# Artificial Neural Network and Kriging Surrogate Model for Embodied Energy Optimization of Prestressed Slab Bridges

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Lightened Slab Bridge Deck Description

^{3}/m

^{2}and an external lighting of 0.51 m

^{3}/m

^{2}.

## 3. Methodology

_{i}is the observed values, and n is the number of observations.

#### 3.1. Kriging Metamodel

#### 3.2. Artificial Neural Network

_{i}, are multiplied by the weighting coefficients, w

_{i}

_{,j}, and then combined linearly with an independent bias term, b

_{j}. The equation governing the behavior of each hidden neuron may be expressed as ∑x

_{i}· w

_{i}

_{,j}+ b

_{j}. Subsequently, each neuron in the hidden layer generates an output by employing a sigmoid tangent function to the linear combination. The output layer employs a linear function.

## 4. Results and Discussion

#### 4.1. Visualization of Observed Data

#### 4.2. Comparison of Predictive Models

#### 4.3. Error Analysis

#### 4.4. Practical Recommendations

^{3}/m

^{2}and 0.60 m

^{3}/m

^{2}for the deck, and a passive reinforcement content of between 100 kg/m

^{3}and 130 kg/m

^{3}, with active reinforcement around 17 kg/m

^{2}of the deck. The characteristic compressive strength of concrete should be at least 40 MPa, the internal lighting should not exceed 0.18 m

^{3}/m

^{2}of the deck area, and the external lighting should be between 0.45 m

^{3}/m

^{2}and 0.55 m

^{3}/m

^{2}of the deck area.

