*4.2. Optimization*

In this section, the cost-optimization of the prestressed concrete precast bridge will be explained. This optimization process consists in the minimization of the cost *C* while some restrictions *gj* are satisfied.

$$\mathcal{C} = f(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \tag{1}$$

$$g\_j(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \le 0 \tag{2}$$

Note that *x*1, *x*2, ... , *xn* are the design variables used for the optimization. The objective function *C* expresses the cost of the bridge and the restrictions *gj* are the serviceability limit states (SLS), the ultimate limit states (ULS), the durability limit states and the geometric and constructability constraints of the problem. There are 40 design variables, including eight variables that define the geometry of the section, two that define the concrete of the slab and the beam, four that define the prestressed steel and 26 that define the reinforcing steel. Furthermore, there are a set of parameters that have no influence on the optimization problem, such as the width, span and web inclination. Structural constraints have been considered according to the Spanish codes [25,26]. The ULSs verify if the ultimate resistance is greater than the ultimate load effect. Besides, the minimum amount of reinforcing steel for the stress requirements and the geometrical conditions are also considered. The SLSs examine different aspects. Cracking limit state requires compliance of the compression and tension cracks, as well as the decompression limit state in the area where the post-tensioned steel is located. Deflections are limited to 1/1000 of the free span length for the quasipermanent combination. In addition, the concrete and steel fatigue has been considered in this study. Table 1 summarizes of the ULSs and SLSs considered.

In this optimization, a hybrid memetic algorithm (MA) is applied. The MA is a population-based approach to stochastic optimization that combines the parallel search used by evolutionary algorithms with a local search of the solutions forming a population [27]. Regarding the local search used, a variable-depth neighbourhood search (VDNS) is used as a variant of the very large-scale neighbourhood search (VLSN) [28]. In this MA-VDNS, a set of 500 random solutions (*n*) is generated as the population. Then each of these solutions is improved by means of a VDNS search to reach a local optimum. To this end, the algorithm begins by changing only one variable and when ten consecutive movements have been performed without improvement (*no\_imp*), there will be an increase

in the number of variables (*var*) that are changed simultaneously, up to eight. Then, with this new improved population, a genetic algorithm is applied. The genetic algorithm develops the population, which is subjected to random movements (mutations and crossovers), preserving the better adapted solutions. The cost assessment takes into account a penalty cost; nevertheless, the VDNS does not consider the penalty cost (only feasible solutions are accepted) in order to avoid the early divergence of the algorithm. The VDNS is applied to the new generation up to 150 generations. Figure 2 shows a flow chart of the hybrid memetic algorithm.


**Figure 2.** Hybrid memetic algorithm flow chart.

The solution obtained for the 40 m-long prestressed concrete precast bridge has a total cost of 108,274.45 €. The geometry of this bridge is shown in Figure 3. The amount of beam concrete used is 0.1117 m3/m2, with a strength of 35 MPa, while the amount of slab concrete used is 0.1797 m3/m2, with a strength of 40 MPa. Furthermore, the precast concrete beams require 6163 kg (12.52 kg/m2) of reinforcing steel and 5184 kg (10.53 kg/m2) of prestressed steel, while the concrete slab is defined by 11,772 kg (23.92 kg/m2) of reinforcing steel.

**Figure 3.** Geometry of optimized bridge.
