Probabilistic analyses of a strip footing on horizontally stratified sandy deposit using advanced constitutive model

An advanced hypoplastic constitutive model is used in probabilistic analyses of a typical geotechnical problem, strip footing. Spatial variability of soil parameters, rather than state variables, is considered in the study. The model, including horizontal and vertical correlation lengths, was calibrated using a comprehensive set of experimental data on sand from horizontally stratified deposit. Some parameters followed normal, whereas other followed lognormal distributions. Monte-Carlo simulations revealed that the foundation displacement u_y for a given load followed closely the lognormal distribution, even though some model parameters were distributed normally. Correlation length in the vertical direction θ_v was varied in the simulation. The case of infinite correlation length was used for evaluation of different approximate probabilistic methods (first order second moment method and several point estimate methods). In the random field Monte-Carlo analyses with finite θ_v, the vertical correlation length was found to have minor effect on the mean value of u_y, but significant effect on its standard deviation. As expected, it decreased with decreasing θ_v due to spatial averaging of soil properties.     

Probabilistic analysis of strip footings resting on a spatially random soil using subset simulation approach

The failure probability of geotechnical structures with spatially varying soil properties is generally computed using Monte Carlo simulation (MCS) methodology. This approach is well known to be very time-consuming when dealing with small failure probabilities. One alternative to MCS is the subset simulation approach. This approach was mainly used in the literature in cases where the uncertain parameters are modelled by random variables. In this article, it is employed in the case where the uncertain parameters are modelled by random fields. This is illustrated through the probabilistic analysis at the serviceability limit state (SLS) of a strip footing resting on a soil with a spatially varying Young's modulus. The probabilistic numerical results have shown that the probability of exceeding a tolerable vertical displacement (P _e) calculated by subset simulation is very close to that computed by MCS methodology but with a significant reduction in the number of realisations. A parametric study to investigate the effect of the soil variability (coefficient of variation and the horizontal and vertical autocorrelation lengths of the Young's modulus) on P _e was presented and discussed. Finally, a reliability-based design of strip footings was presented. It allows one to obtain the probabilistic footing breadth for a given soil variability.     

Seismic hazard evaluation for dams in northern Colorado, U.S.A.

A seismic hazard evaluation for three dams in the Rocky Mountains of northern Colorado is based on a study of the historical seismicity. To model earthquake occurrence as a random process utilizing a maximum likelihood method, the catalog must exhibit random space-time characteristics. This was achieved using a declustering procedure and correction for completeness of recording. On the basis of the resulting a- and b-values, probabilistic epicentral distances for a 2 × 10^–5 annual probability were calculated. For a random earthquake of magnitude M _L 6.0–6.5, this distance is 15 km. Suggested ground motion parameters were estimated using a probabilistic seismic hazard analysis. Critical peak horizontal accelerations at the dams are 0.22g if median values are assumed and 0.39g if variable attenuation and seismicity rates are taken into account. For structural analysis of the dams, synthetic acceleration time series were calculated to match the empirical response spectra. In addition, existing horizontal strong motion records from two Mammoth Lakes, California earthquakes were selected and scaled to fit the target horizontal acceleration response spectra.     

Response of Vertically Loaded Pile in Clay: A Probabilistic Study

This study presents the response of a vertically loaded pile in undrained clay considering spatially distributed undrained shear strength. The probabilistic study is performed considering undrained shear strength as random variable and the analysis is conducted using random field theory. The inherent soil variability is considered as source of variability and the field is modeled as two dimensional non-Gaussian homogeneous random field. Random field is simulated using Cholesky decomposition technique within the finite difference program and Monte Carlo simulation approach is considered for the probabilistic analysis. The influence of variance and spatial correlation of undrained shear strength on the ultimate capacity as summation of ultimate skin friction and end bearing resistance of pile are examined. It is observed that the coefficient of variation and spatial correlation distance are the most important parameters that affect the pile ultimate capacity.     

