Influence of Limit State Function’s Form of Geotechnical Structures on Approximate Analytical Reliability Methods

Abstract

Approximate analytical methods have been frequently used in geotechnical engineering to estimate the reliability of geotechnical structures due to their efficiency and simplicity. The main spirit of these methods is using the moments of the limit state function to estimate the reliability index. However, the moments are strongly dependent on the form of the limit state function, resulting in the fact that these methods are sensitive to the form of limit state functions. This study aims to systematically explore how various equivalent forms of limit state functions affect the performance of several commonly used approximate analytical methods, including the first-order second-moment method, the first-order reliability method, and the point estimation method. The applicable conditions of these methods are illustrated through five typical geotechnical examples. The results indicate that the estimated accuracy for the first-order second-moment method and the point estimation method is affected by the form of the limit state functions. Although the form of the limit state function does not affect the accuracy of the first-order reliability method, it affects computational efficiency. The limit state functions with an equivalent logarithmic form are almost always favorable for the investigated examples and are thus recommended in practice.

1. Introduction

The increasing number of geotechnical engineers have recognized the advantages of applying reliability analysis methods to evaluate the safety of geotechnical structures since these methods can rationally consider the uncertain factors involved [1,2,3,4,5,6,7,8,9]. A large number of geotechnical reliability analysis methods have been developed in the past decades to effectively calculate the reliability index (β) of geotechnical structures. These analysis methods are generally classified into two types: simulation-based methods [10,11,12,13,14] and approximate analytical methods [7,8,15,16,17,18]. The former can generally obtain more accurate reliability analysis results with sufficiently large evaluation times of the limit state function (LSF) of geotechnical structures. However, the LSF of geotechnical structures often needs to be defined by some numerical analysis models with high computational time, such as discrete element models or finite element models, whose evaluation for one time is very time-consuming, let alone the large number of evaluations of the LSFs.

The simulation-based methods often require thousands of evaluations of the LSF to obtain reliable estimation results. In contrast, approximate analytical methods require extremely low computational costs. They only need to evaluate the LSF several times, especially when the number of random variables in the LSF is small and thus favorable to geotechnical engineers. Due to their simplicity and efficiency, approximate analytical methods are also recommended by some international geotechnical codes for evaluating the β of geotechnical systems [19,20,21]. The approximate analytical methods generally use the first four-order moments of LSF to estimate the probability of failure (P_f). This is based on the fundamental idea that the probability distribution of an LSF can be properly approximated by its first four-order moments. Hence, once the relationship between the failure probability and the statistical moments [22] is found, this relationship can provide a bridge to compute the P_f. However, it is noted that the estimated statistical moments of the LSF are strongly dependent on the algebraic form of the LSF. As a result, the results of many approximate analytical methods are prone to being affected by the form of the LSF, although the different forms of the LSF are equivalent and should give the same failure probability.

It is widely accepted that uncertainty in rock and soil properties is an intrinsic feature of geotechnical engineering [23]. Accounting for these uncertainties may lead to a complete description of the damages in geotechnical engineering and an insightful understanding of the failure mechanisms of geotechnical structures [24,25,26]. The uncertainties can be considered using approximate analytical methods. However, approximate analysis methods inherently involve some estimation errors, and the errors to some extent depend on the forms of LSF. Therefore, this paper aims to determine an optimal form of LSF that can reduce the estimation error. In practice, the LSF can be defined with different equivalent forms based on various aspects of failure mechanisms for the same geotechnical structure system [27,28]. In addition, some geotechnical codes [19,29] also suggest different forms of the LSF, such as G = R/L − 1 or G = R − L, in which R denotes the resistance and L denotes the load. Some other equivalent forms of LSFs, such as G = ln(R/L) and G = exp(R − L) − 1 can also be defined by geotechnical engineers for computational convenience. The essence of them being equivalent is that they both achieve G < 0 when R < L (L > 0), i.e., the structure fails. The different forms of LSFs are bound to influence the computational efficiency and estimation accuracy of the approximate analytical methods. However, how much the forms of LSFs influence the performance of the approximate analytical methods and which form of LSFs is most preferable to the approximate analytical methods with more computational efficiency and higher computational accuracy remains unclear.

In order to address this question, this paper seeks to systematically analyze the influence of the forms of LSFs of several typical geotechnical structures on the performance of the commonly used approximate analytical methods, including the first-order second-moment method (FOSM), the first-order reliability method (FORM), and the point estimation method (PEM). The optical form of LSF for conducting reliability analysis by approximate analytical methods can be identified and applied to the practical geotechnical engineering field, which is the main contribution of this study. In addition, since the LSF generally contains multiple uncertain variables, the influence of individual variables’ statistics, such as coefficient of variation and distribution type, is also analyzed. This paper first reviews the computational procedures of FOSM, FORM, and PEM. The influence of the forms of LSFs on the performance of FOSM, PEM, and FORM is then illustrated by four typical geotechnical examples. Finally, some conclusions are drawn.

2. FOSM

FOSM is the simplest approximate analytical method. It has been frequently used in the geotechnical engineering field for calculating the β [1,30,31,32,33,34]. Christian et al. [1] applied the concept of a first-order reliability index to slope stability analysis, providing a more meaningful stability measure than the factor of safety. Suchomel and Mašı [30] investigated the shortcomings of the original version of FOSM and improved it to provide more accurate probability of failure estimates. Wu et al. [31], Löfman and Korkiala [33], and Ma et al. [34] introduced the correlated sampling technique, second-order approximation of the mean, and machine learning, respectively, to propose higher-performance methods based on FOSM.

