Concluding remarks

8.2.1 Behavior of the calibrated n and Γ²

The variation of the calibrated n and Γ² with respect to δL/L, where L is the length of the classical slip curve, and δL is the equivalent SOF along the classical slip curve, is found to exhibit the following behaviors:

1. The Γ²-(δL/L) relationships for all investigated problems follow a unique trend.

Even more remarkably, this trend is consistent with the analytical expression for

² along the classical slip curve.

147

2. The n-(δL/L) relationships have similar behaviors for all problem (except for the friction pile under compression). n is large when δL/L is small, and n approaches 1 when δL/L approaches . The first observation (n is large when δL/L is small) is sensible because the smaller δL/L, the more independence among the PSCs, hence weak-zone seeking is stronger and n is larger. The second observation (n approaches 1 when δL/L approaches ) is also sensible because δL/L =  produces a homogenous problem. For a homogeneous problem, there is only a single independent PSC, hence n = 1. For the friction pile under compression, n is always 1. This is also sensible because the slip curve always coincides with the pile shaft, hence there is only a single independent PSC and n = 1.

3. Although the n-(δL/L) relationships have similar behaviors for all problems, their problem-specific trends can be classified into four groups based on the “degree of constraint” of the slip curve:

a. Low constraint (soil column), where n is the largest: the slip curve is lowly constrained for the soil column because its location can vary all over the entire height of the column.

b. Medium constraint (strip footing and basal heave), where n is the second largest: the slip curve is intermediately constrained because it typically passes through the corners immediately below the footing (strip footing) or passes through the tip of the diaphragm wall (basal heave).

c. High constraint (retaining wall), where n is the second smallest: the slip curve is highly constrained for the retaining wall because it needs to pass through the toe of the wall and its shape is usually close to a straight line.

d. Full constraint (friction pile under compression), where n is always 1: the slip curve is fully constrained because it always coincides with the pile shaft.

4. Coefficient of variation, V, and the ratio (horizontal SOF)/(vertical SOF), δx/δz, can also influence the value of n. n tends to be large for a large V (more variability) and for δx/δz closer to 1 (less anisotropy).

5. The calibrated n values for the stationary-su/σ'v scenario is systematically less than those for the stationary-su scenario, suggesting that the weak-zone seeking for the stationary-su/σ'v scenario (e.g., normally consolidated clay) is less severe.

8.2.2 Performance of the calibrated WPM

For each combination of the ground scenario (stationary-su, stationary-su/σ'v, stationary-tan ) and degree of constraint (low, medium, high, full), a relatively simple empirical equation is constructed that provides a satisfactory fit to the calibrated n. With these n-equations and with the analytical equation for Γ², the WPM is used to simulate mean and variance of the mobilized shear strength for all problems under different ground scenarios. It is found that the calibrated WPM can capture the statistics of the mobilized shear strength simulated by the 2D RFEM. It can also capture the phenomenon of a worst-case scale of fluctuation (SOF) observed by the 2D RFEM. Note that the calibration of the WPM is conducted by code developers, not by design engineers.

Design engineers only need to know how to implement the calibrated n-equations.

8.2.3 Performance of the proposed simple formula

Based on the calibrated WPM, a mobilization-based characteristic value is proposed to determine the characteristic value of shear strength that approximates the

reliability-149

based characteristic value from RFEM. This simple formula is similar to that proposed by Schneider and Schneider (2013). However, the main difference is that the former (current study) addresses the weak zone seeking, while the latter (Schneider and Schneider 2013) does not. In Schneider and Schneider’s formula, a fixed 0.05N = -1.645 is used, while in the proposed simple formula, a variable 0.05WPM is used which has a simple relationship with n. When n = 1 (no week zone seeking), 0.05WPM=0.05N= -1.645.

However, as n increases, the value of 0.05WPM decreases. Therefore, it is fair to say that the proposed formula is a generalization of Schneider and Schneider’s formula by considering weak-zone seeking. For the investigated problems, it is found that the characteristic values predicted by the proposed simple formula are close to those simulated by the RFEM, while Schneider and Schneider’s formula tends to be unconservative, specially for cases with low and medium constraint of slip curve (soil column, footing, and basal heave) and with a relatively large variability (e.g., V = 0.5).

The effectiveness of the proposed formula is further showcased by real case studies. It is shown that weak-zone seeking is insignificant for problems subjected to high constraint of slip curve (such as piles), so the proposed simplified formula does not produce significant improvements over existing simplified formulas. For problems dominated by epistemic uncertainties (e.g., transformation, statistical, and model uncertainties), the effect of weak-zone seeking may be diluted by these uncertainties.

The proposed formula may not produce significant improvements for these problems, either. Nonetheless, for problems with low to medium constraint of slip curve (such as soil columns, slopes, footings, etc.) and with relatively small epistemic uncertainties, the proposed simplified formula may produce significant improvements. This can happen if epistemic uncertainties are reduced by more site investigation data (statistical uncertainty is reduced), more accurate prediction models (model uncertainty is reduced),

and more accurate transformation models (transformation uncertainty is reduced). The reduction of these epistemic uncertainties may be achieved by advanced Bayesian learning methods that have grown in power recently (e.g., Ching and Phoon 2019a, 2020a, 2020b).