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The objective of this paper is to investigate the effect of soil variability on bearing capacity of an embedded foundation in the presence of nonstationary undrained shear strength. The nonstationary undrained shear strength is simulated by a nonstationary random field generator based on the spectral representation method. An embedded foundation buried into the soil to two times of width is presented to investigate the influence of spatially variable undrained shear strength on bearing capacity. Firstly, Monte Carlo simulations are carried out to discuss the effect of distribution type, nonstationary gradient parameter, and horizontal autocorrelation length on the bearing capacity from the standpoint of mean value and standard deviation. Then, the influence of the distribution type on the failure probability of nonstationary random soil is also investigated, with the failure probability for the Beta distribution being demonstrated to be always larger than that for the Lognormal and Gamma distribution.

It is widely recognized that soil exhibits spatial variability in both horizontal and vertical direction due to the effects of chemical and environmental changes and soil depositions. This inherent spatial variability of soil becomes the dominating source of uncertainty for the estimation of soil properties from in situ tests [

The bearing capacity of a shallow foundation is a classical geotechnical problem and has been further studied by probabilistic analysis methods that deal with the spatial variability of soil properties. For example, Popescu et al. [

In the existing literature, the stationary random field has been widely applied to describe the spatial variability of soil parameters. However, it is well known that soil parameters often exhibit nonstationary characteristics; i.e., it was confirmed by some in situ tests. Lumb [

Theoretically, soil properties follow non-Gaussian probability distributions in the probabilistic analysis because physical material properties cannot be assumed as negative values [

This paper aims to investigate the bearing capacity of a deeply embedded foundation in the presence of spatially variable undrained shear strength with different probability distribution functions. A nonlinear finite element model combined with nonstationary random fields, discretized by a random field generator (i.e., spectral representation method), is employed to achieve this aim through Monte Carlo simulation. For comparison with the results obtained from the nonstationary random fields, the results obtained from the corresponding stationary random fields under different PDFs are also provided. With the methodology above, the influences of the distribution type of the undrained shear strength under the condition of both stationarity and nonstationarity on the reliability and bearing capacity of an embedded foundation are illustrated in this paper.

In the current study, a finite element model was performed to compute the bearing capacity of an embedded foundation under the condition of two-dimensional plane strain in this study. As shown in Figure

Finite element model of a foundation embedded at depth

The embedded footing was considered to be rigid and has a rough interface with the soil. The soil was modeled by a linear-elastic perfectly plastic constitutive law. The failure is according to the Tresca yield criterion, with the maximum shear stress in any plane being limited to the undrained shear strength.

The parameters of Young’s modulus (_{u}) for soil model were both considered to be stochastic parameters while only the undrained shear strength was simulated by the random field generator. The undrained shear strength could be assumed to be perfectly correlated to Young’s modulus based on the relationship, i.e., _{u} = 500 [

In the process of Monte Carlo simulation, because of the discrete nature of numerical simulations, a continuous-parameter random filed must be discretized into random variables by a numerical method. Several discretization methods have been developed to attain this aim [

In this study, to investigate the effect of distribution type of undrained shear strength _{u} on the bearing capacity of the embedded foundation, the Lognormal, Gamma, and Beta distributed nonstationary and stationary random fields with the mean value _{h}, and vertical correlation length

The PDFs of standard non-Gaussian distribution with zero mean value and unit variance.

To describe the nonstationary extent of the mean undrained shear strength _{u} at the bottom of the model, where the depth _{u} at the surface level (i.e., ^{3}. In this study, the value of

A squared exponential autocorrelation 2D autocorrelation function was adopted to define the autocorrelation structure of _{u}:_{h} and

Summary of parameters involved in the simulation of random fields of _{u}.

Parameter | Symbol | Value |
---|---|---|

Distribution type | Lognormal, Gamma, Beta | |

Coefficient of variation | COV | 0.3 |

Mean value at surface level | 10 (kPa) | |

The gradient strength parameter | 0, 12 | |

The horizontal autocorrelation length (m) | _{h} | 6, 60 |

The vertical autocorrelation length (m) | 6 |

Based on the coordinate of elements in the finite element model, a series of realizations of the stationary and nonstationary random fields of _{u} are generated by the simulation method and involved parameters mentioned above. In this study, 1000 simulations, which have been enough to ensure the estimated error to be allowed, have been performed for each random case.

