**1. Introduction**

All structural designs involve variability and uncertainty [1,2]. The initial parameters, the structure dimensions, the mechanical characteristics of the materials, and the loads may di ffer from the design values [3,4]. Nevertheless, the design of a structure is made using the nominal value, which has a low probability of occurring (for example, the resistance of concrete is the resistance that has a 5% probability of failure). In addition, safety coe fficients associated with a given probability of failure are assigned. However, a variation of these initial uncertain parameters can influence the variability of the structural behavior. Structural optimization usually uses a deterministic approach that does not consider the e ffects of the associated uncertainty [5–13]. This means that the structure has an optimum behavior only under the conditions initially defined, and the response can vary significantly when the values di ffer from the design values [14,15].

Unlike this approach, robust design has been studied to obtain designs in which the uncertainty of the initial parameters has the lowest possible influence on the objective response. This robust design is reached by a probabilistic optimization. Nowadays, there are two approaches to the optimal probabilistic design of a structure: Reliability-Based Design Optimization (RBDO) [16] and Robust Design Optimization (RDO) [17]. In RBDO, the probability of failure is studied from the variations of the initial parameters. RDO studies a design that is less sensitive to the variation of the initial

parameters. The present paper focuses on the RDO. The concept of robust design was proposed by Taguchi in the 1940s and applied to optimization problems in 1980 [1]. This approach uses the mean and standard deviation to study the variability of the objective response.

The main limitation of RDO is the computational expense because assessing the sensitivity of the objective response of the problem requires a high number of runs [18]. Therefore, it is necessary to find methods that allow carrying out the optimization process more e fficiently [4,19,20]. Metamodels allow the generation of a mathematical approximation of the objective response (an objective surface) from the assessment of points within the design space. Once the response surface has been generated, obtaining the value of the objective response given the inputs is much faster. These mathematical approximations or metamodels have already been used to solve RDO process problems [4]. The most widespread metamodels are polynomial regression, artificial neural networks (ANN) and kriging. ANN has been used in di fferent works related to structural engineering [21,22]. However, the kriging model has been demonstrated to be useful to obtain grea<sup>t</sup> reliability in the assessment of the response due to its predictive accuracy in non-linear functions [23]. Penadés Plà et al. [24] compared standard heuristic optimization and heuristic optimization based on kriging models, demonstrating that the solutions obtained through optimization based on kriging models are close enough to the solutions obtained through standard heuristic optimization, but with high savings in computational costs.

In the present paper, the robust design methodology is applied to a continuous prestressed concrete box-girder footbridge to obtain a bridge that is optimal in terms of its cost objective function and robust in terms of structural stability. Its structural stability is measured by the variability of the vertical displacement in the middle of the bridge [19]. To this end, Latin hypercube sampling is used to obtain the initial sampling, the kriging model is used to obtain the mathematical approximation to the response, and then the simulated annealing optimization algorithm is used to obtain the robust optimum design. All this will be studied for di fferent uncertain design parameters: the modulus of elasticity, the overload, and the prestressing force.

#### **2. Robust Design Optimization**

Robust design studies the variation of the objective response generated by the uncertain initial parameters. Therefore, robust design optimization (RDO) aims to reach the best objective response with the smallest deviation. It implies that the RDO problem is defined as a multi-objective optimization problem in which the objective response is the mean and the standard deviation (Equation (1)).

$$\min \left\| \mu\_{F(\mathbf{x}, \mathbf{z})} (\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \dots, \mathbf{x}\_n), \sigma\_{F(\mathbf{x}, \mathbf{z})} (\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \dots, \mathbf{x}\_n) \right\| \tag{1}$$

where *x*1, *x*2, *x*3, ... ... ,*xn* are the deterministic values of the design variables or the probabilistic function of the uncertain initial parameters.

Commonly, the two objective functions to be minimized in an RDO problem are in conflict. This situation leads to a set of solutions that represent a Pareto frontier. Figure 1 shows an example of the di fference between the optimal solution and the robust optimal solution in a design space of one design variable. Solution A is the optimal solution, point B is the most robust solution and point C is the robust optimal solution. It is possible to see that the same variation of the design variable (*v*) causes a higher variation in the objective function of the solution A (*f* A) than it does in the solution C (*f* C).

**Figure 1.** Robust design optimization.

#### **3. Robust Design Optimization Using Metamodels**

The main goal of metamodels is to obtain results more efficiently by creating a mathematical approximate model of a detailed simulation model (model of a model). This makes it possible to predict output data (objective response) from input data (variables or design parameters) of the design space. There are three main steps to create a metamodel: (1) obtaining the initial points of the input or sampling data set within the design space (size and position), (2) choosing the method to create the approximate mathematical model, and (3) choosing the fitted model. Each of these three steps can be performed using many different options [25]. In this study, Latin hypercube sampling is used to obtain the initial sampling, the kriging model is used to create the approximate mathematical model, and the search for the Best Linear Unbiased Predictor (BLUP) is used as the fitted model. Then, the mathematical approximation created is used to predict the objective functions according to the initial design variables and parameters. In this way, the optimization can be carried out more efficiently, saving a lot of computational costs, which is important in a probabilistic optimization. In addition, the simulated annealing algorithm is used to perform the optimization. Figure 2 shows a flowchart of the robust design optimization using these characteristics. A more detailed description of this approach can be seen in Penadés-Plà et al. [24].

**Figure 2.** Flow diagram of robust design optimization.

#### *3.1. Latin Hypercube Sampling*

McKay et al. proposed in 1979 the Latin hypercube sampling (LHS) [26]. This method is a space-filling type of design of experiments. That means that this type of design of experiment trends to cover all of the design space by the positions of the initial sample points. In this way, local deformation of any area of the design space can be taken into account. For this purpose, a number *N* of non-overlapping intervals must first be determined. These intervals divide each range of the design variables (*v*) into *N* sections, generating a mesh in the design space with *Nv* regions. Then, a combination of *N* random points is generated, so each point is placed in a combination of different intervals of each range of design variables. This guarantees that the initial sampling covers the entire range of each design variable. LHS has been used in several papers, showing its validity for the estimation of metamodel output data [20,27]. For this reason, in the present paper, a uniform distribution of the initial sample points by LHS is employed.
