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Probabilistic methods for unified treatment of geotechnical and geological uncertainties in a geotechnical analysis
Accepted Manuscript Probabilistic methods for unified treatment of geotechnical and geological uncertainties in a geotechnical analysis
C. Hsein Juang, Jie Zhang, Mengfen Shen, Jinzheng Hu PII: DOI: Reference:
S0013-7952(18)31443-1 https://doi.org/10.1016/j.enggeo.2018.12.010 ENGEO 5014
To appear in:
Engineering Geology
Received date: Revised date: Accepted date:
22 August 2018 10 December 2018 12 December 2018
Please cite this article as: C. Hsein Juang, Jie Zhang, Mengfen Shen, Jinzheng Hu , Probabilistic methods for unified treatment of geotechnical and geological uncertainties in a geotechnical analysis. Engeo (2018), https://doi.org/10.1016/j.enggeo.2018.12.010
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ACCEPTED MANUSCRIPT Probabilistic methods for unified treatment of geotechnical and geological uncertainties in a geotechnical analysis C. Hsein Juang1,2 , Jie Zhang3* , Mengfen Shen2 , and Jinzheng Hu3 Abstract: There are typically three types of models involved in a geotechnical analysis, i.e.,
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the geologic model, the ground model, and the geotechnical model, each of which may be associated with uncertainty to a different extent. However, in a typical geotechnical analysis
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the geological uncertainty is seldom characterized and considered explicitly. In this paper, we
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discuss the probabilistic tools developed in the geotechnical profession for tasks such as
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uncertainty characterization, assessment of impact of uncertainties, uncertainty reduction, and evaluation of the value of uncertainty reduction. We postulate that the same tools may be
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used for a unified treatment of various types of uncertainties in a geotechnical analysis, which
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would necessitate the collaboration of the engineering geologists and the geotechnical engineers to comprehensively characterize these uncertainties consistently. To this end, the
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probabilistic tools or methods that are widely used in geotechnical engineering are examined to determine their applicability in the evaluation of geological uncertainties. Examples are
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provided to illustrate various concepts introduced in this paper. Keywords : Geotechnical uncertainty;
Geological uncertainty; Ground
uncertainty;
Reliability; Probabilistic methods __________________ 1
Departmnet of Civil Engineering, National Central University, No. 300, Zhongda Road, Zhongli District, Taoyuan City, Taiwan 320. 2 Glenn Department of Civil Engineering, Clemson University, SC 29634, USA. 3 Department of Geotechnical Engineering and Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, 1239 Siping Road, Shanghai 200092, China. *
Corresponding author: J. Zhang (email: [email protected])
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1.
Introduction Geologists and engineers view the world in complementary but different perspectives.
While geologists tend to seek explanation about the observed details, engineers tend to focus
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on design to meet specific objectives and multiple constraints (Keaton, 2013, 2014). The successful implementation of a geotechnical analysis, however, often requires the joint efforts
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of both engineering geologists and geotechnical engineers. As shown in Fig. 1, a geotechnical
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analysis is typically conducted based on three types o f models, i.e., the geologic model, the
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ground model, and the geotechnical model (Burland, 1987; Morgenstern, 2000; Knill, 2003; Sullivan, 2010; Keaton, 2013). In this figure, the geologic model is a representation of the
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site geologic conditions relevant to the proposed project, the ground model is the geologic
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model expressed in terms of engineering parameters, and the geotechnical model is the ground model with design parameters used to establish predicted performance of the
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proposed project. In practice, the geologic model is often constructed by engineering geologists, the geotechnical model is often built by geotechnical engineers, and the ground
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model is constructed through the joint efforts of both engineering geologists and geotechnical engineers. Consequently, both the geologic and the ground models are essential for any reasonable geotechnical analysis. If either is incorrect, the outcome of the geotechnical analysis is likely incorrect. Given that a model represents only an abstraction of the complex real world, model uncertainties always exist. For example, Einstein and Baecher (1982) found uncertainty as perhaps the most distinctive characteristic of engineering geology compared to other 2
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engineering fields, which is evidenced by the widespread use of engineering judgement. Nevertheless, many practicing geologists are not familiar with methods of quantifying and modeling uncertainty inherent in geology (Bardossy and Fodor 2001), although practically all
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geological statements, measurements, and calculations contain some type and degree of uncertainty. On the other hand, probabilistic methods for modeling uncertainties have been
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extensively studied in geotechnical engineering over the past few decades (e.g., Whitman,
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1984; Baecher and Christian, 2003; Fenton and Griffiths, 2008; Phoon and Ching, 2015).
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While there is an increasing interest on characterizing the uncertainty associated with the soil stratification (e.g., Qi et al., 2016; Wang et al., 2018a), current probabilistic geotechnical
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analysis mainly focuses on the uncertainties associated with the ground model and
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geotechnical model, and the relevant geological uncertainties are seldom addressed. The objective of this paper is to illustrate how the probabilistic methods developed in the
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geotechnical profession may provide a consistent framework for comprehensive treatment of all uncertainties, including the geological uncertainty, in a geotechnical analysis, thus serving
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as a useful guide on how to address the geological uncertainty in a geotechnical analysis. The structure of this paper is as follows. First, the three types of uncertainties are described in detail together with the methods to characterize such uncertainties. Next, the reliability theory is introduced to assess the effect of uncertainties on a geotechnical analysis, followed by the use of additional information to reduce uncertainty. Thereafter, the evaluation of the value of uncertainty reduction in a decision making framework is then discussed. Finally, a discussion of the need for further research is provided. 3
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2.
Geological Uncertainty, Ground Uncertainty, and Geotechnical Uncertainty
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2.1 Uncertainty associated with the geological model Practically all geological statements, measurements, and calculations contain some type
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and degree of uncertainty (Bardossy and Fodor, 2001). The most useful engineering
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geological models characterize uncertainties and unknowns for inclusion into the project
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analyses or for the preparation of a contingency in the project cost estimation to cover the associated risks (Parry et al., 2014). As an example, Fig. 2 shows the infinite-slope model that
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is often used for the assessment of shallow-slope stability and evaluations of regional slope
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hazards (e.g., Crosta and Frattini, 2003; Babu and Murthy, 2005). When the infinite slope model is applied on a regional scale, it is often difficult to determine the slope angle (δ), the
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thickness of the soil layer (h), and the groundwater level (hw ) accurately. In such a case, the uncertainties associated with parameters δ, h, and hw are examples of the geological
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uncertainties. Note that the infinite slope model may not be suitable for each slope in a region. If the infinite slope model is adopted because of the difficulties to determine the soil stratigraphy, the uncertainty caused by the infinite slope can be considered as part of the geological uncertainty. The construction of an engineering geological model requires the specification of two features, namely, the composition of the ground and the geological boundary conditions (Bock 2006). Traditionally, geologists use the line pattern (solid, dashed, dotted, queried) to 4
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represent scientific confidence of stratigraphic or structural contacts that is understood by other geologists, but it is neither quantitative nor understood by non-geologists (Keaton, 2013). A fundamental question in engineering geology is whether the geologic models should
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be constructed using a quantitative method, i.e., whether numbers should be assigned to engineering geologic models. Indeed, many geologists are not comfortable with the
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requirement to assign numbers to geology (Hoek, 1999). Some geologists also argue that
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geological uncertainties are too great for quantification, which is like objections for
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estimating large numbers because of their magnitude (Fookes, 1997). However, the construction of both the ground model and the geotechnical model requires a quantitative
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geological model in which the uncertainties can be quantified in a manner that is consistent
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across the geological, the ground, and the geotechnical model. Consistent treatment of the three types of uncertainties can facilitate the communication of uncertainties among
1997).
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engineers of different fields and other parties involved in the decision making (Fookes,
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Three types of uncertainties are associated with the development of the geological model (Mann, 1993): (1) Type I uncertainty, i.e., imprecision and measurement error associated with the raw data, like the exact position of a formation boundary measured through a borehole with discontinuous measurements; (2) Type II uncertainty, i.e., the uncertainty resulting from the inherent variability of the natural process, such as the variation of the thickness of a soil layer; and (3) Type III uncertainty, i.e., the uncertainty from the lack of scientific data, such as the presence of weak layer or a structural existence. Wellmann et al. (2010) suggested a 5
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method for considering the effect of uncertainty in the raw data (i.e., Type I uncertainty) on the construction of geological models. For Type II uncertainty, which generally results from spatial variability of the natural process, it can often be modeled with geostatistics (e.g.,
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Marinoni, 2003; Tacher et al., 2006). For the third type of uncertainty, Tang (1987), Tang and Halim (1988), and Halim and Tang (1993) used the reliability theory to detect anomalies in
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the ground. Wellmann and Regenauer-Lie (2012) suggested the use of the information
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entropy as a quantitative measure of the quality of a geological model and its geological
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subunits. Despite these efforts, there is as yet no comprehensive approach to assess all the uncertainties in a geological model (Turner, 2006). Consequently, engineering geologists are
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often forced to estimate such uncertainties with engineering judgment even in critical
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applications such as risk assessments of nuclear waste repositories (Mann, 1993). In geotechnical engineering, several probabilistic approaches have been developed to
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identify the soil stratigraphy using data obtained from in-situ testing (e.g., cone penetration test) and/or boreholes, i.e., the conventional statistical analysis based on modified Bartlett
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statistics (Phoon et al., 2003), the Clustering Method (Liao and Mayne, 2007), Bayesian approaches (Cao and Wang, 2013; Li et al., 2013; Wang et al., 2014a), and the Wavelet Transform Modulus Maxima method (Ching et al., 2015). In general, while these approaches provide the “best” estimate on the soil stratigraphy based on prescribed criterion, they provide little information on the uncertainty (or degree-of-belief) in the estimated soil stratigraphy. Recently, Zheng et al. (2018) developed a Bayesian framework for quantifying the identification uncertainty in the soil stratigraphy inferred from the soil behaviour type 6
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index. Using the identified soil stratigraphy at testing locations, it is then possible to estimate the soil stratigraphy at locations without testing data and its associated uncertainty. Traditionally, two types of model are commonly used for such task; they are boundary-based
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and category-based models. The boundary-based model, directly and probabilistically, delineates soil layer boundaries at the locations without testing data (Sitharam et al., 2008;
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Dasaka and Zhang, 2012; Schöbi and Sudret, 2015; Li et al., 2016 a; Xiao et al., 2017). In
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contrast, the category-based model is used to elucidate the soil stratigraphy through
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predicting material types at the locations without testing data (Elfeki and Dekking, 2001; Li, 2007; Park, 2010; Li et al., 2016b; Qi et al., 2016), thus providing a more flexible tool to
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simulate complicated soil stratigraphy (e.g., embedded soil layers). Recently, Wang et al.
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(2018b) suggested a hybrid method which combines the features of the boundary-based and
data.
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category-based methods for soil stratigraphy identification using cone penetration test (CPT)
Geologic maps are often made to satisfy a wide range of needs. An engineering geology
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map is an artistic representation of one interpretation of geologic features and relationships inferred from limited observations of the distribution of rock types, surficial deposits, and geologic structures often with little or no subsurface data or laboratory test results (Keaton 2013). Currently, efforts for quantification of the uncertainty associated with a geologic map which can be directly used by geotechnical engineers are very few. Keaton (2013) suggested a Geologic Model Complexity Rating System (GMCRS) that may be useful in quantifying variability and uncertainty in geologic mapping. GMCRS has nine components, four of which 7
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are related to regional-scale geologic complexity and five of which pertain to site-scale complexity, terrain characteristics, information quality, geologist competency, and time allotted to prepare the model. The Analytic Hierarchy Process method is used to derive a
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cumulative score, which can be converted into a coefficient of variation (COV) to be
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conveniently used by geotechnical engineers.
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2.2. Uncertainty associated with the ground model
The ground model in Fig. 1 is the geologic model expressed in terms of engineering
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parameters. One important feature of the soil properties is that even in the same soil layer, the
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soil properties at different points are not necessarily the same, i.e., the soil properties are spatially variable. In general, the closer the two points are in space, the greater the correlation
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of the soil properties at the two points. The spatial variability of the soil properties can often
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be modeled through the random field theory (e.g., Vanmarcke, 1983; Fenton, 1999a; Griffiths et al., 2009; Zhu and Zhang, 2013; Gong et al., 2018). For example, the spatially variable
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cohesion of the soil, su ( z ) , in Fig. 2 may be modeled as a depth-dependent trend function t(z) and a stationary random field w(z) with a mean of zero as follows: su z t z w z
(1) where z = depth of the soil. Note although Eq. (1) is conceptually simple, it might be difficult to properly divide the trend function and stationary random field from real measurement data as there might be multiple choices for the trend function (e.g., Fenton et al.,2018).
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Three types of uncertainties are associated with the estimated soil property, i.e., inherent variability, measurement error, and transformation uncertainty (Kulhawy and Phoon 2002). The uncertainty associated with the soil property depends on the estimation procedure.
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Following the concept proposed in Kulhawy and Phoon (2002), Fig. 3 shows the results of uncertainties associated with the estimated soil properties when using different types of tests.
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When the soil properties are directly estimated based on laboratory tests, the estimated soil
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property may be subjected to the inherent variability of the soil, the sampling disturbance, the
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measurement error due to imperfect test, and statistical uncertainty due to limited number of tests. When the soil properties are estimated based on in-situ tests through a transformation
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model, the estimated soil property may be further subjected to the error associated with the
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transformation model.
