Chapter 4 Weakest-Path Model for Mobilized Shear Strength
4.2 Concepts of WPM
The WPM is originally developed by Ching and Phoon (2013a). This model has been applied to a retaining wall (Hu and Ching 2015), a 3D soil column (Ching et al. 2016b), and a basal heave problem (Ching et al. 2017a). The WPM is based on the observation that the mobilized shear strength (Xmob) is approximately equal to the spatial average along the critical slip curve (Ching and Phoon 2013a). Note that the critical slip curve is not a prescribed curve but an emergent curve that tends to seek out weak zones. Since the spatial distribution of weak zones changes from realization to realization, the trajectory of the critical slip curve changes as well.
Although the trajectory of the critical slip curve changes from realization to realization, its orientation is typically constrained to fall within a certain range. This is because the critical slip curve has to be mechanically admissible, and Mollon et al.
(2011) pointed out that mechanics tend to play a more dominant role in determining the trajectory of the slip curve rather than the specific spatial variation in a realization. As an example, consider a soil column with a spatially variable undrained shear strength su
that is subjected to axial compression (Figure 4.1a). For the undrained condition (ϕ = 0), the inclination angles of the critical slip curves in different realizations are roughly around 45º degree. If there is a weak layer in the horizontal direction, the trajectory of the critical slip curve will not follow this weak zone because it will not be mechanically admissible. For this soil column, a large number of potential slip curve (PSC), all with an inclination angle of 45º, can be imagined. Each PSC has a spatial average strength Xave, and the PSC with the lowest Xave is the critical slip curve.
In the WPM, two PSCs are said to be “independent” if their Xave values are independent. This can happen when PSCs are sufficiently distant from each other. For example, consider the two PSCc with a separation distance of s in Figure 4.1a. One thousand realizations of su random fields with point mean = 20 kN/m2, coefficient of variation = 0.3, and SOF = 1 m are simulated. For each realization, the Xave values for these two PSCs are calculated. Figure 4.1b shows the results for Δs/SOF = 0.07, while Figure 4.1c shows the results for Δs/SOF = 2.1. It is evident that the two PSCs are not independent when they are very close to each other (Δs/SOF = 0.07, Figure 4.1b).
However, they become independent when they are sufficiently distant from each other (Δs/SOF = 2.1, Figure 4.1c). Note that it is the ratio of Δs/SOF, rather than the absolute distance of Δs, that controls whether two PSCs are independent.
67 (a)
(b) (c)
(d)
Figure 4.1 (a) Two PSCs in a soil column subjected to axial compression, (b) Xave values for the two PSCc when Δs/SOF = 0.07, (c) Xave values for the two PSCc when
Δs/SOF = 2.1, (d) histograms of Xave values for the two PSCc when Δs/SOF = 2.1
Although the two PSCs in Figure 4.1c are independent, they are almost identically distributed. To make this “identically distributed” notion clear, their histograms are plotted in Figure 4.1d. The two histograms look very similar. They have the same mean value, equal to the point mean value = 20 kN/m2. Moreover, they have the same variance, equal to the point variance multiplied by the variance reduction factor (Г2) along these PSCs. Therefore, they are “independent and identically distributed” (IID). The WPM takes advantage of IID property to derive equations shown later in this section.
The main assumption of the WPM is that the critical slip curve finds the weakest slip curve among all potential slip curves (PSCs). The WPM also makes the following two assumptions:
1. Although the degree of variance reduction along different PSCs may be different, all PSCs are assumed to have the same variance reduction. This implies that all PSCs are assumed to have the same size and shape (Figure 4.2a). Note that this can become a poor assumption if the size and shape of the slip curve vary in a wide range (e.g., a slope that has two equally probable failure mechanisms: a deep failure circle and a shallow slip at the crest).
2. There are a large number of PSCs. However, it is assumed that all PSCs can be clustered into n “equivalently independent” representative PSCs. In other words, a cluster of highly correlated PSCs is lumped into one representative PSC (Figure 4.2b).
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(a) (b)
Figure 4.2 Assumptions in the WPM illustrated for the soil column subjected to compression. (a) all PSCs have the same size and shape, (b) a cluster of highly
correlated PSCs is lumped into one representative PSC
With the above assumptions, the mobilized shear strength Xmob can be written as:
(
1 2)
min , ,...,
=
mob ave ave ave
X X X Xn (4.1)
where Xiave is the spatial average along the i-th representative PSC. These n spatial averages (X1ave, X2ave, …, Xnave) are independent and identically distributed (IID). The mean of each Xiave is equal to the point mean , whereas the variance of Xiave is equal to the point variance σ2 multiplied by Γ2. It is reasonable to assume that 2 is the variance reduction factor (Vanmarcke 1977a). However, 2 is treated as one of two unknown parameters of WPM and it is estimated by maximizing the fit to RFEM solutions in this paper. If 2 turns out to be close to the variance reduction factor, one may argue that the PSC are “smooth” in the sense that Xmob does not carry an additional component similar to the dilatancy effect found in dense sand. It is also useful to point out that Xmob is
identical to the first (minimum) order statistics in Eq. (4.1). Pan et al. (2019) observed that the 4th minimum order statistics is more appropriate for tunneling in improved soil, because four hinges need to be formed in the improved circumferential soil to lead to tunnel collapse.
According to the first order statistics (David and Nagaraja 2003), the cumulative density function (CDF) of Xmob can be written as:
(
mob)
1 1( )
nF X = − − F X (4.2)
where F(X) is the CDF of the point shear strength X. Accordingly, the probability density function (PDF) of Xmob can be derived by differentiating the CDF in Eq. (4.2). If the point shear strength X follows the normal distribution with mean = µ and variance = σ2, Ching and Phoon (2013a) showed that the PDF for Xmob based on the first order
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and Eq. (4.4): the number of representative PSCs (n) and the variance reduction factor along each PSC (2). The parameter n is assumed to quantify the weak-zone seeking (seeking is strong when n is large) while 2 is assumed to quantify the spatial averaging effect along each PSC. These assumptions are validated by the geotechnical problems in Section 4.5. The next section describes how to use the maximum likelihood method to estimate n and 2. Note that n is a conceptual parameter in the WPM. In other words, n PSCs may not be observable in the lab. Although the formulation of the WPM [Eq.
(4.1)] looks similar to seeking out the critical failure surface in the limit equilibrium method, the WPM cannot locate the critical failure surface.