**5. Results**

The results are subdivided into two parts according to the initial uncertain design parameter considered: modulus of elasticity and loads (overload and prestressing force). Each one of these sections provides an initial validation of the kriging surfaces generated, the Pareto frontiers obtained, and some comparisons. For this purpose, 200 points are created to verify the accuracy of the kriging surfaces (validation), and another 200 solutions are obtained from the robust design optimization problems carried out (Pareto frontier). After that, the results will be discussed.

#### *5.1. Variation of Modulus of Elasticity*

In this part, the uncertain design parameter studied is the modulus of elasticity. Three di fferent RDO problems are studied. For this purpose, six kriging surfaces are generated depending on the objective function (μcost and σvertical displacement) and the variability considered of the modulus of elasticity (10%, 20%, and 30%). Table 3 shows the di fferent validations of the di fferent kriging surfaces obtained. The accuracy of the kriging surfaces that predict the mean costs are better than the kriging surfaces that predict the variability of the vertical displacement. The di fference between the real and predicted mean value of the cost is lower than 2%, and the di fference between the real and predicted standard deviation of the vertical displacement of the middle of the bridge is lower than 5% in all di fferent uncertainties of the modulus of elasticity considered.


**Table 3.** Validation of the kriging surfaces while varying the modulus of elasticity.

Figure 4 shows the Pareto frontiers for the di fferent uncertainties of the modulus of elasticity considered. This figure represents the mean of the cost against the standard deviation of the vertical displacement of the middle of the bridge. It shows that an increment of the uncertainty of the modulus of elasticity causes a displacement of the Pareto frontier, moving away from the positive ideal point (lowest μcost and lowest σvertical displacement). This is because the design of the structure should resist all the possible values of the uncertain parameter. Therefore, a higher variation of the initial uncertain parameter imposes greater requirements on the design and an increment of the cost.

**Figure 4.** Pareto frontier of modulus of elasticity Robust Design Optimization (RDO) problems.

Table 4 shows a comparison between the designs of the different Pareto frontiers with the same structural behavior. In this case, the reference value taken into account is the standard deviation of the vertical displacement of the cheapest design of the Pareto frontier with the lowest variation of the modulus of elasticity studied. In this way, an imaginary horizontal line will intersect all the Pareto frontiers (dashed line of Figure 4). Solutions S10, S20, and S30 are selected, which correspond to a σvertical displacement lower than 3.82 mm. It shows that to reach similar structural behavior, the price increases with an increment of the uncertainty of the modulus of elasticity and that the design variables that cause this increment of the price are the depth and *f* ck. Both are higher for each increment of the variability of the modulus of elasticity.


**Table 4.** Comparison of design with the same structural behavior in modulus of elasticity RDO problems.

Furthermore, if just one Pareto frontier is studied and three key designs are considered: (A) the optimum or lowest μcost, (B) the robust optimum or shortest to the positive ideal point, and (C) the most robust or lowest σvertical displacement, the same design variables are affected. For example, Table 5 shows these designs for the Pareto frontier with a 20% variability of the modulus of elasticity. As shown in Table 4, the values of depth and *f* ckare higher when more robustness is required.

**Table 5.** Comparison of different designs of the Pareto Frontier with a 20% variation of the modulus of elasticity.


#### *5.2. Variation of Loads: Overload and Prestressing Force*

In this part, the uncertain design parameters studied are two loads. The first one is the overload due to its high uncertainty, and the second one is the prestressing force to know how the variability influences the behavior of the bridge. The overload is defined according to the IAP-11 [37], which corresponds to 5 kN/m2. In this case, due to the higher uncertainty of these parameters, another increment of uncertainty in the loads is considered (40%). Therefore, four RDO problems are studied for each load. For this purpose, eight kriging surfaces are generated for each load depending on the objective function (μcost and σvertical displacement) and the variability considered of the modulus of elasticity (10%, 20%, 30%, and 40%). In these cases, the results discussed are the same as in the previous subsection. In this way, first, the validations of both loads are discussed (Tables 6 and 7) After that, the Pareto frontiers for each di fferent uncertainty of the design parameter are shown (Figures 5 and 6), and finally some solutions are compared following the same rules as in the previous comparison: the overload (Tables 8 and 9), and the prestressing force (Tables 10 and 11).

Tables 6 and 7 show the di fferent validations of the kriging surfaces obtained. As in the previous cases, the discrepancy of the mean value of the cost is lower than 2% in all cases. However, the discrepancy of the standard deviation of the vertical displacement of the middle of the bridge depends on the variability of the displacement, being higher when the vertical displacement variability is higher and lower when the vertical displacement variability is lower. The results show that when the variability of the overload is lower (10%), the kriging method cannot capture the variability of the displacement accurately. Thus, this uncertainty is not considered.


