# Epistemic Uncertainty

## Model Uncertainty

For practical convenience and because of the historical development of the mechanics of deformable solids, the problems in geotechnical engineering are often categorized into two distinct groups - namely, elasticity and stability (Chen 1975). The elasticity problems deal with stress or deformation of soil without failure. We use the term “elasticity” in the sense described by Chen (1975), which covers all elasto-plastic problems prior to ultimate failure. Point stresses beneath a footing or behind an earth retaining wall, deformations around tunnels or excavations and all settlement problems belong in this category (Chen 1975). The stability problems are associated with the determination of a load that will cause the failure of soil in modes such as bearing capacity, passive and active earth pressure and stability of slopes. In many design codes, stability is usually considered in ULS, while elasticity is frequently considered in SLS. The ULS and SLS are commonly calibrated within the framework of limit state design (Becker 1996a, b). Bolton (1981) discussed a more comprehensive range of limit states: (1) unserviceability through soil strain, (2) unserviceability through concrete deformation, (3) collapse of structure through soil failure alone, (4) collapse of structure with both soil and concrete failure and (5) collapse of structure arising without soil failure. Vardanega and Bolton (2016) cited Burland et al. (1977) to describe a category of damage as “disappointing” in that “later develop into serviceability issues, and then ultimately threaten structural collapse only if nothing has been done to interrupt the loading process or enhance the soil-foundation system.” Phoon (2017) noted that demarcation between ULS and SLS is introduced by limit state design, which is distinct from RBD in that it only requires a performance function that is calibrated by load tests, and it can handle ULS, SLS or any intermediate damage limit states.

Figure 2.2 illustrates the basic components in any geotechnical calculation procedure. At one end of the process is the forcing function, which normally consists of loads in conventional foundation engineering. At the other end is the system response, which would be the calculated value in an analysis or design situation. Between the forcing function (load) and the system response (calculated value) is the model invoked to describe the system behaviour, coupled with the properties needed for this particular model. Contrary to popular belief, the quality of geotechnical prediction does not necessarily increase with the level of sophistication in the model

Figure 2.2 Components of geotechnical prediction

(Lambe 1973). A more important criterion on the quality of a geotechnical calculation is whether the model and property are calibrated together for a specific load and subsequent calculation (Kulhawy 1994). Reasonable calculations can be carried out using simple models, even though the type of behaviour to be predicted is beyond the capability of the models, as long as there are sufficient data to calibrate these models empirically. However, these models should be applied within the specific range of conditions in the calibration process (particularly conditions in the databases), unless demonstrated otherwise using more generic databases. Although they lack generality, simplistic models are expected to remain in use for quite some time because of our professional heritage, which is replete with empirical correlations. It is sufficient to note that the art of calculating responses in geotechnical design is somewhat less formal than what is done in structural design. The role of the geotechnical engineer in appreciating the complexities of soil behaviour and geology and recognizing the inherent limitations in the simplified models is clearly of considerable importance. The amount of attention paid to the quality of the input parameters or the numerical sophistication of the calculation procedure is essentially of little consequence if the engineer were to select an inappropriate model for design to begin with. The centrality of engineering experience and judgement in modelling is succinctly illustrated by the geotechnical triangle in Figure 1.1 in Chapter 1, as proposed by Burland (1987, 2007).

In general, it is difficult to determine fully and accurately the field situation and the mechanism that will occur, as well as the selection of input parameters for calculation methods, because of complicated soil-structure interaction (including interface behaviour), complex geological conditions and construction effects (e.g. Lambe 1973; Gibson 1974; Vaughan 1994). To mitigate these problems and develop useful solutions for design, various methods with different levels of sophistication were proposed. Lambe (1973) suggested that (1) predictions are essential to geotechnical engineering on which styles of practice, such as the observational method, rest and

(2) an examination and interpretation of predictions can add considerably to our knowledge. For example, Wilson (1970) analysed the observational data on ground movements that leads to a better understanding of the mechanism of failure and, ultimately, may lead to improvement methods of slope stability. Methods of analysis and design can be classified into three broad categories in Table 2.9, as proposed by Poulos et al. (2001), to assess their relative merits. Category 1 procedures are empirical, probably accounting for a large proportion of foundation design (e.g. direct correlations of capacity and settlement to SPT or CPT data and rock strength). Category 2 procedures have a proper theoretical basis but make significant simplifications, especially with respect to soil behaviour (e.g. total and effective stress methods). Category 3 procedures generally involve the use of a site-specific analysis based on relatively advanced numerical or analytical techniques with the aid of computational software (e.g. EXCEL, MATLAB, ABAQUS, etc.). Many of the Category 2 procedures are derived from Category 3 analyses and then condensed into a simplified or analytical form. The most advanced Category 3 methods (3C) are used relatively sparingly in the past but are becoming more common at the final design stage for complicated projects because of the increasing realism of more sophisticated numerical models (e.g. large deformation finite element analysis).