The basic principle of FOSM is to use the Taylor series to expand the LSF linearly at the mean value point of uncertain parameters. FOSM merely requires the standard deviations and means of the uncertain parameters (i.e., first- and second-order moments) to calculate the β. It does not need to know the specific probability distribution type of uncertain parameters [35]. Take the LSF G(X), for example, where X = (X₁, X₂,…, X_n)^T denotes the n number of uncertain parameters. Assume that the standard deviation and the mean of X are σ_X = (σ_X1, σ_X2, …, σ_Xn)^T and μ_X = (μ_X1, μ_X2, …, μ_Xn)^T, respectively. FOSM first expands the G(X) linearly by the following expression:

$$G\left(\mathit{X}\right)\approx G\left({\mathit{\mu }}_X\right)+{\displaystyle \sum _{i=1}^n{\left(\frac{\partial G\left(\mathit{X}\right)}{\partial X_i}\right)}_{{\mathit{\mu }}_X}\left(X_i-{\mu }_{X_i}\right)}$$

(1)

where (∂G/∂X_i)_μX is the partial derivative of the G(X) relating to the i-th uncertain parameter X_i at the mean point, which is approximately estimated as

$$\frac{\partial G\left(\mathit{X}\right)}{\partial X_i}|{}_{\mathit{X}={\mathit{\mu }}_x}=\frac{G\left({\mu }_1,{\mu }_2,\dots ,{\mu }_i+{\sigma }_i,\dots ,{\mu }_n\right)-G\left({\mu }_1,{\mu }_2,\dots ,{\mu }_i-{\sigma }_i,\dots ,{\mu }_n\right)}{2{\sigma }_i}$$

(2)

The β and P_f are then estimated as

$$\beta =\frac{E\left[G\left(\mathit{X}\right)\right]}{\sqrt{Var\left[G\left(\mathit{X}\right)\right]}}=\frac{G\left({\mathit{\mu }}_X\right)}{\sqrt{\mathit{H}{\mathit{C}}_X{\mathit{H}}^T}}$$

(3)

$$P_f=1-\Phi (\beta )$$

(4)

where C_X represents the covariance matrix and H represents the gradient vector of G(X)

$$\mathit{H}=\{\frac{\partial G\left(\mathit{X}\right)}{\partial X_1}|{}_{\mathit{X}={\mathit{\mu }}_X},\frac{\partial G\left(\mathit{X}\right)}{\partial X_2}|{}_{\mathit{X}={\mathit{\mu }}_X},\dots ,\frac{\partial G\left(\mathit{X}\right)}{\partial X_n}|{}_{\mathit{X}={\mathit{\mu }}_X}\}$$

(5)

It is seen that FOSM is convenient for estimating the β when the random variables’ distribution types are not clear but their variances and mean values are clear. The major limitation of FOSM is that it expands the LSF at the mean point rather than the limit state surface. This expansion leads to the inability to properly approximate the actual LSF and thus affects the accuracy of the estimated β. Moreover, its performance is largely influenced by the various forms of LSFs, as the response and partial derivatives of G(X) differ depending on the LSF form, which are important elements in the reliability index of FOSM. It is interesting to investigate what form of the LSF of geotechnical structures is more preferred by the FOSM and can give more accurate results.

3. FORM

FORM is another commonly used approximate analytical method in the geotechnical engineering field [7,15,36,37,38,39]. Low et al. [36] applied and compared three constrained optimization FORM procedures in the context of a rock slope and an embankment on soft ground. Xiao et al. [7] employed FORM for efficient, full probabilistic design of soil slopes in spatially variable soils. Zhang et al. [15] used a FORM-based approach to study the reliability of a deteriorating reinforced concrete drainage culvert in Shanghai, China. FORM is considered one of the most reliable analysis methods among the approximate analytical methods because of its good estimation accuracy and computational efficiency [40].

Different from FOSM, FORM expands the LSF linearly at the design point. The design point is a special point defined in independent standard normal space with a minimum distance to the origin on the limit-state surface. It is also the most probable failure point (MPP) on the limit-state surface. This distance between the MPP and origin is equal to the β and is also called the Hasofer–Lind index [41]. Based on the definition of β, calculating β can be transformed into an optimization problem as follows:

$$\beta =\underset{G(\mathit{X})\le 0}{\min }\sqrt{\left(\mathit{X}-\mathit{\mu }\right){\mathit{C}}^{-1}\left(\mathit{X}-\mathit{\mu }\right)}$$

(6)

where C denotes the X’s covariance matrix and C⁻¹ represents the inverse matrix of C. FORM is able to consider the effects of random variables’ probability distribution types on the β in the sense that the location of MPP differs for different distributions. As illustrated by previous studies [37,39,42,43], FORM is not sensitive to LSF’s form. This is because FORM’s reliability index is only related to the position of the MMP on the limit-state surface. However, if the same explicit form of LSF is used to describe different interpretive models of geotechnical structures, its results might be affected since the LSF has been implicitly changed. In fact, different equivalent forms of the LSF correspond to the same limit-state surface (G = 0). Therefore, the search for the MMP is the main focus of FORM. There exist many optimization algorithms, such as the well-known HLRF algorithm [37,44,45,46], or optimization tools for searching the MPP. The HLRF algorithm is too complex in terms of mathematical theory and implementation to be used by geotechnical engineers. [47,48] proposed an Excel-based constrained optimization approach for searching the design point. An alternative way is to use the optimization functions provided in other commercial software to search for the design point. This study used the fmincon() function in MATLAB [49] for solving the design point.

