As suggested in other studies [
5,
7,
8], “noise factors” are treated as random variables in this paper; the number of input parameters is therefore reduced and simplified. In the case of insufficient data for a quality statistical analysis, the typical statistical characteristics of geotechnical parameters can be found in literature. The input data is further simplified by applying a sensitivity analysis, based on which the random variables, whose contribution to the considered system response variance is negligible, are “frozen”. As a robustness measure, we use a generalised reliability index
, which is traditionally used as a reliability measure in classic reliability analyses, and by definition fits into the robustness concept. Introducing Eurocode 7 into the procedure additionally ensures the robustness of the designs. Unlike the original procedure, the results are directly applicable in everyday engineering practice. The key terms used in the procedure are defined as follows:
2.2.1. Harmonising the RGD Method Results with Eurocode 7
In the existing RGD method, a feasible region encompasses all designs that meet the pre-defined target failure probability requirement (), i.e., the associated value of the target reliability index (). The connection between and is mathematically defined, and these can therefore be used interchangeably. In this paper we use because it is more user-friendly than . The associated range of for would be .
Eurocode 0 [
12], in Annex C, gives a recommendation on limit values for
, in accordance with the considered limit state, reference period and reliability class. Thus, for a 50-year reference period and reliability class RC2, the following recommendations for
were given:
,
. The same values are used in the foundation optimisation using the original RGD procedure.
Meeting the criteria for the aimed reliability index value does not necessarily ensure meeting the criteria for ultimate and serviceability limit states defined in EC7. Forrest and Orr [
13] study the reliability of shallow foundation designed according to Eurocode 7, in the case of ultimate limit state (ULS). In their study, they conduct a series of analyses, which consider foundation’s reliability indexes according to EC7, and according to the allowable stress design format, with overall Factors of Safety
, and
. Different design situations are defined by the variance of the resultant force position (centric and eccentric load), soil type (granular and fine grain soils), relative load sizes (large and small loads), and the correlation
(with and without correlation). They conduct multi-parameter analyses for all the listed design situations, with the variance of geotechnical parameters: vertical scale of fluctuation (
), tangent of the angle of internal friction (
), characteristic values of
(
and
). The presented results indicate that there are designs that meet the target reliability index, but do not meet the ultimate limit state criterion defined in EC7.
An example of such designs is shown in
Table 2, which lists selected final designs attained through the application of the RGD procedure from the illustrative example given in [
1]. The designs marked grey fail to meet the ULS criterion defined by EC7; therefore, additional assessments of limit states according to EC7 are necessary for the chosen design, along with the RGD procedure.
By introducing additional constraints into the optimisation process, it is possible to exclude all designs that do not meet the ULS and SLS criteria defined in EC7 from the feasible region.
Figure 2 schematically illustrates the design space and the existing and suggested feasible regions in the case of ULS.
According to the existing RGD procedure, the feasible region is composed of regions 1 and 3 from
Figure 2, and is defined as follows:
Region 3 includes designs that meet the criterion concerning the aimed reliability index value, while at the same time failing to meet the criteria defined by EC7. By introducing additional constraints, region 3 is excluded from the feasible region, whose definition is then altered as follows:
2.2.2. Simplification of the RGD Procedure by Substituting the Robustness Measure
The most demanding part of the RGD procedure is the outer loop from
Figure 1, which is composed of M repetitions of FORM with integrated PEM, where M is the total number of the designs in the design domain. Every design requires seven FORM and one PEM analysis, done separately for ULS and SLS. The number seven refers to the number of estimating points in PEM [
14]. The aim of the mentioned procedure is the calculation of the mean value and standard deviation of
, which are used to determine the design robustness measure
. Such a procedure is extremely computationally expensive, and includes differentiations of limit states’ functions, which are mathematically complex in the case of shallow foundations. Furthermore, another shortcoming of applying
is the fact that we need to estimate the probability density function (PDF) of
, which is unknown. Making a wrong assumption considering the PDF of
can lead to a significant mistake in the estimation of
.
This paper explores the possibility of substituting the robustness measure
with a generalised reliability index
[
15], which is defined as follows:
In Equation (4), is the inverse Gaussian distribution and is the probability of failure.
Defining is a part of the RGD procedure within the constraint verification part; consequently, by applying it as the robustness measure, the whole procedure is significantly simplified. We consider to be a good indicator of the design’s robustness, since its value is directly related to the standard deviation of the considered system response. Designs with a smaller standard deviation of the system response are, by definition, more robust.
