Probabilistic Applications in Geotechnical Engineering

Modern geotechnical design codes are migrating towards Load and Resistance Factor Design (LRFD) methodologies. The Danish geotechnical code has been based on LRFD for several decades, but more recently the Eurocode and the Australian Standards have turned in this direction. Where the geotechnical system supports a structure, the load factors are generally determined by the structural codes. The geotechnical resistance factors, typically determined by calibration with traditional working stress (or allowable stress) design, have yet to be clearly defined in geotechnical design codes. Research into the reliability of geotechnical systems is needed in order for resistance factors to be determined. This paper presents the results of a preliminary study into the effect of a soil's spatial variability on the settlement and ultimate load statistics of a pile. The results are used to provide recommendations on approaches to reliability-based deep foundation design at the serviceability and ultimate limit states.

The site investigation phase of any geotechnical design plays a vital role, where inadequate characterization of the subsurface conditions may contribute to either a significantly over designed solution that is not cost-effective, or an under design, which may lead to potential failures. Although it is intuitive to expect that the financial risk of a design will reduce as the site investigation scope increases (i.e. additional sampling), it is not known to what degree the risk is reduced, nor whether other uncertainties have an impact on this relationship. As such, this paper discusses research to measure the impact of varying the scope of a site investigation, on the financial risk of a foundation design project. The financial risk is defined as the total cost, which includes costs associated with undertaking the site investigation, constructing the foundation and superstructure, and any works required to rehabilitate a foundation failure. The analysis is numerically based, where a foundation design simulation model is incorporated into a Monte Carlo framework, in order to generate expected costs, and a measure of the financial risk. Results indicate that the risk of a foundation design is considerably reduced as the scope of a site investigation increases. However, results also indicate that there is an optimal site investigation expenditure, which leads to the least financial risk, and where additional sampling becomes redundant.

In this paper the use of probability theory in reliability based optimum design of reinforced gravity retaining wall is described. The formulation for computing system reliability index is presented. A parametric study is conducted using advanced first order second moment method (AFOSM) developed by Hasofer-Lind and Rackwitz-Fiessler (HL-RF) to asses the effect of uncertainties in design parameters on the probability of failure of reinforced gravity retaining wall. Totally 8 modes of failure are considered, viz overturning, sliding, eccentricity, bearing capacity failure, shear and moment failure in the toe slab and heel slab. The analysis is performed by treating back fill soil properties, foundation soil properties, geometric properties of wall, reinforcement properties and concrete properties as random variables. These results are used to investigate optimum wall proportions for different coefficients of variation of φ (5% and 10%) and targeting system reliability index (β_t) in the range of 3 – 3.2.

There is a need to consider both ultimate limit state and serviceability limit state uniformly within the reliability-based design framework. However, it is more difficult to characterize uncertainties associated with the latter, because load-displacement curves are nonlinear functions that cannot be reduced to a single random variable. This paper demonstrates that a random vector containing two curve-fitting hyperbolic parameters is an adequate probabilistic model for uplift load-displacement curves associated with spread foundations, drilled shafts, and pressure-injected footings. A simple but general computational approach, based on Hermite polynomials, can be used to simulate these hyperbolic parameters, which typically are correlated and may not follow closed-form classical distributions.

This paper presents a reliability-based approach for the analysis and design of a shallow strip footing subjected to an inclined load. The deterministic model used is based on the upper-bound method of limit analysis. Both the punching and sliding modes of failure are considered. The random variables used in the analysis are the soil shear strength parameters and the vertical and horizontal components of the footing load. The reliability index of each mode and the system failure probability were calculated. It was shown that for small values of the vertical component of the footing load, the sliding mode is dominant. When this vertical component increases, the punching mode becomes more critical. The hypothesis of uncorrelated shear strength parameters was found to be conservative in comparison to the one of negatively correlated variables. The coefficients of variation of only the angle of internal friction and the horizontal footing load have a significant effect on the system failure probability. For design, an iterative procedure is performed to determine the breadth of the footing for a target failure probability.