The moment methods (e.g., Tang, 1979; Lacasse and Nadim, 1996; Uzielli et al., 2005),
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maximum likelihood methods (e.g., DeGroot and Baecher, 1993; Fenton, 1999b) and Bayesian approaches (e.g., Wang et al., 2010a; Cao and Wang, 2014; Ching et al., 2016a;
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Tian et al., 2016) have been used to characterize the spatial variability of soil properties. Most previous research entailed decomposing and characterizing the spatial variability into individual vertical and horizontal components (Jaksa, 1995; Stuedlein et al., 2012). Unlike the characterization of vertical spatial variability, a greater challenge lies in characterizing the horizontal spatial variability due to the sparsity of test data in the horizontal direction. Unfortunately, only a few studies have been undertaken to simultaneously address the vertical and horizontal spatial variability using limited test data measured at different 9
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locations in a soil layer (Pardo-Igúzquiza and Dowd, 1997; Xiao et al., 2018). One practical challenge regarding the application of the random field theory is the insufficiency of site investigation data for estimating the parameters of different random
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fields, particularly the cross-correlation among those fields. Recent efforts have been made to calibrate random field models based on sparsely measured geotechnical data (e.g., Wang et al.,
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2018c; Zhao and Wang, 2018). On the other hand, the performance of a geotechnical
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structure is often governed by the average property over a path or a region. As the uncertainty
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caused by the spatial variability tends to cancel out, the uncertainty associated with the average property is generally smaller than the uncertainty at a point (Vanmarcke, 1983). In
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such a case, one may regard the average property as a random variable, and evaluate the
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reliability of a geotechnical system approximately through the average property using the random variable approach. Vanmarcke (1983) provided equations on how to assess the
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statistics of the average property over a prescribed region. As illustrated recently in Ching and Phoon (2013) and Ching et al. (2016b, 2017, 2018), when deriving the statistics of the
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spatial average, the spatial average should be performed along the actual failure path, which in many cases may be uncertain. When the statistics of the average property can be readily established, the random variable approach can be considered as an adequate compromise between the need to consider the spatial variability of the soil and the difficulty involved in reliability analysis directly using the random field theory. In geological and geotechnical engineering, the number of samples tested is often too small to use statistics built based on the law of large samples. Christian (2004) and Keaton 10
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and Ponnaboyina (2014) suggested that small sample statistics can be employed to estimate the uncertain soil properties. From a Bayesian perspective, a small number of samples may also be very valuable in that it may potentially lead to significant reduction in the uncertainty
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associated with the prior knowledge.
(e.g. Wu and Abdel-Latif, 2000):
h hw sat hw sin cos
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(2)
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g θ
c h hw sat w hw cos 2 tan
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For the infinite slope in Fig. 2, its factor of safety (FOS) may be computed as follows
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where γw, γsat, and γ = unit weights of water, saturated soil, and the soil above the water table, respectively; c = effective cohesion of the soil; and φ = effective friction angle of the soil. For
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the infinite slope in Fig. 2, the uncertainty associated with the ground model is represented by
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the uncertainty associated with the unit weights of the soil and the soil strength parameters.
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2.3. Uncertainty associated with the geotechnical model Almost any geotechnical model is associated with some assumptions and limitations, thusly, the model uncertainty almost always exists (Ang and Tang, 1984; Phoon and Kulhawy, 1999). A geotechnical analysis might be biased when the model uncertainty is ignored (Lacasse and Nadim, 1994; Gilbert and Tang, 1995). The model uncertainty can often be characterized by a model factor which represents the difference between, or the ratio of, the actual and predicted performances (Juang et al., 2004, 2011, 2013a; Phoon and Kulhawy,
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2005; Zhang et al., 2009a; Wang et al., 2012), which is applied using either an additive or multiplicative method. Take the geotechnical model given by Eq. (2) as an example. Let g(θ) denote the slope stability model to predict the FOS, in which θ denotes the uncertain
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variables. When a model correction factor is applied in an additive fashion, the FOS, denoted as FS , can be expressed as
(3)
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FS g θ
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In principle, the model factor can be computed by comparing the measured
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performances with the model predictions. The difficulty of an accurate determination of the soil strength parameters almost always leads to an uncertain model prediction. The observed
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geotechnical performance may also be uncertain. For example, if the pile is not loaded to
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failure during a load test, the bearing capacity of the pile foundation is not exactly known. In such a case, the geotechnical model uncertainty is mixed with uncertainty in both the soil
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strength parameters and the geotechnical performance (e.g., Gilbert and Tang 1995). Zhang et al. (2009a, 2012) developed a method for determining the uncertainty associated with a
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geotechnical model in such scenario. Assume that is a normal variable with a mean of and a standard deviation of . Let f(, ) denote the prior probability density function (PDF) of and . Based on Bayes’ theorem, the posterior PDF of and can be evaluated as follows (Zhang et al.,2009a): n
f , | d kf , i 1
1
2 2 g2( i )
1 di g ( i ) exp 2 2 g2( i )
2
(4)
where k is a normalization constant to make the derived posterior PDF valid ; di is the ith 12
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observation; and g( i) and g( i) are the mean and standard deviation of g( i) caused by uncertainties in i. Past studies indicated that a small level of parameter uncertainty and observation
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uncertainty generally results in a negligible impact on the geotechnical model uncertainty characterization (Zhang et al., 2009a, 2012). In such a case, the model uncertainty can be
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characterized approximately without considering the parameter and observation uncertainties.
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For slope stability problems, for example, Duncan and Wright (1980) determined that any
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limit equilibrium method satisfying all conditions of equilibrium could predict the FOS within an accuracy of 5% from what is considered the correct answer. Zhang et al. (2009a)
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also derived similar conclusions through the calibration of the model uncertainty of four
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commonly used limit equilibrium models using centrifuge tests data. Gong et al. (2016a) evaluated and calibrated empirical models from the angle of model
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fidelity (i.e., goodness of fit to the calibration data) versus model robustness (i.e., robustness of prediction for future applications). They found that high fidelity models often lacked the
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desirable level of model robustness, and a balanced approach is desired. Juang et al. (2018) studied the model selection in geological and geotechnical engineering in the face of uncertainty, and showed that simple models might yield more robust predictions compared with complex and sophisticated models. The model factor is often assumed as independent of the model input parameters. However, Zhang et al. (2015a) found that the model factor could be correlated with the model input parameters. The systematic bias of such correlation must be removed prior to 13
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conducting the full probabilistic analysis. The multiple regression equation was found as an efficient solution for characterizing the parameter correlation (Kung et al., 2007; Schuster et al., 2009; Zhang et al., 2015a; Tang and Phoon, 2016; Qi and Zhou, 2017). One may refer to
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Dithinde et al. (2016) for a more comprehensive review on methods for characterization of
Impact of Uncertainties on Performance Predictions
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3.
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the geotechnical model uncertainty.
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3.1. First order reliability analysis (FORM) for explicit performance function The uncertainties involved in a geotechnical analysis lead to uncertain predictions from
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such analysis, making it difficult to determine with certainty the true safety level of the
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geotechnical system. Probabilistic analysis, often through reliability analysis that considers these uncertainties explicitly, could provide complementary information to those obtained
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from the traditional deterministic analysis (Duncan, 2000). Rackwitz (2000) noted that 90% of the reliability problems can be accurately solved with the first order reliability method
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(FORM), in which the safety or performance of a system is measured with a reliability index, which is defined as the minimum distance between the origin and the limit state surface in the uncorrelated reduced space. Let x denote the uncertain variables involved in a geotechnical analysis. For the geotechnical model as given by Eq. (3), x = {, }. Let G(x) = 0 denote the limit state function. With FORM, the failure probability pf is calculated as follows (e.g., Low and Tang, 2007): p f 1
(5) 14
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min yR 1yT
(6)
g x 0
where Φ = the cumulative distribution function (CDF) of the standard normal variable, β= the reliability index, y = the reduced variables of x, and R = the correlation matrix of the
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random variables. Traditionally, the application of probabilistic methods has been quite tedious. Over the
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past decades, significant improvements have been made regarding the practical application of
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reliability methods in geotechnical engineering. When the limit state function is explicit,
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FORM can be efficiently executed in a spreadsheet (Low and Tang 1997, 2007; Juang et al., 2006). Low (2015) provided detailed examples on how the spreadsheet method can be used
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to evaluate the reliability of different types of geotechnical problems.
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As an illustration, we evaluate the failure probability of the slope in Fig. 2 using the spreadsheet method. In this example, γw = 9.8 kN/m3 γsat, = 19 kN/m3 , γ = 19 kN/m3 , and δ =
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35°. The variables c and φ are both assumed to be normally distributed having mean values of 10 kPa and 38°, respectively, and standard deviations of 2 kPa and 2°, respectively. Let m =
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hw/h denote the normalized water table and let m denote the maximum groundwater table within a certain design period. Suppose that h is normally distributed with a mean of 3.0 m and a standard deviation of 0.6 m and that m is normally distributed with a mean of 0.5 and a standard deviation of 0.05. The spreadsheet for evaluating the failure probability of the infinite slope based on FORM is shown in Fig. 4. Solving the optimization problem in Eq. (6) with the Solver in Excel, yields a probability of failure of the slope of 0.033. The minimization point in Eq. (6) is often called the design point. The coordinates of the 15
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design point in the reduced space illustrates the relative importance of different variables to the reliability of the geotechnical system. The greater the deviation from zero, the greater the effect upon the reliability of the geotechnical system. For the current slope example, the
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coordinates of c, φ, m, h, and ε are -1.214, -0.976, 0.501, 0.793, and -0.286, respectively. In this example, h and m characterize the geological uncertainty, c, and φ characterize the
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uncertainty associated with the ground model, and ε characterize the geotechnical model
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uncertainty. From the coordinates of the design point we can see that the reliability of the
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current slope is most affected by the ground model uncertainty, followed by the geological uncertainty, and then by the geotechnical model uncertainty. Therefore, the reliability analysis
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can be used to assess the relative importance of uncertainties of different types.
3.2. FORM for implicit performance function
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When the limit state or performance function is implicit, as in the case of stability analysis of a soil slope using a numerical method-based program, the implementation of
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FORM is challenging due to the coupling between the numerical program and FORM. To deal with such a challenge, response surface methods (RSM) are widely used. The basic idea of the RSM is to approximate the performance function with a surrogate model that is computationally more efficient, followed by reliability analysis based on the surrogate model. For example, a second-order polynomial in the y space can be used to approximate the performance function (Zhang et al., 2015b; Khoshnevisan et al., 2017):
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G(y ) b0 bi yi i 1
2k
by
i k 1
i
2 i
(7)
where bi (i = 0, 1, 2, , 2k + 1) = the coefficients to be determined. In Eq. (7), the coefficients can be determined based on the response of G(y) at the following 2n + 1 points: yc = {yc1 , yc2 , …, ycn }, {yc1 ± y1 , yc2 , …, ycn}, {yc1 , yc2 ± y2 , …, ycn }, …, and {yc1 , yc2 , …, ycr ± yn },
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where yi = standard deviation of yi. The approximated performance function Eq. (7) is then
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used for the reliability analysis. If the performance function cannot be well approximated by
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a second-order polynomial, the reliability index obtained based on Eq. (7) may not be
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accurate. In such a case, a new set of calibration points are used to update the response surface with the centre determined as follows (Zhang et al.,2015b):
y DG μ y
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yc
G μy G y D
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(8)
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Equation (8) is again calibrated using the G(y) responses evaluated at the new calibration points, permitting a reliability analysis based on the updated response surface. Such a process
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is iterated until the obtained reliability index remains stable within a tolerable degree of error. As illustrated in Zhang et al. (2015b), this procedure is easily implemented based on existing deterministic geotechnical programs such as FLAC 3D. The reliability of a reinforced concrete drainage culvert in Shanghai, China is provided as an example, as shown in Fig. 5. The cohesion, the friction angle, and Young’s modulus of the soil are molded as random variables, and their mean values are 22 kPa, 19o , and 5.78 MPa, respectively and their COVs are 0.3, 0.1, and 0.2, respectively. The correlation between the
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cohesion and the friction angle is assumed as -0.5. For simplicity, the three random variables are assumed lognormally distributed. As the structural safety of the culvert is controlled by bending failure at the mid-span of the top slab, the performance function can be written as follows:
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g ( x) M R M S
(9)
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where MR denotes the flexure resistance, and M S denotes the bending moment. In this
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example, FLAC 3D is used to calculate M S. The finite difference mesh of the numerical model
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is shown in Fig. 6(a). Based on the plane strain assumption, only a unit length of the culvert is analyzed. The concrete culvert is modeled as elastic shell element (DKT-CST) embedded
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in FLAC3D that resists both membrane and bending loading. The mesh of the reinforced
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concrete is shown in Fig. 6(b) and the soil was modeled as the Mohr–Coulomb plasticity material. Based on the RSM as mentioned above, the reliability index of the culvert is 3.303,
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and the corresponding design point is {-2.514, -0.134, -1.419} in the reduced space. For the current problem, the cohesion deviates most from the zero at the design point, followed in
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sequence by Young’s modulus and the friction angle. Therefore, the reliability of the culvert is most affected by uncertainty in the cohesion, followed by uncertainty in Young’s modulus, and is least affected by the uncertainty in the friction angle.
3.3. System reliability analysis based on FORM While FORM is versatile, sometimes a geotechnical system may have several competing failure modes. In such a case, the failure probability of the system would be dominated by 18
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more than one design point and thus, the concept of the system reliability is applicable (e.g., Ang and Tang, 1984; Xu et al., 2014; Reale et al.,2016). MCS may be used to determine the system reliability. Alternatively, the system reliability may be determined based on the results
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obtained from FORM. Consider a geotechnical system with r design points. The following equation can be used to calculate the correlation coefficient between design points i and j
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(e.g., Low et al., 2011): y *i R 1 y *j
ij
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T
i j
(10)
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where βi and βj = reliability indexes of the two design points, respectively; and yi* and yj* =
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coordinates of design points i and j in the reduced space, respectively. The system reliability index can then be calculated as follows (e.g., Zeng and Jimenez, 2014): r
r
r
r
r
r
i 1
i 1 j i
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Pfs = Pi Pij Pijk i 1 j i k j
r
r
r
Pi Pij i 1
(11)
i 1 j i
(12)
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P12-n Φn 1， 2， ， n ;R
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where n (-1 , -2 , …, -n ; R) is the cumulative distribution function of a r-dimensional multivariate normal distribution, and R is the correlation matrix with the element in the ith row and jth column being the correlation coefficient (ρij) between design points i and j. As an illustration, Fig. 7 shows a slope in layered soils (Ching et al., 2009). The undrained shear strength of the two clay layers, i.e., cu1 and cu2 , are modeled as independent lognormal random variables. The mean values of cu1 and cu2 are 120 kPa and 160 kPa, respectively and the standard deviations of cu1 and cu2 are 36 kPa and 48 kPa, respectively. 19
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Figs. 8(a) and 8(b) show the two major failure modes controlling the stability of the slope identified based on the shear strength reduction method, respectively (Ma et al.,2017). The reliability indexes of the two slip surfaces are 2.881 and 2.938, respectively and the correlation coefficient between the two failure modes is 0.496. Based on Eqs. (10) & (11), the
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system failure probability of the slope is then 1.198×10-4 .
