In recent decades, following the widespread growth of interest in risk assessment and management, the concept of reliability-based design has developed in civil engineering. The idea of designing a structure in such a way that the calculated probability of failure does not exceed certain specified values (depending on the function of the object, the type of failure or its consequences) has spread and, although rarely used in engineering practice, is nowadays part of European and International standards for construction design (EN 1990, 2002, ISO 2394, 2015). The approach associated with this trend has been developing for several decades also in geotechnics, where spatial variability of soil parameters is considered the most important source of uncertainty (e.g., Cami et al., 2020; Pieczyńska-Kozłowska et al., 2021). It is worth mentioning that variability of soil parameters has often a much greater effect on the structure reliability than variability of other structure or material parameters which are usually (with some exceptions, e.g., Wyjadłowski et al., 2018) easier to control. Furthermore, the problem of spatial variability applies to all structures founded on the soil (i.e., the vast majority of the total number of structures).

Several works in the last two decades have considered modelling the spatial variability of soil parameters using so-called random fields (Vanmarcke, 2010). The application of random fields in geotechnical design is an approach explicitly recommended by Annex D of the latest edition of ISO 2394 (2015). In such an approach, the analysis is performed within the Monte Carlo framework by solving the given problem an appropriate number of times, each time with different ‘realisation’ of random fields (modelling the properties considered) and then analysing the probabilistic characteristics of the results obtained. The numerical variant consists of generating realisations of random fields discretised over a finite element (FE) or a finite difference (FD) mesh, so that in each mesh element, the parameters are constant and usually represent an average of the field values over the area of the element. Despite the facts that the concept of this random finite element method (RFEM) in application to foundation bearing capacity was formulated two decades ago (Griffiths and Fenton, 2001; Fenton and Griffiths, 2003) and that the random fields can also be used in some other semi-analytical approaches (cf. Chwała, 2020a, 2020b), the mentioned numerical method remains very popular (probably due to its universality). So far (together with the analogous random finite difference method), it has been used to analyse a number of geotechnical problems, in particular the random bearing capacity of strip footings (e.g., Fenton and Griffiths, 2003; Vessia et al., 2009; Pieczyńska-Kozłowska et al., 2015), rectangular footings (e.g., Kawa and Puła, 2020; Li et al., 2021), slopes (e.g., Griffiths and Fenton, 2004; Jha and Ching, 2013a; Huang et al., 2019) and different types of retaining walls (e.g., Fenton and Griffiths, 2005; Sert et al., 2018; Kawa et al., 2019; Kawa et al., 2021).

As indicated in the work by Kawa and Puła (2020), little attention has been paid so far to probabilistic modelling of eccentrically loaded strip or rectangular footings. This problem was considered, for example, in the work by Soubra (2009). However, the influence of soil spatial variability was neglected (in the cited work, soil parameters were assumed as random variables, which means that they are constant over the considered space). To the best of authors’ knowledge, the only work that considers the spatial variability of the medium parameters in the analysis of the bearing capacity of an eccentrically loaded strip footing is the conference paper by Ali et al. (2016). In the cited study, the boundary problem is solved using the random finite element limit analysis (RFELA) technique. The effect of the length of the fluctuation scale (correlation radius) on the failure surface is analysed. This surface is presented in the vertical force–bending moment space (the two quantities were assumed as loads for the strip footing). An important simplification adopted in the paper is that the random cohesion field considered is assumed to be isotropic (the fluctuation scales in the horizontal and vertical directions are identical), which is not consistent with the results of other studies (e.g., Lloret-Cabot et al., 2014) and may lead to neoconservative results (cf. Vessia et al., 2009; Pieczyńska-Kozłowska et al., 2015).

