The geological and geotechnical design model in tunnel design: estimation of its reliability through the R-Index

ABSTRACT

In this paper recent improvements of the R-Index method are presented, based on its application on several projects in various geological and geotechnical contexts. The R-Index derives from a probabilistic procedure conceived for estimating the reliability level of the Geological and Geotechnical Design Model used to design underground structures, especially tunnels. The R-Index takes into account the geological complexity of a site and recommended empirical scores (based on expert judgement) for different quality levels of geological surveys and geotechnical and geophysical investigations.     

Role of reliability calculations in geotechnical design

ABSTRACT

This paper adds to the ongoing discussion on the role of reliability calculations in geotechnical design. It situates design calculations, be it verified by a global factor of safety, partial factors, or reliability-based design (RBD), in a larger context of quality management over the life cycle of the structure. It clarifies that uncertainties amenable to probabilistic treatment typically fall under the category of “known unknowns” where some measured data and/or past experience exist for limited site-specific data to be supplemented by both objective regional data and subjective judgement derived from comparable sites elsewhere. Within this category, reliability is very useful in handling complex real-world information (multivariate correlated data) and information imperfections (scarcity of information or incomplete information). It is also very useful in handling real-world design aspects such as spatial variability that cannot be easily treated using deterministic means. Examples are presented to illustrate how reliability calculations could relieve engineering judgement from the unsuitable task of performance verification in the presence of uncertainties so that the engineer can focus on setting up the right lines of scientific investigation, selecting the appropriate models and parameters for calculations, and verifying the reasonableness of the results.     

Defining and selecting characteristic values of geotechnical parameters for designs to Eurocode 7

ABSTRACT

Recent geotechnical design codes (such as Eurocode 7) aim to achieve geotechnical designs with an appropriate target reliability by applying partial factors to characteristic parameter values. Development in Eurocode 7 of the definition and selection of the characteristic values is traced. A significant development occurred with the definition of the characteristic value as a cautious estimate of the value affecting the occurrence of the limit state. Statistical equations to select characteristic values are presented, including one proposed for the revised of Eurocode 7.An alternative equation reduces the amount of subjective judgement and is more likely to achieve the target reliability.     

Probabilistic geotechnical analyses for offshore facilities

Society requires increasingly that the hazard and risk associated with engineered constructions be quantified. The current paper presents geotechnical hazard assessment in the context of a risk framework. Concepts of uncertainties, reliability, safety and risk are briefly reviewed. The use of the approach is exemplified for offshore facilities, including piled foundations, jack-up structures, gravity foundations and underwater slopes. The applications demonstrate that probabilistic analyses complement the conventional deterministic safety factor and deformation-based analyses, and contribute to achieving a safe and optimum design. The probabilistic approach adds value to the results with a modest additional effort. The conclusions emphasize the usefulness of a risk assessment, the importance of engineering judgement in the assessment and the need for involving multi-disciplinary competences to achieve reliable estimates of hazard and risk. The profession can only gain by implementing probabilistic-based thinking and risk-based approaches more systematically than before.     

Probabilistic comparative analysis between design rules of a shallow foundation safety

ABSTRACT

A fact that is generally overlooked in many geotechnical uncertainty analyses is that input data of the model may be correlated. While this correlation may influence the system response, epistemic uncertainties i.e. lack of knowledge of this correlation appears as a risk factor. This paper discusses how a negative correlation between cohesion (c’) and friction angle (Ø’) with their associated uncertainties can influence both the bearing resistance of a shallow strip foundation footing and the estimation of its safety. A probabilistic approach that considers both the negative correlation and the uncertainty is used in this work as a reference. This method is compared to Eurocode 7 variants that do not for the correlation. These variants, resistance and material factoring methods appears to be more or less conservative depending on the negative correlation degree between (c’–Ø), their associated uncertainties and soil configurations. Finally, the proposed probabilistic comparison shows that the material factoring method is more conservative than the resistance one.     