RI
Instead of providing a point estimate of the system failure probability, the outcomes
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from FORM are also used to derive the bounds of the system failure probability using
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methods like Ditlevsen’s bounds (Ditlevsen, 1979). One may refer to Low et al. (2011) and Lü et al. (2013) for examples of estimating bounds of the system failure probability based on
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FORM for different geotechnical problems.
3.4.Other reliability analysis methods
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MCS is a robust and versatile method for reliability analysis. With the rapid development of computing technology and commonly available personal computer, some
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simple geotechnical reliability analyses, such as pile or footing design, can be easily done by MCS even in spreadsheet (e.g., Wang, 2011; Wang and Cao, 2013). However, when the limit state function lacks an explicit form, the application of MCS could be computationally intensive. Many efforts have also made to enhance the computational efficiency of reliability analysis. Among these studies, the RSM has attracted substantial attention. As the second-order polynomial can only fit the limit state function locally, RSMs with global fitting capability have been widely studied, such as those based on artificial neural network (Cho, 20
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2009; Juang et al., 2000; Lü et al., 2012), kriging (Zhang et al., 2011a), Gaussian process regression (Kang et al., 2015), support vector regression (SVR) (Kang and Li, 2015), multivariate adaptive regression splines (Metya et al., 2017), and auxiliary random finite
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element method (Xiao et al., 2016). Li et al. (2016c) assessed various RSMs for system reliability of soil slopes. Sudret (2012) reviewed and compared various RSMs developed for
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structural reliability analysis. Several commonly used RSMs have been implemented in the
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software UQLab for general purpose structural reliability analysis (Marelli and Sudret 2014).
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In general, the more advanced RSMs often need more calibration points and hence computationally more demanding than the simple second-order polynomial based RSM.
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However, when the system failure probability is significantly greater than the failure
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probability evaluated at the most probable failure point, such methods are essential and useful. Other researchers have resorted to more efficient sampling methods for geotechnical
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reliability analysis, such as the importance sampling (e.g., Ching et al., 2009, 2010), subset simulation (e.g., Wang et al., 2010b; Li et al., 2016d, 2017), subdomain sampling (Juang et al.,
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2017), and integration-based method (Gong et al., 2016b&c).
3.5. Reliability analysis considering model selection uncertainty For the same geotechnical problem, it is possible to derive several competing models, particularly geological models. When the data is insufficient to determine the best model uniquely, model selection uncertainty exists. The probabilistic method can be used to consider this selection uncertainty. For example, suppose there are r competing models for 21
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the same problem, z is variable to be predicted, Mi is the ith model, p(z|Mi) is the distribution of z predicted based on model Mi, and P(Mi) is the probability that Mi is the correct model used. Based on the total probability theory, the predicted distribution of z, i.e., p(z),
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considering all competing models can be written as follows (e.g., Hoeting et al.,1999): r
p( z ) p( z | M i ) P( M i )
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i 1
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(13)
The above method has been used in Zhang and Tang (2009) and Zhang et al. (2014a) to
MA
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consider the model selection uncertainty in a geotechnical reliability analysis.
3.6. Reliability-based design
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The reliability theory is increasingly used in geotechnical engineering to calibrate design
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codes, particularly in the format of load and resistance factor design (e.g., Honjo et al., 2002; AASHTO ,2007; Yu et al., 2012; Li et al., 2015; Low, 2017). One major challenge in
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geotechnical reliability-based design (RBD) is that the variabity invovled in a geotechnical analysis varies from site to site. Thus, it difficult to predetermine the appropairte COVs that could be invovled in the calibration process, and the resistance factors calibrated may not be unique. It is increasingly recongized now that the resistance factors used in geotechnical engineering shall depend on the degree of understanding of degree of uncertainty at the site (Fenton et al., 2016; Phoon, 2017). Recently, a robust geotechnical design (RGD) methodology has been developed to complement the existing RBD and to address the
22
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geotechnical uncertainties from a different perspective, namely robust design in the face of uncertainties (e.g., Juang et al., 2013c; Khoshnevisan et al., 2014). Although reliability-based design (RBD) is receiving wider and wider acceptance and adoption among the geotechnical
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engineer, the uncertainty (or variation) in the estimated statistics of input random variables can lead to the variation in the computed system response of a design, which may lead to an
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under-design or over-design relative to the performance requirements. Robust geotechnical
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design (RGD) aims to ensure a design that is robust against, or insensitive to, such variation
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in the input variables by adjusting the design parameters (such as geometry of a geotechnical structure) without seeking additional reduction of the uncertainties in these input variables. A
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design is deemed robust if it has a high design robustness against such unavoidable variation
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(called noise factors herein). RGD achieves its goal through multi-objective optimization, typically by maximizing the design robustness and minimizing the cost while enforcing the
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safety requirements (through meeting target reliability indexes). The RGD methodlogy has been showcased in many geotechnical problems (e.g., Juang and Wang, 2013; Juang et al.,
4.
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2013c; Khoshnevisan et al., 2015; Gong et al., 2016d).
Reducing Uncertainty through Additional Information In practice, there are often information from multiple sources. As each source may
provide partial information about the geotechnical system under investigation, combination of multiple sources of information may result in reduced uncertainty. The reliability of the geotechnical system can also be updated as the uncertainty is reduced, which may result in 23
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improved design and decision making. The information combina tion can be formally conducted based on Bayes’ theorem. Suppose the prior knowledge about , variables of interest (such as the shear strength parameters of a clay), can be denoted by a probability density function (PDF), f(). Let D denote the data from another source, and let L(|D)
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denote the chance to observe D given , which is often called the likelihood function. One
RI
may refer to Juang et al. (2015) on the construction of the likelihood function using various
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types of geotechnical data. Based on Bayes’ theorem, the prior knowledge about and the
L(θ | D) f (θ)
... L(θ | D) f (θ)dθ
(14)
MA
f (θ | D)
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knowledge learned from the data can be combined as follows:
where f(|D) = posterior PDF of representing the combined knowledge. In Eq. (14), both
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the likelihood function L(|D) and prior PDF f() affect f(|D), and the role of the likelihood
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function will become more dominant as the amount of data increases. Various examples for application of the Bayesian method in geotechnical engineering can be found in the literature
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(e.g., Honjo et al., 1994; Zhang et al.,2004; Najjar and Gilbert, 2009; Huang et al., 2014; Wang et al., 2012; Wang et al., 2013; Juang et al., 2013b; Peng et al., 2014; Zhang et al., 2014b; Nishimura et al., 2016; Ering and Sivakumar Babu., 2017). Juang and Zhang (2017) provided a tutorial on the formulation and use of Bayesian methods to solve different types of geotechnical problems. A comprehensive review of application of the Bayesian method in geotechnical engineering can be found in Zhang et al. (2017) and an in-depth discussion on the use of Bayesian thinking to solve different geotechnical problems is available in Baecher
24
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(2017). As an illustration, suppose a technique is available for estimating the depth of the soil layer in Fig. 2 in which hp is the prediction from a soil layer thickness estimation method.
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Assume the actual depth h is normally distributed with a mean of hp and a standard deviation of σp , which describes the measurement error. Before the soil layer thickness estimation
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method is applied, it is known that h is normally distributed with a mean of h and a standard
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deviation of h . In this example, μh = 3 m, and σh = 0.6 m. Therefore, the prior PDF of h can
h h 2 1 exp 2 h2 2 h
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f h
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be written as follows:
(15)
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can be written as follows:
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The chance to observe hp when the value of h is known, i.e., the likelihood function of h
h h 2 p l hp |h exp 2 p2 2 p (16)
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1
Substituting Eqs. (15) & (16) into Eq. (14), the posterior distribution of h is still normal with the following mean and standard deviation (Ang and Tang, 2007):
h p2 hp h2 h h2 p2 (17)
25
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h2 p2 h2 p2
h
(18)
As an example, a comparison of the posterior and prior PDFs of h for the case of hp = 3 m and σp = 0.1 m, 0.2 m, and 0.3 m, respectively is provided in Fig 9. We can see that the
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posterior PDFs of h are more sharply peaked than the prior PDF, indicating the uncertainty in
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the thickness of the soil layer has been reduced. The degree of uncertainty reduction also
SC
increases as the precision of the soil layer thickness estimation method increases. Using the updated distributions of the thickness of the soil layer, the failure probability of the slope can
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again be estimated, which are 0.021, 0.024, and 0.026, respectively. The additional
MA
information is thus shown able to reduce the uncertainty in a geotechnical analysis and improve our knowledge about the reliability of a geotechnical system. As mentioned
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previously, h is an example of geological uncertainty in the infinite slope problem. The
information.
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example here thus illustrates how the geological uncertainty can be reduced through new
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The past performance of a geotechnical system is a very important source of information, which can be viewed as a field test directly performed on the prototype. For the above infinite slope example, the survivability of a slope at a certain state can provide valuable insight into its future performance. Let m1 denote a level of groundwater at which the slope has survived. The presence of observation error, statistical uncertainty, and modeling uncertainty, however may hinder a direct determination of the exact value of m1 . Suppose m 1 is normally distributed with a mean of μm1 and a standard deviation of 0.05. The above performance
26
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information can be used to update the failure probability of the slope when the groundwater is m2 in a future state. Here, let Fs1 and Fs2 denote the FOSs of the slope when the groundwater table is m1 and m2 , respectively. Let pf1 and pf2 denote the failure probabilities of the slope
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calculated based on reliability theory when the groundwater tables are m1 and m 2 respectively without considering the slope performance information. The fact that the slope did not fail for
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the case of m = m1 indicates that Fs1 > 1. The event that the slope will fail at the case of m =
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m2 can be denoted by Fs2 < 1. Let pf2|Fs1>1 denote the probability of failure of the slope for the
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case of m = m 2 in a future state considering such information. Mathematically, pf2|Fs1>1 can be expressed as
P Fs1 1 Fs 2 1
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p f 2|Fs11 P Fs 2 1| Fs1 1
P Fs1 1
p f 1 p f 12 1 pf 1
(19)
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where pf12 = P(Fs1 < 1 ∩ Fs2 < 1), which is the probability of event “Fs1 < 1 and Fs2 < 1”
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without considering survival information. pf12 can be calculated based on pf1 , pf2 , and the correlation between the design points of the slope for the cases of m = m1 and m = m2 (Zhao
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et al., 2007; Zhang et al., 2011b). The updated reliability index at m = m 1 when μm1 takes various values is shown in Fig 10. The reliability index of the slope considering the survival information is generally greater than that without considering the survival information, indicating that knowing its past survival information increases our confidence in the stability of the slope. As the value of μm1 increases, the increase in the reliability index is more obvious, indicating that the updating effect will be more obvious if the slope survived a more critical state. Although the above phenomenon is quite intuitive, the probabilistic method
27
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illustrated here can be used to quantify the effect of past performance on predicting the future performance of a slope. Other types of information such as field monitored data (i.e., Honjo et al.,1994; Vardon
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et al.,2016) and results from model tests (i.e., Zhang et al.,2009b; Wu et al.,2017) may be available and can be used to update our knowledge about of the geotechnical system under
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investigation. For example, Hsiao et al. (2008) proposed a Bayesian based procedure to
SC
update the settlement predictions at subsequent stages in a braced excavation using the
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deformation data, and the serviceability reliability of adjacent buildings was updated stage by stage accordingly. Straub (2011) described a structural reliability method to update the failure
MA
probability of an engineering system based on measured performance in a computationally
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efficient manner. Juang et al. (2013b) proposed a Markov chain Monte Carlo (MCMC) based procedure to update the soil parameters in a staged braced excavation based on field
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observations, and the updated soil parameters were then used to update the deformation predictions in the subsequent stages of excavation. Schweckendiek and Vrouwenvelder (2013)
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applied the method suggested in Straub (2011) to update the reliability of levees based on head monitoring data. Papaioannou and Straub (2015) applied the above approach to update the reliability of a sheet pile wall based on deformation measurement. Wang et al. (2014b) developed a maximum likelihood based procedure to update the predictions of ground deformations and assessment of building damages induced by excavation using the field monitoring data. Lo and Leung (2018) suggested a Bayesian method to update of subsurface spatial variability for improved prediction of braced excavation response. Yang et al. (2018) 28
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suggested a probabilistic back analysis method for characterization of the spatial variability of unsaturated soil properties of a slope based on monitored performance of the slope. As mentioned previously, for the same problem, it is possible that several models may
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be constructed, and yet it is unclear which model is most accurate. The observed performance can also be used to reduce the model selection uncertainty. Let P(Mi ) denote the prior
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probability that model Mi is correct. Let P(D|Mi) denote the probability to observe D if model
SC
Mi is correct. In the absence of prior knowledge, the non-informative prior can be used, i.e.,
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P(Mi) = 1/r where r is the number of competing models. Based on the data D, the posterior
et al., 1999):
P(D | M i ) P( M i ) r
P(D | M ) P( M ) i
(20)
i
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i 1
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P( M i | D)
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probability that model Mi is correct, i.e., P(Mi|D), can be calculated as follows (e.g., Hoeting
One may refer to Zhang et al. (2014a) and Brownlow (2017) on how to determine the model
5.
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selection uncertainty based on the observed data.
Value of Uncertainty Reduction An advantage of probabilistic analysis is that it can explicitly show the trade-off
between investment and reduction of potential losses, and thus facilitating decision- making in the presence of uncertainties. To make an informed decision in geotechnical practice, the decision tree model (e.g., Ang and Tang 1984) is often used. For example, suppose the loss 29
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due to the slope failure for the infinite slope mentioned previously is $2,500,000. Assume that a monitoring system can be installed with a cost of $50,000, which if installed, can reduce the loss due to the slope failure to $250,000 (one tenth of the loss without the monitoring system).