4. PEM

The point estimation method (PEM) was first proposed by Evans [50] for calculating the statistical moments of a function. Many scholars have used PEM for reliability analysis in geotechnical engineering. Fan et al. [17] proposed an improved fourth moment method for reliability analysis of geotechnical engineering by combining PEM with the higher-order moment method. Chen et al. [16,51] demonstrated the applicability of the PEM to the reliability analysis of slopes and retaining walls. Lu et al. [52] used PEM to assess the probability of slope failure under a given rainfall condition. PEM takes advantage of the fact that it can estimate the statistical moment with high computational efficiency but does not require the derivatives of the LSF. There are many point estimation methods available in the literature [53,54,55,56,57]. Fan et al. [58] compared several typical point estimation methods and pointed out that the new probability moment point estimation method suggested by Zhao and Ono [57] is relatively optimal in terms of practicality and accuracy. Therefore, this section focuses on Zhao and Ono’s PEM. With the estimated statistical moments, the β is calculated by applying the fourth-order moment reliability analysis method proposed by Zhao and Ono [22].

4.1. Estimating Statistical Moments

Zhao and Ono [57] suggested a dimensional reduction approximation method to approximate the original LSF G(X) as the summation of n numbers of single variable functions G_i(X_i)

$$G\left(\mathit{X}\right)\approx {\displaystyle \sum _{i=1}^n\left[G_i\left(X_i\right)-G_{\mu }\right]+}{G}_{\mu }$$

(7)

where G_i(X_i) denotes the function of the i-th variable in X by setting the other variables in X as their mean values; G_μ denotes the value of G(X) evaluated at μ_X. Let U = (U₁, U₂, …, U_n)^T denote the independent standard normal random variables. The original variable X can be converted to U by Nataf inverse transformation or Rosenblatt inverse transformation. Let X_i = T⁻¹(U_i) denote the transformation and insert into Equation (7) to obtain Equation (8)

$$G\left(\mathit{X}\right)\approx {\displaystyle \sum _{i=1}^n\left[G_i\left(T^{-1}\left(U_i\right)\right)-G_{\mu }\right]+}{G}_{\mu }$$

(8)

Equation (8) becomes a function of U. Since U_i are mutually independent, so G_i(T⁻¹(U_i)) are also not correlated with each other. Zhao and Ono [57] compute the first fourth moments of the G(X) as follows:

$${\mu }_G={\displaystyle \sum _{i=1}^n\left({\mu }_i-G_{\mu }\right)+}{G}_{\mu }$$

(9)

$${\sigma }_G^2={\displaystyle \sum _{i=1}^n{\sigma }_i^2}$$

(10)

$${\alpha }_{3G}{\sigma }_G^3={\displaystyle \sum _{i=1}^n{\alpha }_{3i}{\sigma }_i^3}$$

(11)

$${\alpha }_{4G}{\sigma }_G^4={\displaystyle \sum _{i=1}^n{\alpha }_{4i}{\sigma }_i^4+6{\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum _{j>1}^n{\sigma }_i^2}}}{\sigma }_j^2$$

(12)

where μ_i, σ_i², α_3i, α_4i are the first fourth probability moments of G_i(T⁻¹(U_i)), which can be computed by the following equations:

$${\mu }_i={\displaystyle \sum _{j=1}^m{P}_j{G}_i\left[T^{-1}\left(U_{i,j}\right)\right]}$$

(13)

$${\sigma }_i^2={{\displaystyle \sum _{j=1}^m{P}_j\left\{G_i\left[T^{-1}\left(U_{i,j}\right)\right]-{\mu }_i\right\}}}^2$$

(14)

$${\alpha }_{ki}{\sigma }_i^k={{\displaystyle \sum _{j=1}^m{P}_j\left\{G_i\left[T^{-1}\left(U_{i,j}\right)\right]-{\mu }_i\right\}}}^k$$

(15)

where U_i,j and P_j are the estimation points and corresponding weights, which can be obtained from Gaussian integral points with a simple transformation. The commonly used five-point and seven-point estimations are listed in

Table 1. Estimation point in PEM [57].

Number of Estimating Points, m	Estimating Points, Uj	Coefficient of Weight, Pj
5	0	8/15
	±1.3556262	0.2220759
	±2.8569700	1.112574 × 10−2
7	0	16/35
	±1.1544054	0.2401233
	±2.3667594	3.07571 × 10−2
	±3.7504397	5.48269 × 10−4

.

