Reliability assessment of three-dimensional bearing capacity of shallow foundation using fuzzy set theory

Abstract

The aim of this study is to investigate the applicability of reliability theory on surface square/rectangular footing against bearing capacity failure using fuzzy set theory in conjunction with the finite element method. Soil is modeled as a three-dimensional spatially varying medium, where its parameters (cohesion, friction angle, unit weight, etc.) are considered as fuzzy variables that maintain some membership functions. Soil is idealized as an elastic-perfectly plastic material obeying the Mohr–Coulomb failure criterion, where both associated and non-associated flow rules are considered in estimating the ultimate bearing capacity of the footing. The spatial variability of the soil is incorporated for both isotropic and anisotropic fields, which are determined by the values of scales of fluctuation in both the horizontal and vertical directions. A new parameter namely, limiting applied pressure at zero failure probability is proposed, and it indirectly predicts the failure probability of the footing. The effect of the coefficient of variation of the friction angle of the soil on the probability of failure is analyzed, and it is observed that the effect is significant. Furthermore, the effect of the scale of fluctuation on the probability of failure is investigated, and the necessity for considering spatial variability in the reliability analysis is well proven.

Graphical abstract

Keywords

finite element method / square footing / reliability analysis / fuzzy set theory / coefficient of variation / spatial variability

Introduction

Generally, two types of foundation failures, i.e., settlement and bearing capacity failure, are considered in any foundation design. Failure due to the bearing capacity of soil is an issue in many civil engineering structures. This study focused on the bearing capacity failure of square/rectangular footing. As the safety of the foundation is a major issue for most civil engineering structures, reliability analysis must be performed to provide caution regarding the failure of any structure caused by foundation failure. In addition, the identifying parameter for the failure intensity, i.e., the probability of failure, is systematically indicated via reliability analysis.

To estimate the bearing capacity of square or rectangular footing, the parameter that is typically computed along with the bearing capacity factor is the shape factors. The governing equation for estimating the bearing capacity of surface square or rectangular footing is as follows [1]:

(1)q_{\mathrm{u}\mathrm{l}\mathrm{t}}=c{N}_{\mathrm{c}}{s}_{\mathrm{c}}+q_{\mathrm{s}}{N}_{\mathrm{q}}{s}_{\mathrm{q}}+0.5\mathrm{\gamma }B{N}_{\gamma }{s}_{\mathrm{\gamma }},

where q_ult is the ultimate bearing capacity of the footing; c and γ are cohesion and unit weight of the soil, respectively; q_s is the surcharge pressure; B is the width of the footing; N_c, N_q, and N_γ are the bearing capacity factors; s_c, s_q, and s_γ are the shape modifiers or shape factors.

Previously, some empirical and semi-empirical approaches [1–3] have been used to propose the shape factors of footing for different friction angles of soil. Some conflicting results were discovered for the formulation proposed by Meyerhof [2] and de Beer [3]. In de Beer’s formulation, the shape factors are functions of only the aspect ratio (L/B), whereas in Meyerhof’s equation, they are a function of both the aspect ratio and soil friction angle (φ). Subsequently, some numerical approaches such as limit analysis and finite element (FE) analysis were deployed to determine the shape factors of square or rectangular footing resting on both cohesive and cohesion-frictional soil deposits. Michalowski [4] used limit analysis in conjunction with the plane deformation mechanism to calculate the shape factors for both square and rectangular footings. This approach predicts higher values of shape factors than the values proposed earlier via semi-empirical approaches. Zhu and Michalowski [5] determined the shape factors for both square and rectangular footings resting on c-φ soil medium using the finite element method (FEM). The results showed that the shape factors for cohesion and surcharge terms, i.e., s_c and s_q, respectively, agree well with the results obtained from the formulation proposed by Meyerhof [2] and de Beer [3]. However, the s_γ values differed from those of earlier investigations. Based on the FE analysis results, Gourvenec et al. [6] proposed a formulation to determine the s_c values as a function of L/B of the footing. Li et al. [7] investigated the effects of shape factors on the bearing capacity of footing using the three-dimensional (3D) finite difference method, and the results showed that the bearing capacity of a rectangular footing decreased as the footing aspect ratio increased. Gupta et al. [8] proposed a closed-form solution for estimating the bearing capacity of square/rectangular footings on a layered soil system using the punching shear failure mechanism at the ultimate load. The study implied that the top layer thickness, aspect ratio of the footing, and shear strength parameters of the soil significantly affected the bearing capacity of the footing. Moreover, to investigate the effect of the embedment depth of a rectangular footing on the bearing capacity, Liu et al. [9] performed a study to investigate various aspects and embedment ratios of footing using small strain FE analysis and proposed a formulation to calculate the bearing capacity factor for rectangular footing with different aspect ratios at any embedment depth. Recently, Osman [10] presented a rigid block mechanism to estimate the shape factors of smooth square/rectangular footings using upper bound limit analysis.