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The decision tree model in such a case (to determine whether to install the monitoring system) is shown in Fig. 11. Here, two decision alternatives are involved, i.e., installing or not
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installing the monitoring system. Let a1 and a2 denote the alternative that a monitoring
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system will be and will not be installed, respectively. Let Ri denote the risk associated with
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the ith decision alternative. Based on the decision tree, Ri can be computed with the following equation:
MA
Ri Ci1 p f Ci 2 1 p f
(21)
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where Ci1 and Ci2 are the total costs associated slope failure and no slope failure when ai is adopted, respectively. Note that the total cost of a decision alternative here includes the cost
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of installing the monitoring system, if installed, and the loss due to the slope failure. Thus, C11 = ($250,000 + $50,000) = $300,000, C12 = $50,000, C21 = $2,500,000, and C22 = 0. The
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risk associated with the two decision alternatives are R1 = $300,000 × 0.0.033 + ($50,000) × (1 − 0.033) = $58,250, and R2 = ($2,500,000) × 0.033 + $0 × (1 − 0.033) = $82,500, respectively. If the maximum expected monetary value (EMV) criterion is followed, the decision maker would choose to install the monitoring system, i.e., a1 , as it has a smaller risk. As mentioned previously, the availability of new information may lead to the reduction of the uncertainty, which could result in a more informed engineering decision. As gaining information usually requires additional investment, a decision- maker frequently encounters 30
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the dilemma of determining the worthwhile nature of collecting more information. For the infinite slope example mentioned above, the use of a new technique to determine the thickness of soil layer could reduce the uncertainty regarding this thickness. Before the new
PT
technique is used, the outcome from this technique is uncertain, making the effect of using the new technique also uncertain. To this end, a value of information (VOI) analysis could be
RI
conducted to determine whether the use of the new technique is appropriate (e.g. Yokota and
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Thompson, 2004; Quiggin, 2016).
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For example, suppose that the precision of the new technique is σp = 0.1 m and the outcome from the test is hp = 3.0 m. Based on Eqs. (17) and (18), the updated knowledge
MA
about h is that h is normally distributed with a mean of 3.0 m and a standard deviation of
ED
0.0986 m. With the updated distribution of h, the failure probability of the slope evaluated based on Eqs. (5) and (6) is Pf = 0.020. In such a case, the risk associated with a1 and a2 are
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$55,000 and $50,000, respectively, and the optimal alternative is a2 with a R2 = $50,000. Compared with the case of not exploring the soil layer thickness further, the risk of slope
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failure has been reduced by $8,250, i.e., the VOI is $8,250. In the above analysis, the VOI is assessed for the case of hp = 3 m. However, before the new technique is used, the value of hp is uncertain. The uncertainty in hp should also be considered in the VOI analysis, which can be formally examined through the decision tree model as shown in Fig. 12. Let a = {a1 , a2 }. Mathematically, the VOI for further investigating the thickness of the soil layer can be computed as follows:
31
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VOI max Ci1Pf Ci 2 1 Pf f hp dhp max Ci1Pf Ci 2 1 Pf hp ai a ai a
(21)
Fig. 13 shows how the VOI varies with σp . As expected, the VOI increases as the precision of the thickness estimation method increases. For the cases of σp = 0.1 m and 1.0 m,
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the values of VOI are $20,407 and $8,421, respectively. The VOI indicates the expected risk reduction achieved through uncertainty reduction using the new soil layer thickness
SC
economically not feasible to adopt the new technique.
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estimation technique. If the cost employing the new technique exceeds its VOI, then it is
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In this example, the uncertainty in the thickness of the soil layer is an example of the geological uncertainty. It illustrates how the probabilistic theory can be used to assess the
MA
value of reducing the geological uncertainty. Similarly, it can also be used to assess the value
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of reducing the uncertainties associated with the ground model and the geotechnical model. Research in terms of determining the worthwhile nature of this effort has been undertaken by
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Tang (1971) who pioneered the evaluation of VOI for design of shallow foundations, and Angulo and Tang (1999) who designed the optimal groundwater detection monitoring system
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based on VOI analysis. Gilbert and Habibi (2014) assessed the VOI in the design of site investigation
and
construction
quality
assurance
programs.
Schweckendiek
and
Vrouwenvelder (2015) studied the VOI in the retrofitting of flood defenses. Najjar et al. (2017) suggested a decision- making framework to evaluate the value of load test on the design of pile foundations. Yoshida et al. (2018) studied the placement of additional borings as a liquefaction countermeasure for an embankment along a river. These studies provide excellent examples on evaluating the value of reducing various types of uncertainties in 32
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geotechnical decision-making.
6. Needs for Future Research
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In the previous sections, we have illustrated how probabilistic methods can be used to characterize uncertainty, evaluate the impact of uncertainty on geotechnical prediction,
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reduce uncertainty with additional data, and assess the value of uncertainty reduction.
SC
Nevertheless, these methods are mainly developed in the geotechnical profession, where the
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geological uncertainty has not been adequately considered or treated with the same degree of rigor as in the case of ground and geotechnical uncertainties. Indeed, the geologic
MA
uncertainties are seldom considered in the current reliability-based geotechnical design
ED
(Keaton 2013), which has been an important missing element in the probabilistic geotechnical analysis. Further, although the probabilistic methods have been widely used in
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geotechnical engineering, the use of these methods in geological engineering is still in its infancy. In light of these situations, a few urgent research needs are summarized below.
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(1) Geologists are encouraged to develop quantitative engineering geologic models with explicit consideration of the geologic uncertainties in a way that is easy-to-use and to access by geotechnical engineers. In the infinite slope example illustrated previously, for example, the geologic uncertainty is considered through the uncertain thickness of the soil layer. In reality, the geologic uncertainty may be much more complex. Most of the studies reported in the literature, however, are not relevant to geotechnical applications, and their use in the geotechnical analysis has not been clearly defined. 33
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There is an urgent need to characterize the geologic uncertainties for use in geotechnical engineering. (2) The probabilistic methods are relatively well established in the geotechnical
PT
profession to analyze the impact of uncertainties on the geotechnical design and performance predictions. In principle, these probabilistic methods should provide
RI
convenient tools for engineering geologists to consider relevant uncertainties in the
SC
geologic model. Nevertheless, the effectiveness of these methods in treating the
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geologic uncertainties has not yet been proven. Research to determine whether these methods are equally applicable to the analysis of geologic uncertainties is needed. If
MA
they are deemed unsuitable, new probabilistic methods for the analysis of geologic
ED
uncertainties should be developed, and the role of geological uncertainty in a probabilistic analysis must be further elucidated.
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(3) There is a need to collect and document case histories that demonstrate the use of probabilistic methods in the analysis of geologic uncertainties and the risk
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quantification and decision making in engineering geology. The detailed and well-documented case histories may offer a basis for confirming and/or enhancing the applicability of probabilistic methods in the fields of geotechnical and geological engineering. (4) In recent years, the machine learning (ML) technology, which focuses on automatically learning from machine readable data by computational and statistical methods, is empowering the research and practice in many fields (LeCun et al.,2015; 34
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Lary et al.,2016). Uncertainty quantification, uncertainty reduction and reliability assessment in geological and geotechnical engineering requires learning from data in different forms, such as engineering geology map, site investigation report, and
PT
monitored data. ML may potentially significantly enhance data learning process for such purposes. Indeed, the multiple point geostatistics has been shown as an effective
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ML tool to model subsurface heterogeneity (e.g., de Vries et al.,2009). Recently, a
SC
pioneering work for uncertainty characterizing of soil parameters through Bayesian
NU
ML was reported in Ching and Phoon (2019). Nevertheless, the ML related research in geological and geotechnical engineering significantly lags behind those in many
MA
other fields, and how to leverage the full potential of ML such that uncertainties in
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geotechnical and geological engineering can be more efficiently and more effectively
Concluding Remarks
The research reported in this paper can be summarized with the following remarks:
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7.
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addressed needs further research.
(1) The geologic model, the ground model, and the geotechnical model are typically involved in a geotechnical analysis. As they are usually associated with considerable level of uncertainty, any geotechnical analysis would benefit from a careful consideration of the uncertainties of all three. (2) The probabilistic methods developed in the geotechnical profession are versatile tools for uncertainty characterization, assessment of impact of uncertainties, 35
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uncertainty reduction, and evaluation of the value of uncertainty reduction. It is quite possible to use these methods for a unified treatment of various types of uncertainties, from geotechnical to geological, in a geotechnical analysis. In this
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regard, the current probabilistic methods that are widely used in geotechnical engineering should be further examined to determine their applicability in the
RI
evaluation of geologic uncertainties.
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(3) Geologists are encouraged to develop quantitative engineering geologic models with
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explicit consideration of the geologic uncertainties in a way that is easy-to-use and to access by geotechnical engineers. To this end, engineering geologists and
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geotechnical engineers are encouraged to collaborate in an effort to comprehensively
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Acknowledgements
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characterize various types of uncertainties consistently.
The first co-author wishes to acknowledge the support of National Science Foundation
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through grants CMMI-0300198 & CMMI-1200117; the second co-author and fourth co-author wish to acknowledge the support of the National Science of Foundation of China (NSFC) through Project No. 41672276; and the third co-author wishes to acknowledge the financial support from the China Scholarship Council (CSC) and the Glenn Department of Civil Engineering, Clemson University. The authors are grateful to the information provided by Dr. Zijun Cao of Wuhan University, Dr. Dongming Zhang of Tongji University, Dr. Peng Zeng of Chengdu University of Technology, and Mr. Xiangrui Duan of Tongji University. 36
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Finally, but not the least, the following individuals are thanked for their contributions to the studies of geotechnical safety and reliability at Clemson University: Drs. Jinxia Chen, Tao Jiang, Haiming Yuan, Hui Yang, Yu-Ting Su, Kun Li, Ye Fang, Cheng-Liang Hsiao, Matthew
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Schuster, Raul Flores, Zhe Luo, Lei Wang, Wenping Gong, and Sara Khoshnevisan.
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Cao, Z., Wang, Y., 2014. Bayesian model comparison and selection of spatial correlation functions for soil parameters. Struct. Saf. 49, 10-17.
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Ching, J. Y., Phoon, K.K., Pan, Y.K., 2017. On characterizing spatially variable soil Young’s Modulus using spatial average, Struct. Saf. 66, 106–117.
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List of Figures
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Fig. 1. Relationship between the geological model, ground model, and geotechnical model [Adapted from Keaton (2013)] Fig. 2. An infinite slope and distribution of the cohesion Fig. 3. Uncertainties in the estimated soil properties Fig. 4. Spreadsheet template for reliability analysis of infinite slope Fig. 5. The typical cross section of the drainage culvert Fig. 6. The numerical model of the drainage culvert: (a) finite difference mesh; and (b) mesh of the reinforced concrete Fig. 7. Geometry of a slope in layered soils [Adapted from Ching et al. (2009)] Fig. 8. Failure mechanisms at two design points: (a) At point {68.725 kPa, 81.068 kPa}; (b) At point {47.795kPa, 146.328 kPa} Fig. 9. The prior and posterior PDFs of h for the case of hp = 3 m as σp varies Fig. 10. The reliability index at m = m1 when μm2 takes various values Fig. 11. Decision tree to determine if the monitoring system should be installed Fig. 12. Decision tree model for the value of information analysis Fig. 13. Impact of the measurement error on the value of information
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Geologic Materials & Properties Surface Water & Ground Water
Geologic Model Landscape Evolution Extreme Events
Site Investigation Ground Description
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Geologic Processes Geologic History
Uncertainties
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Performance Safety & Risk
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Fig. 1. Relationship between geological model, ground model, and geotechnical model [Adapted from Keaton (2013)]
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c (kPa) 5.0 6.0 7.0 8.0 9.0 10.0 0.0 ε0
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Deterministic variables 3 δ (o) h w (m) γ s (kN/m ) 35 1.825 19 Random variables o c (kPa) φ () x 7.572 36.049 μx 10 38 σx 2 2 y* -1.214 -0.976
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m 0.525 0.5 0.05 0.501
h (m) 3.476 3 0.6 0.793
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Notes: (1)This spreadsheet calculates the reliability index of the infinite slope. (2)The setting in Solver is "minimize N6 by changing values in cells E13:I13, subjected to M6 = 0"
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cu1 , Unit weight = 19.5 kN/m3
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Fig. 7. The geometry of a slope in layered soils [Adapted from Ching et al. (2009)]
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(b)
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Decision alternatives
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Outcome of investigation
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Highlights
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The uncertainties associated with geologic, ground and geotechnical models are analyzed Probabilistic methods can be used to characterize these uncertainties Reliability methods can be used to consider the effect of these uncertainties The Bayesian method can be used for uncertainty reduction The value of uncertainty reduction can be assessed through decision analysis
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1. 2. 3. 4. 5.