The present paper complements and expands the study mentioned above. Similar to the work of Ali (2016), it was assumed that the soil medium is purely cohesive and that cohesion is the only parameter modelled by a random field. However, in the present work, the field of cohesion was assumed to be an anisotropic random field with the horizontal fluctuation scale larger than the vertical one. For easier scaling of the problem solution, it was assumed that the soil is weightless. The analysis was carried out using a typical RFEM; the individual random field realisations were generated using the Fourier series method (FSM; Jha and Ching, 2013b) and the considered boundary value problems were solved in ABAQUS software. For eight combinations of typical values of the fluctuation scale (representing both weak and strong anisotropy), the effect of the eccentricity of the vertical foundation load on the mean, standard deviation (SD) and coefficient of variation (CoV) of the bearing capacity of the soil was investigated. In the final part of the paper, an example reliability analysis was also performed showing the influence of eccentricity on the reliability index.

This paper is organised as follows. The next section briefly describes the basic assumptions for the study. In the following section, the numerical model, the preliminary deterministic tests and the probabilistic analysis performed are described. The results of the analysis are then presented. A brief section of conclusions closes the article.

As mentioned in the ‘Introduction’ section, the RFEM in geotechnical applications involves modelling soil parameters using a random field discretised over a FE mesh. The phrase ‘random field’ refers to a spatial generalisation of a stochastic process (see, e.g., Vanmarcke, 2010) and is associated with a whole group of different field classes. The class considered in this paper is restricted to ergodic and weakly stationary random fields, which is a typical assumption in RFEM applications. For this class of fields (Doob, 1953), the probability distribution does not depend on the position of a point, where it is tested, and similarly, the autocorrelation function defining the correlation between values of field in two different points does not depend on the position of these points, but only on their relative positions, that is, the direction and length of the vector between them (e.g., Kawa et al., 2021). Thus, the point probability distribution and the stationary (relative position dependent) autocorrelation function with specified parameters, that is, the so-called scales of fluctuations (SOFs), form a complete set of information needed to define such a field. A satisfactory description of probability distribution can usually be obtained by choosing the distribution type (the normal or log-normal distributions are most commonly accepted) and assuming or determining its first two moments, that is, mean and SD. It is much more difficult to credibly estimate the values of SOF. Identification of these parameters, which can be understood as the average dimensions of a cluster within which the field values are strongly correlated, is a problem widely described in the literature (Lloret-Cabot et al., 2006; Cami et al., 2020; Bagińska et al., 2018; Ching et al., 2018). It is generally known that the value of SOF in the horizontal direction is larger than the value of SOF in the vertical direction. Due to the way in which soil investigations are carried out, horizontal SOF is considered more difficult to identify and is often the subject of parametric studies.

If bearing capacity of footings is considered, only the strength parameters of the soil medium are usually modelled by random fields (cf. Puła and Zaskórski, 2015). In the present paper, it is assumed that the soil under consideration is a purely cohesive medium, and thus, the only parameter modelled by the random field is its cohesion

It was assumed that the random field of cohesion considered is described by a normal probability distribution with mean _{c}_{c}_{c} = _{c}_{c}_{c} range allowed for the normal distribution). Therefore, the results obtained in this study for _{c} = 10% can be scaled to the other values of _{c} (e.g., _{c}_{c} and 10% (in the case of _{c}

The autocorrelation function describing the internal correlation structure of the field was assumed to be a two-dimensional Markov function, that is,
_{v}_{h}_{v}_{h}_{v}_{h}

The boundary value problem of the bearing capacity of the strip foundation was defined and solved using the Abaqus FE code under the assumption of a plane strain. The FE model used is presented in Fig. 1. The model consists of two parts: a 10.8 m × 3.6 m soil domain and a 2 m × 0.5 m concrete strip footing. These dimensions were verified in multivariate numerical simulations (including numerous realisations of the cohesion random field) as sufficient to determine the bearing capacity for the considered case (plastic zones for none of the simulations performed were near the edge of the domain). The contact between the footing and the soil has been assumed to be perfectly smooth. To discretise the continuum model, the soil domain has been divided into 54 ×18 squared FEs and the footing into 10 × 3 FEs, resulting in an element of size

Assumed model parameters.