With respect to the reliability analysis of footing against bearing capacity failure under a probabilistic framework, different investigations [11–21] were attempted using conventional reliability theory (First Order Reliability Method (FORM), Monte Carlo simulation (MCS), etc.). In most reliability analyses, the variables (soil parameter) were assumed to be random variables [22–24] that maintained a specified probability distribution. To define the appropriate distribution patterns of the random variables, researchers have encountered problems due to the non-availability of large numbers of datasets for conducting statistical analysis. Limited datasets cannot provide appropriate results from which one can infer the distribution pattern of the variables. In addition, in conventional reliability analysis, statistical or modeling uncertainties are considered, but the vagueness/fuzziness and ambiguity of the parameters are debarred from the analysis. By contrast, in fuzzy reliability analysis, the first shortcoming can be overcome by considering the variables as fuzzy numbers described by some membership functions, and fuzziness is considered in the reliability analysis when the parameters belong to a fuzzy set. Therefore, a few researchers have implemented reliability analysis using fuzzy set theory [25–31], where variables are assumed as fuzzy numbers. Most studies regarding fuzzy reliability analysis are primarily confined to slope, rock stability, and braced excavation problems. Fuzzy reliability in geotechnical foundation problems (bearing capacity and settlement) is yet to be investigated. Therefore, researchers have begun to perform some investigations in this area. Recently, some applications of reliability analysis using fuzzy set theory in two-dimensional settlement and bearing capacity problems of shallow footing have been published by the authors [32–35], where spatial variability [36–44] of soil properties was considered. Pramanik et al. [45] investigated the applicability of fuzzy reliability analysis to compute the settlement of square footing resting on the surface of a homogeneous cohesionless soil mass. However, as square or rectangular foundations are used more frequently than strip footing, the 3D bearing capacity of those foundations must be investigated. Therefore, the 3D computation of bearing capacity is more essential; furthermore, safety against 3D bearing capacity failure is necessitated from the total structural safety standpoint. Therefore, the risk assessment of the bearing capacity of square or rectangular footings cannot be disregarded in the field of foundation reliability.

The purpose of this study is to perform a reliability analysis of the bearing capacity of square/rectangular (3D analysis) footing resting on the surface of homogeneous soil deposits using fuzzy set theory in conjunction with the displacement-based FEM [46–48]. Both associated (φ = ψ) and non-associated (φ≠ ψ) flow rules were considered for the plasticity model. The spatial variability of the soil parameters was incorporated into the analysis to capture the inherent randomness of the parameters. Both isotropic and anisotropic scales of fluctuation were considered in the analysis, and their effects on the failure probability of footings were assessed. Furthermore, the effects of the friction angle (φ) and dilation angle (ψ) of the soil on the probability of failure (P_f) were analyzed.

Reliability analysis using fuzzy set theory

The main difference between conventional reliability and fuzzy reliability analyses is the assumption of the characteristics of the parameters. In conventional reliability analysis, parameters (variables) are considered as random variables, whereas in fuzzy reliability analysis, variables are considered as fuzzy variables or fuzzy numbers. Detailed theoretical background regarding fuzzy reliability analysis was discussed in the authors’ recent publications [33,34]. However, a brief description is presented herein.

