Kok-Kwang Phoon

Implementation of Karhunen -Loeve expansion for simulation using a wavelet-Galerkin scheme

Abstract The feasibility of implementing Karhunen–Loeve (K–L) expansion as a practical simulation tool hinges crucially on the ability to compute a large number of K–L terms accurately and cheaply. This study presents a simple wavelet-Galerkin approach to solve the Fredholm integral equation for K–L simulation. The proposed method has significant computational advantages over the conventional Galerkin method. Wavelet bases provide localized compact support, which lead to sparse representations of functions and integral operators. Existing efficient numerical scheme to obtain wavelet coefficients and inverse wavelet transforms can be taken advantage of solving the integral equation. The computational efficiency of the wavelet-Garlekin method is illustrated using two stationary covariance functions (exponential and squared exponential) and one non-stationary covariance function (Wiener–Levy). The ability of the wavelet-Galerkin approach to compute a large number of eigensolutions accurately and cheaply can be exploited to great advantage in implementing the K–L expansion for practical simulation.

Reliability-Based Design in Geotechnical Engineering : Computations and Applications

1. Numerical recipes for reliability analysis - a primer Kok-Kwang Phoon 2. Spatial variability and geotechnical reliability Gregory B. Baecher and John T Christian 3. Practical reliability approach using spreadsheet Bak-Kong Low 4. Monte Carlo simulation in reliability analysis Yusuke Honjo 5. Practical application of reliability-based design in decision making R. B. Gilbert, S. Najjar, Y. J. Choi and S. J. Gambino 6. Randomly heterogeneous soils under static and dynamic loads Radu Popescu, George Deodatis and Jean-Herve Prevost 7. Stochastic finite element methods in geotechnical engineering Bruno Sudret and Marc Berveiller 8. Eurocode 7 and reliability-based design Trevor L. L. Orr and Denys Breysse 9. Serviceability limit state reliability-based design Kok-Kwang Phoon and Fred H Kulhawy 10. Reliability verification using pile load tests Limin Zhang 11. Reliability analysis of slopes Tien H. Wu 12. Application to levees and hydraulic structures Thomas F. Wolff 13. Reliability analysis of liquefaction potential of soils using Standard Penetration Test C. Hsein Juang, Sunny Ye Fang, and David Kun Li

Simulation of second-order processes using Karhunen–Loeve expansion

Abstract A unified and practical framework is developed for generating second-order stationary and non-stationary, Gaussian and non-Gaussian processes with a specified marginal distribution function and covariance function. It utilizes the Karhunen–Loeve expansion for simulation and an iterative mapping scheme to fit the target marginal distribution function. The proposed method has three main advantages: (a) processes with Gaussian-like marginal distribution can be generated almost directly without iteration, (b) distributions that deviate significantly from the Gaussian case can be handled efficiently and (c) non-stationary processes can be generated within the same unified framework. Four numerical examples are used to demonstrate the validity and convergence characteristics of the proposed algorithm. Based on these examples, it was shown that the proposed algorithm is more robust and general than the commonly used spectral representation method.

Efficient System Reliability Analysis of Slope Stability in Spatially Variable Soils Using Monte Carlo Simulation

Abstract Monte Carlo simulation (MCS) provides a conceptually simple and robust method to evaluate the system reliability of slope stability, particularly in spatially variable soils. However, it suffers from a lack of efficiency at small probability levels, which are of great interest in geotechnical design practice. To address this problem, this paper develops a MCS-based approach for efficient evaluation of the system failure probability P f , s of slope stability in spatially variable soils. The proposed approach allows explicit modeling of the inherent spatial variability of soil properties in a system reliability analysis of slope stability. It facilitates the slope system reliability analysis using representative slip surfaces (i.e., dominating slope failure modes) and multiple stochastic response surfaces. Based on the stochastic response surfaces, the values of P f , s are efficiently calculated using MCS with negligible computational effort. For illustration, the proposed MCS-based system reliab...

Efficient Evaluation of Reliability for Slopes with Circular Slip Surfaces Using Importance Sampling

Evaluating the reliability of a slope is a challenging task because the possible slip surface is not known beforehand. Approximate methods via the first-order reliability method provide efficient ways of evaluating failure probability of the “most probable” failure surface. The tradeoff is that the failure probability estimates may be biased towards the unconservative side. The Monte Carlo simulation (MCS) is a viable unbiased way of estimating the failure probability of a slope, but MCS is inefficient for problems with small failure probabilities. This study proposes a novel way based on the importance sampling technique of estimating slope reliability that is unbiased and yet is much more efficient than MCS. In particular, the critical issue of the specification of the importance sampling probability density function will be addressed in detail. Three examples of slope reliability will be used to demonstrate the performance of the new method.