**Table 6.** Validation of the kriging surfaces varying the overload.



Figures 5 and 6 represent the Pareto frontiers for the di fferent variations of the loads. In both cases, the Pareto frontiers have the same behavior as before, moving away from the positive ideal point according to the increment of the uncertainty of the loads. In addition, the comparisons made (Tables 8–11) have similar behavior to the above.

Tables 8 and 10 show a comparison between di fferent designs with the same structural behavior of the di fferent Pareto frontiers. Table 8 corresponds to the RDO problems in which the overload is the uncertain parameter, and the σvertical displacement of reference corresponds to 2.93 mm (dashed line of Figure 5). Table 9 corresponds to the RDO problems in which the prestressing force is the uncertain parameter, and the σvertical displacement of reference corresponds to 11.06 mm (dashed line of Figure 6). In both cases, to reach a similar structural behavior the price increases with an increment of the uncertainty of the loads. As well as in the case of the RDO problems in which the modulus of elasticity is the uncertain parameter, the increment of the price is due to the increment of the depth and *f* ck. The di fference is that in the case (where the modulus of elasticity is the uncertain parameter) the depth and the value of *f* ck increase in each increment of variability, and in the case where the uncertain parameter is the load, the increment of the depth and *f* ck is not simultaneous. In these cases, a balance between these two design variables is achieved to reach a similar structural behavior. In addition, this increment of depth and *f* ck is less significant in the case of the overload, due to the low di fferences among the di fferent uncertainties. The same occurs when the comparison is made between the optimum or cheapest (A), the robust optimum or shortest to the positive ideal point (B), and the most robust or lowest variation of the vertical displacement (C) (Tables 9 and 11). As above, the key design variables to modify the structural behavior change are the depth and *f* ck. These variables tend to be higher when higher robustness is required.

**Figure 5.** Pareto frontier of overload RDO problems.

**Figure 6.** Pareto frontier of prestressing force RDO problems.

**Table 8.** Comparison of designs with the same structural behavior in overload RDO problems.



**Table 9.** Comparison of different designs of the Pareto Frontier with a 20% variation of the overload.

**Table 10.** Comparison of designs with the same structural behavior in prestressing force RDO problems.


**Table 11.** Comparison of different designs of the Pareto Frontier with a 20% variation of the prestressing force.


## **6. Conclusions**

Currently, the design of structures is made according to a deterministic design. This approach has the result that when the design is optimized according to a conventional objective function, the behavior of the structure is really dependent on the initial values considered. This paper uses a probabilistic approach to consider the variation of the design parameters. In addition, to reduce the large computational cost of the probabilistic optimization, Latin hypercube sampling and kriging metamodels are used. Each point of the Latin hypercube sampling is calculated 20 times varying the initial uncertain parameters (modulus of elasticity, overload, and prestressing force) obtaining the mean of the cost and the standard deviation of the vertical displacement in the middle of the bridge. These values are used to create the kriging surface that predicts the objective response depending on the initial design variables. These surfaces have an error lower than 2% in the mean of the cost for all cases, lower than 5% in the standard deviation of the vertical displacement when the modulus of elasticity is the uncertain parameter, and an accuracy dependent on the value of the vertical displacement when the loads are the uncertain parameters. After that, 200 solutions have been calculated for each case to obtain the different Pareto frontiers.

The Pareto frontiers show that, for all RDO problems, an increment of the uncertainty causes a displacement of the Pareto frontier, moving away from the positive ideal point. That means that to obtain specific robustness when the uncertainty of the parameter is higher, the cost of the design will be higher. In addition, when just one Pareto frontier is taken into account, a more robust design implies an expensive design. In all cases, this increment of the price is due to an increment of two specific design variables: depth (*h*) and *f* ck. Therefore, to obtain a robust design, it is necessary to increment the depth (*h*) and/or *f* ck. However, these Pareto frontiers allow obtaining a compromise design between cost and robustness: the optimum robust design. This solution is the design closest to the positive ideal point.

This work shows that a probabilistic optimization can be carried out to obtain an optimum robust design. Nevertheless, the robust design optimization of complex problems requires a high computational cost. Therefore, the use of metamodels is necessary to carry out probabilistic optimization. In previous works, the computational cost saved and the validity of kriging metamodels were proven. This work shows that the kriging metamodel has an appropriate behavior to carry out the robust design optimization, and therefore can be used to carry out optimization where there is uncertain information. **Author Contributions:** This paper represents a result of teamwork. The authors jointly designed the research. V.P.-P. drafted the manuscript. T.G.-S. and V.Y. edited and improved the manuscript until all authors are satisfied with the final version. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Ministerio de Economía, Ciencia y Competitividad and FEDER funding gran<sup>t</sup> number [BIA2017-85098-R].

**Conflicts of Interest:** The authors declare no conflict of interest.
