# Robust Design Optimization for Low-Cost Concrete Box-Girder Bridge

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Robust Design Optimization

_{1}, x

_{2}, x

_{3},……,x

_{n}are the deterministic values of the design variables or the probabilistic function of the uncertain initial parameters.

_{A}) than it does in the solution C (f

_{C}).

## 3. Robust Design Optimization Using Metamodels

#### 3.1. Latin Hypercube Sampling

#### 3.2. Kriging

^{2}and a non-zero covariance. The first term of the equation, f(x), offers a global approach to the design space that is similar to a regression model (Equation (3)). The second term, Z(x), generates local deviations to interpolate the initial sample points using the kriging model (Equation (4)).

^{2}scales the spatial correlation function R(x

_{i},x

_{j}) between two data points. The Gaussian correlation function (Equation (5)) is widely used in engineering design [29]. It can be defined with a single parameter (θ) that determines the area of influence of the adjacent points [30]. When the sample points have a high correlation, then θ is low, thus Z(x) will be similar throughout the design space. As the θ grows, the closest points will have the greatest correlation, thus Z(x) will vary according to the point in the design space:

#### 3.3. The Fitted Model

#### 3.4. Mean and Variance

#### 3.5. Optimization

## 4. Problem Design

#### 4.1. Description of the Box-Girder Footbridge

#### 4.2. Description of the Robust Design Optimization Problem

_{1}, x

_{2}, x

_{3},……, x

_{n}are the design variables.

_{i}) relating to the construction units are calculated as defined by the design variables. The variation of the vertical displacement in the middle of the bridge has been obtained according to the standard deviation of 20 different cases varying the initial uncertain parameter. Each one of these vertical displacements has been calculated in accordance with Spanish regulations [36,37] along with the Eurocodes [38,39].

_{1}and w

_{2}are weights with values in the range [0,1] such that w

_{1}+ w

_{2}= 1.

_{1}runs from 0 to 1 with increasing 1/N and w

_{2}corresponds to 1-w

_{1}. In this way, 200 different optimizations are made and all the possible solutions of the Pareto frontier are covered.

## 5. Results

#### 5.1. Variation of Modulus of Elasticity

_{cost}and σ

_{vertical displacement}) and the variability considered of the modulus of elasticity (10%, 20%, and 30%). Table 3 shows the different validations of the different 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 difference between the real and predicted mean value of the cost is lower than 2%, and the difference between the real and predicted standard deviation of the vertical displacement of the middle of the bridge is lower than 5% in all different uncertainties of the modulus of elasticity considered.

_{cost}and lowest σ

_{vertical 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.

_{vertical 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.

_{cost}, (B) the robust optimum or shortest to the positive ideal point, and (C) the most robust or lowest σ

_{vertical 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

_{ck}are higher when more robustness is required.

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

^{2}. 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 (μ

_{cost}and σ

_{vertical 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 (Table 6 and Table 7) After that, the Pareto frontiers for each different uncertainty of the design parameter are shown (Figure 5 and Figure 6), and finally some solutions are compared following the same rules as in the previous comparison: the overload (Table 8 and Table 9), and the prestressing force (Table 10 and Table 11).

_{vertical 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 σ

_{vertical 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 difference 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 differences among the different 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) (Table 9 and Table 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.

## 6. Conclusions

_{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.

## Author Contributions

## Funding

## Conflicts of Interest

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Design Variable | Min. Value (m) | Max. Value (m) | Precision (m) |
---|---|---|---|

Depth (h) | 1.25 | 2.5 | 0.05 |

Width (b) | 1.2 | 1.8 | 0.05 |

Inclination width (d) | 0 | 0.4 | 0.05 |

Top slab thickness (es) | 0.15 | 0.4 | 0.05 |

External cantilever section thickness (ev) | 0.15 | 0.4 | 0.05 |

Bottom slab thickness (ei) | 0.15 | 0.4 | 0.05 |

Webs slab thickness (ea) | 0.3 | 0.6 | 0.05 |

Unit Measurements | Cost (€) |
---|---|

m^{3} of scaffolding | 10.2 |

m^{2} of formwork | 33.81 |

m^{3} of lighting | 104.57 |

kg of steel (B-500-S) | 1.16 |

kg of post-tensioned steel (Y1860-S7) | 3.40 |

m^{3} of concrete HP-35 | 104.57 |

m^{3} of concrete HP-40 | 109.33 |

m^{3} of concrete HP-45 | 114.10 |

m^{3} of concrete HP-50 | 118.87 |

m^{3} of concrete HP-55 | 123.64 |

m^{3} of concrete HP-60 | 128.41 |

m^{3} of concrete HP-70 | 137.95 |

m^{3} of concrete HP-80 | 147.49 |

m^{3} of concrete HP-90 | 157.02 |

m^{3} of concrete HP-100 | 166.56 |

Uncertainty of E (%) | 10 | 20 | 30 |
---|---|---|---|

μ Cost discrepancy | 1.21% | 1.28% | 1.07% |

σ Displacement discrepancy | 4.63% | 4.75% | 4.03% |

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

b (mm) | h (mm) | d (mm) | e_{v} (mm) | e_{s} (mm) | e_{a} (mm) | e_{i} (mm) | f_{ck} (MPa) | c (mm) | μ_{cost} (€) | σ_{v,displacement} (mm) | |
---|---|---|---|---|---|---|---|---|---|---|---|