Probabilistic assessment of soil liquefaction considering spatial variability of CPT measurements

The conventional liquefaction potential assessment methods (also known as simplified methods) profoundly rely on empirical correlations based on observations from case histories. A probabilistic framework is developed to incorporate uncertainties in the earthquake ground motion prediction, the cyclic resistance prediction, and the cyclic demand prediction. The results of a probabilistic seismic hazard assessment, site response analyses, and liquefaction potential analyses are convolved to derive a relationship for the annual probability and return period of liquefaction. The random field spatial model is employed to quantify the spatial uncertainty associated with the in-situ measurements of geotechnical material.     

Effect of sampling plan and trend removal on residual uncertainty

ABSTRACT

The ground is one of the most highly variable of all engineering materials. As a result, geotechnical designs depend upon a site investigation to estimate the ability of the ground to perform acceptably. For example, when a shallow foundation is being proportioned to avoid a bearing capacity failure under a certain applied load, the frictional and cohesive properties of the ground under the foundation must first be estimated through a site investigation. Questions which arise are: How does the quality and intensity of the site investigation affect the design? Is more investigation cost effective? If the site is sampled at one location and the foundation placed at a different location, how does this mismatch affect the target design and the reliability of the final foundation? By modelling the ground as a spatially variable material, questions such as the above can be investigated through Monte Carlo simulation and sometimes theoretical probabilistic models. Using such tools, this paper looks specifically at how the intensity (frequency and spatial distribution) of a site sampling plan, and how the samples are used, affects the understanding of the ground properties under a foundation. Interestingly, it is found that removing the sample mean outperforms removing the best linear unbiased estimate (BLUE) when the actual field correlation length is small but the BLUE correlation length is assumed equal to the field size. Recommendations are made regarding number of samples and the type of trend to best characterise the field.

Abbreviations: BLUE: best linear unbiased estimate; MCS: Monte Carlo simulation; LAS: local average subdivision     

Genetic algorithms in probabilistic finite element analysis of geotechnical problems

In application to numerical analysis of geotechnical problems, the limit-state surface is usually not known in any closed form. The probability of failure can be assessed via the so-called reliability index. A minimization problem can naturally be formed with an implicit equality constraint defined as the limit-state function and optimization methods can be used for such problems. In this paper, a genetic algorithm is proposed and incorporated into a displacement finite element method to find the Hasofer–Lind reliability index. The probabilistic finite element method is then used to analyse the reliability of classical geotechnical systems. The performance of the genetic algorithm (GA) is compared with simpler probability methods such as the first-order-second-moment Taylor series method. The comparison shows that the GA can produce the results fairly quickly and is applicable to evaluation of the failure performance of geotechnical problems involving a large number of decision variables.     

Application of the dynamic bounds method in the safety assessment of flood defences,a case study: 17th Street flood wall,New Orleans

In this paper we provide a computational framework for evaluation of reliability and safety assessment of infrastructures. It is based on the combined application of the dynamic bounds (DB) method and a probabilistic finite element model (FEM). The DB improves the computational efficiency of the FEM when calculating time-dependent failure analyses of coastal and offshore structures, and can speed up the simulation process by several orders of magnitude.

Our approach is demonstrated here for an example problem, and shown to be the most efficient method in applications with a limited number of influential variables, which is true for geotechnical and coastal flood defence systems. It is applied to the 17th Street flood wall, a failing component of the flood defence system in New Orleans during Hurricane Katrina. The variation in soil parameters is a critical input in the reliability estimation of this structure, and the calculated probability of failure depends on these assumed values.     

Full probabilistic design of slopes in spatially variable soils using simplified reliability analysis method

ABSTRACT

A simplified reliability analysis method is proposed for efficient full probabilistic design of soil slopes in spatially variable soils. The soil slope is viewed as a series system comprised of numerous potential slip surfaces and the spatial variability of soil properties is modelled by the spatial averaging technique along potential slip surfaces. The proposed approach not only provides sufficiently accurate reliability estimates of slope stability, but also significantly improves the computational efficiency of soil slope design in comparison with simulation-based full probabilistic design. It is found that the spatial variability has considerable effects on the optimal slope design.     