In general, fuzzy numbers [49,50] are expressed by their membership function (MF), which is typically triangular or trapezoidal in shape. In addition, several types of MFs exist that express the fuzzy number, e.g., Gaussian, generalized bell, sigmoid, sharp gamma, and Cauchy [30,50]. In most previous studies [25–29] regarding the fuzzy reliability analysis of geotechnical structures, soil parameters are assumed as fuzzy numbers or variables following either triangular or trapezoidal MFs, owing to the conservativeness of these two types of MFs in terms of uncertainty. In this study, an exercise was first performed for both the triangular and trapezoidal MFs of the input variables; thereafter, a triangular MF was used. The possibility of occurrence of any number within a fuzzy set is defined by this MF (μ_F), which varies between a closed interval of 0.0 to 1.0 ({\mu }_{\mathrm{F}}\in [0.0,1.0]). The triangular MF is adopted herein to represent a fuzzy number. The fuzzy FE approach [27] is used in this study, where the fuzzy input numbers are incorporated into the FEM code via the vertex method [51]. In the vertex method, at each α–cut level, two values are determined for each input fuzzy number (soil parameters) that are incorporated into the FE code to obtain the final fuzzy number (ultimate bearing capacity). Six α–cut levels between 0.0 and 1.0, with an interval of 0.2, were considered in this study. Furthermore, the probability of failure (P_f) was determined using the area ratio method (see Fig. 1). Using the area ratio method, the P_f values were calculated from the positions of the lower and upper limits of the ultimate fuzzy number (ultimate bearing capacity) and the applied limiting pressure (q_lim). As shown in Fig. 1, when the lower limit of the bearing capacity at 0.0 membership grade was greater than q_lim, the P_f value became zero (Fig. 1 [Case I]). If the upper limit of q_ult at 0.0 membership grade is less than q_lim, then P_f becomes 1.0 (Fig. 1 [Case II]). If the limiting applied pressure (q_lim) lies between the lower and upper limits of the bearing capacity, then the P_f value is calculated as follows (Fig. 1 [Case III]):

(2)

where A₁ and A₂ are the areas exceeding and not exceeding q_lim, respectively. A is the total area under the MF of q_ult.

Moreover, the spatial variability of the soil parameters was incorporated into the present fuzzy reliability analysis. The spatial correlation function [r(τ)] was introduced to define the spatial variability of the parameters, which is a function of a vector of absolute distance between two points (\mathit{\tau }=\left|a_{\mathrm{1}}-a_{\mathrm{2}}\right|) (where a₁ and a₂ are two points along any direction). For the present study, an exponentially decaying Markovian spatial correlation function [52,53] expressed as follows was used:

(3)\rho ({\tau }_x,{\tau }_y,{\tau }_z)=\exp \left(-\mathrm{2}\sqrt{{\left({\displaystyle \frac{{\tau }_x}{{\delta }_x}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{{\tau }_y}{{\delta }_y}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{{\tau }_z}{{\delta }_z}}\right)}^{\mathrm{2}}}\right),

where τ_x, τ_y, and τ_z are the distances between any two points in the x-, y-, and z-directions, respectively. δ_x, δ_y, and δ_z are scales of fluctuation in the x-, y-, and z-directions, respectively.

Consequently, the variance reduction function and the reduced variance for 3D analysis were determined as follows [53]:

{\gamma }_{\mathrm{v}\mathrm{r}}\left(d\right)=\frac{8}{d^2}{\displaystyle {\int }_0^{d_{\mathrm{x}}}{\displaystyle {\int }_0^{d_{\mathrm{y}}}{\displaystyle {\int }_0^{d_{\mathrm{z}}}(d_x-{\tau }_x)(d_y-{\tau }_y)(d_z-{\tau }_z)}}}

(4)\rho ({\tau }_x,{\tau }_y,{\tau }_z)\mathrm{d}{\tau }_z\mathrm{d}{\tau }_y\mathrm{d}{\tau }_x,

(5){\sigma }_{{\mathrm{A}}_{\mathrm{R}}}^2={\gamma }_{\mathrm{v}\mathrm{r}}\left(d\right){\sigma }_{\mathrm{A}}^2,

where γ_vr(d) is the variance reduction function; d_x, d_y, and d_z are the dimensions of a single 3D element; {\sigma }_{{\mathrm{A}}_{\mathrm{R}}}^2and {\sigma }_{\mathrm{A}}^{\mathrm{2}} are the reduced and actual variances of the parameters, respectively.

