packages to model material interaction and how to select model parameters to give the same numerical outcomes. The numerical results quantitatively demonstrate the conditions that give good agreement between the two programs for the same steel strip reinforced soil–structure problem and the situations where they do not. For example, the paper demonstrates that results can be very different depending on the type of structure element used to model horizontal reinforcement layers that are discontinuous in the plane-strain direction. [ABSTRACT FROM AUTHOR]