Incorporating spatial variability of soil and rock masses has recently become an important subject for geotechnical engineers’ interest. In recent times, probabilistic approaches are being used in many fields of geotechnical engineering like tunnelling (e.g., Lu et al., 2018; Pan and Dias, 2018; Chen et al., 2019) or deep excavations (e.g., Ching et al., 2017; Goh et al., 2019). Popularity of probabilistic approaches is also visible in the case of shallow foundations bearing capacity estimations. The direct reason for this interest is the popularity of cone penetrometer test soundings (CPT), and consequently, obtaining enough data to estimate the vertical scale of fluctuation (that characterized the considered soil strength spatial variability). On the other hand, the recent development and increase of experience in the field of estimating fluctuation scales techniques accelerate the development towards the use of random field theory to assess bearing capacity (Ching et al., 2018). However, most of the existed studies on this subject use two-dimensional simplification of the problem by assuming plane strain conditions (e.g., Fenton and Griffiths 2008; Pieczyńska-Kozłowska et. al., 2015; Ali et. al., 2016, Johari et al., 2017; Puła and Chwała 2018). Due to the ignorance of soil spatial variability in one direction (perpendicular to the assumed plane), this simplification may cause conservative estimation of variation coefficient of bearing capacity. To avoid this effect, it is necessary to perform numerical analyses in three dimensions. However, the three-dimensional analyses in the case of a finite element approach require a lot of computational effort. As a result, there is a lack of such analyses in the literature, one of the first papers in this area was published by Kawa and Puła (2019) and uses the finite difference method. Contrary to the finite element method approaches, the method proposed by Puła and Chwała (2015, 2018) that uses conjunction of Vanmarcke’s spatial averaging (1983) and kinematical failure mechanisms may result in significantly improved computational efficiency. The application of this approach to the three-dimensional issue of bearing capacity estimation of square or rectangular foundations in the case of undrained conditions was developed recently by Chwała (2019a). In this study, the above-mentioned method is modified in a way to ensure further increase in computational efficiency. The proposed modification uses the constant covariance matrix instead of individually determined as in Chwała (2019a). The constant covariance matrix is received by using the expected values of soil strength parameters. Therefore, in the case of this study, an expected value of undrained shear strength is used. The covariance matrix is determined based on the optimal failure geometry that is the result of optimization procedure. Due to the usage of upper bound analysis, the geometry for the specified undrained shear strength is searched in a way to minimize the resulting bearing capacity. The above briefly described method was used for analysing numerous scenarios to enable the creation of graphs that allow an immediate read of the approximate value of the variation coefficient of undrained bearing capacity (BC). In the proposed method, as the prerequisites, the undrained shear strength mean value (
In this study, a Prandtl-type three-dimensional failure mechanism for a rough foundation base is assumed (Gourvenec et al, 2006). However, for the purpose of this study, a probabilistic version of the failure mechanism is needed (to calculate the bearing capacity in the case of different values of undrained shear strength are applicate to different dissipation regions). Therefore, the results obtained in the study by Chwała (2019a) were used as a basis for the proposed approach and numerical analyses performed in this study. The geometry of the failure mechanism is shown in Fig 1.
Failure geometry for a rough foundation base in the case of rectangular foundation. More details on failure geometry can be found in Chwała (2019a).
A set of failure geometry parameters shown in Fig. 1 is responsible for describing failure geometry. However, for unique failure geometry determination, only eight of those parameters are necessary, namely: six angles α1, α2, α3, α4,
Value of bearing capacity during simulation within optimization procedure
The kinematical failure mechanism can be used in conjunction with random field theory by using Vanmarcke’s (1983) spatial averaging. By this procedure, the average values of undrained shear strength are determined for each dissipation region resulting from failure geometry. The application of spatial averaging provides no necessity of direct random field sample generation. Instead of that, the averaged values of undrained shear strength are generated. The generation takes place from single random variables obtained after discretization of the initial random field. Each random variable is dedicated to the corresponding dissipation region. Therefore, in this study, there are 30 random variables (because the failure mechanism consists of 30 dissipation regions). However, those variables are not independent, they are correlated in accordance with their relative positions, assumed correlation structure and values of fluctuation scales. Each random variable is obtained by integrating over the dissipation region geometry; thus, its shape, size and position with respect to the coordinate system is taken into account. To obtain full information about correlation between each of those 30 random variables, the matrix of size 30 over 30 is necessary. This matrix is called as the covariance matrix and denoted as
The numerical algorithm that uses the constant covariance matrix is described below. The algorithm consists of two main parts. The first one is the determination of the covariance matrix that takes place before the Monte Carlo loop started. The second part is the actual Monte Carlo loop with the generation of averaged undrained shear strengths and bearing capacity calculation. Below, the detailed algorithm is described. Moreover, the flow chart of the procedure is shown in Fig. 3.