This reduced variance was used for the analysis. The flow diagram of the fuzzy reliability analysis is shown in Fig. 2.

FE model and boundary conditions

In this study, 20-noded isoparametric brick elements were used to discretize the soil domain into a finite number of meshes. Reduced integration (Gauss quadrature) with 2 × 2 × 2 integrating points per element was adopted for calculation. The FE mesh comprised 3388 twenty-noded elements (22 × 22 × 7) for the footing dimension of 1 m × 1 m. Owing to the higher stress concentration, mesh refinement was performed near the footing position. The size of the problem domain for a square footing measuring 1m × 1m (L× B) was 15 m × 15 m × 5 m. As the length of the footing increased, the domain size in the x-direction (length was along the x-direction) was increased to minimize the boundary effect in the analysis. A smooth rigid square/rectangular footing resting on the surface of a c–φ soil underlain by a rigid layer was considered for the analysis. A schematic diagram of the position of the footing and the boundary conditions are shown in Fig. 3. Among the different failure criteria available in [54] for the plasticity model, the modified Mohr–Coulomb failure criterion was deployed in this study, where soil was idealized as an elastic-perfectly plastic material. The viscoplastic technique [48,55] was used for the plasticity model owing to its advantages. Table 1 shows the properties of the soil used in this study, where only the angle of internal friction of the soil (φ) was varied from 5º to 20º, whereas the remaining parameters were fixed. The surcharge pressure (q_s) was set to 27.0 kN/m². The associated flow rule (ψ = φ) was considered to calculate the bearing capacity factors and shape factors. The non-associated flow rule for ψ = 0 (corresponding to the volume preserving deformation while in shear, i.e., shearing in the soil occurs at a constant volume) is applied at the later part of this paper. The L/B ratios of the footing were varied from 1 (square) to 10 (rectangular). The analysis was performed in MATLAB version R2015a. The procedure for calculating the shape factors is explained in the next section.

Evaluation of shape factors

The bearing capacity of the rectangular footing is expressed as

(6)q_{\mathrm{u}\mathrm{l}\mathrm{t}}=c{N}_{\mathrm{c}}^{\prime }+q_{\mathrm{s}}{N}_{\mathrm{q}}^{\prime }+0.5\gamma B{N}_{\gamma }^{\prime },

where N_c′, N_q′, and N_γ′ are the factors that represent both the bearing capacity factors and shape factors.

In the FE analysis, the bearing capacity factors (in addition to the shape factors) were calculated using the superposition method. As an example, the soil was assumed to be weightless and the surcharge pressure was set as zero for calculating N_c′. Subsequently, the value of N_c′ was calculated as N_c′ = q_ult/c (by substituting q_s= γ = 0). Similarly, N_q′ was calculated by substituting c = γ = 0, and N_γ′ was calculated by substituting c = q_s = 0. Next, the shape factors were calculated by dividing the bearing capacity factors of the rectangular footing (N_c′, N_q′, and N_γ′) by those of the strip footing (N_c, N_q, and N_γ). As an example, s_c was calculated as follows:

(7)s_{\mathrm{c}}=\frac{N_{\mathrm{c}}^{\text{'}}}{N_{\mathrm{c}}},

where N_c (= 5.14) is the bearing capacity factor for the exact plane strain condition. Furthermore, s_q and s_γ were calculated similarly.

Validation of present model

First, to validate the present FE model, the present results were compared with previously published results for the undrained condition (φ = 0) of the soil. Table 2 shows a comparison of the shape factor values (s_c) for square and rectangular footings (L/B ranging from 1 to 10) obtained from the present analysis and previously published results [6]. As shown, the s_c values were on the lower side as compared with the reported results because the present analysis was performed for smooth footing–soil interface conditions, whereas the reported results were for the rough interface condition. Because the roughness of the interface increased the bearing capacity factors, the present results differed slightly from those obtained by Gourvenec et al. [6] on the lower side.