Figure 3 illustrates the relation of
to different values of the standard deviation of system response in shallow foundations in cohesionless soil. In the given example, a factor of safety (FS) was chosen for system response, with the assumption that it is lognormally distributed.
The statistical distribution of the factor of safety in shallow foundations was studied by Dodigović et al. [
16]; based on comprehensive statistical analyses, they concluded that such an assumption was justified.
Figure 3 clearly shows that designs with higher
favour lesser values of the standard deviation of system response, making them less sensitive to the variations of input parameters—i.e., more robust.
In order to compare the results yielded from the application of the original robustness measure, we analysed the relation
for different foundation widths and coefficients of variations of the soil’s internal friction angle (
). The results of the analysis are shown in
Figure 4. Due to a linear relation with a high correlation coefficient between
and
, we conclude that the design within the Pareto front—but the shape of the Pareto front as well—will be very similar for examples where the robustness measure is
or
.
It is possible to calculate the reliability index by using the PEM method, which is significantly simpler and computationally less demanding than the other reliability methods (FORM, SORM, FOSN, Monte Carlo). Zhao and Ono [
14] suggest PEM with seven estimating points, used in the outer loop of the RGD procedure, whose accuracy is significantly greater than the earlier instances of the method using 2, 3 and 5 estimating points.
The main shortcoming of PEM with seven estimating points is the fact that the random variables are transformed into standard normal space using the Rosenblatt transformation. In the example of correlated random variables, the transformation of random variables requires having total probabilistic information data, which is almost never the case in engineering. By substituting the Rosenblatt transformation with the Nataf transformation in PEM, we enable the consideration of correlated variables with the knowledge of the correlation matrix, their marginal distributions, mean values and standard deviations. The data on correlations between geotechnical parameters, their statistical distribution and standard deviation are available in literature, and the mean values can be determined from the geotechnical investigation results. The modification of PEM by introducing the Nataf transformation was suggested by Yu et al. [
16], and they named the modified method IPEM. The Nataf transformation and its inverse are expressed in the following equations:
where
is the marginal cumulative probability density function (CDF) of
X,
is the standard normal CDF of
X,
is the lower triangle matrix yielded from the Cholesky decomposition of the correlation matrix
. The procedure of determining the correlation matrix
is composed of a series of complex function integrations [
17], but its approximation is possible, based on a set of semiempirical equations suggested by Kiureghian and Liu [
18].
is approximated based on the known correlations of
and the ratio of
F in the following way:
The value of the ratio of F is given for different sets of marginal distributions, divided into two groups. According to the distribution pair, tables are given which provide the values/equations of F.
The IPEM procedure is conducted using simpler mathematical operations. The inverse Nataf transformation, which is a part of IPEM, is required only at the estimating points, and can be done simply, e.g., by using the built-in function of Microsoft Excel, or the Python program language used with the “SciPy” package. Since the accuracy of the calculations of
is highly dependent on the method used [
19], the accuracy of IPEM for ULS and SLS of shallow foundation was examined. The Monte Carlo method was chosen as the control method, and the results are given in
Table 3 and
Table 4.
The average error in the estimation of the reliability index using the IPEM method for ULS and SLS of shallow foundations, for different variation coefficients of the internal friction angle, is approximately 1%—which is negligible.
2.2.3. The Optimisation of the Number of Random Variables Using a Sensitivity Analysis
Obtaining non-dominated designs by applying the genetic algorithm in the RGD procedure can be optimised by reducing the number of random variables included in the reliability analysis. In the reliability calculation, failure is a probabilistic event, and its probability is given by:
where
X is a random vector and
g(.) is a limit state function.
In the case of a shallow foundation loaded by permanent and variable vertical action in cohesionless soil, the function of the limit state for ULS can be expressed as follows:
where
i
are load capacity coefficients,
and
are the coefficients of foundation shape,
is the effective unit weight of soil,
is the effective foundation width,
is the effective overburden pressure at the foundation base level,
is the permanent vertical load,
is the variable vertical load. The function of the serviceability limit state can be defined by applying a formula developed by Akbas and Kulhawy [
20], which is derived from load-settlement behaviour of a shallow foundation under axial compression loading
where
is the tolerated settlement,
is the foundation width, and coefficients
and
are the parameters of the hyperbolic model that fit the normalised-settlement curve. Random vectors from Equation (8) for the ultimate limit state can be expressed as
; for the serviceability limit state, the following applies:
.