_{c} |
_{c} |
|||||
---|---|---|---|---|---|---|

Soil | 200 MPa | 0.33 | 20 kPa | 2 kPa | 0° | 0° |

Footing | 32 GPa | 0.00 | — |

During the computations, six positions of the vertical force

Before running the Monte Carlo simulations (MCSs), some preliminary deterministic computations were performed for all the values of eccentricity considered. In these calculations, a constant cohesion value equal to the mean value _{c}

As seen, the reaction after reaching the maximum value exhibits some small decreases, which can be an effect of a large sliding assumption. The peak decreases obtained for larger eccentricities (which also appear after reaching the maximum value) are related to the passing of the foundation and the soil nodes (during larger foundation slides).

To verify the correctness of the numerical computations performed, the model was validated based on Prandtl's solution of the strip-bearing capacity problem (for cohesive and weightless soil), that is,
_{an}_{FEM} (presented in Fig. 2) in Table 2. The table also shows the relative differences between the numerical method and the analytical solution. As seen, the respective values are close to each other, although for larger eccentricity values, the differences between the values obtained are significant.

Comparison of semi-analytical and numerical values of bearing capacity.

_{FEM} (kN/m) |
217.28 | 197.43 | 176.81 | 156.12 | 135.44 | 114.68 | |

_{an} |
205.66 | 185.10 | 164.53 | 143.96 | 123.40 | 102.83 | |

Relative difference (%) | 5.65 | 6.66 | 7.46 | 8.45 | 9.76 | 11.52 | |

_{an} |
217.28 | 196.72 | 176.15 | 155.58 | 135.02 | 114.45 | |

Relative difference (%) | 0.00 | 0.36 | 0.38 | 0.34 | 0.31 | 0.20 |

It is worth mentioning that the bearing capacity obtained numerically may correspond to a footing width

After the validation of the model, the actual MCSs were carried out. The bearing capacity for each of the six-load eccentricity values considered was analysed for eight pairs of SOF values assumed to generate a cohesion random field. These pairs were combinations of the two considered vertical SOF values _{v}_{h}_{v}_{h}

As mentioned in the previous section, the analysis included eight pairs of SOF values and six eccentricity values, that is, 48 problems in total. For each of these problems, 1000 MCSs with different realisations of random field modelling cohesion were carried out. The time needed to solve a single series of 1000 MCSs on a modern PC is about 24 hours.

The bearing capacity values obtained for each problem were collected and analysed probabilistically. First, the assumed normal distribution of the results was estimated and tested with the Kolmogorov–Smirnov goodness-of-fit test. The test results obtained were in the range of 30%–70%. Exemplary probability density functions (PDF) for the estimated normal distribution for _{v}_{h}

The mean values _{Q}_{Q}_{Q}_{Q}_{Q}_{Q}_{v}_{v}_{h}

Obtained and interpolated values of _{Q} and _{Q}

_{v} |
_{h} |
_{Q}/δ_{Q} |
||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.5 | 2.0 | _{Q} |
210.19 | - | - | 194.21 | 173.95 | 153.71 | 133.45 | 113.14 |

_{Q} |
3.65 | - | - | 4.08 | 4.33 | 4.63 | 4.99 | 5.42 | ||

5.0 | _{Q}(kN/m) |
210.13 | - | - | 193.88 | 173.78 | 153.66 | 133.52 | 113.29 | |