Cantilever retaining walls have been the most common type of retaining walls. Proper drainage behind cantilever retaining walls is mostly provided and ground water table remains at or below the foundation subgrade during the wall design lifetime. Therefore, it is important to account for the unsaturated conditions of the retained soils in the wall design. AASHTO has recently adopted Load and Resistance Factor Design (LRFD) method for the design of retaining walls. In this paper, an external stability including sliding, overturning, and soil bearing for a typical reinforced concrete cantilever wall retaining unsaturated soils is evaluated using AASHTO LRFD method and incorporating the unsaturated soil mechanics theory in estimating the lateral earth pressures acting on the wall. The paper also shows that matric suction, an independent stress variable of unsaturated soils, is a source of resistance in the retaining wall design using AASHTO LRFD and results in more economic design.

An approach is developed to locally estimate the failure probability of a system under various design values by using simple Monte Carlo simulation. Although it seems to require numerous reliability analysis runs to locally estimate the failure probability function, which is a function of the design variables, the approach only requires simple Monte Carlo simulations. It is also possible to find the confidence interval of the failure probability function as well as estimate the gradient of the logarithm of that function with respect to the design variables. The use of the new approach is demonstrated with two simulated examples. The results show that the new approach can effectively locally estimate the entire failure probability function and the confidence interval. The approach should be valuable for reliability-based design and reliability sensitivity analysis.

Paleoliquefaction features can be used to estimate lower bounds on the magnitude and ground motion associated with the earthquake that caused the liquefaction feature. The engineering back-analysis of paleoliquefaction features is usually conducted using state of the practice liquefaction triggering analysis methodologies. Recent studies have shown that these methodologies are associated with variable probabilities of liquefaction depending on the soil parameters. This would imply that estimates of magnitude and ground motion intensity obtained from these methodologies would not be consistent for all soil sites. Moreover, these estimates could be unconservative. In this paper, the use of a probabilistic methodology for the back-analysis of paleoliquefaction features is proposed. The proposed methodology permits the incorporation of model and parameter uncertainty into the analysis. Moreover, the methodology permits the user to directly account for the likelihood of liquefaction associated with each paleoliquefaction site. Previously published paleoliquefaction data are used to demonstrate the applicability of the proposed method.

A probabilistic framework for simulation of problems with uncertain material properties in geomechanics has been outlined. The methodology is based on spectral stochastic finite element method and on the Eulerian-Lagrangian form of Fokker-Planck-Kolmogorov equation recently developed for probabilistic elasto-plasticity. The proposed methodology is demonstrated on low and high OCR modified Cam Clay models.

Spatial variability in soils gives rise to various phenomena that do not take place in homogeneous media. We report in this document emergent phenomena related to variability in stiffness, strength, conduction and diffusion. Stiffness variability triggers stress focusing along stiffer percolating zones, and alters elastic wave propagation causing ray bending, mode conversion, diffraction healing, and coda. The heterogeneous distribution of strength promotes localized shear failure along interconnected weaknesses. Excess pore water pressure generation and dissipation are highly sensitive to the variability in stiffness and hydraulic conductivity. The effective hydraulic conductivity decreases with increasing variability in k-fields. Spatial correlation plays a secondary role in the absence of high-k percolating paths or low-k transverse seams. Most analytical closed-form solutions for effective stiffness, strength, conduction or diffusion are based on volume fractions and fail to capture the spatial distribution and correlation that are inherent in natural sediments.

Spatial coherency is estimated from seismic data recorded at the unique Parkway array in the Wainuiomata Valley, New Zealand, a small alluvial valley surrounded by greywacke outcrops. This dense array consisted of stations on both the sediment basin and the surrounding rock, with station separation distances pertinent for earthquake engineering applications. It is shown that coherency at the site is generally lower than that at flat alluvial sites, and depends on the type od wave dominating the motions, the component (E-W or N-S) of the motions, and the station pair orientation. These observations can have a significant effect on the seismic response evaluation of lifelines, especially bridges, constructed at sites with irregular subsurface topography.