Results and discussions

Effect of MF

To investigate the effect of the MF shape of the input fuzzy variables on the ultimate bearing capacity and the failure probability of the footing, an exercise was performed for both the triangular and trapezoidal MFs. Figure 4 presents the MFs of the ultimate bearing capacity for both the MFs, as discussed above, and the corresponding failure probability for different scales of fluctuation (δ). As shown, the variations in the failure probability with different levels of q_lim were similar for both MFs (Figs. 4(a) and 4(b)). In fact, for higher q_lim, the triangular MF predicted higher P_f than the trapezoidal MF. For example, for δ = 1.0 m and q_lim = 300.0 kPa, the triangular MF predicted a P_f that was 7.5% higher than that of the trapezoidal MF. This proves the conservativeness of adopting the triangular MF of the input variables for higher q_lim. Furthermore, it was observed that the q_lim for the lowest (0.0) and highest (1.0) probability of failure were similar for both the MFs. For example, for δ = 1.0 m, these values were approximately 170.0 and 350.0 kPa for the lowest (0.0) and highest (1.0) probabilities of failure, respectively.

Triangular MF of q_ult

First, an exercise was performed to obtain the MF of the final fuzzy number, i.e., the ultimate bearing capacity (q_ult) of the square footing resting on c–φ soil. The fuzzy variables (soil parameters) and their statistical parameters are presented in Table 3. In most reliability analyses regarding the bearing capacity of footing resting on c–φ soil, the variation in the soil friction angle (φ) exhibits the most prominent effect on the failure probability [34]. Therefore, the coefficient of variation of the soil friction angle (COV_ϕ) was set as 0.1 (a lower value) and 0.4 (a significantly higher value) to obtain the support widths of the ultimate bearing capacity. However, the COV of the other parameters were fixed at a particular value, as shown in Table 3. Figure 5 shows the membership functions of the ultimate bearing capacity for φ = 20°. A non-associated flow rule for ψ = 0 was imposed to construct the MF of q_ult. The membership function of q_ult was obtained for five δ values varying from 1.0 to 100.0 m. It was observed that the dispersion of the MF was significantly less for δ = 1.0 m, whereas the highest dispersion was achieved for δ = 100.0 m. This phenomenon indicates that the variance reduction was significantly smaller for δ = 100.0 m than that for d = 1.0 m. As less variance was reduced for higher δ values, the variables would perform near the same variance, as in the beginning of the analysis or before accounting for the spatial variability. This would result in a higher variance for a higher δ value (d = 100.0 m) than for a smaller one (δ = 1.0 m). Therefore, a smaller amount of dispersion would occur for smaller δ values than that for higher δ values. However, the difference in dispersion for δ = 50.0 and δ = 100.0 m is negligible.

Effect of variation in φ

In this section, the effect of φ on the P_f of a square footing is analyzed. To conduct this study, the variation in the P_f values was estimated by varying the q_lim for five different spatial correlation lengths (δ) and φ values varying from 5° to 20°. The COV of the φ values were set as 0.1 and 0.4 to investigate the effect of the variation in φ on P_f. This study was performed without considering the effect of the dilation angle (non-associated flow rule for ψ = 0). Figure 6 presents the variation in P_f for varying q_lim and different δ. It was observed that the smaller δ predicted a smaller P_f up to the deterministic bearing capacity of the footing (the value changed for different φ values), but the trend reversed beyond this limit. This phenomenon is related to the formation of the MF of the ultimate bearing capacity. As shown in Fig. 5, the dispersion of the MF was much higher beyond the mean value of the ultimate bearing capacity (i.e., q_ult for mean value of the soil parameters) than that below the mean value of q_ult. This dispersion trend intensified as δ increased. Therefore, the area ratio (Eq. (2)) or the probability of failure increased more significantly for smaller δ values than the higher ones when q_lim crossed the deterministic or mean value of q_ult. Furthermore, a close observation of the results indicated that as φ increased, the q_lim at zero failure probability (q⁰_lim) increases as well, consistent with the increase in the COV of φ. For example, for COV_φ = 0.1, as φ increased from 5° to 20°, q⁰_lim increased by 63.4%, and for COV_φ = 0.4, q⁰_lim increased by 40% for the same increment in φ. However, the increase in q⁰_lim was higher for a lower COV_f than the higher value. Meanwhile, as φ increased, a higher q_lim was required to reach the highest failure probability (P_f = 100%) for both COV_φ = 0.1 and 0.4. For example, for COV_⌽ = 0.1, as φ increased from 5° to 20°, q_lim increased by 69.7%, whereas for COV_⌽ = 0.4, q_lim increased by 80.8% for the same increment in φ. In this case, the higher COV_⌽ exhibited a more significant effect on the P_f value than the lower one.