The majority of available analysis and calculation methods fall into the Category 1 and 2 procedures (often in analytical form). Analytical solutions

Table 2.9 Categories of methods of analysis and design

 Category Sub division Characteristics Method of parameter estimation 1 — Empirical - not based on soil mechanics principles Simple in situ or laboratory tests with correlations 2 2A Based on simplified theory or charts - uses soil mechanics principles, amenable to hand calculation, simple linear elastic or rigid plastic soil models Routine relevant in situ or laboratory tests - may require some correlations 2B With respect to 2A, the theory is non-linear (deformation) or elasto-plastic (stability) 3 ЗА Based on the theory using site-specific analysis and uses soil mechanics principles. Theory is linear elastic (deformation) or rigid plastic (stability) Careful laboratory and/or in situ tests, which follow the appropriate stress paths ЗВ With respect to ЗА, non-linearity is allowed for in a relatively simple manner ЗС With respect to ЗА, non-linearity is allowed for via proper constitutive soil models

(Source: Table 2.6 in Poulos et al. 2001)

• (link between theory and practice) are the clearest language through which engineering systems educate us in respect to controlling behaviour in geotechnical design (e.g. Lambe 1973; Gibson 1974; Vaughan 1994; Randolph
• 2013). To apply in practice, these methods usually entail making assumptions of material properties and boundary conditions, as well as simplifications of problem geometries and interface characteristics (e.g. Vaughan 1994; Dithinde et al. 2016). For example, solutions to stability problems are generally obtained by the theory of perfect plasticity, which ignores work softening (or hardening) and assumes a continuing plastic flow at constant stress (Chen 1975). The perfect-plastic assumption differs from the reality, where soil often exhibits some degree of work softening (or hardening). Solutions to elasticity problems are commonly solved with the theory of linear elasticity. Nevertheless, soil behaviour is highly non-linear, even at small strain, which has an important influence on the selection of geotechnical parameters in routine design (e.g. Burland 1989; Atkinson 2000). Besides, construction disturbance is hard to model. As a consequence, the predicted response (e.g. internal stresses, deformations and stability for a geotechnical structure) will deviate from the measured one (typically on the safe side).

### Model Factor

This deviation can be directly captured by a ratio of the measured value (Xm) to the calculated value (Xc) that is also known as a model factor M.

The method based on Eq. (2.1) is practical, familiar to engineers and grounded realistically on a load test database. The quantity X could be a load, a resistance, a displacement, etc. The model factor itself is not constant but takes a range of values that may depend on the scenarios covered in a load test database. It is customary to model M as a random variable, although it is important to validate that the range of values is random - i.e. the variation in M is not explainable by other known variables (e.g. pile length and soil properties). This important aspect is deserving of more attention, as discussed in Section 2.4.1.2. The probabilistic distribution of the model factor is arguably the most common and simplest complete representation of model uncertainty. In Annex D of ISO 2394:2015, the characterization of model uncertainty is identified as the second important element in the geotechnical RBD process (Phoon et al. 2016a). The simplest method for characterizing a random variable is to calculate the mean and COV. A mean and COV of M close to 1 and 0, respectively, would represent a near-perfect calculation method that matches measured responses for all scenarios in the calibration database. It goes without saying that such methods do not exist in geotechnical engineering, regardless of their numerical sophistication.

More sophisticated calculation methods typically require more input parameters, and some of these parameters are not measured in routine site investigations or commercial laboratories.