β分布的参数确定及其在岩土工程中的应用

对岩土工程中随机变量的空间概率特征进行了统计分析,介绍了确定β分布各参数的迭代方法,提出了经验公式简化迭代过程,有效地处理了随机变量分布范围的估计问题,并阐明了β分布在可靠度领域中的适用性。作为对比,对同一样本采用了其他分布进行拟合。分析结果表明,β分布拟合精度高于其他分布。     

Reliability based assessment of axial pile bearing capacity: static analysis,SPT and CPT-based methods

ABSTRACT

Since piles are one of the major geotechnical foundation systems, estimation of their axial bearing capacity is of great importance. Employing different design methods, resulting in a wide range of bearing capacity estimations, complicates the selection of an appropriate design scheme and confirms the existence of model error along with the inherent soil variability in bearing capacity prediction. This paper tends to evaluate different predictive methods in Reliability-Based Design (RBD) framework. In this regard, different static analyses, SPT and CPT-based methods are considered to evaluate which approaches collectively and which method individually, have more reliable predictions for compiled data bank. In order to assess reliability indices and resistance factors, two approaches have been considered, i.e. First Order Second Moment method (FOSM) and First Order Reliability Method (FORM). To investigate the reliability indices for different methods in both RBD approaches, various safety factors and loading ratios have been considered. Also, the Load and Resistance Factor Design (LRFD) resistance factors are calibrated for different target reliability indices and loading ratios. Results show that CPT-based methods are more reliable among other methods. Furthermore, the estimated efficiency ratio, i.e. the ratio of resistance factor to resistance bias factor, confirms this agreement.     

Influence of random heterogeneity of the friction angle on bearing capacity factor Nγ

ABSTRACT

Probabilistic methods in geotechnical engineering have received a lot of attention during the last decade and different methodologies are used to capture the inherent variability of soil in different geotechnical engineering problems. In this paper, numerical simulations are conducted to obtain the bearing capacity factor, N_γ, for a purely frictional heterogenous soil where the friction angle is modelled as randomly distributed throughout the domain and the effect of its spatial variability on N_γ is investigated. A finite element method, based on the upper bound limit analysis was combined with random field theory and linear programming to develop a probabilistic analysis. Monte Carlo simulations were performed and the effect of the variability of the friction angle defined by statistical parameters on the bearing capacity factor was investigated. Results show that the mean bearing capacity factor N_γ of a footing on a spatially variable cohesionless soil is generally higher than the deterministic N_γ obtained from a constant mean value. Increasing the heterogeneity of the friction angle by an increase in the coefficient of variation generally increases this deviation. This can be explained by the nonlinearity of the relationship between N_γ and the friction angle.     

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.     

Uncertainty and Reliability Analysis Applied to Slope Stability: A Case Study From Sungun Copper Mine

Nowadays, there are many new methods for slope stability analysis; including probabilistic methods assessing geotechnical uncertainties to develop safety factors. In this paper, a reliability index analysis for the Sungun copper mine slope stability is evaluated based on three methods of uncertainties consisting Taylor series method, Rosenblueth point estimate method and Monte-Carlo simulation method. Sungun copper mine will be one of the Iran’s biggest mines with final pit’s height of 700 meters. For this study two of its main slopes were assessed, one dipping to the NE (030) and the other to the SE (140). Probability density function of cohesion and angle of friction for the slopes were developed using limit equilibrium methods. These shear strengths were then used to determine the probability density function of safety factor and reliability index using the probabilistic methods. Results of the probabilistic analysis indicate that with ascending values of the uncertainties the reliability index decreases. Furthermore, it was determined that with the Monte Carlo simulation the seed number used has little effect on the reliability index of the safety factor especially with seed numbers in excess of 1200. Variations in the overall reliability index of safety factor were observed between the two slopes and this difference is explained by the differences in complexities of the geology within the cross-section.     

