Comparison between baseline technique design and partial factor design in slope engineering

Abstract

In slope engineering, the allowable safety factor design (ASFD) is widely used. However, it includes all sources of uncertainties and does not truly account for risk. Therefore, a more robust design method is required. This study compares the actual reliability levels that ASFD, PFD (partial factor design), and BTD (baseline technique design) provide. The results indicate BTD provides the most stable reliability levels. The major findings are as follows. (1) A sampling strategy is provided to evaluate the performance that ASFD, PFD, and BTD achieve specific target reliability indexes. It provides reliability-based calibration and evaluation. (2) BTD provides more stable reliability levels, especially when the target reliability index is high. (3) The performance of PFD is influenced by the selection of quantile; BTD uses optimal quantiles and achieves the most stable reliability levels. This study provides a numerical investigation of design factors for shear strength parameters, and it uses a sampling strategy considering typical soils from sand to clay. Preliminary results show that BTD is attractive in numerical aspects.

Appendices

Appendix 1. Approximating performance functions for the calibration

In this study, an explicit performance function is used to accelerate all the calibration procedure.

Given friction angle φ and slope angle α, the $\frac{c}{\rho gh}$ meets the limit state was solved by the commercial software Geo-slope. Then, the limit state Z = 0 can be described by sufficient data, and the probability that Z > 0 can be solved using an explicit performance function. As shown in Fig. 9, the limit state Z = 0 for different α were approximated, and a unified fitting curve was found to be

$$ Z=\frac{c}{\rho gh}+0.028-\frac{1}{6.02-\frac{\alpha }{22.6}}\frac{1}{\exp \left(\varphi /\left(1.70+\frac{\alpha }{2.43}\right)\right)} $$

(15)

Fig. 9

An explicit approximation to the performance function

Also, the performance function can be approximated by the response surface method (RSM). However, an explicit function could be simpler and more efficient in the calibration, in which 80 cases need to meet the design criterion and then solve reliability indexes. Besides, there are other reasons that RSM is not used. (1) Sometimes, convergence issues arise in RSM (e.g., jumping back and forth between two points), and iterative skills are required. (2) It is inconvenient in RSM to ensure that non-circular critical slip surfaces are correctly obtained without local extremum. For the above reason, RSM is only used for the designs in Fig. 8, where the two-step slope’s performance function cannot be Eq. 15.

Appendix 2. Implementing reliability-based calibration

The calibration in the “Reliability-based evaluation on limit state design methods” section needs to solve the percentile β₅₀ and β₁₀ for slope cases meeting the design criterion Z(x(X), y) = 0. Based on the levels in Table 2, there are a total of 80 slope cases. In each case, the random variables X = (μ_c, μ_f, δ_c, δ_f) are given, and we need to find non-random variables y = (ρg, h, α) that meet Z(x(X), y) = 0.

The detailed procedure is listed as follows.

• Step 1. Set function handle x(X) based on PFD (Eq. 13) or BTD (Eq. 14).

• Step 2. For each combination [X, y], (which represents a single slope case), solve the slope angle: α satisfying Z(x(X), y) = 0. (There must be a unique solution for α since Z is monotonic to the slope angle). Then, record all slope cases in {Ω}

• Step 3. For each slope case in {Ω}, solve the reliability index by FORM. Then, record all reliability indexes in {β}.

• Step 4. Solve the median reliability level β₅₀ and the lower reliability level β₁₀. (E.g., the Matlab code reads “beta10 = prctile (betas,10,'method’, ‘approximate’)”).

The above only evaluates a single design factor. In the calibration, we need to evaluate many design factors, plot the contour, and finally find the optimal one. These take many computation efforts, and the explicit performance function in Appendix 1 is essential for efficiency.