The mean of M would provide an engineer with a sense of the hidden FS that either adds or subtracts from the nominal global FS, depending on whether the capacity calculation method is conservative (mean > 1) or unconservative (mean < 1) in the average sense. If X is a displacement, then the opposite is true (i.e. the calculation method is unconservative for mean > 1 and conservative for mean < 1 on the average). It should not be inferred that a calculation method is conservative or otherwise for a specific case because M takes a range of values in actuality (hence it is random). This random nature is practically significant because it implies that a calculation method can be unconservative when applied to a specific case, even though the method is conservative on average. Therefore, it is also necessary to consider the degree of scatter (dispersion) in M and to ensure the probability of a measured value being lower than the calculated value is capped at a known value, say, p% (Lesny 2017a). This idea is conceptually similar to EN 1997-1:2004, 2.4.5.2 (11), which recommends that a cautious estimation (or characteristic value) for a geotechnical design parameter can be “derived such that the calculated probability of a worse value governing the occurrence of the limit state under consideration is not greater than 5%.” As discussed previously, it is possible to handle a random model factor more rationally in a partial factor design format by selecting the first-order reliability method (FORM) design point for the model partial factor (Phoon 2017).

To provide some context on how model uncertainty is considered/applied in practice, several design codes/guidelines are reviewed. Broadly, two review approaches exist. The first is to introduce a deterministic partial factor to ensure that the design calculation is sufficiently safe, as given in Annex A of Eurocode 7 - that is, EN 1997-1:2004 (CEN 2004). On the other hand, Table 6.2 in the Canadian Highway Bridge Design Code-CAN/CSA-S6-14 (CSA 2014) recommended the ULS and SLS resistance factors for various geostructures associated with the degree of “site and prediction model understanding.” Site understanding is more or less related to the consideration of natural variability through effective site characterization, while model understanding is closely associated with the consideration of model uncertainty through model calibration. CAN/CSA-S6-14 applies the semi- probabilistic LRFD, which is the most popular simplified RBD format in North America. Several DOTs in the United States are increasingly calibrating the LRFD resistance factors for pile foundations using rigorous reliability analyses (Seo et al. 2015). The second approach is to introduce a random variable to characterize the model factor. This is the basis of direct probability-based design methods. Section 2.2.3.5 in DNVGL-RP-C207 (DNV

2017) stated, “Model uncertainty involves two elements, viz.: (1) a bias if the model systematically leads to overprediction or underprediction of a quantity in question and (2) a randomness associated with the variability in the predictions from one prediction of that quantity to another.” Table 2.10 presents the indicative values for the mean and COV of the model factors for several geostructures (e.g. embankments, sheet pile walls, shallow foundations and driven piles; JCSS 2006). Phoon (2017) opined that the design point corresponding to the model factor in the FORM can bridge the deterministic and random approaches; i.e. it can be a rational choice for the model partial factor described in EN 1990:2002 (CEN 2002). The need for assessment of model uncertainty has been emphasized and considered in the current revision process of the Eurocodes (Lesny 2017a). Some discussions on the impact of model uncertainty on design can be found in Bauduin (2003), Phoon (2005), Forrest and Orr (201 l),Teixeira et al. (2012), Burlon et al. (2014), Lesny (2017a), Abchir et al. (2016) and Plaque and Abu- Farsakh (2018).

Equation (2.1) has been widely used to evaluate the model uncertainty in

(1) stability problems (e.g. capacity of foundations, pipelines, anchors and mechanically stabilized earth structures or FS for soil slopes) and (2) elasticity problems (e.g. settlement of foundations, wall and ground movements in braced excavations). A comprehensive review of the statistics was given by Phoon and Tang (2019), which is a significant update of the JCSS Probabilistic Model Code (JCSS 2006) in Table 2.10. One should be mindful of past studies that define the model factor as the ratio of the calculated value to the measured value - reciprocal of Eq. (2.1). The model factor statistics reported in this book are based on Eq. (2.1) only. Where necessary, the model factor statistics reported in past studies are re-calculated based on the original measured and calculated values to conform to Eq. (2.1). Phoon and Tang (2019) compiled some of the model factor statistics based on what was

Table 2.10 Indicative computation model uncertainty factors

 Type of problem Type of calculation model Mean SD Embankment Stability Homogeneous soil LEM (e.g. Bishop, Spencer); l.l 0.05 Nonhomogeneous soil 2D FEM l.l 0.1 Settlement — 1 0.2 Shallow foundation Stability Homogeneous soil Brinch Hansen 1 0.15 Nonhomogeneous soil 1 0.2 Settlement 1 0.2 - 0.3 Driven pile Point bearing capacity CPT based empirical 1 0.25 Shaft resistance design rules 1 0.15

(Source:Table 3.7.5.1 in Section 3.7 of 2006 JCSS Probabilistic Model Code)

Note: SD = standard deviation, LEM = limit equilibrium method and FEM = finite element method.

reported in the original paper. Hence, both definitions of the model factors appear in their summary. This inconsistency has been addressed in this book.