Finite element analysis of consolidation of layered clay soils using an elastic visco-plastic model

This paper presents a general one-dimensional (1-D) finite element (FE) procedure for a highly non-linear 1-D elastic visco-plastic (1-D EVP) model proposed by Yin and Graham for consolidation analysis of layered clay soils. In formulating the 1-D FE procedure, a trapezoidal formula is used to avoid the unsymmetry of the stiffness matrix for a Newton (modified Newton) iteration scheme. Unlike many other 1-D FE approaches in which the initial in situ stresses (or stress/strain states) are considered indirectly or even not considered, the initial in situ stress/strain states are taken into account directly in this paper. The proposed FE procedure is used for analysis of 1-D consolidation of a clay with published test results in the literature. The FE modelling results are in good agreement with the measured results. The FE model and procedure is then used to analyse the consolidation of a multi-layered clay soils with a parametric study on the effects of the variations of creep parameters in Yin and Graham's 1-D EVP model. It is found that the creep parameters ψ/V and t₀ have significant influence on the compression and porewater pressure dissipation. For some boundary conditions, changes of parameters in one layer will have some effects on the consolidation behaviour of another layer due to the different consolidation rates. Finally, the importance of initial stress/strain states is illustrated and discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Magnitude conversion problem for the Turkish earthquake data

Earthquake catalogues which form the main input in seismic hazard analysis generally report earthquake magnitudes in different scales. Magnitudes reported in different scales have to be converted to a common scale while compiling a seismic data base to be utilized in seismic hazard analysis. This study aims at developing empirical relationships to convert earthquake magnitudes reported in different scales, namely, surface wave magnitude, M _S, local magnitude, M _L, body wave magnitude, m _b and duration magnitude, M _d, to the moment magnitude (M _w). For this purpose, an earthquake data catalogue is compiled from domestic and international data bases for the earthquakes occurred in Turkey. The earthquake reporting differences of various data sources are assessed. Conversion relationships are established between the same earthquake magnitude scale of different data sources and different earthquake magnitude scales. Appropriate statistical methods are employed iteratively, considering the random errors both in the independent and dependent variables. The results are found to be sensitive to the choice of the analysis methods.     

Analytical theory for one‐dimensional consolidation of clayey soils exhibiting rheological characteristics under time‐dependent loading

This paper presents analytical solutions to the one‐dimensional consolidation problem taking into consideration the rheological properties of clayey soil under variable loadings. A four‐element rheological model is introduced, and different loading types are involved, i.e. constant loading, one‐step loading, triangular loading, rectangular loading, and isosceles–trapezoidal cyclic loading. The differential equations governing consolidation are solved by the Laplace transform. Based on the solutions obtained, the influences of the rheological parameters and loading conditions on the consolidation process are investigated. It has been shown that the consolidation behavior is mainly governed by four dimensionless parameters, a₁, a₂, b, and T_v0. Load shape has a great influence on the rate of consolidation. A decrease either in the modulus of the spring in the Kelvin body or in the viscosity coefficient of independent dashpot will slow down the rate of consolidation. An increase in the viscosity coefficient of the dashpot in the Kelvin body will make the rate of consolidation increase at an early stage but decrease at a later stage. For isosceles–trapezoidal cyclic loading, the consolidation rate in each cycle reaches a maximum at the end of the constant loading phase and the minimum at the end of this cycle. Copyright © 2008 John Wiley & Sons, Ltd.     

Secondary consolidation of clay as an anomalous diffusion process

Secondary consolidation of clay soil is considered as a result of anomalous diffusion of pore water pressure from the micropores to the macropores. By using simplified pore geometry, a heuristic approach allows us to infer the expression of the associated rate of vertical secondary deformation written as a fractional derivative of the pore pressure. The insertion of this expression into the 1D Terzaghi's theory leads to a particular type of time‐fractional diffusion equation of the pore pressure that is solved semi‐analytically. The advantage of such theoretical approach stems from the concise and compact way of treating the secondary consolidation. Only two additional parameters are needed: the fractional order, ν, and the fractional viscosity factor, θ, both accounting for the physicochemical interactions between pore fluid and clay particles. This theoretical approach is tested on experimental data obtained from the Cubzac‐les‐Ponts clay soil intensively studied for secondary consolidation. This application shows a good agreement between the data and the predicted values confirming the interest of the initial assumption and the use of the fractional derivatives formalism. Moreover, good correlations between the inverted fractional parameters and the empirical secondary consolidation index C_α measured independently are obtained: the fractional order ν, if experimentally calibrated, can be used as a reasonable estimator of the slope of the secondary consolidation portion of consolidation curve. Copyright © 2014 John Wiley & Sons, Ltd.     

