Soil and Rock Parametric Uncertainties
Journal
Uncertainty, Modeling, and Decision Making in Geotechnics
ISBN
9781003801252
Date Issued
2023-01-01
Author(s)
Abstract
The purpose of Chapter 2 is to present practical tools and guidelines for the uncertainty quantification of soil and rock properties with a primary focus on enhancing decision-making in geotechnical design. The context underlying uncertainty quantification is emphasized in this chapter. A conceptual framework is presented to guide the engineer to consider uncertainty quantification in both the absence and the presence of a geotechnical structure (limit state) separately. This will provide conceptual clarity on the distinctive role of statistics (spatial variability), physics, and their interactions, which is somewhat lacking at present. The conceptual framework contains three building blocks: (1) point value, (2) spatial average, and (3) mobilized value. The first two blocks do not involve a geotechnical structure. The point value is the most basic building block. It refers to a value measured by one laboratory/field test (univariate case) at a given location/depth. It can also refer to a set of values from multiple tests conducted in close proximity (multivariate case). The classical model to describe spatial variability is to assume that two point values are independent and identically distributed, regardless of their measurement locations. Under this classical model, the engineer needs to characterize a single (univariate or multivariate) probability distribution function. The typical uncertainty quantification approach for the univariate case is to select a lognormal distribution and determine its mean and coefficient of variation (COV) based on site data. Extensive COV guidelines are provided in this chapter for clays, sands, rocks, and rock masses. These guidelines are immediately useful for the selection of a characteristic value based on the 5% fractile, firstorder second-moment reliability analysis, and reliability-based design. The second building block is the spatial average. It is defined as the value affecting the occurrence of a slip along a prescribed trial line. To apply the spatial average, it may be sufficient to quantify a scale of fluctuation (roughly the characteristic length between strongly correlated measurements) in addition to the COV. The spatial correlation between two point values is not zero in this random field model. Extensive guidelines on the scale of fluctuation are provided in this chapter. The scale of fluctuation is typically measured in the vertical direction. Actual spatially varying soils are usually anisotropic in the sense that the vertical and horizontal scales of fluctuation are different. The latter is roughly 10 to 20 times longer than the former. Very little is known about the spatial variability of cement-admixed soils at present. When a limit state is governed by a spatial average, its uncertainty reduces with the characteristic length of the geotechnical structure. This important effect can be quantified using Vanmarcke’s variance reduction function (VRF), which is available in simple analytical forms for the classical one-parameter autocorrelation models. This chapter provides a more general VRF (approximate) that is applicable to classical and non-classical Cosine Whittle–Matérn (CosWM) autocorrelation models. The CosWM model can address a broader class of spatial variability that exhibit roughness and pseudo-periodicity in addition to a scale of fluctuation. The new VRF is also applicable to inclined slip lines. The random field theory underlying the spatial average generalizes easily to a vector random field (at least theoretically) in the presence of multivariate data. The correlation between two soil properties produced by two different tests (called a cross correlation) is distinct from spatial correlation which is the correlation between the same property measured at different locations. The most familiar application of cross correlation is the transformation model that relates a field or laboratory measurement to a design parameter. This chapter illustrates the reduction in the uncertainty of one or more design parameters when information in one or more tests is available using useful closed-form solutions from a conditional multivariate normal distribution. The key task for an engineer is to assemble a correlation matrix from the cross correlations between all pairs of soil parameters. It is not sufficiently emphasized that the matrix must be positive definite to be valid. Although classical transformation models constructed in this manner can be multivariate, they are unable to address the full set of MUSIC-3X (Multivariate, Uncertain and Unique, Sparse, Incomplete, potentially Corrupted, 3D spatially variations) attributes commonly found in geotechnical databases. The third building block is the mobilized value. It is closest to the “value affecting the occurrence of the limit state” in Clause 2.4.5.2(2) of Eurocode 7. The characteristic value is defined as a cautious estimate of this value. Note that the mobilized value used in this chapter refers to the probability distribution of this value. It is not the same as the spatial average. The reason is that the spatial average is defined over a prescribed trial surface that is not the critical surface affecting the occurrence of a limit state. The reliability-based characteristic value is fully compatible with physics and statistics. However, it is very costly to compute because it requires the random finite element method. An approximation called a mobilization-based characteristic value may be more applicable to practice. It has the same “look and feel” as the current spatial average-based characteristic and it is only slightly more costly.
SDGs
Type
book part