The Sobol sensitivity analysis is conducted, with the aim of determining the contribution of the variance of each individual random variable to the total variance of limit state functions for both ULS and SLS. The Sobol sensitivity analysis is a variance-based method which determines the influence of uncertainties in the model input factors on the uncertainty in the output of the model [
21]. By determining the first order Sobol indices, we determine the effect of the variation of individual random variables on the variance of system response. The Sobol indices were calculated in the Python program, using the open source package “OpenTURNS” [
22]. The variables whose contribution to the system response variance is negligible can be “frozen”, which will make the RGD procedure less computationally demanding.
Table 5 and
Table 6 show the results of sensitivity analyses for different
values, for both ULS and SLS. The presented results demonstrate that the soil’s internal friction angle is the dominant variable and contributes most to the total variance of system responses. The value of the first order Sobol index is similar for ULS and SLS, and for different
values it is within in the
range, depending on the value of the internal friction angle.
Due to the high values of first order Sobol indices of the soil’s internal friction angle, reliability analyses of the ultimate and serviceability limit states were conducted for two groups of random vectors, which differ in the number of random variables. The first group of random vectors is composed solely of the soil’s internal friction angle, and can be expressed as follows:
and
. In the second group the random vectors include all the random variables defined by limit state functions for ULS and SLS:
and
The aim of the analysis is to determine the influence of reducing the number of random variables on the reliability analyses results, in order to optimise the RGD procedure. Reliability analyses were conducted using the IPEM method, and the results are presented in
Table 7 and
Table 8.
The average difference in reliability indexes
and
of the results presented in
Table 7 and
Table 8 for ULS is 1.3%, and 1.42% for SLS, with maximum deviations of 2.36% and 3.25%. For the value of
, as suggested in literature, deviations are smaller—0.56% for ULS and 0.41% for SLS. Such small deviations are in line with the results of sensitivity analyses presented in
Table 3 and
Table 4. We therefore conclude that the errors in calculating the reliability index, stemming from freezing random variables
,
and
in the ULS reliability analysis and
,
,
,
and
in the SLS reliability analysis, are negligible. Consequently, we propose conducting reliability analyses for ULS and SLS in which the internal friction angle is the only random variable. An analysis of the influence of ODF values on first order Sobol indices was also conducted. The results of this analysis are not presented, but it was confirmed that ODF has no significant influence on the value of the first order Sobol indexes for both ULS and SLS.
We would like to point out that the limit state functions for shallow foundations can also be determined by using other calculation models. This paper used the analytical expression for estimating the bearing capacity of a shallow foundation which is recommended in Eurocode 7, Annex D [
23] (Equation (9)). This expression is based on a theory extrapolated from the results of laboratory tests on the behaviour of small footings in dense sand and under 1 g condition [
24]. Altaee and Fellenius [
25] are researching soil behaviour in such conditions and conclude that it has little relevance to the behaviour of a full-scale prototype. The main reasons for this are inadequate stress levels in soil and non-linear stress–strain soil behaviour. With small-scale tests, the depth of model influence is relatively small, so soil behaves as if unconsolidated—after an initial volume assessment, soil dilates and then contracts. This results in a stress–strain curve which suggests ultimate resistance; this has not been proven in full-scale tests to this day. Due to all of the above, Fellenius [
26] recommends bearing capacity analysis to be used only as a simple estimate for comparing the design with previous designs.
Despite the mentioned shortcomings and simplifications, Eurocode 7 recommends using the expression from Equation (9) for determining the bearing capacity of a shallow foundation; it is consequently used in everyday engineering practice. Since the aim of this paper is to present the procedure of the modified RGD method whose results have been harmonised with EC7, the bearing capacity of a shallow foundation is determined according to the recommendations from this technical standard. Limit state functions can also be defined using other calculation models following the same principle, which will result in different mathematical expressions of limit state functions.
2.2.4. Optimisation Using the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)
Genetic algorithms (GA) are the heuristic optimisation and search techniques that mimic nature’s evolutionary principles. The concept of the genetic algorithm was developed by John Holland, and it is used in solving problems from various problem domains, including sciences, commerce, and engineering [
27]. The non-dominated sorting genetic algorithm II (NSGA-II) was first suggested by Deb et al. [
28] with the aim of improving the existing multi-objective evolutionary algorithms that use non-dominated sorting. A flowchart of the NSGA-II algorithm is shown in
Figure 5.