4.2. Fourth Moment Methods

Three fourth-order moment methods were proposed by Zhao and Ono [22] to calculate β, which are represented by FM1, FM2, and FM3 in the paper, respectively. The FM1 is based on the high-order moment standardization method [59]. With the first fourth statistical moments of G(X) obtained by PEM, FM1 estimates the β by

$${\beta }_{\mathrm{FM}1}=\frac{3\left({\alpha }_{4G}-1\right){\beta }_{SM}+{\alpha }_{3G}\left({\beta }_{SM}^2-1\right)}{\sqrt{\left(9{\alpha }_{4G}-5{\alpha }_{3G}^2-9\right)\left({\alpha }_{4G}-1\right)}}$$

(16)

where β_SM = μ_G/σ_G is the second-order moment β. Different from FM1, FM2 is based on the Edgeworth expansion method [60,61]. It obtains the β by

$$P_{f,\mathrm{FM}2}=\Phi \left(-{\beta }_{SM}\right)-\phi \left({\beta }_{SM}\right)\left[\frac{1}{6}{\alpha }_{3G}{H}_2\left(-{\beta }_{SM}\right)+\frac{1}{24}\left({\alpha }_{4G}-3\right)H_3\left(-{\beta }_{SM}\right)+\frac{1}{72}{\alpha }_{3G}^2{H}_5\left(-{\beta }_{SM}\right)\right]$$

(17)

$${\beta }_{FM2}=-{\Phi }^{-1}\left(P_{f,FM2}\right)$$

(18)

where H₂(x) = x² − 1, H₃(x) = x³ − 3x and H₅(x) = x⁵ − 10x³ + 15x, respectively. The FM3 is based on the Pearson system method [61]. It uses the Pearson system to solve the probability density function f(z_u) of standardized variables z_u = [G(X)−μ_G]/σ_G and then integrate f(z_u) to calculate the β as follows:

$$\frac{1}{f}\frac{df}{d{z}_u}=-\frac{a{z}_u+b}{c+b{z}_u+d{z}_u^2}$$

(19)

$${\beta }_{\mathrm{FM}3}=-{\Phi }^{-1}\left({\displaystyle {\int }_{-\infty }^{-{\beta }_{SM}}f\left(z_u\right)}d{z}_u\right)$$

(20)

where a = 10α_4G − 12α_3G² − 18, b = α_3G(α_4G + 3), c = 4α_4G – 3α_3G² and d = 2α_4G − 3α_3G² − 6.

Although fourth-order moment methods perform well in terms of computational efficiency and accuracy, they are still sensitive to the form of the LSF. The different responses of G(X) can lead to differences in the calculation results of the first four probability moments, which ultimately result in differences in the estimated reliability indexes. In addition, some limitations are also necessary to be noted, which have already been studied by Zhao and Ono [22,62]. On the one hand, FM1, FM2, and FM3 are not applicable to the LSF, in which random variables exceed the fourth power. On the other hand, when the values of α²_3G and α_4G are within a certain range, the Pearson system will have no solution, resulting in the failure of FM3. These limitations of the three fourth-moment methods are further demonstrated in the following example III.

5. Illustrative Examples

This section compares the performance of FOSM, FORM, FM1, FM2, and FM3 using four typical geotechnical examples, including a pile, a square footing, and two retaining walls. The LSF of these geotechnical structures is artificially expressed in four forms for investigating the effects of different algebraic forms of LSF: G = R − L, G = R/L − 1, G = ln(R/L), and G = exp(R − L) − 1. The statistical moments required for the fourth-moment methods are obtained from seven-point estimates. The β estimated by Monte Carlo simulation (MCS) is considered the reference solution. The MCS is implemented with 10⁶ samples for all the investigated examples.

5.1. Example I: Floating Pile Embedded in Two-Layer Soil

This section employs a floating pile example to demonstrate the performance of the investigated methods. This example has been investigated by [63]. The schematic of the pile is depicted in Figure 1. The pile is installed in a clay layer with a thickness of L_c = 9 m and a sand layer with a thickness of L_s = 36 m. Let the DL and LL denote the vertical dead and live loads, respectively. The DL and LL exert force on the top of the pile. The pile tip resistance is minor and thus neglected in the analysis. According to [63], four equivalent forms of the performance function are defined as follows:

$$G=\ln \left(\frac{R_{\mathrm{c}}+R_{\mathrm{s}}}{DL+LL}\right)=\ln \left(\frac{14.9\pi L_{c}B{\epsilon }_{\alpha }{s}_u^{0.3}+2.71\pi L_{s}B{\epsilon }_{N}N}{DL+LL}\right)$$

(21)

$$\begin{array}{ll}G& =\exp \left[\left(R_c+R_s\right)-\left(DL+LL\right)\right]-1\hfill \\ & =\exp [\left(14.9\pi L_{c}B{\epsilon }_{\alpha }{s}_u^{0.3}+2.71\pi L_{s}B{\epsilon }_{N}N\right)-\left(DL+LL\right)]-1\hfill \end{array}$$

(22)

$$G=\frac{R_{\mathrm{c}}+R_{\mathrm{s}}}{DL+LL}-1=\frac{14.9\pi L_{c}B{\epsilon }_{\alpha }{s}_u^{0.3}+2.71\pi L_{s}B{\epsilon }_{N}N}{DL+LL}-1$$

(23)

$$G=\left(R_{\mathrm{c}}+R_{\mathrm{s}}\right)-\left(DL+LL\right)=\left(14.9\pi L_{c}B{\epsilon }_{\alpha }{s}_u^{0.3}+2.71\pi L_{s}B{\epsilon }_{N}N\right)-\left(DL+LL\right)$$

(24)

where B denotes the diameter of the pile; R_c and R_s are resistances provided by the clay and sand layers, respectively; s_u represents the clay layer’s undrained shear strength; N represents the sand layer’s SPT-N value; ε_α and ε_N are model transformation uncertainties. Herein, the s_u and SPT-N values represent the average values within the clay layer and sand layer if the spatial variability of soil properties were considered. The DL, LL, s_u, N, ε_α and ε_N are considered to be uncertain and mutually independent.