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C. Hsein Juang, Jie Zhang, Mengfen Shen, Jinzheng Hu PII: DOI: Reference:
S0013-7952(18)31443-1 https://doi.org/10.1016/j.enggeo.2018.12.010 ENGEO 5014
To appear in:
Engineering Geology
Received date: Revised date: Accepted date:
22 August 2018 10 December 2018 12 December 2018
Please cite this article as: C. Hsein Juang, Jie Zhang, Mengfen Shen, Jinzheng Hu , Probabilistic methods for unified treatment of geotechnical and geological uncertainties in a geotechnical analysis. Engeo (2018), https://doi.org/10.1016/j.enggeo.2018.12.010
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ACCEPTED MANUSCRIPT Probabilistic methods for unified treatment of geotechnical and geological uncertainties in a geotechnical analysis C. Hsein Juang1,2 , Jie Zhang3* , Mengfen Shen2 , and Jinzheng Hu3 Abstract: There are typically three types of models involved in a geotechnical analysis, i.e.,
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the geologic model, the ground model, and the geotechnical model, each of which may be associated with uncertainty to a different extent. However, in a typical geotechnical analysis
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the geological uncertainty is seldom characterized and considered explicitly. In this paper, we
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discuss the probabilistic tools developed in the geotechnical profession for tasks such as
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uncertainty characterization, assessment of impact of uncertainties, uncertainty reduction, and evaluation of the value of uncertainty reduction. We postulate that the same tools may be
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used for a unified treatment of various types of uncertainties in a geotechnical analysis, which
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would necessitate the collaboration of the engineering geologists and the geotechnical engineers to comprehensively characterize these uncertainties consistently. To this end, the
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probabilistic tools or methods that are widely used in geotechnical engineering are examined to determine their applicability in the evaluation of geological uncertainties. Examples are
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provided to illustrate various concepts introduced in this paper. Keywords : Geotechnical uncertainty;
Geological uncertainty; Ground
uncertainty;
Reliability; Probabilistic methods __________________ 1
Departmnet of Civil Engineering, National Central University, No. 300, Zhongda Road, Zhongli District, Taoyuan City, Taiwan 320. 2 Glenn Department of Civil Engineering, Clemson University, SC 29634, USA. 3 Department of Geotechnical Engineering and Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, 1239 Siping Road, Shanghai 200092, China. *
Corresponding author: J. Zhang (email: [email protected])
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1.
Introduction Geologists and engineers view the world in complementary but different perspectives.
While geologists tend to seek explanation about the observed details, engineers tend to focus
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on design to meet specific objectives and multiple constraints (Keaton, 2013, 2014). The successful implementation of a geotechnical analysis, however, often requires the joint efforts
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of both engineering geologists and geotechnical engineers. As shown in Fig. 1, a geotechnical
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analysis is typically conducted based on three types o f models, i.e., the geologic model, the
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ground model, and the geotechnical model (Burland, 1987; Morgenstern, 2000; Knill, 2003; Sullivan, 2010; Keaton, 2013). In this figure, the geologic model is a representation of the
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site geologic conditions relevant to the proposed project, the ground model is the geologic
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model expressed in terms of engineering parameters, and the geotechnical model is the ground model with design parameters used to establish predicted performance of the
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proposed project. In practice, the geologic model is often constructed by engineering geologists, the geotechnical model is often built by geotechnical engineers, and the ground
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model is constructed through the joint efforts of both engineering geologists and geotechnical engineers. Consequently, both the geologic and the ground models are essential for any reasonable geotechnical analysis. If either is incorrect, the outcome of the geotechnical analysis is likely incorrect. Given that a model represents only an abstraction of the complex real world, model uncertainties always exist. For example, Einstein and Baecher (1982) found uncertainty as perhaps the most distinctive characteristic of engineering geology compared to other 2
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engineering fields, which is evidenced by the widespread use of engineering judgement. Nevertheless, many practicing geologists are not familiar with methods of quantifying and modeling uncertainty inherent in geology (Bardossy and Fodor 2001), although practically all
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geological statements, measurements, and calculations contain some type and degree of uncertainty. On the other hand, probabilistic methods for modeling uncertainties have been
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extensively studied in geotechnical engineering over the past few decades (e.g., Whitman,
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1984; Baecher and Christian, 2003; Fenton and Griffiths, 2008; Phoon and Ching, 2015).
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While there is an increasing interest on characterizing the uncertainty associated with the soil stratification (e.g., Qi et al., 2016; Wang et al., 2018a), current probabilistic geotechnical
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analysis mainly focuses on the uncertainties associated with the ground model and
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geotechnical model, and the relevant geological uncertainties are seldom addressed. The objective of this paper is to illustrate how the probabilistic methods developed in the
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geotechnical profession may provide a consistent framework for comprehensive treatment of all uncertainties, including the geological uncertainty, in a geotechnical analysis, thus serving
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as a useful guide on how to address the geological uncertainty in a geotechnical analysis. The structure of this paper is as follows. First, the three types of uncertainties are described in detail together with the methods to characterize such uncertainties. Next, the reliability theory is introduced to assess the effect of uncertainties on a geotechnical analysis, followed by the use of additional information to reduce uncertainty. Thereafter, the evaluation of the value of uncertainty reduction in a decision making framework is then discussed. Finally, a discussion of the need for further research is provided. 3
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2.
Geological Uncertainty, Ground Uncertainty, and Geotechnical Uncertainty
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2.1 Uncertainty associated with the geological model Practically all geological statements, measurements, and calculations contain some type
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and degree of uncertainty (Bardossy and Fodor, 2001). The most useful engineering
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geological models characterize uncertainties and unknowns for inclusion into the project
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analyses or for the preparation of a contingency in the project cost estimation to cover the associated risks (Parry et al., 2014). As an example, Fig. 2 shows the infinite-slope model that
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is often used for the assessment of shallow-slope stability and evaluations of regional slope
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hazards (e.g., Crosta and Frattini, 2003; Babu and Murthy, 2005). When the infinite slope model is applied on a regional scale, it is often difficult to determine the slope angle (δ), the
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thickness of the soil layer (h), and the groundwater level (hw ) accurately. In such a case, the uncertainties associated with parameters δ, h, and hw are examples of the geological
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uncertainties. Note that the infinite slope model may not be suitable for each slope in a region. If the infinite slope model is adopted because of the difficulties to determine the soil stratigraphy, the uncertainty caused by the infinite slope can be considered as part of the geological uncertainty. The construction of an engineering geological model requires the specification of two features, namely, the composition of the ground and the geological boundary conditions (Bock 2006). Traditionally, geologists use the line pattern (solid, dashed, dotted, queried) to 4
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represent scientific confidence of stratigraphic or structural contacts that is understood by other geologists, but it is neither quantitative nor understood by non-geologists (Keaton, 2013). A fundamental question in engineering geology is whether the geologic models should
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be constructed using a quantitative method, i.e., whether numbers should be assigned to engineering geologic models. Indeed, many geologists are not comfortable with the
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requirement to assign numbers to geology (Hoek, 1999). Some geologists also argue that
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geological uncertainties are too great for quantification, which is like objections for
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estimating large numbers because of their magnitude (Fookes, 1997). However, the construction of both the ground model and the geotechnical model requires a quantitative
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geological model in which the uncertainties can be quantified in a manner that is consistent
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across the geological, the ground, and the geotechnical model. Consistent treatment of the three types of uncertainties can facilitate the communication of uncertainties among
1997).
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engineers of different fields and other parties involved in the decision making (Fookes,
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Three types of uncertainties are associated with the development of the geological model (Mann, 1993): (1) Type I uncertainty, i.e., imprecision and measurement error associated with the raw data, like the exact position of a formation boundary measured through a borehole with discontinuous measurements; (2) Type II uncertainty, i.e., the uncertainty resulting from the inherent variability of the natural process, such as the variation of the thickness of a soil layer; and (3) Type III uncertainty, i.e., the uncertainty from the lack of scientific data, such as the presence of weak layer or a structural existence. Wellmann et al. (2010) suggested a 5
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method for considering the effect of uncertainty in the raw data (i.e., Type I uncertainty) on the construction of geological models. For Type II uncertainty, which generally results from spatial variability of the natural process, it can often be modeled with geostatistics (e.g.,
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Marinoni, 2003; Tacher et al., 2006). For the third type of uncertainty, Tang (1987), Tang and Halim (1988), and Halim and Tang (1993) used the reliability theory to detect anomalies in
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the ground. Wellmann and Regenauer-Lie (2012) suggested the use of the information
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entropy as a quantitative measure of the quality of a geological model and its geological
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subunits. Despite these efforts, there is as yet no comprehensive approach to assess all the uncertainties in a geological model (Turner, 2006). Consequently, engineering geologists are
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often forced to estimate such uncertainties with engineering judgment even in critical
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applications such as risk assessments of nuclear waste repositories (Mann, 1993). In geotechnical engineering, several probabilistic approaches have been developed to
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identify the soil stratigraphy using data obtained from in-situ testing (e.g., cone penetration test) and/or boreholes, i.e., the conventional statistical analysis based on modified Bartlett
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statistics (Phoon et al., 2003), the Clustering Method (Liao and Mayne, 2007), Bayesian approaches (Cao and Wang, 2013; Li et al., 2013; Wang et al., 2014a), and the Wavelet Transform Modulus Maxima method (Ching et al., 2015). In general, while these approaches provide the “best” estimate on the soil stratigraphy based on prescribed criterion, they provide little information on the uncertainty (or degree-of-belief) in the estimated soil stratigraphy. Recently, Zheng et al. (2018) developed a Bayesian framework for quantifying the identification uncertainty in the soil stratigraphy inferred from the soil behaviour type 6
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index. Using the identified soil stratigraphy at testing locations, it is then possible to estimate the soil stratigraphy at locations without testing data and its associated uncertainty. Traditionally, two types of model are commonly used for such task; they are boundary-based
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and category-based models. The boundary-based model, directly and probabilistically, delineates soil layer boundaries at the locations without testing data (Sitharam et al., 2008;
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Dasaka and Zhang, 2012; Schöbi and Sudret, 2015; Li et al., 2016 a; Xiao et al., 2017). In
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contrast, the category-based model is used to elucidate the soil stratigraphy through
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predicting material types at the locations without testing data (Elfeki and Dekking, 2001; Li, 2007; Park, 2010; Li et al., 2016b; Qi et al., 2016), thus providing a more flexible tool to
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simulate complicated soil stratigraphy (e.g., embedded soil layers). Recently, Wang et al.
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category-based methods for soil stratigraphy identification using cone penetration test (CPT)
Geologic maps are often made to satisfy a wide range of needs. An engineering geology
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map is an artistic representation of one interpretation of geologic features and relationships inferred from limited observations of the distribution of rock types, surficial deposits, and geologic structures often with little or no subsurface data or laboratory test results (Keaton 2013). Currently, efforts for quantification of the uncertainty associated with a geologic map which can be directly used by geotechnical engineers are very few. Keaton (2013) suggested a Geologic Model Complexity Rating System (GMCRS) that may be useful in quantifying variability and uncertainty in geologic mapping. GMCRS has nine components, four of which 7
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are related to regional-scale geologic complexity and five of which pertain to site-scale complexity, terrain characteristics, information quality, geologist competency, and time allotted to prepare the model. The Analytic Hierarchy Process method is used to derive a
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cumulative score, which can be converted into a coefficient of variation (COV) to be
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conveniently used by geotechnical engineers.
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2.2. Uncertainty associated with the ground model
The ground model in Fig. 1 is the geologic model expressed in terms of engineering
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parameters. One important feature of the soil properties is that even in the same soil layer, the
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soil properties at different points are not necessarily the same, i.e., the soil properties are spatially variable. In general, the closer the two points are in space, the greater the correlation
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of the soil properties at the two points. The spatial variability of the soil properties can often
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be modeled through the random field theory (e.g., Vanmarcke, 1983; Fenton, 1999a; Griffiths et al., 2009; Zhu and Zhang, 2013; Gong et al., 2018). For example, the spatially variable
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cohesion of the soil, su ( z ) , in Fig. 2 may be modeled as a depth-dependent trend function t(z) and a stationary random field w(z) with a mean of zero as follows: su z t z w z
(1) where z = depth of the soil. Note although Eq. (1) is conceptually simple, it might be difficult to properly divide the trend function and stationary random field from real measurement data as there might be multiple choices for the trend function (e.g., Fenton et al.,2018).
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Three types of uncertainties are associated with the estimated soil property, i.e., inherent variability, measurement error, and transformation uncertainty (Kulhawy and Phoon 2002). The uncertainty associated with the soil property depends on the estimation procedure.
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Following the concept proposed in Kulhawy and Phoon (2002), Fig. 3 shows the results of uncertainties associated with the estimated soil properties when using different types of tests.
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When the soil properties are directly estimated based on laboratory tests, the estimated soil
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property may be subjected to the inherent variability of the soil, the sampling disturbance, the
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measurement error due to imperfect test, and statistical uncertainty due to limited number of tests. When the soil properties are estimated based on in-situ tests through a transformation
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model, the estimated soil property may be further subjected to the error associated with the
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transformation model.
The moment methods (e.g., Tang, 1979; Lacasse and Nadim, 1996; Uzielli et al., 2005),
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maximum likelihood methods (e.g., DeGroot and Baecher, 1993; Fenton, 1999b) and Bayesian approaches (e.g., Wang et al., 2010a; Cao and Wang, 2014; Ching et al., 2016a;
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Tian et al., 2016) have been used to characterize the spatial variability of soil properties. Most previous research entailed decomposing and characterizing the spatial variability into individual vertical and horizontal components (Jaksa, 1995; Stuedlein et al., 2012). Unlike the characterization of vertical spatial variability, a greater challenge lies in characterizing the horizontal spatial variability due to the sparsity of test data in the horizontal direction. Unfortunately, only a few studies have been undertaken to simultaneously address the vertical and horizontal spatial variability using limited test data measured at different 9
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locations in a soil layer (Pardo-Igúzquiza and Dowd, 1997; Xiao et al., 2018). One practical challenge regarding the application of the random field theory is the insufficiency of site investigation data for estimating the parameters of different random
PT
fields, particularly the cross-correlation among those fields. Recent efforts have been made to calibrate random field models based on sparsely measured geotechnical data (e.g., Wang et al.,
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2018c; Zhao and Wang, 2018). On the other hand, the performance of a geotechnical
SC
structure is often governed by the average property over a path or a region. As the uncertainty
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caused by the spatial variability tends to cancel out, the uncertainty associated with the average property is generally smaller than the uncertainty at a point (Vanmarcke, 1983). In
MA
such a case, one may regard the average property as a random variable, and evaluate the
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reliability of a geotechnical system approximately through the average property using the random variable approach. Vanmarcke (1983) provided equations on how to assess the
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statistics of the average property over a prescribed region. As illustrated recently in Ching and Phoon (2013) and Ching et al. (2016b, 2017, 2018), when deriving the statistics of the
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spatial average, the spatial average should be performed along the actual failure path, which in many cases may be uncertain. When the statistics of the average property can be readily established, the random variable approach can be considered as an adequate compromise between the need to consider the spatial variability of the soil and the difficulty involved in reliability analysis directly using the random field theory. In geological and geotechnical engineering, the number of samples tested is often too small to use statistics built based on the law of large samples. Christian (2004) and Keaton 10
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and Ponnaboyina (2014) suggested that small sample statistics can be employed to estimate the uncertain soil properties. From a Bayesian perspective, a small number of samples may also be very valuable in that it may potentially lead to significant reduction in the uncertainty
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associated with the prior knowledge.