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- IEA; UNEP. Global Status Report: Towards a Zero-Emission, Efficient and Resilient Buildings and Construction Sector; International Energy Agency and the United Nations Environment Programme: Paris, France, 2018. [Google Scholar]
- Wang, T.; Lee, I.S.; Kendall, A.; Harvey, J.; Lee, E.B.; Kim, C. Life cycle energy consumption and GHG emission from pavement rehabilitation with different rolling resistance. J. Clean Prod.
**2012**, 33, 86–96. [Google Scholar] [CrossRef] - Wang, E.; Shen, Z. A hybrid Data Quality Indicator and statistical method for improving uncertainty analysis in LCA of complex system—Application to the whole-building embodied energy analysis. J. Clean Prod.
**2013**, 43, 166–173. [Google Scholar] [CrossRef] - Halder, A.; Batra, S. Application of Predictive Analytics in Built Environment Research: A Comprehensive Bibliometric Study to Explore Knowledge Domains and Future Research Agenda. Arch. Computat. Methods Eng.
**2023**, 30, 4299–4324. [Google Scholar] [CrossRef] - Opoku, D.G.J.; Agyekum, K.; Ayarkwa, J. Drivers of environmental sustainability of construction projects: A thematic analysis of verbatim comments from built environment consultants. Int. J. Constr. Manag.
**2022**, 22, 1033–1041. [Google Scholar] [CrossRef] - Leonhardt, F. Prestressed Concrete: Design and Construction; Ernst & Sohn: Berlin, Germany, 1982. [Google Scholar]
- Warner, R.F.; Rangan, B.V.; Hall, A.S.; Faulkes, K.A. Concrete Structures; Longman: North Lakes, QLD, Australia, 1998. [Google Scholar]
- Yeo, D.; Gabbai, R.D. Sustainable design of reinforced concrete structures through embodied energy optimization. Energ. Buildings
**2011**, 43, 2028–2033. [Google Scholar] [CrossRef] - Quaglia, C.P.; Yu, N.; Thrall, A.P.; Paolucci, S. Balancing energy efficiency and structural performance through multi-objective shape optimization: Case study of a rapidly deployable origami-inspired shelter. Energ. Buildings
**2014**, 82, 733–745. [Google Scholar] [CrossRef] - Cabeza, L.F.; Boquera, L.; Chàfer, M.; Vérez, D. Embodied energy and embodied carbon of structural building materials: Worldwide progress and barriers through literature map analysis. Energy Build
**2021**, 231, 110612. [Google Scholar] [CrossRef] - Miller, D.; Doh, J.H.; Mulvey, M. Concrete slab comparison and embodied energy optimisation for alternate design and construction techniques. Constr. Build Mater.
**2015**, 80, 329–338. [Google Scholar] [CrossRef] - Foraboschi, P.; Mercanzin, M.; Trabucco, D. Sustainable structural design of tall buildings based on embodied energy. Energ. Buildings
**2014**, 68, 254–269. [Google Scholar] [CrossRef] - Alcalá, J.; González-Vidosa, F.; Yepes, V.; Martí, J.V. Embodied energy optimization of prestressed concrete slab bridge decks. Technologies
**2018**, 6, 43. [Google Scholar] [CrossRef] - Martí, J.V.; García-Segura, T.; Yepes, V. Structural design of prescast-prestressed concrete U-beam road bridges based on embodied energy. J. Clean. Prod.
**2016**, 120, 231–240. [Google Scholar] [CrossRef] - Minunno, R.; O’Grady, T.; Morrison, G.M.; Gruner, R.L. Investigating the embodied energy and carbon of buildings: A systematic literature review and meta-analysis of life cycle assessments. Renew. Sustain. Energy Rev.
**2021**, 143, 110935. [Google Scholar] [CrossRef] - Penadés-Plà, V.; García-Segura, T.; Yepes, V. Accelerated optimization method for low-embodied energy concrete box-girder bridge design. Eng. Struct
**2019**, 179, 556–565. [Google Scholar] [CrossRef] - Cressie, N. The origins of Kriging. Math. Geol.
**1990**, 22, 239–252. [Google Scholar] [CrossRef] - Martínez-Frutos, J.; Martí, P. Diseño óptimo robusto utilizando modelos Kriging: Aplicación al diseño óptimo robusto de estructuras articuladas. Rev. Int. Métodos Numér. Cálc. Diseño Ing.
**2014**, 30, 97–105. [Google Scholar] [CrossRef] - YiFei, L.; MaoSen, C.; Hoa, T.N.; Khatir, S.; Minh, H.L.; SangTo, T.; Cuong-Le, T.; Wahab, M.A. Metamodel-assisted hybrid optimization strategy for model updating using vibration response data. Adv. Eng. Softw.
**2023**, 185, 103515. [Google Scholar] [CrossRef] - Sánchez-Zabala, V.F.; Gómez-Acebo, T. Building energy performance metamodels for district energy management optimisation platforms. Energy Convers. Manag. X
**2024**, 21, 100512. [Google Scholar] [CrossRef] - Yepes-Bellver, L.; Brun-Izquierdo, A.; Alcalá, J.; Yepes, V. CO
_{2}-optimization of post-tensioned concrete slab-bridge decks using surrogate modeling. Materials**2022**, 15, 4776. [Google Scholar] [CrossRef] - Yepes-Bellver, L.; Brun-Izquierdo, A.; Alcalá, J.; Yepes, V. Embodied energy optimization of prestressed concrete road flyovers by a two-phase Kriging surrogate model. Materials
**2023**, 16, 6767. [Google Scholar] [CrossRef] - Zhang, Y.; Wu, G. Seismic vulnerability analysis of RC bridges based on Kriging model. J. Earthq. Eng.
**2019**, 23, 242–260. [Google Scholar] [CrossRef] - Wu, J.; Cheng, F.; Zou, C.; Zhang, R.; Li, C.; Huang, S.; Zhou, Y. Swarm intelligent optimization conjunction with Kriging model for bridge structure finite element model updating. Buildings
**2022**, 12, 504. [Google Scholar] [CrossRef] - Cheng, A.; Low, Y.M. A new metamodel for predicting the nonlinear time-domain response of offshore structures subjected to stochastic wave current and wind loads. Comput. Struct.
**2024**, 297, 107340. [Google Scholar] [CrossRef] - Martí-Vargas, J.R.; Ferri, F.J.; Yepes, V. Prediction of the transfer length of prestressing strands with neural networks. Comput. Concr.
**2013**, 12, 187–209. [Google Scholar] [CrossRef] - Hong, W.K.; Nguyen, M.C.; Pham, T.D. Pre-tensioned concrete beams optimized with a unified function of objective (UFO) using ANN-based Hong-Lagrange method. J. Asian Archit. Build Eng.
**2023**, 23, 1573–1595. [Google Scholar] [CrossRef] - Lophaven, N.S.; Nielsen, H.B.; Sondergaard, J. MATLAB Kriging Toolbox DACE (Design and Analysis of Computer Experiments) Version 2.0. 2002. Available online: http://www2.imm.dtu.dk/pubdb/p.php?1460 (accessed on 11 July 2024).
- McKay, M.D.; Beckman, R.J.; Conover, W.J. A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code. Technometrics
**1979**, 21, 239–245. [Google Scholar] [CrossRef] - LeCun, Y.; Bengio, Y.; Hinton, G. Deep Learning. Nature
**2015**, 521, 436–444. [Google Scholar] [CrossRef] - Zhang, G.; Patuwo, B.E.; Hu, M.Y. Forecasting with artificial neural networks: The state of the art. Int. J. Forecast
**1998**, 14, 35–62. [Google Scholar] [CrossRef] - Hornik, K.; Stinchcombe, M.; White, H. Multilayer feedforward networks are universal approximator. Neural Netw.
**1989**, 2, 359–366. [Google Scholar] [CrossRef] - Rumelhart, D.E.; McClelland, J.L.; PDP Research Group. Parallel Distributed Processing: Explorations in the Microstructure of Cognition; MIT Press: Cambridge, UK, 1986; Volume 1. [Google Scholar] [CrossRef]
- Rumelhart, D.E.; Hinton, G.E.; Williams, R.J. Learning Representations by Back-Propagating Errors. Nature
**1986**, 323, 533–536. [Google Scholar] [CrossRef] - Dirección General de Carreteras. Obras de paso de Nueva Construcción: Conceptos Generales; Ministerio de Fomento, Centro de Publicaciones: Madrid, Spain, 2000. (In Spanish)
- SETRA. Ponts-Dalles. Guide de Conception; Ministère de l’Equipement, du Logement des Transports et de la Mer: Bagneux, France, 1989. (In French)