In the original RGD procedure, objectives are calculated for every design in the design space, followed by non-dominated sorting to determine the Pareto front (
Figure 1). Such an approach results in a high number of reliability analyses, which depends on the number of designs in the design space, as well as on the number of noise factors. In the example from [
1], the number of designs is 450, and the number of noise factors is 4 and 5 for ULS and SLS, respectively. In that case, the RGD procedure includes 7 × 4 × 450 = 12,600 FORM and 450 PEM analyses, and 7 × 5 × 450 = 15,750 FORM and 450 PEM analyses for SLS (the number 7 refers to the 7 estimating points according to the PEM procedure).
We suggest calculating objectives only for the designs chosen in the procedures included in the NSGA-II algorithm. In that case, the number of computations for every objective depends on the given termination criterion, which can be, e.g., the number of function evaluations, the number of iterations, or an advanced criterion based on the defined performance metric.
Due to a number of reasons, applying any of the algorithmic methods will rarely enable determining the true Pareto front, with only its approximation being possible. The approximation quality can be measured using various indicators, among which the hypervolume indicator (HI) [
29] is highly important. An increase in the value of HI is an indicator of convergence towards the true Pareto front. When its value ceases to change significantly, in relation to the number of evaluations of objective functions, the algorithm is considered to have converged. An example of the hypervolume indicator for a two-objective case is shown in
Figure 6, and includes a hatched region bound by the reference point
r.
We analysed the performance of the NSGA-II algorithm in the case of the RGD procedure, modified as described in
Section 2.2.1,
Section 2.2.2 and
Section 2.2.3. An optimisation was conducted, with three objectives defined as follows: minimise foundation cost, maximise ULS and SLS robustness with the decision variables being foundation width (B) and depth (D).
Since the choice of genetic algorithm parameters directly influences the quality of solutions and convergence [
30], we conducted a parametric analysis of crossover and mutation of operator parameters, with the aim of determining their optimum values. The analysis was conducted for different
values. The work included the usage of a simulated binary crossover (SBX) operator, which was shown to be efficient for real variables [
31]. The parameters of the SBX operator are crossover probability
and distribution index (
. Crossover probability is the number of realised crossovers in one generation. If its value is 0%, then the entire new generation equals the preceding one; if the crossover rate is 100%, the entire generation is substituted with new offsprings, yielded from the crossover of units in the previous generation. The distance of the offsprings from the parent solution will depend on the value of
: if
is large, the resulting offsprings will be near the parent solution, with the opposite being the case for smaller values. The mutation operator ensures the maintaining of genetic diversity in the genetic algorithm population. Deb and Deb [
32] analyse the usage of polynomial and Gaussian mutation operators for real-parameter genetic algorithms. They conclude that both operators are equally efficient, and this work used the polynomial mutation operator with a defined mutation probability (
) and distribution index (
). The value of the
parameter was chosen according to the proven efficient expression
, where
is the number of decision variables [
30]. The parametric analysis yielded optimal parameters of the NSGA-II algorithm:
,
,
and
. The relation between normalised hypervolume and the number of evaluations of objective functions, for the mentioned parameters, is shown in
Figure 7 and
Table 9.
Figure 7 and
Table 9 both show the quick convergence of the NSGA-II algorithm. Compared to the original RGD procedure, which requires 450 evaluations for the same input parameters, the suggested approach reduces that number by 40–50%.
By adopting the modification suggested in the
Section 2.2.1,
Section 2.2.2,
Section 2.2.3,
Section 2.2.4, the RGD procedure is simplified, and its flow diagram is shown in
Figure 8. During the first step, it is necessary to determine the statistic characteristics (mean value, standard deviation, statistical distribution) for all random variables and their mutual correlations. The number of random variables depends on the chosen geotechnical model of ultimate and serviceability limit state. The next step is conducting a sensitivity analysis with the aim of freezing the variables whose contribution to the variance of system response is negligible. When all the random variables are defined, we need to choose a decision variable, which makes the optimisation problem completely defined. The final step is composed of conducting a multi-objective optimisation using the NSGA-II algorithm. Unlike the original procedure, the inner loop (
Figure 1) is not conducted and determining the robustness measures from the outer loop is done within the NSGA-II algorithm, which reduces the required number of objective functions evaluations. The result of the optimisation is a set of non-dominated solutions, out of which the final design is chosen.