(e.g. Wu and Abdel-Latif, 2000):
h hw sat hw sin cos
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(2)
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g θ
c h hw sat w hw cos 2 tan
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For the infinite slope in Fig. 2, its factor of safety (FOS) may be computed as follows
MA
where γw, γsat, and γ = unit weights of water, saturated soil, and the soil above the water table, respectively; c = effective cohesion of the soil; and φ = effective friction angle of the soil. For
ED
the infinite slope in Fig. 2, the uncertainty associated with the ground model is represented by
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the uncertainty associated with the unit weights of the soil and the soil strength parameters.
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2.3. Uncertainty associated with the geotechnical model Almost any geotechnical model is associated with some assumptions and limitations, thusly, the model uncertainty almost always exists (Ang and Tang, 1984; Phoon and Kulhawy, 1999). A geotechnical analysis might be biased when the model uncertainty is ignored (Lacasse and Nadim, 1994; Gilbert and Tang, 1995). The model uncertainty can often be characterized by a model factor which represents the difference between, or the ratio of, the actual and predicted performances (Juang et al., 2004, 2011, 2013a; Phoon and Kulhawy,
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2005; Zhang et al., 2009a; Wang et al., 2012), which is applied using either an additive or multiplicative method. Take the geotechnical model given by Eq. (2) as an example. Let g(θ) denote the slope stability model to predict the FOS, in which θ denotes the uncertain
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variables. When a model correction factor is applied in an additive fashion, the FOS, denoted as FS , can be expressed as
(3)
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FS g θ
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In principle, the model factor can be computed by comparing the measured
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performances with the model predictions. The difficulty of an accurate determination of the soil strength parameters almost always leads to an uncertain model prediction. The observed
MA
geotechnical performance may also be uncertain. For example, if the pile is not loaded to
ED
failure during a load test, the bearing capacity of the pile foundation is not exactly known. In such a case, the geotechnical model uncertainty is mixed with uncertainty in both the soil
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strength parameters and the geotechnical performance (e.g., Gilbert and Tang 1995). Zhang et al. (2009a, 2012) developed a method for determining the uncertainty associated with a
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geotechnical model in such scenario. Assume that is a normal variable with a mean of and a standard deviation of . Let f(, ) denote the prior probability density function (PDF) of and . Based on Bayes’ theorem, the posterior PDF of and can be evaluated as follows (Zhang et al.,2009a): n
f , | d kf , i 1
1
2 2 g2( i )
1 di g ( i ) exp 2 2 g2( i )
2
(4)
where k is a normalization constant to make the derived posterior PDF valid ; di is the ith 12
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observation; and g( i) and g( i) are the mean and standard deviation of g( i) caused by uncertainties in i. Past studies indicated that a small level of parameter uncertainty and observation
PT
uncertainty generally results in a negligible impact on the geotechnical model uncertainty characterization (Zhang et al., 2009a, 2012). In such a case, the model uncertainty can be
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characterized approximately without considering the parameter and observation uncertainties.
SC
For slope stability problems, for example, Duncan and Wright (1980) determined that any
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limit equilibrium method satisfying all conditions of equilibrium could predict the FOS within an accuracy of 5% from what is considered the correct answer. Zhang et al. (2009a)
MA
also derived similar conclusions through the calibration of the model uncertainty of four
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commonly used limit equilibrium models using centrifuge tests data. Gong et al. (2016a) evaluated and calibrated empirical models from the angle of model
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fidelity (i.e., goodness of fit to the calibration data) versus model robustness (i.e., robustness of prediction for future applications). They found that high fidelity models often lacked the
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desirable level of model robustness, and a balanced approach is desired. Juang et al. (2018) studied the model selection in geological and geotechnical engineering in the face of uncertainty, and showed that simple models might yield more robust predictions compared with complex and sophisticated models. The model factor is often assumed as independent of the model input parameters. However, Zhang et al. (2015a) found that the model factor could be correlated with the model input parameters. The systematic bias of such correlation must be removed prior to 13
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conducting the full probabilistic analysis. The multiple regression equation was found as an efficient solution for characterizing the parameter correlation (Kung et al., 2007; Schuster et al., 2009; Zhang et al., 2015a; Tang and Phoon, 2016; Qi and Zhou, 2017). One may refer to
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Dithinde et al. (2016) for a more comprehensive review on methods for characterization of
Impact of Uncertainties on Performance Predictions
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3.
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the geotechnical model uncertainty.
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3.1. First order reliability analysis (FORM) for explicit performance function The uncertainties involved in a geotechnical analysis lead to uncertain predictions from
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such analysis, making it difficult to determine with certainty the true safety level of the
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geotechnical system. Probabilistic analysis, often through reliability analysis that considers these uncertainties explicitly, could provide complementary information to those obtained
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from the traditional deterministic analysis (Duncan, 2000). Rackwitz (2000) noted that 90% of the reliability problems can be accurately solved with the first order reliability method
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(FORM), in which the safety or performance of a system is measured with a reliability index, which is defined as the minimum distance between the origin and the limit state surface in the uncorrelated reduced space. Let x denote the uncertain variables involved in a geotechnical analysis. For the geotechnical model as given by Eq. (3), x = {, }. Let G(x) = 0 denote the limit state function. With FORM, the failure probability pf is calculated as follows (e.g., Low and Tang, 2007): p f 1
(5) 14
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min yR 1yT
(6)
g x 0
where Φ = the cumulative distribution function (CDF) of the standard normal variable, β= the reliability index, y = the reduced variables of x, and R = the correlation matrix of the
PT
random variables. Traditionally, the application of probabilistic methods has been quite tedious. Over the
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past decades, significant improvements have been made regarding the practical application of
SC
reliability methods in geotechnical engineering. When the limit state function is explicit,
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FORM can be efficiently executed in a spreadsheet (Low and Tang 1997, 2007; Juang et al., 2006). Low (2015) provided detailed examples on how the spreadsheet method can be used
MA
to evaluate the reliability of different types of geotechnical problems.
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As an illustration, we evaluate the failure probability of the slope in Fig. 2 using the spreadsheet method. In this example, γw = 9.8 kN/m3 γsat, = 19 kN/m3 , γ = 19 kN/m3 , and δ =
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35°. The variables c and φ are both assumed to be normally distributed having mean values of 10 kPa and 38°, respectively, and standard deviations of 2 kPa and 2°, respectively. Let m =
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hw/h denote the normalized water table and let m denote the maximum groundwater table within a certain design period. Suppose that h is normally distributed with a mean of 3.0 m and a standard deviation of 0.6 m and that m is normally distributed with a mean of 0.5 and a standard deviation of 0.05. The spreadsheet for evaluating the failure probability of the infinite slope based on FORM is shown in Fig. 4. Solving the optimization problem in Eq. (6) with the Solver in Excel, yields a probability of failure of the slope of 0.033. The minimization point in Eq. (6) is often called the design point. The coordinates of the 15
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design point in the reduced space illustrates the relative importance of different variables to the reliability of the geotechnical system. The greater the deviation from zero, the greater the effect upon the reliability of the geotechnical system. For the current slope example, the
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coordinates of c, φ, m, h, and ε are -1.214, -0.976, 0.501, 0.793, and -0.286, respectively. In this example, h and m characterize the geological uncertainty, c, and φ characterize the
RI
uncertainty associated with the ground model, and ε characterize the geotechnical model
SC
uncertainty. From the coordinates of the design point we can see that the reliability of the
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current slope is most affected by the ground model uncertainty, followed by the geological uncertainty, and then by the geotechnical model uncertainty. Therefore, the reliability analysis
ED
MA
can be used to assess the relative importance of uncertainties of different types.
3.2. FORM for implicit performance function
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When the limit state or performance function is implicit, as in the case of stability analysis of a soil slope using a numerical method-based program, the implementation of
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FORM is challenging due to the coupling between the numerical program and FORM. To deal with such a challenge, response surface methods (RSM) are widely used. The basic idea of the RSM is to approximate the performance function with a surrogate model that is computationally more efficient, followed by reliability analysis based on the surrogate model. For example, a second-order polynomial in the y space can be used to approximate the performance function (Zhang et al., 2015b; Khoshnevisan et al., 2017):
16
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G(y ) b0 bi yi i 1
2k
by
i k 1
i
2 i
(7)
where bi (i = 0, 1, 2, , 2k + 1) = the coefficients to be determined. In Eq. (7), the coefficients can be determined based on the response of G(y) at the following 2n + 1 points: yc = {yc1 , yc2 , …, ycn }, {yc1 ± y1 , yc2 , …, ycn}, {yc1 , yc2 ± y2 , …, ycn }, …, and {yc1 , yc2 , …, ycr ± yn },
PT
where yi = standard deviation of yi. The approximated performance function Eq. (7) is then
RI
used for the reliability analysis. If the performance function cannot be well approximated by
SC
a second-order polynomial, the reliability index obtained based on Eq. (7) may not be
NU
accurate. In such a case, a new set of calibration points are used to update the response surface with the centre determined as follows (Zhang et al.,2015b):
y DG μ y
MA
yc
G μy G y D
ED
(8)
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Equation (8) is again calibrated using the G(y) responses evaluated at the new calibration points, permitting a reliability analysis based on the updated response surface. Such a process
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is iterated until the obtained reliability index remains stable within a tolerable degree of error. As illustrated in Zhang et al. (2015b), this procedure is easily implemented based on existing deterministic geotechnical programs such as FLAC 3D. The reliability of a reinforced concrete drainage culvert in Shanghai, China is provided as an example, as shown in Fig. 5. The cohesion, the friction angle, and Young’s modulus of the soil are molded as random variables, and their mean values are 22 kPa, 19o , and 5.78 MPa, respectively and their COVs are 0.3, 0.1, and 0.2, respectively. The correlation between the
17
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cohesion and the friction angle is assumed as -0.5. For simplicity, the three random variables are assumed lognormally distributed. As the structural safety of the culvert is controlled by bending failure at the mid-span of the top slab, the performance function can be written as follows:
PT
g ( x) M R M S
(9)
RI
where MR denotes the flexure resistance, and M S denotes the bending moment. In this
SC
example, FLAC 3D is used to calculate M S. The finite difference mesh of the numerical model
NU
is shown in Fig. 6(a). Based on the plane strain assumption, only a unit length of the culvert is analyzed. The concrete culvert is modeled as elastic shell element (DKT-CST) embedded
MA
in FLAC3D that resists both membrane and bending loading. The mesh of the reinforced
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concrete is shown in Fig. 6(b) and the soil was modeled as the Mohr–Coulomb plasticity material. Based on the RSM as mentioned above, the reliability index of the culvert is 3.303,
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and the corresponding design point is {-2.514, -0.134, -1.419} in the reduced space. For the current problem, the cohesion deviates most from the zero at the design point, followed in
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sequence by Young’s modulus and the friction angle. Therefore, the reliability of the culvert is most affected by uncertainty in the cohesion, followed by uncertainty in Young’s modulus, and is least affected by the uncertainty in the friction angle.
3.3. System reliability analysis based on FORM While FORM is versatile, sometimes a geotechnical system may have several competing failure modes. In such a case, the failure probability of the system would be dominated by 18
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more than one design point and thus, the concept of the system reliability is applicable (e.g., Ang and Tang, 1984; Xu et al., 2014; Reale et al.,2016). MCS may be used to determine the system reliability. Alternatively, the system reliability may be determined based on the results
PT
obtained from FORM. Consider a geotechnical system with r design points. The following equation can be used to calculate the correlation coefficient between design points i and j
RI
(e.g., Low et al., 2011): y *i R 1 y *j
ij
SC
T
i j
(10)
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where βi and βj = reliability indexes of the two design points, respectively; and yi* and yj* =
MA
coordinates of design points i and j in the reduced space, respectively. The system reliability index can then be calculated as follows (e.g., Zeng and Jimenez, 2014): r
r
r
r
r
r
i 1
i 1 j i
ED
Pfs = Pi Pij Pijk i 1 j i k j
r
r
r
Pi Pij i 1
(11)
i 1 j i
(12)
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P12-n Φn 1， 2， ， n ;R
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where n (-1 , -2 , …, -n ; R) is the cumulative distribution function of a r-dimensional multivariate normal distribution, and R is the correlation matrix with the element in the ith row and jth column being the correlation coefficient (ρij) between design points i and j. As an illustration, Fig. 7 shows a slope in layered soils (Ching et al., 2009). The undrained shear strength of the two clay layers, i.e., cu1 and cu2 , are modeled as independent lognormal random variables. The mean values of cu1 and cu2 are 120 kPa and 160 kPa, respectively and the standard deviations of cu1 and cu2 are 36 kPa and 48 kPa, respectively. 19
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Figs. 8(a) and 8(b) show the two major failure modes controlling the stability of the slope identified based on the shear strength reduction method, respectively (Ma et al.,2017). The reliability indexes of the two slip surfaces are 2.881 and 2.938, respectively and the correlation coefficient between the two failure modes is 0.496. Based on Eqs. (10) & (11), the
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system failure probability of the slope is then 1.198×10-4 .
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Instead of providing a point estimate of the system failure probability, the outcomes
SC
from FORM are also used to derive the bounds of the system failure probability using
NU
methods like Ditlevsen’s bounds (Ditlevsen, 1979). One may refer to Low et al. (2011) and Lü et al. (2013) for examples of estimating bounds of the system failure probability based on
ED
MA
FORM for different geotechnical problems.