Table 2. Statistical information of uncertain parameters for the pile example.

Uncertain Parameters	Mean	COV	Distribution
DL (kN)	2500	10%	Lognormal
LL (kN)	1250	20%	Lognormal
su (kN/m2)	125	30%	Lognormal
N	30	30%	Lognormal
εα	1.05	32%	Lognormal
εN	1.22	70%	Lognormal

lists the statistical information for the five uncertain parameters.

Figure 2 shows the effect of different equivalent forms of the LSF on the β obtained by different approximated analytical reliability analysis methods. The pile diameter B takes the values of 0.5 m, 0.6 m, 0.7 m, …, 1.9 m, and 2.0 m to show the effect of the pile diameter. It is shown that the accuracy of the estimated β for different equivalent forms of the LSF gradually decreases as the pile diameter increases. The result of FORM for only one form of LSF is shown in Figure 2a because the result of FORM is not affected by the form of LSF. For the other investigated methods, the β obtained from the logarithmic form of LSF (i.e., G = ln(R/L)) is much closer to the reference solution compared to the other forms. For FOSM and PEM methods, the β obtained from the LSF with the forms G = R/L − 1 and G = R − L are close and have larger estimation errors with increasing pile diameter. As shown in Figure 2c, FM2 cannot give a solution when the pile diameter is greater than 1.2 m, which is caused by the applicability of the method. Similarly, the FM3 method also has an inherent no-solution region, which will be illustrated in the subsequent examples. The LSF with the exponential form such as G = exp(R − L) − 1 for this example will cause some numerical issues because some response values may exceed the maximum real value of the double precision algorithm in the MATLAB software.

Figure 3 compares the number of evaluations of LSF for different methods. The computational efficiency of FOSM and PEM methods is not sensitive to the form of the LSF. This is because, for the FOSM and PEM, the evaluation number of LSF is only related to the number of uncertain parameters and the number of estimating points. For FORM, the evaluation number of LSF with an exponential form is much greater than other forms. It indicates that although the β computed by FORM is not impacted by the form of the LSF, the computing efficiency of FORM may be affected by the different forms of the LSF.

5.2. Example II: Gravity Retaining Wall

The example of a gravity retaining wall was initially studied by [64] and later slightly modified by [8]. The schematic of the retaining wall is depicted in Figure 4. The retaining wall’s total height is H_T = 5 m. The depth of the retaining wall embedded in the base sand is 0.5 m. The width of the concrete base is W. The thickness of both the concrete base and the concrete stem is t = 0.5 m. The concrete’s unit weight is γ_c = 25 kN/m³. Let ϕ_b′ denote the base sand’s effective friction angle. The effective friction angle and unit weight of the backfill sand are denoted as ϕ_f and γ_f, respectively. q exerts a surcharge load on the ground surface behind the wall. Five random variables γ_fS, γ_fD, ϕ_f, ϕ_b′, and q, are mutually independent, and their statistics and distribution types are listed in

Table 3. Statistics of uncertain parameters for Example II.

Uncertain Parameters	Mean	COV	Distribution
γfS (kN/m3)	17	10%	Normal
γfD (kN/m3)	17	10%	Normal
ϕf (°)	35	10%	Lognormal
ϕb′ (°)	35	10%	Lognormal
q (kN/m2)	10	10%	Gumbel

(denoted as Case 1). The four equivalent forms of LSF are defined as

$$G=\ln \left(\frac{\left[\left(W+H_T-t\right)t{\gamma }_c+\left(W-\mathrm{t}\right)\left(H_T-t\right){\gamma }_{fS}\right]\tan \left({\varphi }_b^{\prime }\right)}{0.5{\gamma }_{fD}{H}_T{}^2{K}_a+q{H}_T{K}_a}\right)$$

(25)

$$G=\exp \left\{\left[\left(W+H_T-t\right)t{\gamma }_c+\left(W-\mathrm{t}\right)\left(H_T-t\right){\gamma }_{fS}\right]\tan \left({\varphi }_b^{\prime }\right)-\left(0.5{\gamma }_{fD}{H}_T{}^2{K}_a+q{H}_T{K}_a\right)\right\}-1$$

(26)

$$G=\frac{\left[\left(W+H_T-t\right)t{\gamma }_c+\left(W-\mathrm{t}\right)\left(H_T-t\right){\gamma }_{fS}\right]\tan \left({\varphi }_b^{\prime }\right)}{0.5{\gamma }_{fD}{H}_T{}^2{K}_a+q{H}_T{K}_a}-1$$

(27)

$$G=\left[\left(W+H_T-t\right)t{\gamma }_c+\left(W-\mathrm{t}\right)\left(H_T-t\right){\gamma }_{fS}\right]\tan \left({\varphi }_b^{\prime }\right)-\left(0.5{\gamma }_{fD}{H}_T{}^2{K}_a+q{H}_T{K}_a\right)$$

(28)

where K_a = tan (45° − ϕ_f/2)² is the coefficient of active earth pressure.