水泥土支护体稳定的可靠度分析

以土层剪切强度指标为基本变量, 对水泥土支护体稳定性设计内容建立了可靠度分析方法, 依据上海软土地区 21 例地质资料, 利用一次二阶矩验算点法( JC 法)对有关的失效模式进行了可靠度指标的核算与分析, 探讨了基本变量的敏感性。     

Quasi-Stochastic Analysis of Uncertainty for Modelling Structurally Controlled Failures

In one approach to predicting the behaviour of rock masses, effort is being devoted to the use of probabilistic methods to model structures interior to a rock mass (sometimes referred to as ‘inferred’ or ‘stochastic’ structures). The physical properties of these structures (e.g. position, orientation, size) are modelled as random parameters, the statistical properties of which are derived from the measurements of a sample of the population (sometimes referred to as ‘deterministic’ structures). Relatively little attention has been devoted to the uncertainty associated with the deterministic structures. Typical geotechnical analyses rely on either an entirely stochastic analysis, or deterministic analyses representing the structures with a fixed shape (i.e. disc), position, size, and orientation. The simplifications assumed for this model introduce both epistemic and stochastic uncertainties. In this paper, it is shown that these uncertainties should be quantified and propagated to the predictions of behaviour derived from subsequent analyses. We demonstrate a methodology which we have termed quasi-stochastic analysis to perform this propagation. It is shown that relatively small levels of uncertainty can have large influence on the uncertainties associated with geotechnical analyses, such as predictions of block size and block stability, and therefore this methodology can provide the practitioner with a method for better interpretation of these results.     

Learning soil parameters and updating geotechnical reliability estimates under spatial variability – theory and application to shallow foundations

ABSTRACT

Field data is commonly used to determine soil parameters for geotechnical analysis. Bayesian analysis allows combining field data with other information on soil parameters in a consistent manner. We show that the spatial variability of the soil properties and the associated measurements can be captured through two different modelling approaches. In the first approach, a single random variable (RV) represents the soil property within the area of interest, while the second approach models the spatial variability explicitly with a random field (RF). We apply the Bayesian concept exemplarily to the reliability assessment of a shallow foundation in a silty soil with spatially variable data. We show that the simpler RV approach is applicable in cases where the measurements do not influence the correlation structure of the soil property at the vicinity of the foundation. In other cases, it is expected to underestimate the reliability, and a RF model is required to obtain accurate results.     

Reliability analysis of shallow foundation in the vicinity of the existing buried conduit

ABSTRACT

The present study proposes reliability-based approach for assessing the performance of shallow foundation placed in the vicinity of an existing buried flexible pipe or utility tunnel. Performance function for the reliability analysis is defined in terms of % bearing capacity loss in the load carrying capacity of the shallow foundation due to the presence of buried flexible pipe or utility tunnel, and, allowable bearing capacity loss in load carrying capacity that can be tolerated. For the reliability analysis, an explicit functional relationship between input variables, such as geotechnical parameters of in situ soil as well as material properties of pipe, and, output response, i.e. % bearing capacity loss in load carrying capacity of foundation soil is needed. Using concept of response surface methodology (RSM) combined with the results of the numerical analysis; such an explicit functional relationship is easily established. Thereafter, reliability analysis can be performed, conveniently, using standard First Order Second Moment (FOSM) approach and performance of the foundation soil system with buried flexible pipe, present in the vicinity, can be assessed in terms of an index, popularly known as ‘reliability index (β)’.     

Uncertainty in spatial average undrained shear strength with a site-specific transformation model

ABSTRACT

Transformation models are used to infer geotechnical properties from indirect measurements. A site-specific transformation model can be calibrated with direct and indirect measurements from a site. When such a model is used, then spatial variability, measurement errors and statistical uncertainty propagate into the uncertainty of the spatial average, which is the variable of interest in most geotechnical analyses. This research shows how all components enter the total uncertainty of a transformation model for undrained shear strength from cone resistance. A method is proposed to estimate the uncertainty in the spatial average undrained shear strength, particularly focusing on the role of averaging of all spatially variable error components. The main finding is that if a considerable share of the measurement and transformation errors is random or spatially variable, the uncertainty estimates can be considerably lower compared to methods proposed earlier, and hence, characteristic values can be considerably higher.