Effect of flow rule

In this section, the effect of the flow rule on the failure probability is demonstrated. Both associated (ψ = φ) and non-associated (ψ≠ φ) flow rules were considered in the analysis, and the variation in the probability of failure with the variation in the q_lim was investigated. Two δ values were used: one on the lower side (1.0 m), and the other on the higher side (100.0 m), based on the fact that the P_f values for other δ values were within this wide range of δ. Figure 7 presents the variation in P_f with varying q_lim for both the associated and non-associated (ψ = 0 condition) flow rules for φ = 10° and 20°. As shown in the figure, a higher P_f value was obtained for the non-associated flow rule for ψ = 0. In other words, for the same level of q_lim, the P_f obtained by considering the non-associated flow rule for ψ = 0 was higher than that obtained by considering the associated flow rule. When the effect of ψ was considered for the bearing capacity calculation, the effective φ increased until the soil strength reached its peak. The maximum value was attained for the associated flow rule i.e., for ψ = φ, and the bearing capacity factors attained their highest values at this condition [35]. Furthermore, as φ increased, the soil strength increased and the failure probability decreased, as shown in Fig.7. For example, for φ = 10°, COV_⌽ = 0.4, δ = 1.0 m, and q_lim = 300.0 kPa, if the flow rule assumption changes from associated (ψ = φ) to non-associated (ψ = 0 condition), then the P_f value increases by 15.2%. Similarly, for φ = 20°, COV_⌽ = 0.4, δ = 1.0 m, and q_lim= 700.0 kPa, it increased up to 25.8%. However, P_f varied similarly as discussed earlier for both the non-associated and associated flow rules. Furthermore, to verify the effect of the ψ value on the failure probability, a similar type of analysis was performed for two additional ψ values (ψ = φ/3 and 2φ/3) along with ψ = 0 and φ. Figure 8 shows the variation in the failure probability with the varying q_lim for four ψ values and δ = 1.0 m. It was observed that the probability of failure for ψ = 0 was much larger than those for the other ψ values for a particular q_lim. As an example, for q_lim = 600.0 kPa, P_f decreased by 19.1%, 29.5%, and 36.7% as ψ increased from 0 to φ/3, 2φ/3, and φ, respectively.

Effect of δ on P_f

In this section, the effect of δ on the P_f of the system is investigated. P_f is not directly quantified herein; instead, the q_lim at zero failure probability (q⁰_lim) is determined. It is noteworthy that the larger q⁰_lim indicates a smaller P_f.