Figure 5 shows the β estimated by FORM, FOSM, and FM1/2/3 based on different forms of LSF. The width of concrete base B varies from 2 m to 3 m with an increment of 0.5 m to investigate the influence of B. The β estimated by FORM with different initial iteration points is shown in Figure 5a, where FORM^a takes several random values as the initial value of the iterations in the fmincon() function, while FORM^b only takes the mean value of the random variables as the initial value of the iterations. It is seen that the β estimated by FORM may be affected by the initial iteration point, especially when B is greater than 2.6 m. This influence might be dependent on the nonlinear optimization algorithm. When the FORM is used for estimating the β of geotechnical structures, one should exercise caution on the initial iteration point to avoid obtaining unreasonable results. A practical suggestion is to try different initial iteration points to cross-validate the result of FORM.

Due to the same reason as in Example 1, the results for the exponential form for the methods used are not always attainable. The LSF of a geotechnical structure with an exponential form is easy to make some numerical errors with, and thus the LSF with this form will not be discussed in the following examples. Similar to Example 1, for FOSM and PEM methods, the β obtained from the logarithmic form of LSF (i.e., G = ln(R/L)) is much closer to the reference solution compared to the other forms. For FOSM and PEM methods, the β has larger estimation errors with increasing B. The type of parameter distribution also affects the performance of the different approximate analytical methods. It is interesting to investigate the influence of different forms of LSFs under different distribution types. Case 2 adopts the normal distributions of soil parameters to estimate the reliability index of the gravity retaining wall. Case 2 is compared with Case 1, where the soil parameters follow lognormal distributions, to demonstrate the influence of distribution type on the selection of the optimal form of LSF. The impact of the probability distribution is demonstrated by comparing the estimation error. The comparison results are summarized in

Table 4. Estimated error of reliability index compared to the reference solution for the Example II.

Estimation Error	FOSM		FM1		FM2		FM3
	Case 1	Case 2	Case 1	Case 2	Case 1	Case 2	Case 1	Case 2
G = ln(R/L)	8.4%	1.1%	5.0%	0.7%	27.1%	21.4%	22.6%	17.6%
G = R/L − 1	34.7%	29.1%	29.9%	24.1%	40.6%	36.8%	34.1%	29.3%
G = R − L	13.0%	5.4%	8.7%	3.3%	31.5%	25.8%	23.8%	19.0%

. Compared with Case 1, the estimated error of the reliability index for each form of the LSF is reduced for each method. Although the distribution type is different, the estimation error of the reliability index with the lognormal form of the LSF is still the smallest among the three investigated forms of the LFS.

5.3. Example III: Square Footing

This section further uses the square footing investigated by [18] to illustrate the performance of the approximate analytical methods. As shown in Figure 6, the square footing’s size is B × B. The footing is embedded in a clay layer with an embedment depth of D = 1 m. The clay’s unit weight is γ = 17 kN/m³. The vertical load V and horizontal load H are imposed on the footing base and considered uncertain.

Table 5. Statistics of uncertain parameters for Example III.

Uncertain Parameters	Mean	COV	Distribution
c (kN/m2)	19	30%	Lognormal
ϕ (°)	25	20%	Lognormal
H (kN)	200	15%	Lognormal
V (kN)	800	10%	Lognormal

summarizes the statistical information for uncertain parameters c, ϕ, H, and V. The c, ϕ, H, and V are independent except that c and ϕ have a negative correlation with ρ = −0.5. The water table is so low that it has no effect on the footing capacity. According to Hansen’s ultimate bearing capacity formula, the four performance functions in different forms are defined as

$$G=\ln \left[\frac{\left(\mathrm{c}{N}_1{\mathrm{s}}_1{d}_1{i}_1+q{N}_2{s}_2{d}_2{i}_2+0.5\gamma B{N}_3{s}_3{d}_3{i}_3\right)B^2}{V+W}\right]$$

(29)

$$G=\exp \left[\left(\mathrm{c}{N}_1{\mathrm{s}}_1{d}_1{i}_1+q{N}_2{s}_2{d}_2{i}_2+0.5\gamma B{N}_3{s}_3{d}_3{i}_3\right)B^2-\left(V+W\right)\right]-1$$

(30)

$$G=\frac{\left(\mathrm{c}{N}_1{\mathrm{s}}_1{d}_1{i}_1+q{N}_2{s}_2{d}_2{i}_2+0.5\gamma B{N}_3{s}_3{d}_3{i}_3\right)B^2}{V+W}-1$$

(31)

$$G=\left(\mathrm{c}{N}_1{\mathrm{s}}_1{d}_1{i}_1+q{N}_2{s}_2{d}_2{i}_2+0.5\gamma B{N}_3{s}_3{d}_3{i}_3\right)B^2-\left(V+W\right)$$

(32)

where W= γ_cB²D denotes the footing concrete’s weight γ_c = 23 kN/m³ denotes the concrete’s unit weight. The bearing capacity factors, shape factors, depth factors, and inclination factors are summarized in

Table 6. Bearing capacity factors, Shape factors, Depth factors and Inclination factors for Example III.