3.4.Other reliability analysis methods
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MCS is a robust and versatile method for reliability analysis. With the rapid development of computing technology and commonly available personal computer, some
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simple geotechnical reliability analyses, such as pile or footing design, can be easily done by MCS even in spreadsheet (e.g., Wang, 2011; Wang and Cao, 2013). However, when the limit state function lacks an explicit form, the application of MCS could be computationally intensive. Many efforts have also made to enhance the computational efficiency of reliability analysis. Among these studies, the RSM has attracted substantial attention. As the second-order polynomial can only fit the limit state function locally, RSMs with global fitting capability have been widely studied, such as those based on artificial neural network (Cho, 20
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2009; Juang et al., 2000; Lü et al., 2012), kriging (Zhang et al., 2011a), Gaussian process regression (Kang et al., 2015), support vector regression (SVR) (Kang and Li, 2015), multivariate adaptive regression splines (Metya et al., 2017), and auxiliary random finite
PT
element method (Xiao et al., 2016). Li et al. (2016c) assessed various RSMs for system reliability of soil slopes. Sudret (2012) reviewed and compared various RSMs developed for
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structural reliability analysis. Several commonly used RSMs have been implemented in the
SC
software UQLab for general purpose structural reliability analysis (Marelli and Sudret 2014).
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In general, the more advanced RSMs often need more calibration points and hence computationally more demanding than the simple second-order polynomial based RSM.
MA
However, when the system failure probability is significantly greater than the failure
ED
probability evaluated at the most probable failure point, such methods are essential and useful. Other researchers have resorted to more efficient sampling methods for geotechnical
EP T
reliability analysis, such as the importance sampling (e.g., Ching et al., 2009, 2010), subset simulation (e.g., Wang et al., 2010b; Li et al., 2016d, 2017), subdomain sampling (Juang et al.,
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2017), and integration-based method (Gong et al., 2016b&c).
3.5. Reliability analysis considering model selection uncertainty For the same geotechnical problem, it is possible to derive several competing models, particularly geological models. When the data is insufficient to determine the best model uniquely, model selection uncertainty exists. The probabilistic method can be used to consider this selection uncertainty. For example, suppose there are r competing models for 21
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the same problem, z is variable to be predicted, Mi is the ith model, p(z|Mi) is the distribution of z predicted based on model Mi, and P(Mi) is the probability that Mi is the correct model used. Based on the total probability theory, the predicted distribution of z, i.e., p(z),
PT
considering all competing models can be written as follows (e.g., Hoeting et al.,1999): r
p( z ) p( z | M i ) P( M i )
RI
i 1
SC
(13)
The above method has been used in Zhang and Tang (2009) and Zhang et al. (2014a) to
MA
NU
consider the model selection uncertainty in a geotechnical reliability analysis.
3.6. Reliability-based design
ED
The reliability theory is increasingly used in geotechnical engineering to calibrate design
EP T
codes, particularly in the format of load and resistance factor design (e.g., Honjo et al., 2002; AASHTO ,2007; Yu et al., 2012; Li et al., 2015; Low, 2017). One major challenge in
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geotechnical reliability-based design (RBD) is that the variabity invovled in a geotechnical analysis varies from site to site. Thus, it difficult to predetermine the appropairte COVs that could be invovled in the calibration process, and the resistance factors calibrated may not be unique. It is increasingly recongized now that the resistance factors used in geotechnical engineering shall depend on the degree of understanding of degree of uncertainty at the site (Fenton et al., 2016; Phoon, 2017). Recently, a robust geotechnical design (RGD) methodology has been developed to complement the existing RBD and to address the
22
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geotechnical uncertainties from a different perspective, namely robust design in the face of uncertainties (e.g., Juang et al., 2013c; Khoshnevisan et al., 2014). Although reliability-based design (RBD) is receiving wider and wider acceptance and adoption among the geotechnical
PT
engineer, the uncertainty (or variation) in the estimated statistics of input random variables can lead to the variation in the computed system response of a design, which may lead to an
RI
under-design or over-design relative to the performance requirements. Robust geotechnical
SC
design (RGD) aims to ensure a design that is robust against, or insensitive to, such variation
NU
in the input variables by adjusting the design parameters (such as geometry of a geotechnical structure) without seeking additional reduction of the uncertainties in these input variables. A
MA
design is deemed robust if it has a high design robustness against such unavoidable variation
ED
(called noise factors herein). RGD achieves its goal through multi-objective optimization, typically by maximizing the design robustness and minimizing the cost while enforcing the
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safety requirements (through meeting target reliability indexes). The RGD methodlogy has been showcased in many geotechnical problems (e.g., Juang and Wang, 2013; Juang et al.,
4.
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2013c; Khoshnevisan et al., 2015; Gong et al., 2016d).
Reducing Uncertainty through Additional Information In practice, there are often information from multiple sources. As each source may
provide partial information about the geotechnical system under investigation, combination of multiple sources of information may result in reduced uncertainty. The reliability of the geotechnical system can also be updated as the uncertainty is reduced, which may result in 23
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improved design and decision making. The information combina tion can be formally conducted based on Bayes’ theorem. Suppose the prior knowledge about , variables of interest (such as the shear strength parameters of a clay), can be denoted by a probability density function (PDF), f(). Let D denote the data from another source, and let L(|D)
PT
denote the chance to observe D given , which is often called the likelihood function. One
RI
may refer to Juang et al. (2015) on the construction of the likelihood function using various
SC
types of geotechnical data. Based on Bayes’ theorem, the prior knowledge about and the
L(θ | D) f (θ)
... L(θ | D) f (θ)dθ
(14)
MA
f (θ | D)
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knowledge learned from the data can be combined as follows:
where f(|D) = posterior PDF of representing the combined knowledge. In Eq. (14), both
ED
the likelihood function L(|D) and prior PDF f() affect f(|D), and the role of the likelihood
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function will become more dominant as the amount of data increases. Various examples for application of the Bayesian method in geotechnical engineering can be found in the literature
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(e.g., Honjo et al., 1994; Zhang et al.,2004; Najjar and Gilbert, 2009; Huang et al., 2014; Wang et al., 2012; Wang et al., 2013; Juang et al., 2013b; Peng et al., 2014; Zhang et al., 2014b; Nishimura et al., 2016; Ering and Sivakumar Babu., 2017). Juang and Zhang (2017) provided a tutorial on the formulation and use of Bayesian methods to solve different types of geotechnical problems. A comprehensive review of application of the Bayesian method in geotechnical engineering can be found in Zhang et al. (2017) and an in-depth discussion on the use of Bayesian thinking to solve different geotechnical problems is available in Baecher
24
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(2017). As an illustration, suppose a technique is available for estimating the depth of the soil layer in Fig. 2 in which hp is the prediction from a soil layer thickness estimation method.
PT
Assume the actual depth h is normally distributed with a mean of hp and a standard deviation of σp , which describes the measurement error. Before the soil layer thickness estimation
RI
method is applied, it is known that h is normally distributed with a mean of h and a standard
SC
deviation of h . In this example, μh = 3 m, and σh = 0.6 m. Therefore, the prior PDF of h can
h h 2 1 exp 2 h2 2 h
MA
f h
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be written as follows:
(15)
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can be written as follows:
ED
The chance to observe hp when the value of h is known, i.e., the likelihood function of h
h h 2 p l hp |h exp 2 p2 2 p (16)
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1
Substituting Eqs. (15) & (16) into Eq. (14), the posterior distribution of h is still normal with the following mean and standard deviation (Ang and Tang, 2007):
h p2 hp h2 h h2 p2 (17)
25
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h2 p2 h2 p2
h
(18)
As an example, a comparison of the posterior and prior PDFs of h for the case of hp = 3 m and σp = 0.1 m, 0.2 m, and 0.3 m, respectively is provided in Fig 9. We can see that the
PT
posterior PDFs of h are more sharply peaked than the prior PDF, indicating the uncertainty in
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the thickness of the soil layer has been reduced. The degree of uncertainty reduction also
SC
increases as the precision of the soil layer thickness estimation method increases. Using the updated distributions of the thickness of the soil layer, the failure probability of the slope can
NU
again be estimated, which are 0.021, 0.024, and 0.026, respectively. The additional
MA
information is thus shown able to reduce the uncertainty in a geotechnical analysis and improve our knowledge about the reliability of a geotechnical system. As mentioned
ED
previously, h is an example of geological uncertainty in the infinite slope problem. The
information.
EP T
example here thus illustrates how the geological uncertainty can be reduced through new
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The past performance of a geotechnical system is a very important source of information, which can be viewed as a field test directly performed on the prototype. For the above infinite slope example, the survivability of a slope at a certain state can provide valuable insight into its future performance. Let m1 denote a level of groundwater at which the slope has survived. The presence of observation error, statistical uncertainty, and modeling uncertainty, however may hinder a direct determination of the exact value of m1 . Suppose m 1 is normally distributed with a mean of μm1 and a standard deviation of 0.05. The above performance
26
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information can be used to update the failure probability of the slope when the groundwater is m2 in a future state. Here, let Fs1 and Fs2 denote the FOSs of the slope when the groundwater table is m1 and m2 , respectively. Let pf1 and pf2 denote the failure probabilities of the slope
PT
calculated based on reliability theory when the groundwater tables are m1 and m 2 respectively without considering the slope performance information. The fact that the slope did not fail for
RI
the case of m = m1 indicates that Fs1 > 1. The event that the slope will fail at the case of m =
SC
m2 can be denoted by Fs2 < 1. Let pf2|Fs1>1 denote the probability of failure of the slope for the
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case of m = m 2 in a future state considering such information. Mathematically, pf2|Fs1>1 can be expressed as
P Fs1 1 Fs 2 1
MA
p f 2|Fs11 P Fs 2 1| Fs1 1
P Fs1 1
p f 1 p f 12 1 pf 1
(19)
ED
where pf12 = P(Fs1 < 1 ∩ Fs2 < 1), which is the probability of event “Fs1 < 1 and Fs2 < 1”
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without considering survival information. pf12 can be calculated based on pf1 , pf2 , and the correlation between the design points of the slope for the cases of m = m1 and m = m2 (Zhao
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et al., 2007; Zhang et al., 2011b). The updated reliability index at m = m 1 when μm1 takes various values is shown in Fig 10. The reliability index of the slope considering the survival information is generally greater than that without considering the survival information, indicating that knowing its past survival information increases our confidence in the stability of the slope. As the value of μm1 increases, the increase in the reliability index is more obvious, indicating that the updating effect will be more obvious if the slope survived a more critical state. Although the above phenomenon is quite intuitive, the probabilistic method
27
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illustrated here can be used to quantify the effect of past performance on predicting the future performance of a slope. Other types of information such as field monitored data (i.e., Honjo et al.,1994; Vardon
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et al.,2016) and results from model tests (i.e., Zhang et al.,2009b; Wu et al.,2017) may be available and can be used to update our knowledge about of the geotechnical system under
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investigation. For example, Hsiao et al. (2008) proposed a Bayesian based procedure to
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update the settlement predictions at subsequent stages in a braced excavation using the
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deformation data, and the serviceability reliability of adjacent buildings was updated stage by stage accordingly. Straub (2011) described a structural reliability method to update the failure
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probability of an engineering system based on measured performance in a computationally
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efficient manner. Juang et al. (2013b) proposed a Markov chain Monte Carlo (MCMC) based procedure to update the soil parameters in a staged braced excavation based on field
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observations, and the updated soil parameters were then used to update the deformation predictions in the subsequent stages of excavation. Schweckendiek and Vrouwenvelder (2013)
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applied the method suggested in Straub (2011) to update the reliability of levees based on head monitoring data. Papaioannou and Straub (2015) applied the above approach to update the reliability of a sheet pile wall based on deformation measurement. Wang et al. (2014b) developed a maximum likelihood based procedure to update the predictions of ground deformations and assessment of building damages induced by excavation using the field monitoring data. Lo and Leung (2018) suggested a Bayesian method to update of subsurface spatial variability for improved prediction of braced excavation response. Yang et al. (2018) 28
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suggested a probabilistic back analysis method for characterization of the spatial variability of unsaturated soil properties of a slope based on monitored performance of the slope. As mentioned previously, for the same problem, it is possible that several models may
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be constructed, and yet it is unclear which model is most accurate. The observed performance can also be used to reduce the model selection uncertainty. Let P(Mi ) denote the prior
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probability that model Mi is correct. Let P(D|Mi) denote the probability to observe D if model
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Mi is correct. In the absence of prior knowledge, the non-informative prior can be used, i.e.,
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P(Mi) = 1/r where r is the number of competing models. Based on the data D, the posterior
et al., 1999):
P(D | M i ) P( M i ) r
P(D | M ) P( M ) i
(20)
i
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i 1
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P( M i | D)
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probability that model Mi is correct, i.e., P(Mi|D), can be calculated as follows (e.g., Hoeting
One may refer to Zhang et al. (2014a) and Brownlow (2017) on how to determine the model
5.
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selection uncertainty based on the observed data.
Value of Uncertainty Reduction An advantage of probabilistic analysis is that it can explicitly show the trade-off
between investment and reduction of potential losses, and thus facilitating decision- making in the presence of uncertainties. To make an informed decision in geotechnical practice, the decision tree model (e.g., Ang and Tang 1984) is often used. For example, suppose the loss 29
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due to the slope failure for the infinite slope mentioned previously is $2,500,000. Assume that a monitoring system can be installed with a cost of $50,000, which if installed, can reduce the loss due to the slope failure to $250,000 (one tenth of the loss without the monitoring system).