Sensitive bounding surface constitutive model for structured clay

The main purpose of the paper is to present a relatively simple, yet realistic, constitutive model for simulations of structured sensitive clays. The proposed constitutive model can simulate 1‐D and isotropic consolidation, and drained and undrained shear response of sensitive structured clay. The proposed sensitive bounding surface model is based on concepts from the modified Cam clay model 8 and bounding surface plasticity 27 , with the addition of a simple degradation law. The key material parameters are M, λ, κ, and ν from the modified Cam clay framework, h from the bounding surface framework to model a smoothed elasto‐plastic transition, and ω_v, ω_q, and S_sr to model softening associated with destructuration. The model has separate parameters to model destructuration caused by volumetric strain and deviatoric strain. The model is capable of modeling unusual behavior of strain softening during 1‐D compression (i.e., a reduction of effective stress as void ratio decreases). A good match between test results and the model simulation is demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.     

Consolidation in spatially random unsaturated soils based on coupled flow‐deformation simulation

This paper integrates random field simulation of soil spatial variability with numerical modeling of coupled flow and deformation to investigate consolidation in spatially random unsaturated soil. The spatial variability of soil properties is simulated using the covariance matrix decomposition method. The random soil properties are imported into an interactive multiphysics software COMSOL to solve the governing partial differential equations. The effects of the spatial variability of Young's modulus and saturated permeability together with unsaturated hydraulic parameters on the dissipation of excess pore water pressure and settlement are investigated using an example of consolidation in a saturated‐unsaturated soil column because of loading. It is found that the surface settlement and the pore water pressure profile during the process of consolidation are significantly affected by the spatially varying Young's modulus. The mean value of the settlement of the spatially random soil is more than 100% greater than that of the deterministic case, and the surface settlement is subject to large uncertainty, which implies that consolidation settlement is difficult to predict accurately based on the conventional deterministic approach. The uncertainty of the settlement increases with the scale of fluctuation because of the averaging effect of spatial variability. The effects of spatial variability of saturated permeability k_sat and air entry parameters are much less significant than that of elastic modulus. The spatial variability of air entry value parameters affects the uncertainties of settlement and excess pore pressure mostly in the unsaturated zone. Copyright © 2016 John Wiley & Sons, Ltd.     

Reliability analysis for bearing capacity of surface strip footing using fuzzy finite element method

ABSTRACT

This paper presents the reliability analysis on the basis of the foundation failure against bearing capacity using the concept of fuzzy set theory. A surface strip footing is considered for the analysis and the bearing capacity is estimated using the conventional Finite Element Method (FEM). The spatial variability of the variables is taken into consideration to capture the physical randomness of the soil parameters for an isotropic field. A variation of the probability of failure (P_f) against a varying limiting applied pressure (q) is presented for different Coefficient of Variation (COV) of the variables and different scale of fluctuation (θ). The results reveal that the friction angle of soil (?) is the most influencing parameter among the other variables. Further, the influence of the scale of fluctuation (θ) on the probability of failure (P_f) is also examined. It is observed that for a particular COV of ?, higher value of θ predicts higher P_f whereas, P_f increases as COV of ? increases for a particular θ value. Later, a comparison study is accomplished to verify the viability of the present method and it can be noticed that the present method compares well with the other reliability method (First Order Reliability Method) to a reasonably good extent.     