The MCS was employed to conduct probability analysis and evaluate the statistical properties of the bearing capacity of the embedded foundation in both stationary and nonstationary soils. In the framework of MSC, the 1000 simulations of _{u} for each type of random field, coupled with the fixed finite element model, were recalled by a batch program to compute the bearing capacity on each of them. It is worth noting that, according to the predecessors’ researches [

It is worth noting that, in the statistical analysis, the normalized bearing capacity is usually discussed rather than the actual bearing capacity. For a stationary case, the bearing capacity normalized by the mean undrained shear strength is widely accepted as done by Griffiths and Fenton [

A series of computed bearing capacity of the embedded foundation is illustrated in Figures _{u} of the foundation soil on the bearing capacity from the view of mean value and standard variation. It can be seen from Figures _{u} is smaller than the bearing capacity for deterministic analysis. Note that _{u} of the deterministic analysis corresponding to the nonstationary random fields linearly increases with depth according to

The mean and standard deviation (SD) of the bearing capacity for different cases of PDFs with (a, c)

The mean and standard deviation (SD) of the bearing capacity for different cases of horizontal correlation length with (a, d) Lognormal distribution, (b, e) Gamma distribution, and (c, f) Beta distribution.

It is obvious that the distribution type of _{u} has a significant influence on the bearing capacity, although the different distribution types are assigned identical mean value and standard variation of _{u} in the process of random field simulation. Particularly, from Figures _{h} = 60 m, the statistics of bearing capacity for the Beta distributed random soil exhibits the smallest mean value and largest standard deviation among the three distribution cases. A similar observation about the difference between the estimated mean bearing capacity and corresponding standard variation of Beta distribution type and that of Lognormal and Gamma distribution was also found for shallow foundation [_{u} than Beta distribution. Hence, _{u} of Lognormal and Gamma distribution produces larger and more concentrative bearing capacity than _{u} of the Beta distribution. However, for the random fields with _{h} = 6 m, the standard deviation of bearing capacity is demonstrated to not conform to the above rule.

It can be seen from Figures

The development of failure plane for the embedded foundation ((a, e) Step = 50; (b, f) Step = 60; (c, g) Step = 70; (d, h) Step = 113). (a–d)

According to Li et al. [

In this section, the statistical distribution characteristics of bearing capacity are studied. Figure _{h} = 60 m, the CDFs of the bearing capacity for a foundation embedded at 2_{u} in this paper is much larger than that in Wu et al. [

The CDFs of the bearing capacity factor _{c} of the foundation for stationary and nonstationary cases ((a, d) Lognormal distribution; (b, e) Gamma distribution; (c, f) Beta distribution). (a–c) _{h} = 60 m; (d–f) _{h} = 6 m.

The CDFs of the bearing capacity factor _{c} of the foundation for different PDFs of shear strength ((a, c) _{h} = 60 m; (c, d) _{h} = 6 m.

To better investigate the influence of the distributions of _{u} on the foundation’s bearing capacity, the differential bearing capacity

The CDFs of differential bearing capacity _{h} is set to 60 m, in most realizations, the bearing capacity _{h} = 6 m. In addition, _{u} with Beta distribution produces a larger amount of higher _{u} with Lognormal and Gamma distribution, which is more obviously observed in the cases of _{h} = 60 m. Therefore, if we simply use the value of soil strength at depth of 2_{u} with Beta distribution than _{u} with Lognormal and Gamma distribution.

The CDFs of the differential bearing capacity factor Δ_{c} for _{h} = 60 m; (b) _{h} = 6 m.

In practice, it is essential to know the potential failure probability of soil foundation from a design standpoint. According to Griffiths and Fenton [_{u} linearly increasing with depth according to

Cumulative probability of the normalized bearing capacity for foundations with different PDFs of shear strength ((a, c) _{h} = 60 m; (c, d) _{h} = 6 m.