Bearing Capacity Factors	Shape Factors	Depth Factors	Inclination Factors
N1 = (N2 − 1)cot(ϕ)	s1 = 1 + N2/N1	d1 = 1 + 0.4D/B	i1 = [1 − tan−1(H/V)/(π/2)]2
N2 = eπtan(ϕ)tan2(45° + ϕ/2)	s2 = 1+tan(ϕ)	d2 = 1 + 2tan(ϕ)[1 – sin(ϕ)]2D/B	i2 = i1
N3 = 2(N2 + 1)tan(ϕ)	s3= 0.6	d3= 1	i3 = [1 − tan−1(H/V)/(πϕ/180)]2

.

Figure 7 shows the estimated β by FORM, FOSM, and FM1/2/3 based on the different LSFs. The footing size B takes values = 1.5 m, 1.6 m, 1.7 m, …, 2.9 m, and 3 m to investigate the influence of reliability levels. The finding is similar to examples I and II: The logarithmic form of LSF can be more accurate β compared with the other forms for the FOSM and PEM methods. Different from examples I and II, Figure 7c,d indicate that FM2 and FM3 can only estimate the β of the footing example based on the logarithmic form of LFS. For the exponential form, the reason is the same as in the previous two examples. For the other two forms of performance functions, FM2 has inherent limitations that may result in a negative failure probability. Take the form G = R – L and B = 1.5 m, for example, and FM2 obtains the first four order probability moments of the LSF, i.e., the mean, variance, skewness, and kurtosis = (1471.45, 2556.16, 18.00, 547.04). Based on these statistical moments, FM2 gives a P_f = −0.4778 using Equation (17). The Equation (17) is able to give some negative failure probability values when the first fourth-order moments of the performance function satisfy certain relations. The FM3 also has some inherent limitations because the Pearson system contains an impossible area, as shown in Figure 8. FM3 is not applicable when the α²_3G and α_4G of the LSF are in the impossible area.

5.4. Example IV: Active Lateral Forces on Retaining Walls

This section further employs a retaining wall, considering the spatially variable soil properties, to demonstrate the performance of approximate analytical methods. This example is adopted from the study of [65]. Figure 9 shows the geometry of the retaining wall and one typical random field realization of the undrained shear strength parameter s_u. The s_u is lognormally distributed, with a mean value of 20 kPa and a coefficient of variation (COV) of 30%. A stationary lognormal random field s_u(x, z) is utilized to simulate the spatial variability of s_u. The spatial autocorrelation of s_u is modeled by the single exponential auto-correlation model, i.e., ρ(Δx, Δz) = exp(−2|Δx|/δ_x − 2|Δz|/δ_z), where Δx and Δz represent the horizontal and vertical distances between two points, and δ_x and δ_z denote the horizontal and vertical scales of fluctuations. This study considers δ_x = δ_z = δ for simplification. Assume that the potential slip lines are straight lines through the toe. According to the force equilibrium of the wedge above the potential slip lines, the equation for the lateral force F can be deduced as

$$F=\frac{1}{2}\gamma H^2-s_u^{LA}\cdot \frac{H}{\sin \left(\alpha \right)\cos \left(\alpha \right)}$$

(33)

where γ = 20 kN/m³ is the unit weight of soil; s_u ^LA is the averaged value of s_u(x, z) along a potential slip line; α denotes the potential slip line’s inclination angle.

In this example, the potential slip lines with inclination angles varying from 30° to 60° are considered. Each potential slip line can obtain a lateral force F. The active lateral force P_a is noted as the maximum value of the F(α) function as follows:

$$P_a=\underset{\alpha }{\max }F\left(\alpha \right)=\underset{\alpha }{\max }\left(\frac{1}{2}\gamma H^2-s_u^{LA}\cdot \frac{H}{\sin \left(\alpha \right)\cos \left(\alpha \right)}\right)$$

(34)

The retaining wall’s ultimate bearing capacity is taken as 100 kN, and the LSF can be defined as the following four forms:

$$G=\ln \left(\frac{100}{P_a}\right)$$

(35)

$$G=\exp \left(100-P_a\right)-1$$

(36)

$$G=\frac{100}{P_a}-1$$

(37)

$$G=100-P_a$$

(38)

Figure 10 shows the effect of four equivalent forms of the performance functions on the β obtained by FORM, FOSM, and FM1/2/3 for scales of fluctuation δ = 0.1 m, 0.5 m, 2.5 m, 10 m, 50 m, 250 m, and 1500 m. It is shown from Figure 10a that FORM can still obtain approximately accurate results compared with the reference solution obtained by the MCS. For FOSM, the β corresponding to the LSF in logarithmic form is no longer the most accurate compared with the previous three examples. The form G = R − L gives the most accurate result for FOSM. This is because Equations (35) and (37) are not exactly equivalent to Equations (36) and (38) in the sense that some negative values of P_a may appear when the uncertain factors involved in the retention system are considered. Specifically, Equation (35) cannot accept negative P_a, and Equation (37) gives G < 0 for negative P_a. A similar issue was also noted in the study by [62]. One should exercise caution when reformulating the forms of LSF to ensure that the reformulated forms of LSF are equivalent to each other.

The investigated point estimation methods are all not applicable to this example. This may be because this example contains 5000 random variables. As the dimension of the random variable increases, the estimation errors of statistical moments obtained by point estimation methods become larger. In addition, the correlation and larger COV of random variables can also have a considerable influence on the accuracy of the point estimates [58]. Therefore, for high-dimensional problems with complex correlations and a large COV of random variables, the point estimation method is strongly not recommended.