**Figure 3.**Aerial image of the overpass at kilometer 441 of the A-7 highway in Cocentaina (Alicante). Image: Google Maps.

**Figure 8.**Response surface of the 42 observed slab bridge deck data (Table 2).

**Figure 9.**Contour plot of the 42 observed slab data (Table 2).

**Figure 10.**ANN energy cost prediction as a function of deck depth, considering a 3.70 m base width and 40 MPa concrete grade.

**Figure 11.**ANN energy cost prediction as a function of deck depth width, considering a 1.20 m depth and 40 MPa concrete grade.

**Figure 12.**ANN energy cost prediction as a function of the concrete grade, considering a 1.20 m depth and 3.35 m base width.

**Table 1.**Energy cost of the deck [16].

Material | kWh/kg | kWh/m^{3} | kWh/m^{2} |
---|---|---|---|

Y-1860-S7 steel | 5.64 | ||

B-500-St steel | 3.03 | ||

Lighting | 604.42 | ||

Slab formwork | 2.24 | ||

C-30 concrete | 227.01 | ||

C-35 concrete | 263.96 | ||

C-40 concrete | 298.57 | ||

C-45 concrete | 330.25 | ||

C-50 concrete | 358.97 | ||

Lighting | 604.42 | ||

Slab formwork | 2.24 |

**Table 2.**Values of design variables obtained within the specified ranges [22].

Deck | Deck Depth (m) | Base Width (m) | Concrete Grade (MPa) | Energy Cost (MW·h) |
---|---|---|---|---|

