1.1 Background
In design calculations, be it using a global factor of safety, partial factors, or otherwise, the selection of suitable values for soil parameters is a challenging task for engineers.
The challenge stems from special features of geotechnical practice, such as the remarkable variability of soils, and the fact that the soil volume tested is much smaller than the soil volume involved in a design. Due to difficulties in finding a detailed and prescriptive advice, the process of selecting soil parameters is sometimes viewed as a
“black art” (Simpson and Driscoll 1998). The European standard for geotechnical design, Eurocode 7 (CEN 2004), attempts to provide some guidelines on the selection of soil parameters, under Clause 2.4.5.2 “Characteristic values of geotechnical parameters”.
Eurocode 7 defines the characteristic value of a soil parameter as “a cautious estimate of the value affecting the occurrence of the limit state” [Clause 2.4.5.2(2)].
There are two main aspects in the definition of characteristic values: (a) a cautious estimate, and (b) the value affecting the occurrence of the limit state. The first aspect is related to uncertainties or statistics (e.g., spatial variability, transformation uncertainty, statistical uncertainty), while the second aspect is related to mechanics (e.g., spatial variability, force equilibrium, boundary conditions). Note that spatial variability (i.e., point to point variation of soil properties) is influential in both aspects, statistical and mechanical. It is often the spatial average, rather than the property at a certain point (point property), of soil properties over some influence zone that governs the performance of a geotechnical structure (e.g., spatial average of shear strength along a critical slip curve).
The current study focuses on the influence of spatial variability (both statistical and mechanical aspects) on characteristic values of shear strength for ultimate limit states. The term “point shear strength” is adopted to refer to the shear strength measured at a certain point of a spatially variable soil mass, whereas the term “mobilized shear strength” is adopted to refer to the shear strength that is effectively exhibited by the spatially variable soil mass as a whole. In other words, the mobilized shear strength is a value of shear strength that emerges from a back-analysis of a failure, if the failure occurs. In an ideal and deterministic world, the mobilized shear strength is a single (deterministic) value. However, in the real world, the mobilized shear strength takes a range of values in the presence of spatial variability and other sources of uncertainties, making it a random variable. Considering that the mobilized shear strength affects the occurrence of the limit state, it is sensible to adopt a cautious estimate of the mobilized shear strength as the characteristic value of shear strength, following Eurocde 7’s
definition.
Spatial variability can have both beneficial and detrimental impacts on the mobilized shear strength. There is also an added complexity arising from interactions between these types of impacts. Recall that the performance of a geotechnical structure is often governed by the spatial average over some influence zone. Along this influence zone, high values of point shear strength tend to be balanced by low values of point shear strength, leading to a lower variation of the spatial average. This effect, known as variance reduction (Vanmarcke 1977a), reduces the variance of the mobilized shear strength. This is a beneficial effect of spatial variability, because it reduces uncertainty (smaller variance), which in turn, allows engineers to select a less conservative value for their cautious estimates of the mobilized shear strength. Spatial variability, however, can also lead to failure mechanisms that deviate from the theoretical ones (derived from
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homogenous soils) to seek out weak zones. This reduces the mean of the mobilized shear strength. This is a detrimental effect of spatial variability, because it shifts the mean of the mobilized shear strength to the unconservative side.
Although Eurocode 7 provides a sensible definition for the characteristic values, it does not provide any detailed procedure. Therefore, due to the subjective nature of engineers’ judgment, a wide range of characteristic values may be selected for the same limit state, resulting in designs with different degrees of reliability. To address this, several simple formulas have been proposed to help engineers selecting more objective and consistent characteristic values (e.g., Schneider 1997; Schneider and Schneider 2013; Orr 2017). However, most existing formulas consider only the statistical aspect of characteristic values, and do not fully consider the mechanical aspect. In particular, they almost ignore the detrimental effect of spatial variability (weak-zone seeking), which is potentially unconservative.
1.2 Objectives of the thesis
Since existing simple formulas are not fully compliant with Eurocode 7, it is of practical value to develop a simple formula that respects both statistical and mechanical aspects of characteristic values. The weakest-path model (WPM), originally developed by Ching and Phoon (2013a), is a simple model that can simulate the probability distribution of the mobilized shear strength in the presence of spatial variability. The WPM considers both beneficial and detrimental effects of spatial variability. However, the WPM requires calibration using advanced methods such as random finite element method (RFEM). Therefore, the main objectives of the current study are:
1. To calibrate the WPM for several geotechnical problems.
2. To propose a simple formula for determining the characteristic value of shear strength based on the calibrated WPM.
3. To compare the performance of the proposed formula with existing formulas using numerical examples and real cases studies.
1.3 Structure of the thesis
The contents of the current thesis are divided into eight chapters, which are briefly described as follows:
Chapter 2 provides an overview of the literature relevant to characteristic values.
The random filed theory, which is usually used for modelling spatial variability, and the phenomenon of the worst-case scale of fluctuation are also reviewed in this chapter.
Chapter 3 is devoted to the issue of the discretization error in the RFEM. Recall that the WPM requires calibration using advanced methods such as the RFEM. It is evident that if implementation errors in the RFEM itself are not controlled properly, they can propagate to the calibration of the WPM. Hence, the aim of Chapter 3 is to make sure that the RFEM is implemented properly.
Chapter 4 presents the concepts of the WPM. It also presents six geotechnical problems, together with three scenarios of ground properties to be used for the calibration of the WPM.
Chapter 5 presents the calibration results for the investigated geotechnical problems. After calibration, the WPM is used to simulate the mean and variance of the mobilized shear, and the results are compared with those from the RFEM. This chapter also demonstrates the performance of the WPM in capturing the worst-case scale of fluctuation.
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Chapter 6 proposes a simple formula based on the WPM for determining the characteristic value of the shear strength. The performance of the proposed formula is verified using the RFEM for the investigated problems. This chapter also provides a comparison with past works, and shows the improvements of the proposed formula over past works in the literature.
Chapter 7 illustrates and verifies the use of the proposed simple formula for three real case studies. The case studies include a pile, a footing, and several cement-treated soil columns.
Chapter 8 concludes this thesis by highlighting the main findings of the current study, and provides some suggestions for future research.