Flow chart of the numerical algorithm. Detailed descriptions of the steps are in the text.
The concept of using a constant covariance matrix may introduce differences in the obtained random bearing capacity estimations in comparison with the situation where the covariance matrix is determined individually to the generated undrained shear strengths. However, earlier experiences in the case of two dimensional Prandtl’s failure mechanism indicated that those differences are very small, for more details please see Puła and Chwała (2015). Despite this, for the purpose of this study, the comparison analyses are conducted to verify the impact on accuracy for the three-dimensional case. For this reason, a series of numerical analyses were performed to examine the variation coefficients of the undrained bearing capacity obtained by the standard version of the algorithm (Chwała, 2019a) and by the approach presented in this study. The analyses cover variety of foundation lengths; namely,
Comparison of bearing capacity variation coefficients obtained by the method proposed in this study (constant covariance matrix) with standard algorithm (individual covariance matrix, Chwała (2019a)). The relative differences between both approaches are below 5%, which is sufficient for the purpose of this study. A detailed description is in the text.
In Fig. 4, the results obtained by random finite limit analysis are also shown for the same soil conditions. Those results were obtained by Huang et al. (2013) for plane strain conditions (see green dashed line in Fig. 4); therefore, they cannot be compared directly with those obtained by the approach proposed in this study. Nevertheless, Fig. 4 illustrates the importance of considering spatial variability in three dimensions – for longer foundations, plane strain assumption may provide more conservative BC coefficient of variation estimates.
Moreover, by using the algorithm proposed in this study, the same scenarios were examined as in the study by Simoes et al. (2014), where the three dimensional spatial variability was considered for 8 m foundation section (the authors used random finite limit analysis). Note that in the study by Simoes et al. (2014) plane strain conditions were assumed. Therefore, the obtained failure mechanism was not fully three-dimensional; however, due to its proportions 8.0 m over 1.0 m, this impact is not very significant. The comparison is shown in Table 1. The results obtained by both approaches are consistent; however, in the study by Simoes et al. (2014), lower bearing capacity mean values and standard deviations were obtained. Certain parts of these differences can be explained by the fully three-dimensional mechanism considered in this paper and different covariance functions assumed in both studies.
Comparison of the results obtained in this study with those obtained by Simoes et al. (2014) by random finite limit analysis.
COV | RFLA (Simoes et al. 2014) | 4.77 | 1.68 | 0.35 |
This study | 5.19 | 2.14 | 0.41 | |
COV | RFLA (Simoes et al. 2014) | 3.72 | 1.23 | 0.33 |
This study | 4.51 | 1.77 | 0.39 |
As mentioned above, a number of 2000 simulations was used for the estimation of bearing capacity mean value and standard deviation. This number is found to be sufficient for the objectives of this study. To illustrate that the mean value and standard deviations are stabilized for
Mean value, standard deviation and variation coefficient of bearing capacity as a function of simulation number
To construct graphs that are sufficiently accurate for the approximate estimation of the coefficient of variation of undrained bearing capacity in a wide range of soil and foundation geometry scenarios, numerous numerical simulations have to be performed. The results are present in a dimensionless coordinate system to cover almost all possible scenarios (from practical application viewpoint in the case of standard foundation sizes). As the vertical axis, the dimensionless coefficient of variation of undrained bearing capacity
The resulting coefficients of variation of undrained bearing capacity for the scenarios described in the previous section are used here to construct the graphs for reading approximate values of undrained bearing capacity (without the necessity of conducting numerical calculations). The description of the graphs is given below; however, the examples of their application are shown in section 5.1. The final results were divided into two graphs shown in Fig. 6 and Fig. 7. This separation is to ensure their clarity. From Fig. 6, the values of coefficients of variations for ratios
Graph for reading
Graph for reading
Results shown in Fig. 6 and Fig. 7 are represented as functions of
Graph for reading
To demonstrate the possible usage of the graphs in Fig. 6, Fig. 7 and Fig. 8, the undrained bearing capacity coefficient of variation for a set of scenarios is estimated based on these figures. To verify the outcomes, the same scenarios were analysed by the proposed algorithm individually. Finally, the obtained results are compared and juxtaposed in Table 2. The locations of the considered scenarios are shown in Fig. 9 and Fig. 10. Let’s discuss the first row of Table 2. According to the initial data for the first scenario, the following dimensionless coordinates of point 1 can be found:
Illustration of example scenarios. Note that the colour in the legend indicates
Illustration of example scenarios. Note that the colour in the legend indicates
According to Table 2, the obtained differences are below 10% and they are caused by the finite number of considered scenarios and interpolation errors. Note that the positioning of scenarios from Table 1 is made without the use of drafting equipment. This level of accuracy can be accepted for the approximate estimations of undrained bearing capacity coefficient of variations. Note that the errors in estimation of the fluctuation scale and COV
Exemplary usage of the graphs proposed in this study. Detailed descriptions are in the text.