5.5. Example V: A Homogeneous Clay Slope

Geotechnical structures are easily subject to shear damage [66]. Numerical analysis methods such as the finite element method, the finite difference method, and the discrete element method provide effective tools for solving geotechnical stability problems involving shear damage. When these numerical methods are used, the LSF of the geotechnical structure is implicit. This section further considers a homogeneous soil slope with implicit LSF to demonstrate the performance of the investigated methods. The slope example has been previously investigated by [67]. The finite element method is used to estimate the factor of safety (FS) of the slope. Figure 11 shows the slope finite element model established in ABAQUS. The height of the slope is 13 m, and the slope angle is 45°. The unit weight of clay is considered to be deterministic, with a value of 20 kN/m³. The friction angle ϕ and cohesion c are considered to be two independent lognormal random variables with mean values of 20°and 25.4 kPa, respectively. Suppose that the coefficient of variation (COV) for ϕ has a fixed value of 0.3. However, the COV of c varies from 0.1 to 0.3 with an increment of 0.05 to analyze the effect of different levels of COV on the performance of LSFs. It is noted that although the LSF of the slope example is implicit, it can still be formulated with the following two equivalent forms:

$$G=\ln (FS)$$

(39)

$$G=FS-1$$

(40)

where the FS is estimated by the strength reduction method through the finite element model.

Figure 12 shows the influence of COV of cohesion on the reliability indexes of the slope for different forms of LSFs. It is seen that the performance of various methods on different forms of LSFs is not affected by the varying COV of cohesion. The logarithmic form is more preferred by the FOSM and point estimation methods.

Table 7. Estimated error of reliability index compared to the reference solution for the Example V.

Estimation Error	FOSM		FM1		FM2		FM3
	G1	G2	G1	G2	G1	G2	G1	G2
COV = 0.10	9.1%	23.7%	3.7%	9.4%	1.4%	1.5%	0%	8.2%
COV = 0.15	3.4%	18.8%	0.8%	10.5%	1.8%	4.6%	1.8%	10.2%
COV = 0.20	5.0%	19.5%	1.5%	12.2%	1.7%	6.5%	2.1%	10.9%
COV = 0.25	3.4%	18.2%	4.4%	12.9%	4.3%	10.9%	4.3%	11.9%
COV = 0.30	0.1%	15.2%	1.0%	9.4%	0.6%	6.1%	0.8%	7.7%

Note: G₁ and G₂ represent the forms of G₁ = ln(FS) and G₂ = FS − 1, respectively.

also presents the estimation errors of reliability indexes estimated by the four investigated methods compared to the reference solution obtained by the MCS method. For the same form of LSF, there is no clear trend in the estimation error as the coefficient of variation of cohesion increases. The logarithmic form of LSF generally gives a smaller estimation error for the investigated approximate analytical method, regardless of the level of COV of the cohesion. Therefore, the logarithmic form of LSF is also recommended to be used for the approximate analytical reliability analysis methods for the slope example with implicit LSF.

6. Conclusions

This study systematically investigates the influence of LSF form on the performance of commonly used approximate analytical methods, including FOSM, FORM, and three PEMs, including FM1, FM2, and FM3. Four equivalent forms of LSF, including G = ln(R/L), G = exp(R − L) − 1, G = R/L − 1 and G = R − L are considered. The comparison conclusions are summarized as follows:

• The form of LSF does not affect the accuracy of β obtained by FORM; however, it affects the computational efficiency of FORM. The accuracy of β obtained by FORM may be affected by the initial iteration point used in the optimization algorithms. When FORM is used for estimating the β of geotechnical structures, one should exercise caution on the initial iteration point to avoid obtaining unreasonable results. A practical suggestion is to try different initial iteration points to cross-validate the result of FORM.
• FOSM is sensitive to forms of LSF. Generally, FOSM can obtain more accurate estimation results with the logarithmic form of LSF compared with other forms. The computational efficiency of FOSM is independent of the form of LFS but dependent on the dimension of random variables. It is recommended to use the logarithmic form of LSF when FOSM is used to estimate the β of the geotechnical systems.
• The investigated PEMs are also sensitive to forms of LSF. Similar to FOSM, PEMs can also obtain more accurate estimation results with the logarithmic form of LSF compared with other forms. The computational efficiency of PEMs is dependent on the dimension of the random variable. PEMs may yield extremely large estimation errors due to the large number of random variables or complex correlations among random variables. Choosing a different but equivalent form of LSFs can sometimes avoid the inherent limitations of FM2 and FM3, where the Pearson system does not have a solution.
• All investigated examples show that the logarithmic form G = ln(R/L) is always the optimal choice among the four equivalent forms for the approximate reliability methods that are sensitive to the forms of LSFs. The probability distribution type, number of variables involved in the LSF, and level of coefficient of variation of the random variables have effects on the reliability indexes estimated by different forms of LSFs; however, they do not affect the result that the logarithmic form of the LSF is superior to the other forms. It should be noted that this result is not guaranteed theoretically but is supported by analyzing the five typical geotechnical examples. There may be some rare exceptions or specific limits that violate the result. For general cases, the logarithmic form G = ln(R/L) is recommended for the approximate reliability analysis methods.