The consolidation characteristics of an unsaturated compacted soil

The compacted soils are extensively employed as mineral liners or a sealing system, constructed under municipal solid waste and other containment hazardous materials to prevent or minimize the leakage of pollutant liquids and gases into ground water and sublayers. This article presents a detailed experimental study on settlement characteristics, which are required for the construction of a sealing system. For the experimental study where a compacted unsaturated soil taken from Adana in Turkey has been used, a special equipped consolidation cell has been designed and constructed. The matric suction adopted as independent stress-state variables has been controlled by using the axis translation technique. The consolidation tests have been performed under constant matric suction. Furthermore, the pressure plate tests for various net vertical stresses have been conducted. The test results have indicated that the settlement characteristics of a compacted soil as mineral liners and covers are highly affected by matric suction. The water retention capacity of mineral liners, an important factor for hydraulic conductivity, also changes with the vertical load. The compressibility decreases as the matric suction increases. It has been observed that there exists an exponential relationship between the compressive indexes (C _t, D _t) and matric suction.     

Nonlinear consolidation in randomly heterogeneous highly compressible aquitards

Severe land subsidence due to groundwater extraction may occur in multiaquifer systems where highly compressible aquitards are present. The highly compressible nature of the aquitards leads to nonlinear consolidation where the groundwater flow parameters are stress-dependent. The case is further complicated by the heterogeneity of the hydrogeologic and geotechnical properties of the aquitards. The effect of realistic vertical heterogeneity of hydrogeologic and geotechnical parameters on the consolidation of highly compressible aquitards is investigated by means of one-dimensional Monte Carlo numerical simulations where the lower boundary represents the effect of an instant drop in hydraulic head due to groundwater pumping. Two thousand realizations are generated for each of the following parameters: hydraulic conductivity (K), compression index (C _c), void ratio (e) and m (an empirical parameter relating hydraulic conductivity and void ratio). The correlation structure, the mean and the variance for each parameter were obtained from a literature review about field studies in the lacustrine sediments of Mexico City. The results indicate that among the parameters considered, random K has the largest effect on the ensemble average behavior of the system when compared to a nonlinear consolidation model with deterministic initial parameters. The deterministic solution underestimates the ensemble average of total settlement when initial K is random. In addition, random K leads to the largest variance (and therefore largest uncertainty) of total settlement, groundwater flux and time to reach steady-state conditions.     

Probabilistic analysis of rock slope stability and random properties of discontinuity parameters, Interstate Highway 40, Western North Carolina, USA

Probabilistic analysis has been used as an effective tool to evaluate uncertainty so prevalent in variables governing rock slope stability. In this study a probabilistic analysis procedure and related algorithms were developed by extending the Monte Carlo simulation. The approach was used to analyze rock slope stability for Interstate Highway 40 (I-40), North Carolina, USA. This probabilistic approach consists of two parts: analysis of available geotechnical data to obtain random properties of discontinuity parameters; and probabilistic analysis of slope stability based on parameters with random properties. Random geometric and strength parameters for discontinuities were derived from field measurements and analysis using the statistical inference method or obtained from experience and engineering judgment of parameters. Specifically, this study shows that a certain amount of experience and engineering judgment can be utilized to determine random properties of discontinuity parameters. Probabilistic stability analysis is accomplished using statistical parameters and probability density functions for each discontinuity parameter. Then, the two requisite conditions, kinematic and kinetic instability for evaluating rock slope stability, are determined and evaluated separately, and subsequently the two probabilities are combined to provide an overall stability measure. Following the probabilistic analysis to account for variation in parameters, results of the probabilistic analyses were compared to those of a deterministic analysis, illustrating deficiencies in the latter procedure. Two geometries for the cut slopes on I-40 were evaluated, the original 75° slope and the 50° slope which has developed over the past 40 years of weathering.     

A generalized probabilistic approach for slope analysis

Summary A new probabilistic approach is introduced for slope stability analysis, which is general in types of variable distributions and correlations or dependency between variables, and flexible enough to include any adverse impact analysis for blasting vibrations and groundwater conditions.The material strength within a slope area, given in terms of the internal friction angle (ø) and cohesion (c), is randomized in the bivariate joint probability analysis. To be a completely general engineering method, the new probabilistic approach employs the random variable transformation technique: the Hermite model of the Gaussian transformation function, which transforms the experimental histogram of shear strength parameters to the standard Gaussian distribution (=0, ²=1.0).Because a binormal joint probability is analysed on the true probability region projected on the plane of the Gaussian transformed variables, it is an exact solution of slope stability based on the available sample data. No assumption on the shape of the experimental histogram or independency between two random variables is made as in the current probability methods of slope analysis.     