The general stochastic problem involves the propagation of input uncertainties through a computation model to arrive at a random output vector. This paper presents the application of the multi-dimensional Hermite polynomials to reduce an unknown random output vector into a significantly simpler unknown vector of numbers. The unknown numbers are evaluated using a collocation method because it has the important practical advantage of allowing existing deterministic numerical codes to be used as "black boxes". A simple laterally loaded pile example involving two input random variables demonstrated that a third- or fourth-order Hermite expansion is adequate to reproduce probabilities of failure between 10^–3 and 10^–4. A simple and efficient 2-term recurrence method for obtaining Hermite polynomials of any order in the case of two random dimensions is proposed. To our knowledge, this proposal appears to be original.

Estimating potential construction-induced building damage is critical for underground construction projects. "Angular distortion" and "deflection ratio", sometimes coupled with horizontal strain, have been used as criteria for assessing settlement-induced damage but these do not reflect the diversity of urban structures or various modes of displacement that contribute to damage. Neither do they support a rational and probabilistic approach to risk management. Choosing applicable underground construction methods necessitates an examination of risks to surface facilities. Assumptions regarding complex variables, including ground conditions, construction technologies, and construction workmanship can influence decisions of significant technical and economical consequence. By combining construction-induced ground deformation patterns, well-known damage category criteria, strain superposition and critical strain concepts, and stochastic modeling techniques, the potential effects of ground deformations on buildings and their root causes can be better understood allow for more informed decision-making.

This paper presents an overview of methods that can be used to predict damage to buildings as a result of excavation-induced ground movements. To make such a prediction, estimates are needed of the distributions of ground movements at the support level of a building foundation. These estimates can be made based on semi-empirical methods or results of finite element analyses. Semi-empirical methods that describe ground movement distributions both perpendicular and parallel to an excavation wall are summarized. Key issues related to the ability of finite element results to adequately predict such ground movement distributions are discussed. Damage predictions also must consider the type of structure that is impacted by the ground movements. In urban areas, one usually must evaluate the impact on the architectural finishes of structures or load bearing walls supported on shallow foundations. Methods to evaluate these impacts are reviewed and evaluated in light of their applicability.

As part of the investigation of the serviceability reliability of a braced excavation system, KJHH model has recently been developed for estimating wall deflection and ground surface settlement in the braced excavation. In this study excavation-induced settlement determined by the KJHH model is used as an indicator to assess the serviceability reliability of adjacent buildings. The calculated settlement, considered as the load in the context of reliability analyses in this study, is compared with the tolerable limits of settlement proposed by Skempton and MacDonald (1956), considered as the resistance herein. The reliability of the building against "damage" (defined herein as the violation of serviceability requirements, or specifically in this case, whenever resistance is less than load) is analyzed during the design phase. Further, with the proceeding of excavation during the construction, observed settlement is used to update the KJHH model, and the predictions of settlement at subsequent stages are updated accordingly.

Load-displacement analysis of a single deep foundation element can be accomplished by utilizing a soil-structure interaction model, such as the "t-z" model. By combining the soil-structure interaction model with a probabilistic analysis technique, methods to rationally incorporate design variability can be provided. The service limit state load capacity of a single deep foundation element can then be computed utilizing the probabilistic results and an allowable total head displacement. However, in design, differential settlement between individual foundation elements is often the event of interest. This paper develops a reliability-based design methodology for differential settlement of deep foundations.

The serviceability limit state for excavation-induced building damage, interpreted from the chart and database developed by Son and Cording, is analyzed probabilistically herein with the application of Bayesian mapping and reliability methods. The damage is referred to herein as the violation of serviceability requirements such as exceeding the threshold principal strain. The probability of damage is obtained using both reliability methods and a Bayesian mapping function. The Bayesian mapping function is established empirically based on the distributions of principal strains in the cases examined by Son and Cording, and is used to calibrate the uncertainty of the limit state model. With known model and parameter uncertainties, reliability analysis using the first order reliability method can be performed and the probability of damage can be determined. Criteria for interpreting the calculated probability of damage are presented.