_{Q} |
4.70 | - | - | 5.24 | 5.51 | 5.81 | 6.17 | 6.60 | ||

10.0 | _{Q} |
210.24 | - | - | 193.95 | 173.79 | 153.64 | 133.44 | 113.18 | |

_{Q} |
5.27 | - | - | 5.69 | 5.93 | 6.20 | 6.48 | 6.80 | ||

30.0 | _{Q} |
211.68 | - | - | 194.51 | 174.27 | 154.02 | 133.72 | 113.33 | |

_{Q} |
5.37 | - | - | 5.67 | 5.90 | 6.16 | 6.47 | 6.85 | ||

1.0 | 2.0 | _{Q} |
209.72 | - | - | 194.14 | 173.97 | 153.81 | 133.59 | 113.31 |

_{Q} |
4.51 | - | - | 5.07 | 5.34 | 5.63 | 5.96 | 6.31 | ||

5.0 | _{Q} |
209.70 | - | - | 194.03 | 173.92 | 153.78 | 133.60 | 113.33 | |

_{Q} |
5.77 | - | - | 6.13 | 6.38 | 6.65 | 6.95 | 7.25 | ||

10.0 | _{Q} |
209.88 | 208.32 | 206.75 | 194.25 | 174.18 | 154.07 | 133.89 | 113.60 | |

_{Q} |
6.26 | 6.30 | 6.34 | 6.64 | 6.88 | 7.14 | 7.42 | 7.74 | ||

30.0 | _{Q} |
211.35 | - | - | 194.66 | 174.53 | 154.38 | 134.19 | 113.90 | |

_{Q} |
6.60 | - | - | 6.86 | 7.06 | 7.27 | 7.52 | 7.79 |

Values for these eccentricities were interpolated (only for _{v}_{h}

As seen in the figures, for small eccentricity values, the stabilisation of the SD of the bearing capacity is slower than in the case of larger eccentricity values. The graphs also show that as the eccentricity increases, not only the mean value but also the SD of the bearing capacity decreases. However, the resulting CoV increases with eccentricity. All these effects are due to a reduction in the effective width of the foundation. This reduction is associated with the smaller size of the failure mechanism, and thus a smaller averaging area of the cohesion values (cf. Chwała and Kawa, 2021), which, according to local averaging theory (Vanmarcke, 2010), means a smaller reduction in variance. For larger eccentricities, this results in both faster stabilisation of SD (when the mechanism size is smaller than SOF, the variance reaches its maximum value) and relatively larger variation in bearing capacity (smaller reduction of variance). Although easy to explain, the increase in CoV of the bearing capacity _{Q}

An interesting behaviour regarding small eccentricity values can be observed in Fig. 8. For both the considered vertical SOF values (Fig. 8a, b), the graph for _{Q}_{Q}

In the last step of the analysis, the effect of eccentricity on the value of reliability index _{f}_{v}_{h}_{Q}_{Q}_{Q}_{Q}_{v}_{h}_{f}

The values of _{f} calculated (based on Table 3) for _{v}_{h}

3.80 | 3.69 | 3.57 | 2.66 | 1.19 | −0.53 | −2.62 | −5.27 | |

_{f} |
7.2×10^{−5} |
1.1×10^{−4} |
1.8×10^{−4} |
3.9×10^{−3} |
1.2×10^{−1} |
7.0×10^{−1} |
≈1.0 | ≈1.0 |

As seen in the table, an increase in eccentricity causes a significant decrease in the reliability index _{f}_{Q}_{f}

In this paper, a probabilistic analysis of the bearing capacity of an eccentrically loaded shallow foundation on a purely cohesive soil was carried out. The soil was assumed to be weightless. The cohesion was modelled by an anisotropic random field, for which eight pairs of different vertical and horizontal SOFs were considered. For each of these cases, six different values of load eccentricity were assumed, resulting in a total of 48 problems. For each of these problems,

Risk management for geotechnical structures requires modelling that includes the effect of spatial variability of soil parameters. A frequently used approach in this case is the RFEM, which allows the modelling of various structures located in soil, including strip footings (which were among the first to be modelled using this method). Although in typical structures, the loading of foundations almost always occurs with a certain eccentricity, this has, so far, rarely been taken into account in probabilistic studies. To the authors’ knowledge, this paper is the first study to analyse the bearing capacity of eccentrically loaded strip footings on a soil modelled by an anisotropic random field.

The presented results show that the mean, SD or CoV of the bearing capacity increases and stabilises with increasing values of SOFs, which is similar to the behaviour reported in the literature for random bearing capacity values. The linear decrease of the mean value of the bearing capacity with increasing eccentricity is consistent with Meyerhof's theory. However, from the point of view of the reliability of the structure, it is important that the CoV of the bearing capacity increases with eccentricity. As shown above, the dependence of SD or (to a slightly lesser extent) CoV of the bearing capacity on eccentricity is nonlinear. However, this effect, especially visible for small eccentricities, needs further investigation.