S10 | 1200 | 1450 | 0 | 150 | 150 | 350 | 225 | 45 | 225 | 167,370.9 | 3.811 |

S20 | 1200 | 1800 | 125 | 150 | 150 | 350 | 250 | 60 | 250 | 192,570.6 | 3.778 |

S30 | 1200 | 1950 | 0 | 150 | 150 | 350 | 225 | 80 | 225 | 208,111.9 | 3.548 |

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

b (mm) | h (mm) | d (mm) | e_{v} (mm) | e_{s} (mm) | e_{a} (mm) | e_{i} (mm) | f_{ck} (MPa) | c (mm) | μ_{cost} (€) | σ_{v,displacement} (mm) | |
---|---|---|---|---|---|---|---|---|---|---|---|

A | 1200 | 1800 | 125 | 150 | 150 | 350 | 250 | 60 | 250 | 192,570.6 | 3.778 |

B | 1200 | 1900 | 50 | 150 | 150 | 350 | 150 | 80 | 150 | 201,479.9 | 2.794 |

C | 1800 | 2000 | 200 | 150 | 150 | 350 | 175 | 100 | 220 | 269,128.5 | 1.684 |

Uncertainty of Overload (%) | 10 | 20 | 30 | 40 |
---|---|---|---|---|

μ Cost discrepancy | 1.32% | 1.19% | 1.17% | 1.28% |

σ Displacement discrepancy | 38.61% | 15.78% | 11.53% | 15.18% |

Uncertainty of P0 (%) | 10 | 20 | 30 | 40 |
---|---|---|---|---|

μ Cost discrepancy | 1.34% | 1.09% | 1.06% | 1.21% |

σ Displacement discrepancy | 13.5% | 7.16% | 3.47% | 4% |

b (mm) | h (mm) | d (mm) | e_{v} (mm) | e_{s} (mm) | e_{a} (mm) | e_{i} (mm) | f_{ck} (MPa) | c (mm) | μ_{cost} (€) | σ_{v,displacement} (mm) | |
---|---|---|---|---|---|---|---|---|---|---|---|

S20 | 1200 | 1250 | 0 | 150 | 150 | 350 | 200 | 60 | 200 | 164,594.2 | 2.924 |

S30 | 1200 | 1250 | 200 | 150 | 150 | 350 | 175 | 70 | 175 | 174,467.1 | 2.991 |

S40 | 1200 | 1700 | 25 | 175 | 175 | 350 | 250 | 50 | 250 | 184,821.6 | 2.917 |

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

b (mm) | h (mm) | d (mm) | e_{v} (mm) | e_{s} (mm) | e_{a} (mm) | e_{i} (mm) | f_{ck} (MPa) | c (mm) | μ_{cost} (€) | σ_{v,displacement} (mm) | |
---|---|---|---|---|---|---|---|---|---|---|---|

A | 1200 | 1350 | 100 | 150 | 150 | 350 | 175 | 80 | 175 | 180,240.5 | 1.913 |

B | 1200 | 1850 | 200 | 175 | 175 | 350 | 225 | 60 | 225 | 198,687.3 | 0.971 |

C | 1600 | 1800 | 150 | 275 | 275 | 350 | 225 | 70 | 225 | 238,573.8 | 0.753 |

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

b (mm) | h (mm) | d (mm) | e_{v} (mm) | e_{s} (mm) | e_{a} (mm) | e_{i} (mm) | f_{ck} (MPa) | c (mm) | μ_{cost} (€) | σ_{v,displacement} (mm) | |
---|---|---|---|---|---|---|---|---|---|---|---|

S20 | 1200 | 1350 | 0 | 150 | 150 | 350 | 200 | 60 | 200 | 168,833.9 | 11.058 |

S30 | 1200 | 1400 | 200 | 150 | 150 | 350 | 150 | 80 | 150 | 181,276.4 | 9.552 |

S40 | 1200 | 1750 | 125 | 150 | 150 | 350 | 200 | 55 | 200 | 186,380.7 | 10.497 |

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

b (mm) | h (mm) | d (mm) | e_{v} (mm) | e_{s} (mm) | e_{a} (mm) | e_{i} (mm) | f_{ck} (MPa) | c (mm) | μ_{cost} (€) | σ_{v,displacement} (mm) | |
---|---|---|---|---|---|---|---|---|---|---|---|

A | 1200 | 1350 | 0 | 150 | 150 | 350 | 200 | 60 | 200 | 168,833.9 | 11.058 |

B | 1200 | 1650 | 0 | 150 | 150 | 350 | 175 | 80 | 175 | 190,734.7 | 5.510 |

C | 1300 | 2000 | 0 | 225 | 300 | 350 | 275 | 80 | 275 | 231,832.0 | 3.772 |

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**MDPI and ACS Style**

Penadés-Plà, V.; García-Segura, T.; Yepes, V. Robust Design Optimization for Low-Cost Concrete Box-Girder Bridge. *Mathematics* **2020**, *8*, 398.
https://doi.org/10.3390/math8030398

**AMA Style**

Penadés-Plà V, García-Segura T, Yepes V. Robust Design Optimization for Low-Cost Concrete Box-Girder Bridge. *Mathematics*. 2020; 8(3):398.
https://doi.org/10.3390/math8030398

**Chicago/Turabian Style**

Penadés-Plà, Vicent, Tatiana García-Segura, and Víctor Yepes. 2020. "Robust Design Optimization for Low-Cost Concrete Box-Girder Bridge" *Mathematics* 8, no. 3: 398.
https://doi.org/10.3390/math8030398