Characterization of Model Uncertainty in the Static Pile Design Formula

Level 1 reliability methods have been internationally accepted as the basis for development of the new generation of geotechnical design codes. A key requirement of this design approach is the identification and quantification of uncertainties associated with the geotechnical design under consideration. This paper presents four load test databases from South Africa for driven piles in noncohesive soils (29 tests), bored piles in noncohesive soils (33 tests), driven piles in cohesive soils (59 tests), and bored piles in cohesive soils (53 tests). The capacity model factor is defined as the ratio of the interpreted capacity (Chin-Davisson approach) and the predicted capacity (static pile design formula). The uncertainty in the capacity model factor is modeled as a lognormal random variable. The model factor statistics reported in this study are required for reliability-based ultimate limit state design. The uncertainty in the load-settlement behavior is characterized by fitting measured load-settlement data to a hyperbolic equation and then normalizing the hyperbolic curve with the interpreted capacity. The resulting exercise reduces uncertainties in a set of nonlinear continuous curves to uncertainties in two hyperbolic curve-fitting parameters. This approach is practical and grounded realistically on the load test database with minimal assumptions. The hyperbolic parameter statistics reported in this study are required for reliability-based serviceability limit state design.

Reducing shear strength uncertainties in clays by multivariate correlations.

Quantifications of uncertainties in soil shear strengths, including undrained shear strength of clay, are essential for geotechnical reliability-based design. In particular, how to reduce the uncertainties in undrained shear strengths based on all available information by correlation is a practical research subject, given the considerable cost of a typical site investigation. Although it is simple to reduce the uncertainties by correlation when the information is one dimensional (or univariate), it is quite challenging to reduce the uncertainties by using multivariate information through multiple correlations. This study proposes a systematic way of achieving multivariate correlations on undrained shear strengths. A set of simplified equations are obtained through Bayesian analysis for the purpose of reducing uncertainties: the inputs to the equations are the results of in situ or laboratory tests and the outputs are the updated mean values and coefficients of variation (c.o.v.s) of the undrained shear st...

Probabilistic Analysis of Soil-Water Characteristic Curves

Direct measurement of the soil-water characteristic curve (SWCC) is costly and time consuming. A first-order estimate from statistical generalization of experimental data belonging to soils with similar textural and structural properties is useful. A simple approach is to fit the data with a nonlinear function and to construct an appropriate probability model of the curve-fitting parameters. This approach is illustrated using sandy clay loam, loam, loamy sand, clay, and silty clay data in Unsaturated Soil Database. This paper demonstrates that a lognormal random vector is suitable to model the curve-fitting parameters of the SWCC. Other probability models using normal, gamma, Johnson, and other distributions do not provide better fit than the proposed lognormal model. The engineering impact of adopting a probabilistic SWCC is briefly discussed by studying the uncertainty of unsaturated shear strength due to the uncertainty of SWCC.

Validation of a New 2D Failure Mechanism for the Stability Analysis of a Pressurized Tunnel Face in a Spatially Varying Sand

A new two-dimensional 2D limit analysis failure mechanism is presented for the determination of the critical collapse pressure of a pressurized tunnel face in the case of a soil exhibiting spatial variability in its shear strength parameters. The proposed failure mechanism is a rotational rigid block mechanism. It is constructed in such a manner to respect the normality condition of the limit analysis theory at every point of the velocity discontinuity surfaces taking into account the spatial variation of the soil angle of internal friction. Thus, the slip surfaces of the failure mechanism are not described by standard curves such as log-spirals. Indeed, they are determined point by point using a spatial discretization technique. Though the proposed mechanism is able to deal with frictional and cohesive soils, the present paper only focuses on sands. The mathematical formulation used for the generation of the failure mechanism is first detailed. The proposed kinematical approach is then presented and validated by comparison with numerical simulations. The present failure mechanism was shown to give results in terms of critical collapse pressure and shape of the collapse mechanism that compare reasonably well with the numerical simulations at a significantly cheaper computational cost. DOI: 10.1061/ASCEEM.1943-7889.0000196 CE Database subject headings: Tunnels; Limit analysis; Failures; Shear strength; Parameters. Author keywords: Tunnels; Active pressure; Limit analysis; Spatial variability; Local weakness.

Reliability-based design for transmission line structure foundations

Abstract This paper presents one of the geotechnical initiatives in reliability-based code development that has been sponsored by the Electric Power Research Institute for transmission line structure foundations. The framework for the development of a practical reliability-based design approach is illustrated using the design of drilled shafts (bored piles) for uplift under undrained loading. A target reliability index of 3.2 is selected based on the reliability indices implied by existing working stress designs. Two simple design formats (load and resistance factor design and multiple resistance factor design) are rigorously calibrated using the first-order reliability method to produce designs that achieve a known level of reliability consistently.