1 | 2.0 | 2.0 | 0.6 | 12.0 | 0.6 | 0.41 | 0.435 | −6.1% |
2 | 10.0 | 1.5 | 1.0 | 5.0 | 0.4 | 0.226 | 0.210 | +7.1% |
3 | 20.0 | 0.9 | 0.8 | 3.0 | 1.0 | 0.29 | 0.311 | −7.2% |
4 | 25.0 | 1.8 | 1.2 | 1.2 | 0.5 | 0.057 | 0.052 | +8.7% |
5 | 2.0 | 1.0 | 1.5 | 2.0 | 0.7 | 0.451 | 0.458 | −1.5% |
6 | 25.0 | 3.0 | 0.5 | 10.0 | 1.0 | 0.29 | 0.280 | +3.4% |
7 | 3.0 | 3.0 | 0.40 | ∞ | 0.24 | 0.13 | 0.130 | 0.0% |
8 | 20.0 | 1.0 | 0.47 | ∞ | 0.51 | 0.40 | 0.403 | −0.7% |
This paper presents a very efficient algorithm for the three-dimensional analysis of the undrained bearing capacity of shallow foundations in the case of spatially variable undrained shear strength. The algorithm uses constant covariance matrix approach and is based on the Vanmarcke’s spatial averaging in conjunction with upper bound approach (a kinematical failure mechanism is used). The algorithm was used in the study to analyse a series of scenarios. The objective of this extensive analysis is to create graphs that allow reading the approximate value of undrained bearing capacity coefficient of variation. The ranges of foundation geometries and fluctuation scales are assumed in a manner to cover almost all scenarios possible to exist for standard shallow foundations’ geometries and for the reported values of fluctuation scales. Square and rectangular foundations were analysed. The proposed algorithm is very efficient; namely, one three-dimensional bearing capacity evaluation for spatially variable undrained shear strength takes below 1s for one core of a standard notebook. According to the presented algorithm and performed numerical analyses, the following conclusions can be drawn:
As indicated in Fig. 4, the resulting coefficient of variation of undrained bearing capacity obtained by using constant covariance matrix (the matrix computed for the expected value of undrained shear strength) is very close to the values obtained for individually determined covariance matrix (Chwała, 2019a). As a result of using the proposed approach, a similar accuracy can be achieved, but the numerical efficiency is improved dramatically (the computation time for one simulation is reduced about 100 times). The observed efficiency is far better than the observed for the methods based on finite element method, or finite difference method (Kawa and Puła, 2019). The problem of poor numerical efficiency is the reason why there is a lack of studies concerning three-dimensional bearing capacity for spatially variable soil. The paper by Kawa and Puła is one of the first in this area. Nevertheless, the efficiency of the method proposed in this study allows for its usage for the applications that required many simulations to be analysed, like the analyses performed in this study or the recently proposed approach for searching optimal borehole locations (Chwała, 2019b). The proposed algorithm was used to create graphs that allow fast reading of approximate coefficient of variation of undrained bearing capacity. The usage of the graphs is demonstrated in the study, the obtained results indicate that the final accuracy can be accepted for this purpose. The graphs can be used if vertical fluctuation scale and coefficient of variation of undrained shear strength are known. However, those values can be relatively easily determined from CPT sounding. The graphs can be used to make a comparison with the other three-dimensional methods for random bearing capacity estimation. As it can be observed in Fig. 6, Fig. 7 and Fig. 8, there are no worst-case scenario (e.g., Fenton and Griffiths, 2008; Ching et. al., 2017) in the coefficients of variation of bearing capacity exists. This is in agreement with earlier experiences where the worst-case is observed mostly in mean values (e.g., Puła et al., 2017). The study indicates the importance of performing three-dimensional analyses if the soil spatial variability is taken into account. A three-dimensional algorithm is necessary to consider realistically the soil spatial variability that has a three-dimensional nature. This phenomenon is visible in Fig. 6 and Fig. 7, where a significant decreasing tendency in variation coefficient of undrained bearing capacity is observed with an increase of foundation length