To develop reliability-based serviceability criteria for foundation design, the probability distributions of limiting tolerable displacements should be available. In this study, displacement information of 380 buildings, including .124 cases from Hong Kong and Mainland China, is collected for analysis. The background information of these buildings is described. The buildings are divided into four categories for study: soil type, foundation type, structural type, and building usage. Based on the observed performance of these buildings, distributions of the limiting tolerable settlement and angular distortion for each building category are established following a probability-based procedure. The statistics of the limiting tolerable settlement and angular distortion are obtained. It is found that the limiting tolerable displacements of structures of different categories differ significantly. The tolerable displacement distributions can be used directly for full probability-based design or for determining allowable displacements for limit-state or conventional design.

The paper contrasts results obtained by the traditional First Order Reliability Method (FORM) and a more advanced Random Finite Element Method (RFEM) in a benchmark problem of slope stability analysis with random shear strength parameters. The key difference between the methods is that RFEM takes into account spatial correlation in a rigorous way allowing slope failure to occur naturally along the path of least resistance. Both methods lead to predictions of the "probability of slope failure" as opposed to the more traditional "factor of safety" measure of slope safety, however they give significant different results depending on the value of the correlation length. For small correlation lengths FORM is generally conservative, however it is shown that there is a "worst case" correlation length for which FORM leads to unconservative predictions of slope reliability.

The presence of cracks can significantly decrease the stability of slopes. Yet the capability of predicting the effect of cracks on slope performance is still limited. This paper studies the hydraulic properties of random cracks in soil. First, the statistical parameters for the geometry of cracks in soil are reviewed. Next, a method of generating a random crack network based on the statistical parameters is presented. Then water flow through the generated random crack network is studied through a numerical model. Finally the hydraulic properties of the random crack network are investigated. Results show that a random crack network consisting of two orthogonal sets of cracks behaves like an isotropic porous medium as expected when the two sets of cracks have similar mean length, mean aperture and density. If the relative orientation is not orthogonal, then the permeability of the random crack network will be anisotropic. The anisotropic ratio increases as the relative orientation decreases. The directional permeability increases with the mean crack aperture and length as expected. On the other hand, the changes in aperture do not affect the shape of permeability ellipses and the equivalent porous medium assumption.

Recent events in New Orleans have shown the magnitude of life loss and property damage that can occur due to the failure of man-made levees. The California Bay Delta and Sacramento levee systems in California pose a similar or greater level of risk to life and property. In order to effectively mitigate the hazard associated with levee failure a systematic evaluation of risk must be carried out. This paper presents preliminary research into the risk associated with the California Bay Delta. A comprehensive list of failure modes for man-made levees is presented. Preliminary empirical data on the temporal frequency of failure and the consequences of failure in the Bay Delta has been compiled. Also presented is the frequency of high water conditions, frequency of strong ground shaking, and a discussion of possible correlation of failure with these loading conditions. Based on preliminary empirical data, the distribution of the risk function is estimated using Monte Carlo simulations. The general objectives of this paper are to present an approach to levee risk analysis for the California Bay Delta, stimulate discussion, outline the data gaps that exist, and push for continued research on mitigating this hazard.

A 1:25,000 scale topographic map of Maki Village of Japan was digitized. Based on the Digital Elevation Model (DEM) of 2 m cell size that was prepared with the digitized contours, spatial distribution of slope gradient was calculated for the entire village. About 70% of the slopes in the Village are formed in tertiary mudstone, having varying degree of weathering. Soil samples were collected from 150 locations of different geological formations and degree of weathering. Residual shear strength, plasticity limits, USCS classification, chemical composition, and mineralogical composition of the soil specimen were measured. A distribution map of existing shallow landslides was also prepared for the entire area. Based on the average dimensions of the shallow slides, and the "layers" of other slope stability parameters, 3D slope stability analysis was done covering the entire village. Assuming a safety factor of unity for the existing slides, the proportion of mobilized shear strength to the residual shear strength was back calculated. Those proportions were used to predict the hazard potential.