For a normally distributed bearing capacity, the probability of failure can be estimated by the following formula:

Note that the probability of failure utilized in this study is based on normalized bearing capacity rather than the actual bearing capacity. This defined probability of failure only reflects the comparison between bearing capacity from probabilistic analyses and those from the deterministic analysis. It does not reflect the real safety of the foundation.

In Figure _{h} = 60 m as an example, the failure probabilities are 86.4, 86.3, and 87.7% for the stationary _{u} satisfied with Lognormal, Gamma, and Beta distribution, respectively, and 85.9, 85.8, and 86.6% for the nonstationary _{u} satisfied with Lognormal, Gamma, and Beta distribution, respectively. It is obvious that the failure probability of the Beta distribution is the largest one among the probabilities of three distributions and the failure probability decreases when the nonstationary factor _{u} and the nature of nonstationarity on the failure probability.

However, the above failure probabilities are too large to accept when directly using the bearing capacity of the corresponding uniform model to evaluate the failure probability. The probability that the bearing capacity is less than a certain level of applied load is widely utilized by introducing the factor of safety (FS). Therefore, the probability of failure in the form of

In Figure _{h} = 60 m are illustrated. The failure probabilities are 2.5, 1.85, and 4% for the stationary _{u} satisfied with Lognormal, Gamma, and Beta distribution, respectively, and 6.4, 5.4, and 9.2% for the nonstationary _{u} satisfied with Lognormal, Gamma, and Beta distribution, respectively. It was found that the failure probability for _{u} with Beta distribution is still larger than that with Lognormal and Gamma distribution when FS is 1.3. However, it is interesting to find that, in contrast to the failure probability obtained from Figure

Failure probabilities of foundation with FS = 1.3 for different types of distributions: (a)

To investigate the variation of failure probability for different distribution types with the value of FS, the curves of relations between the failure probability of nonstationary random soil and the value of FS are shown in Figure _{h} = 60 m, it is found that the failure probability decreases from about 10^{0} to 10^{−3} when the factor of safety decreases from 1.0 to 1.5; meanwhile, _{u} with Beta distribution always exhibits higher failure probability than _{u} with Lognormal and Gamma distribution. In addition to the expected trend showing the influence of the PDFs on the failure probability when _{h} = 6 m, Figure _{h} = 60 m. Hence, it can be concluded that from the view of probability analysis, the Beta distributed shear strength may result in an overconservative failure probability, i.e., higher failure probability, of soil foundation than the Lognormal and Gamma distributed shear strength. Although under a certain factor of safety can the failure probability of soil for Lognormal and Gamma distribution satisfy the target probability range of 10^{−2} to 10^{−3}, the failure probability for Beta distribution under the same FS may fail to meet the target probability.

Failure probability for nonstationary random soils under different factors of safety: (a) _{h} = 60 m; (b) _{h} = 6 m.

This paper focuses on investigating the stochastic results of bearing capacity for a deeply embedded foundation, buried at depth of 2_{u} with Lognormal, Gamma, and Beta distribution and different vertical correlation lengths were generated by a generator based on the spectral representation method. Parametric studies were studied to evaluate the influence of the probability of function (PDF) of _{u} in both stationary and nonstationary stochastic soils. According to this study, the following conclusions were drawn:

The PDF of _{u} has a significant effect on the estimated mean value and standard deviation of the computed bearing capacity of the 2

The bearing capacity in nonstationary soils exhibits greater differences than that in stationary soils. It was found that, compared to a stationary spatial variable soil foundation, the failure plane of nonstationary soils extends more easily to soil surface rather than around the foundation and thus results in a smaller mean bearing capacity.

Compared with the Lognormal and Gamma distributed random fields of undrained shear strength, the Beta distributed random field may result in an overconservative failure probability in the bearing capacity. If we simply use the value of soil strength at depth of 2_{u} with Beta distribution than _{u} with Lognormal and Gamma distribution.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The support by the Foundation of Inner Mongolia Power (Group) Co., Ltd. (Grant no. 2019097) is greatly acknowledged.