First, the study was performed for a square footing (L/B = 1.0) by considering the isotropic δ (δ_x = δ_y = δ_z = δ). Figure 9 presents the variation in q⁰_lim with varying δ/B ratios for different φ values. It was observed that irrespective of the φ values, as the δ/B ratio increased, q⁰_lim decreased, and this trend was more prominent for the higher φ values compared with the lower values. For example, for φ = 5°, as the δ/B ratio increased by ten times (from 1.0 to 10.0), q⁰_lim decreased by 5.3%, whereas, for φ = 20°, q⁰_lim decreased by 16.3% for the same increment in the δ/B ratio. Furthermore, it was observed that as φ increased, q⁰_lim increased as well, indicating a lower P_f. Similarly, for a particular φ, as the COV_⌽ increased, the q⁰_limdecreased, thereby confirming a higher P_f. For example, for δ/B = 5.0 and COV_⌽ = 0.4, as φ increased by four times (from 5° to 20°), q⁰_lim increased by 43.5%. Meanwhile, for φ = 20° and δ/B = 5.0, as COV_φ increased by four times (from 0.1 to 0.4), q⁰_lim decreased by 35.6%. Moreover, the effect of the COV_⌽ was more significant for higher φ than for lower ones. For example, for φ = 5° and δ/B = 10.0, as COV_⌽ increased by four times (from 0.1 to 0.4), q⁰_lim decreased by only 3.8%, whereas for φ = 20°, it decreased by 36.2%. However, the change in q⁰_lim value was significant up to a δ/B ratio of 10.0; beyond that, the variation was negligible. Furthermore, q⁰_lim values were determined for different L/B values ranging from 1.0 to 10.0. In this case, a fully cohesive soil (φ = 0) deposit was considered, and the cohesion value (c) shown in Table 3 was used, whereas the COV of c (COV_c) was assumed to be 0.4. Figure 10 shows the variation in q⁰_lim with various L/B values and for different δ. As shown in Fig.10, q⁰_lim decreased abruptly when L/B changed from 1.0 to 2.0; beyond that, the reduction in q⁰_lim was insignificant. This fact is related to the values of the bearing capacity factors for different aspect ratios (L/B) of the footing. As shown in Table 2, the reduction in the bearing capacity factor (N_c′) was the highest when L/B changed from 1.0 to 2.0 (N_c′ decreased by 12.3%). Beyond that, the reduction was significantly lower (N_c′ only decreased by 2.7% and 1.1% for L/B changing from 2.0 to 5.0 and 5.0 to 10.0, respectively) compared with the former case. As the reduction in the bearing capacity factors was much higher when L/B was changed from 1.0 to 2.0, the reduction in q⁰_lim was the highest within this range. This phenomenon is reflected in Fig. 10, where an abrupt decrease in q⁰_lim occurred when L/B changed from 1.0 to 2.0; beyond that, the reduction was minimal. This phenomenon was consistent for all δ values. As an example, for δ = 1.0 m, if L/B increased by 100% (from 1.0 to 2.0), then the q⁰_lim values decrease by 12.2%. Similarly, the q⁰_lim value is only decreased by 2.8% for a further 100% increment in L/B (from 2.0 to 4.0). Furthermore, for d = 50.0 m, if L/B is increased by 100% (from 1.0 to 2.0), then q⁰_lim decreases by 12.4%. In addition, q⁰_lim is only decreased by 3.3% for a further 150% increment in L/B (from 2.0 to 5.0). However, the smaller δ predicted a higher q⁰_lim value, confirming a lower P_f.

Furthermore, an analysis was performed for the anisotropic δ, where the variation in q⁰_lim with the variation in δ in two directions were considered by fixing the δ values in the third direction. In this study, the anisotropy was incorporated by fixing the δ values in the x- and y-directions (horizontal) and varying it in the z-direction (vertical). This analysis was also conducted for the square footing (L/B = 1.0). Figure 11 presents the variation in q⁰_lim with the variation in δ_x/B for different δ_z values, and the δ_y values were set equal to the δ_x values. The φ and COV_⌽ were set as 20° and 0.4, respectively. As shown in the figure, for a particular δ_x/B or δ_y/B value, an increase in the δ_z resulted in a decrease in q⁰_lim. Similarly, for a particular δ_z value, an increase in δ_x/B resulted in a decrease in q⁰_lim. Moreover, the decrease in q⁰_lim was more significant up to δ_z of 5.0 m and δ_x/B of 10.0. For example, for δ_x/B = δ_y/B = 10.0, if δ_z is increased by 10 times (from 1.0 to 10.0 m), then q⁰_lim is decreased by 8.0%. Meanwhile, if δ_z is increased subsequently by 100 times (from 10.0 to 1000.0 m), then q⁰_lim is only decreased by 1.8%. However, for δ_z = 10.0 m, if δ_x/B is increased by 10 times (from 1.0 to 10.0), then q⁰_lim is decreased by 11.2%, whereas if δ_x/B is increased subsequently by 100 times (from 10.0 to 1000.0), then q⁰_lim is only decreased by 0.9%. From the discussion above, it can be concluded that the δ in the horizontal direction (either axial or lateral) is more influential than that in the vertical direction. However, a significant change in q⁰_lim was observed when the anisotropic δ was considered. As an example, for δ_x/B = 10.0, a considerable increase in q⁰_lim(8.7%) occurred for δ_z= 1.0 m, whereas only a 0.5% reduction was observed for δ_z= 50.0 m.