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The decision tree model in such a case (to determine whether to install the monitoring system) is shown in Fig. 11. Here, two decision alternatives are involved, i.e., installing or not
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installing the monitoring system. Let a1 and a2 denote the alternative that a monitoring
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system will be and will not be installed, respectively. Let Ri denote the risk associated with
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the ith decision alternative. Based on the decision tree, Ri can be computed with the following equation:
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Ri Ci1 p f Ci 2 1 p f
(21)
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where Ci1 and Ci2 are the total costs associated slope failure and no slope failure when ai is adopted, respectively. Note that the total cost of a decision alternative here includes the cost
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of installing the monitoring system, if installed, and the loss due to the slope failure. Thus, C11 = ($250,000 + $50,000) = $300,000, C12 = $50,000, C21 = $2,500,000, and C22 = 0. The
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risk associated with the two decision alternatives are R1 = $300,000 × 0.0.033 + ($50,000) × (1 − 0.033) = $58,250, and R2 = ($2,500,000) × 0.033 + $0 × (1 − 0.033) = $82,500, respectively. If the maximum expected monetary value (EMV) criterion is followed, the decision maker would choose to install the monitoring system, i.e., a1 , as it has a smaller risk. As mentioned previously, the availability of new information may lead to the reduction of the uncertainty, which could result in a more informed engineering decision. As gaining information usually requires additional investment, a decision- maker frequently encounters 30
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the dilemma of determining the worthwhile nature of collecting more information. For the infinite slope example mentioned above, the use of a new technique to determine the thickness of soil layer could reduce the uncertainty regarding this thickness. Before the new
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technique is used, the outcome from this technique is uncertain, making the effect of using the new technique also uncertain. To this end, a value of information (VOI) analysis could be
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conducted to determine whether the use of the new technique is appropriate (e.g. Yokota and
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Thompson, 2004; Quiggin, 2016).
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For example, suppose that the precision of the new technique is σp = 0.1 m and the outcome from the test is hp = 3.0 m. Based on Eqs. (17) and (18), the updated knowledge
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about h is that h is normally distributed with a mean of 3.0 m and a standard deviation of
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0.0986 m. With the updated distribution of h, the failure probability of the slope evaluated based on Eqs. (5) and (6) is Pf = 0.020. In such a case, the risk associated with a1 and a2 are
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$55,000 and $50,000, respectively, and the optimal alternative is a2 with a R2 = $50,000. Compared with the case of not exploring the soil layer thickness further, the risk of slope
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failure has been reduced by $8,250, i.e., the VOI is $8,250. In the above analysis, the VOI is assessed for the case of hp = 3 m. However, before the new technique is used, the value of hp is uncertain. The uncertainty in hp should also be considered in the VOI analysis, which can be formally examined through the decision tree model as shown in Fig. 12. Let a = {a1 , a2 }. Mathematically, the VOI for further investigating the thickness of the soil layer can be computed as follows:
31
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VOI max Ci1Pf Ci 2 1 Pf f hp dhp max Ci1Pf Ci 2 1 Pf hp ai a ai a
(21)
Fig. 13 shows how the VOI varies with σp . As expected, the VOI increases as the precision of the thickness estimation method increases. For the cases of σp = 0.1 m and 1.0 m,
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the values of VOI are $20,407 and $8,421, respectively. The VOI indicates the expected risk reduction achieved through uncertainty reduction using the new soil layer thickness
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economically not feasible to adopt the new technique.
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estimation technique. If the cost employing the new technique exceeds its VOI, then it is
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In this example, the uncertainty in the thickness of the soil layer is an example of the geological uncertainty. It illustrates how the probabilistic theory can be used to assess the
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value of reducing the geological uncertainty. Similarly, it can also be used to assess the value
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of reducing the uncertainties associated with the ground model and the geotechnical model. Research in terms of determining the worthwhile nature of this effort has been undertaken by
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Tang (1971) who pioneered the evaluation of VOI for design of shallow foundations, and Angulo and Tang (1999) who designed the optimal groundwater detection monitoring system
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based on VOI analysis. Gilbert and Habibi (2014) assessed the VOI in the design of site investigation
and
construction
quality
assurance
programs.
Schweckendiek
and
Vrouwenvelder (2015) studied the VOI in the retrofitting of flood defenses. Najjar et al. (2017) suggested a decision- making framework to evaluate the value of load test on the design of pile foundations. Yoshida et al. (2018) studied the placement of additional borings as a liquefaction countermeasure for an embankment along a river. These studies provide excellent examples on evaluating the value of reducing various types of uncertainties in 32
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geotechnical decision-making.
6. Needs for Future Research
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In the previous sections, we have illustrated how probabilistic methods can be used to characterize uncertainty, evaluate the impact of uncertainty on geotechnical prediction,
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reduce uncertainty with additional data, and assess the value of uncertainty reduction.
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Nevertheless, these methods are mainly developed in the geotechnical profession, where the
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geological uncertainty has not been adequately considered or treated with the same degree of rigor as in the case of ground and geotechnical uncertainties. Indeed, the geologic
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uncertainties are seldom considered in the current reliability-based geotechnical design
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(Keaton 2013), which has been an important missing element in the probabilistic geotechnical analysis. Further, although the probabilistic methods have been widely used in
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geotechnical engineering, the use of these methods in geological engineering is still in its infancy. In light of these situations, a few urgent research needs are summarized below.
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(1) Geologists are encouraged to develop quantitative engineering geologic models with explicit consideration of the geologic uncertainties in a way that is easy-to-use and to access by geotechnical engineers. In the infinite slope example illustrated previously, for example, the geologic uncertainty is considered through the uncertain thickness of the soil layer. In reality, the geologic uncertainty may be much more complex. Most of the studies reported in the literature, however, are not relevant to geotechnical applications, and their use in the geotechnical analysis has not been clearly defined. 33
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There is an urgent need to characterize the geologic uncertainties for use in geotechnical engineering. (2) The probabilistic methods are relatively well established in the geotechnical
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profession to analyze the impact of uncertainties on the geotechnical design and performance predictions. In principle, these probabilistic methods should provide
RI
convenient tools for engineering geologists to consider relevant uncertainties in the
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geologic model. Nevertheless, the effectiveness of these methods in treating the
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geologic uncertainties has not yet been proven. Research to determine whether these methods are equally applicable to the analysis of geologic uncertainties is needed. If
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they are deemed unsuitable, new probabilistic methods for the analysis of geologic
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uncertainties should be developed, and the role of geological uncertainty in a probabilistic analysis must be further elucidated.
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(3) There is a need to collect and document case histories that demonstrate the use of probabilistic methods in the analysis of geologic uncertainties and the risk
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quantification and decision making in engineering geology. The detailed and well-documented case histories may offer a basis for confirming and/or enhancing the applicability of probabilistic methods in the fields of geotechnical and geological engineering. (4) In recent years, the machine learning (ML) technology, which focuses on automatically learning from machine readable data by computational and statistical methods, is empowering the research and practice in many fields (LeCun et al.,2015; 34
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Lary et al.,2016). Uncertainty quantification, uncertainty reduction and reliability assessment in geological and geotechnical engineering requires learning from data in different forms, such as engineering geology map, site investigation report, and
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monitored data. ML may potentially significantly enhance data learning process for such purposes. Indeed, the multiple point geostatistics has been shown as an effective
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ML tool to model subsurface heterogeneity (e.g., de Vries et al.,2009). Recently, a
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pioneering work for uncertainty characterizing of soil parameters through Bayesian
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ML was reported in Ching and Phoon (2019). Nevertheless, the ML related research in geological and geotechnical engineering significantly lags behind those in many
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other fields, and how to leverage the full potential of ML such that uncertainties in
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geotechnical and geological engineering can be more efficiently and more effectively
Concluding Remarks
The research reported in this paper can be summarized with the following remarks:
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7.
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addressed needs further research.
(1) The geologic model, the ground model, and the geotechnical model are typically involved in a geotechnical analysis. As they are usually associated with considerable level of uncertainty, any geotechnical analysis would benefit from a careful consideration of the uncertainties of all three. (2) The probabilistic methods developed in the geotechnical profession are versatile tools for uncertainty characterization, assessment of impact of uncertainties, 35
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uncertainty reduction, and evaluation of the value of uncertainty reduction. It is quite possible to use these methods for a unified treatment of various types of uncertainties, from geotechnical to geological, in a geotechnical analysis. In this
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regard, the current probabilistic methods that are widely used in geotechnical engineering should be further examined to determine their applicability in the
RI
evaluation of geologic uncertainties.
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(3) Geologists are encouraged to develop quantitative engineering geologic models with
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explicit consideration of the geologic uncertainties in a way that is easy-to-use and to access by geotechnical engineers. To this end, engineering geologists and
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geotechnical engineers are encouraged to collaborate in an effort to comprehensively
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Acknowledgements
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characterize various types of uncertainties consistently.
The first co-author wishes to acknowledge the support of National Science Foundation
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through grants CMMI-0300198 & CMMI-1200117; the second co-author and fourth co-author wish to acknowledge the support of the National Science of Foundation of China (NSFC) through Project No. 41672276; and the third co-author wishes to acknowledge the financial support from the China Scholarship Council (CSC) and the Glenn Department of Civil Engineering, Clemson University. The authors are grateful to the information provided by Dr. Zijun Cao of Wuhan University, Dr. Dongming Zhang of Tongji University, Dr. Peng Zeng of Chengdu University of Technology, and Mr. Xiangrui Duan of Tongji University. 36
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Finally, but not the least, the following individuals are thanked for their contributions to the studies of geotechnical safety and reliability at Clemson University: Drs. Jinxia Chen, Tao Jiang, Haiming Yuan, Hui Yang, Yu-Ting Su, Kun Li, Ye Fang, Cheng-Liang Hsiao, Matthew
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Schuster, Raul Flores, Zhe Luo, Lei Wang, Wenping Gong, and Sara Khoshnevisan.
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List of Figures
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Fig. 1. Relationship between the geological model, ground model, and geotechnical model [Adapted from Keaton (2013)] Fig. 2. An infinite slope and distribution of the cohesion Fig. 3. Uncertainties in the estimated soil properties Fig. 4. Spreadsheet template for reliability analysis of infinite slope Fig. 5. The typical cross section of the drainage culvert Fig. 6. The numerical model of the drainage culvert: (a) finite difference mesh; and (b) mesh of the reinforced concrete Fig. 7. Geometry of a slope in layered soils [Adapted from Ching et al. (2009)] Fig. 8. Failure mechanisms at two design points: (a) At point {68.725 kPa, 81.068 kPa}; (b) At point {47.795kPa, 146.328 kPa} Fig. 9. The prior and posterior PDFs of h for the case of hp = 3 m as σp varies Fig. 10. The reliability index at m = m1 when μm2 takes various values Fig. 11. Decision tree to determine if the monitoring system should be installed Fig. 12. Decision tree model for the value of information analysis Fig. 13. Impact of the measurement error on the value of information
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Geologic Materials & Properties Surface Water & Ground Water
Geologic Model Landscape Evolution Extreme Events
Site Investigation Ground Description
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Geologic Processes Geologic History
Uncertainties
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Geotechnical Model
Performance Safety & Risk
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Ground Model
Idealization Evaluation
Analytical Modeling Physical Modeling
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Fig. 1. Relationship between geological model, ground model, and geotechnical model [Adapted from Keaton (2013)]
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c (kPa) 5.0 6.0 7.0 8.0 9.0 10.0 0.0 ε0
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Deterministic variables 3 δ (o) h w (m) γ s (kN/m ) 35 1.825 19 Random variables o c (kPa) φ () x 7.572 36.049 μx 10 38 σx 2 2 y* -1.214 -0.976
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γ (kN/m3) 17
γ w (kN/m3) 9.8
m 0.525 0.5 0.05 0.501
h (m) 3.476 3 0.6 0.793
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Performance function Reliability analysis g (θ) Fs F s -1 β pf 1.000 1 -6E-09 1.841 0.033 Ry 1 0 0 0 0
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Notes: (1)This spreadsheet calculates the reliability index of the infinite slope. (2)The setting in Solver is "minimize N6 by changing values in cells E13:I13, subjected to M6 = 0"
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Fig. 4. Spreadsheet template for the reliability analysis of infinite slope
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Fig. 5. The typical cross section of the drainage culvert
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cu1 , Unit weight = 19.5 kN/m3
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cu2 , Unit weight = 19.5 kN/m3
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Fig. 7. The geometry of a slope in layered soils [Adapted from Ching et al. (2009)]
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At point {47.795kPa, 146.328 kPa}
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5.0 4.0
Posterior, σp = 0.1 m
3.0
PT
2.0 1.0 0.0 3.0 h (m)
SC
2.0
NU
1.0
RI
Posterior, σp = 0.2 m Posterior, σp = 0.3 m Prior
4.0
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Fig. 9. The prior and posterior PDFs of h for the case of hp = 3 m as σp varies
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5.0
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2.1
PT
2.0
1.9
SC
RI
Reliability index
Considering survial information Without considering survial information
1.8
0.2
0.3 μm1 (m)
NU
0.1
0.4
0.5
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Fig. 10. The reliability index at m = m 1 when μm2 assumes the various values
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Decision alternatives
Landslide ?
Losses
Yes pf
$300,000
Install monitoring system
$2500,000
SC
Do not install monitoring system
$50,000
RI
Yes pf
PT
No 1-pf
MA
NU
No 1-pf
$0
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Fig. 11. Decision tree to determine if the monitoring system should be installed
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Outcome of investigation
Updated distribution of h
Decision alternatives
Landslide ？
Losses
Yes pf
$300,000
No 1-pf
$50,000
Yes pf
$2500,000
No 1-pf
$0
Yes pf = 0.033
$300,000
Install monitoring system (a1)
Decision alternatives
. . . . . .
Install monitoring system (a1)
NU
Do not investigate the thickness of the soil
Do not install monitoring system ( a 2)
RI
hp
SC
Investigate the thickness of the soil
PT
h
ED
MA
Do not install monitoring system ( a 2)
No
Yes pf = 0.033 No 1-pf = 0.967
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Fig. 12. Decision tree model for the value of information analysis
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$50,000
1-pf = 0.967 $2500,000 $0
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PT
2.0
1.0
RI
VOI (104 $)
3.0
0.1
SC
0.0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Standard deviation of measurement error, σp (m)
1.0
AC C
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MA
NU
Fig. 13. Impact of the measurement error on the value of information
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Highlights
EP T
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MA
NU
SC
RI
PT
The uncertainties associated with geologic, ground and geotechnical models are analyzed Probabilistic methods can be used to characterize these uncertainties Reliability methods can be used to consider the effect of these uncertainties The Bayesian method can be used for uncertainty reduction The value of uncertainty reduction can be assessed through decision analysis
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1. 2. 3. 4. 5.
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