Analysis of the effect of eccentricity on the reliability index value, carried out in the final part of the paper, shows that even small load eccentricities (of the order of 0.01–0.02 m) may cause a significant decrease in the reliability index (or increase in the probability of failure). Although this effect is not surprising, as it is associated with a decrease in the mean value of bearing capacity according to Meyerhof's theory, it can pose a problem if the allowable load of a strip footing was assessed without considering load eccentricity. Eccentricities of a few centimetres are common in practice, and with the typical accuracy assumed in civil engineering, they occur even if they were not planned. This shows the necessity of taking into account at least accidental random eccentricities in the probabilistic analysis of the foundation bearing capacity treated as a design method for real structures.

#### Comparison of semi-analytical and numerical values of bearing capacity.

_{FEM} (kN/m) |
217.28 | 197.43 | 176.81 | 156.12 | 135.44 | 114.68 | |

_{an} |
205.66 | 185.10 | 164.53 | 143.96 | 123.40 | 102.83 | |

Relative difference (%) | 5.65 | 6.66 | 7.46 | 8.45 | 9.76 | 11.52 | |

_{an} |
217.28 | 196.72 | 176.15 | 155.58 | 135.02 | 114.45 | |

Relative difference (%) | 0.00 | 0.36 | 0.38 | 0.34 | 0.31 | 0.20 |

#### Assumed model parameters.

_{c} |
_{c} |
|||||
---|---|---|---|---|---|---|

Soil | 200 MPa | 0.33 | 20 kPa | 2 kPa | 0° | 0° |

Footing | 32 GPa | 0.00 | — |

#### The values of β and pf calculated (based on Table 3) for θv = 1 m θh = 10 m and Q = 159.95 kN/m.

3.80 | 3.69 | 3.57 | 2.66 | 1.19 | −0.53 | −2.62 | −5.27 | |

_{f} |
7.2×10^{−5} |
1.1×10^{−4} |
1.8×10^{−4} |
3.9×10^{−3} |
1.2×10^{−1} |
7.0×10^{−1} |
≈1.0 | ≈1.0 |

#### Obtained and interpolated values of μQ and δQ.

_{v} |
_{h} |
_{Q}/δ_{Q} |
||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.5 | 2.0 | _{Q} |
210.19 | - | - | 194.21 | 173.95 | 153.71 | 133.45 | 113.14 |

_{Q} |
3.65 | - | - | 4.08 | 4.33 | 4.63 | 4.99 | 5.42 | ||

5.0 | _{Q}(kN/m) |
210.13 | - | - | 193.88 | 173.78 | 153.66 | 133.52 | 113.29 | |

_{Q} |
4.70 | - | - | 5.24 | 5.51 | 5.81 | 6.17 | 6.60 | ||

10.0 | _{Q} |
210.24 | - | - | 193.95 | 173.79 | 153.64 | 133.44 | 113.18 | |

_{Q} |
5.27 | - | - | 5.69 | 5.93 | 6.20 | 6.48 | 6.80 | ||

30.0 | _{Q} |
211.68 | - | - | 194.51 | 174.27 | 154.02 | 133.72 | 113.33 | |

_{Q} |
5.37 | - | - | 5.67 | 5.90 | 6.16 | 6.47 | 6.85 | ||

1.0 | 2.0 | _{Q} |
209.72 | - | - | 194.14 | 173.97 | 153.81 | 133.59 | 113.31 |

_{Q} |
4.51 | - | - | 5.07 | 5.34 | 5.63 | 5.96 | 6.31 | ||

5.0 | _{Q} |
209.70 | - | - | 194.03 | 173.92 | 153.78 | 133.60 | 113.33 | |

_{Q} |
5.77 | - | - | 6.13 | 6.38 | 6.65 | 6.95 | 7.25 | ||

10.0 | _{Q} |
209.88 | 208.32 | 206.75 | 194.25 | 174.18 | 154.07 | 133.89 | 113.60 | |

_{Q} |
6.26 | 6.30 | 6.34 | 6.64 | 6.88 | 7.14 | 7.42 | 7.74 | ||

30.0 | _{Q} |
211.35 | - | - | 194.66 | 174.53 | 154.38 | 134.19 | 113.90 | |

_{Q} |
6.60 | - | - | 6.86 | 7.06 | 7.27 | 7.52 | 7.79 |

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