1 | 1.65 | 3.65 | 35 | 1149.88 |

2 | 1.70 | 3.80 | 45 | 1182.89 |

3 | 1.20 | 3.85 | 40 | 1065.87 |

4 | 1.55 | 3.60 | 45 | 1140.79 |

5 | 1.20 | 4.85 | 50 | 1170.72 |

6 | 1.15 | 4.50 | 50 | 1199.59 |

7 | 1.35 | 3.95 | 30 | 1103.18 |

8 | 1.30 | 4.45 | 30 | 1180.31 |

9 | 1.35 | 4.25 | 45 | 1132.71 |

10 | 1.50 | 4.55 | 30 | 1138.00 |

11 | 1.60 | 4.20 | 40 | 1267.85 |

12 | 1.25 | 4.70 | 40 | 1191.65 |

13 | 1.50 | 4.05 | 45 | 1183.17 |

14 | 1.45 | 4.35 | 35 | 1119.17 |

15 | 1.65 | 3.45 | 45 | 1145.07 |

16 | 1.55 | 4.10 | 35 | 1162.92 |

17 | 1.25 | 3.50 | 45 | 1073.75 |

18 | 1.40 | 3.30 | 40 | 1152.33 |

19 | 1.45 | 3.90 | 45 | 1145.21 |

20 | 1.35 | 3.60 | 35 | 1094.86 |

21 | 1.50 | 3.35 | 45 | 1134.93 |

22 | 1.50 | 4.50 | 45 | 1189.53 |

23 | 1.55 | 3.20 | 30 | 1103.41 |

24 | 1.25 | 3.00 | 50 | 1101.04 |

25 | 1.40 | 3.45 | 45 | 1201.73 |

26 | 1.50 | 3.55 | 35 | 1105.44 |

27 | 1.70 | 3.85 | 45 | 1165.47 |

28 | 1.20 | 3.60 | 40 | 1083.41 |

29 | 1.30 | 4.90 | 40 | 1215.82 |

30 | 1.45 | 4.75 | 35 | 1163.59 |

31 | 1.20 | 3.40 | 40 | 1059.87 |

32 | 1.15 | 3.90 | 35 | 1129.22 |

33 | 1.05 | 3.50 | 35 | 1237.89 |

34 | 1.10 | 3.80 | 45 | 1178.72 |

35 | 1.15 | 3.35 | 45 | 1074.77 |

36 | 1.25 | 3.60 | 45 | 1078.71 |

37 | 1.10 | 3.45 | 40 | 1124.21 |

38 | 1.20 | 3.35 | 45 | 1065.44 |

39 | 1.25 | 3.40 | 45 | 1084.92 |

40 | 1.15 | 3.60 | 45 | 1104.77 |

41 | 1.15 | 3.35 | 40 | 1051.00 |

42 | 1.15 | 3.70 | 40 | 1038.28 |

**Table 3.**Observed value and prediction for local optima in diversification (#41) and intensification phase (#42), as well as their absolute and relative errors.

#41 | #42 | Absolute Error #41 | Relative Error #41 | Absolute Error #42 | Relative Error #42 | |
---|---|---|---|---|---|---|

Observed | 1051.00 | 1038.28 | 0.00 | 0.00% | 0.00 | 0.00% |

Kriging 1 | 1130.68 | 1091.95 | 79.68 | 7.58% | 53.67 | 5.17% |

Kriging 2 | 1073.98 | 1085.84 | 22.98 | 2.19% | 47.56 | 4.58% |

Kriging 3 | 1060.58 | 1079.81 | 9.58 | 0.91% | 41.53 | 4.00% |

ANN average | 1073.06 | 1091.85 | 22.06 | 2.10% | 53.57 | 5.16% |

Predictive Models | MSE | RMSE |
---|---|---|

Kriging 1 | 2212.98 | 47.04 |

Kriging 2 | 3923.49 | 62.64 |

Kriging 3 | 4976.80 | 70.55 |

ANN average | 1037.22 | 30.95 |

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**MDPI and ACS Style**

Yepes-Bellver, L.; Brun-Izquierdo, A.; Alcalá, J.; Yepes, V.
Artificial Neural Network and Kriging Surrogate Model for Embodied Energy Optimization of Prestressed Slab Bridges. *Sustainability* **2024**, *16*, 8450.
https://doi.org/10.3390/su16198450

**AMA Style**

Yepes-Bellver L, Brun-Izquierdo A, Alcalá J, Yepes V.
Artificial Neural Network and Kriging Surrogate Model for Embodied Energy Optimization of Prestressed Slab Bridges. *Sustainability*. 2024; 16(19):8450.
https://doi.org/10.3390/su16198450

**Chicago/Turabian Style**

Yepes-Bellver, Lorena, Alejandro Brun-Izquierdo, Julián Alcalá, and Víctor Yepes.
2024. "Artificial Neural Network and Kriging Surrogate Model for Embodied Energy Optimization of Prestressed Slab Bridges" *Sustainability* 16, no. 19: 8450.
https://doi.org/10.3390/su16198450