In addition to the analyses above, the effect of the anisotropic δ on q⁰_lim was quantified by the values of the influence factor (I_f), which is the ratio of the difference between the q⁰_lim value for the anisotropic and isotropic δ to q⁰_lim for the anisotropic δ only (see Fig. 12). Figure 12 presents the variation in I_f with the variation in δ_x values for different δ_z values based on setting δ_y equal to δ_x. The φ and COV_⌽ were kept same as taken in Fig. 11. As shown, as δ_x increased, I_f increased for all δ_z. For δ_z = 1.0 m, I_f increased from 0.0% (q⁰_lim for anisotropic case becomes equal to the isotropic case) to 9.4% when δ_x changed from 1.0 to 100.0 m. A similar variation was observed for other δ_z values. On the contrary, I_f reduced slightly for δ_z = 50.0 and 100.0 m when δ_x increased from 1.0 to 5.0 m. This phenomenon is related to the formation of the MF of the ultimate bearing capacity (see subsection 6.2). As the δ reached a higher value, the dispersion on the right half of the MF of q_ult became higher than that on the left half (see Fig. 5). Consequently, P_f increased to a higher magnitude for higher δ. Hence, as the δ_z values were much higher (50.0 or 100.0 m) compared with the δ_x or δ_y values (5.0 m), the increment in P_f was much higher for the combination of δ_x= δ_y=5.0 m and δ_z= 50.0 or 100.0 m compared with the combination of δ_x = δ_y =1.0 m and δ_z = 50.0 or 100.0 m. Similarly, the higher the increment in P_f, the higher was the reduction in q⁰_lim. Consequently, the influence factor decreased initially for smaller δ_x or δ_y values, and as δ_x or δ_y increased, it increased as well. Furthermore, it was observed that the I_f values were much higher for δ_z = 1.0 m than those for other δ_z values.

Conclusions

The following conclusions were obtained in this study.

1) The MF of the final fuzzy number (ultimate bearing capacity) indicated that the effect of COV_⌽ was significant for the construction of the final MF. A higher COV resulted in a right skewed MF. Moreover, the support width on both the right and left sides varied in the final MF.

2) The effect of φ on P_f was evident in this study. A higher φ predicted a higher q⁰_lim, indirectly signifying a lower P_f. In addition, a higher COV_⌽ predicted a smaller amount of q⁰_lim (higher P_f).

3) It was observed that a reduction in P_f occurred when the associated flow rule was considered in the reliability analysis. This phenomenon highlighted that the inclusion of ψ increased φ until the soil strength reached its peak.

4) The role of δ in determining the P_f was discovered to be significant. The higher the δ, the higher was the P_f. This phenomenon is related to the determination of the reduced variance. The variance reduction function increased (up to a value of approximately 1.0) as the δ increased; subsequently, the variance reduced at a lower rate for the larger δ than that for the smaller one. Less variance reduction implies that the variables will perform with a variance similar to the original variance, whereas a high variance reduction implies that the variables will perform with a variance that deviates significantly from their original values. Therefore, it is clear that the system will perform with more uncertainty if the variance reduction is less.

5) The variability is expected to be higher in the horizontal direction as the failure surface propagates in the horizontal direction more widely than in the vertical direction. This was supported by the reliability analysis, as the P_f values were more sensitive to the δ in the horizontal direction.

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