Approximate up-scaling of geo-spatial variables applied to deep foundation design

We present a series of simple approximate methods for up-scaling the cumulative distribution function of spatially correlated variables by using an effective number n _e of independent variables. Methods are based on the property of distribution permanence of the gamma and inverse Gaussian distributions under averaging, bootstrap sampling and expansions about the normal and gamma distributions. A stochastic simulation study is used to validate each method, and simple parameters are defined to identify respective ranges of applicability. A practical example is presented where core sample rock strength data are up-scaled to shaft size for probabilistic (risk-based) deep foundation design. Supplemental material is available online.     

Bearing capacity of spatially random rock masses obeying Hoek–Brown failure criterion

This paper presents a probabilistic analysis to compute the probability density function of the bearing capacity of a strip footing resting on a spatially varying rock mass. The rock is assumed to follow the generalised Hoek–Brown failure criterion. The uniaxial compressive strength of the intact rock (σc) was considered as a random field and the geological strength index was modelled as a random variable. The uncertainty propagation methodology employed in the analysis is the sparse polynomial chaos expansion. A global sensitivity analysis based on Sobol indices was performed. Some numerical results were presented and discussed.     

Quantitative Parameters for Rock Joint Surface Roughness

Summary The morphologies of two artificial granite joints (sanded and hammered surfaces), one artificial regularly undulated joint and one natural schist joint, were studied. The sanded and hammered granite joints underwent 5 cycles of direct shear under 3 normal stress levels ranging between 0.3–4 MPa. The regularly undulated joint underwent 10 cycles of shear under 6 normal stress levels ranging between 0.5–5 MPa and the natural schist replicas underwent a monotonous shear under 5 normal stress levels ranging between 0.4–2.4 MPa. In order to characterize the morphology of the sheared joints, a laser sensor profilometer was used to perform surface data measurements prior to and after each shear test. Rather than describing the morphology of the joints from the single profiles, our characterization is based on a simultaneous analysis of all the surface profiles. Roughness was viewed as a combination of a primary roughness and a secondary roughness. The surface angularity was quantified by defining its three-dimensional mean angle, θ_s, and the parameter Z2_s. The surface anisotropy and the secondary roughness were respectively quantified by the degree of apparent anisotropy, k _a, and the surface relative roughness coefficient, R _s. The surface sinuosity was quantified by the surface tortuosity coefficient, T _s.  Comparison between the means of the classical linear parameters and those proposed shows that linear parameters underestimate the morphological characteristics of the joint surfaces. As a result, the proposed bi-dimensional and tri-dimensional parameters better describe the evolution of the joints initial roughness during the course of shearing.     

Bearing capacity of strip footings on spatially random soils using sparse polynomial chaos expansion

A probabilistic model is presented to compute the probability density function (PDF) of the ultimate bearing capacity of a strip footing resting on a spatially varying soil. The soil cohesion and friction angle were considered as two anisotropic cross‐correlated non‐Gaussian random fields. The deterministic model was based on numerical simulations. An efficient uncertainty propagation methodology that makes use of a non‐intrusive approach to build up a sparse polynomial chaos expansion for the system response was employed. The probabilistic numerical results were presented in the case of a weightless soil. Sobol indices have shown that the variability of the ultimate bearing capacity is mainly due to the soil cohesion. An increase in the coefficient of variation of a soil parameter (c or φ) increases its Sobol index, this increase being more significant for the friction angle. The negative correlation between the soil shear strength parameters decreases the response variability. The variability of the ultimate bearing capacity increases with the increase in the coefficients of variation of the random fields, the increase being more significant for the cohesion parameter. The decrease in the autocorrelation distances may lead to a smaller variability of the ultimate bearing capacity. Finally, the probabilistic mean value of the ultimate bearing capacity presents a minimum. This minimum is obtained in the isotropic case when the autocorrelation distance is nearly equal to the footing breadth. However, for the anisotropic case, this minimum is obtained at a given value of the ratio between the horizontal and vertical autocorrelation distances. Copyright © 2012 John Wiley & Sons, Ltd.