# A Parametric Study of Optimum Road Modular Hinged Frames by Hybrid Metaheuristics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Optimization Problem

_{2}emissions $AE\left(\overrightarrow{x}\right)$ and embodied energy $EE\left(\overrightarrow{x}\right)$ were evaluated through Equations (2) and (3). Similar to the objective function, emissions are obtained as a factor of each material’s quantity ${m}_{i}\left(\overrightarrow{x}\right)$, the unitary emissions $a{e}_{i}$ and unitary embodied energy $e{e}_{i}$, respectively. The designs must comply with a series of constraints $R\left(\overrightarrow{x}\right)$ detailed in Section 2.3. The compliance of the optimal designs with the constraints is generally expressed through Equation (4).

_{2}emissions $a{e}_{i}$ and embodied energy $e{e}_{i}$ for each of the materials can be consulted in Table 1. These values were obtained from the Construction Technology Institute of Catalonia by the BEDEC database [29].

#### 2.1. Variables

#### 2.2. Parameters

#### 2.3. Constrains

## 3. Proposed Hybrid Metaheuristic Strategies

#### 3.1. Hybrid Simulated Annealing

#### 3.2. Hybrid Threshold Accepting

#### 3.3. Hybrid Old Bachelor’s Acceptance

#### 3.4. Design of Experiments

## 4. Results of the Parametric Study and Regression Analysis

#### 4.1. Final Cost Analysis

#### 4.2. Sustainability Analysis

_{2}emissions and embodied energy of the optimum RMHF. The regression analysis allowed identifying similar characteristics to those mentioned for the final cost in section ref. In both cases, there is a quadratic relationship with the span. In addition to a linear relation to the earth cover depth.

_{2}emissions and embodied energy as impact measuring tools associated with the optimal RMHF obtained. However, these were not considered objective functions. Thus, the particularized study of the characteristics of optimal frames as a function of such variables is beyond the scope of the study. Figure 5 and Figure 6 show the results obtained in the regression analysis. With ${R}^{2}$ correlation coefficients close to one, the expressions form rough impact measuring tools for the design of RMHFs.

#### 4.3. Geometrical Characteristics Analysis

#### 4.4. Materials Analysis

_{2}emitted into the atmosphere.

_{2}emissions.

## 5. Conclusions

- The hybridization of local search-based algorithms with GA mutation operators gives rise to hybrid metaheuristics. These techniques are applicable in automating the optimal design of precast structures. The SAMO presents the best performance in solving the problem posed. The calibrated method has Markov chain lengths of 10,000 iterations, a cooling coefficient of 0.8 and a stopping criterion of 5 chains without improvement. In addition, the mutation operator affects one variable with a standard deviation of 0.1.
- The cost and environmental impact meters present an excellent fitting quadratic relationship when studied as a function of the horizontal span. This relationship is linear when considered a function of the earth cover depth. The expressions obtained are representative and form a valuable tool for the approximate calculation of the final cost, the associated CO
_{2}emissions and the embodied energy of RMHFs. - The RMHF design depends on a large number of variables. The study of each particular variable lands out of the scope of the present work. However, optimal structures present reduced depths with dense reinforcement designs. This density increases with both span and earth cover depth. In addition, the mid-span upper slab reinforcement area shows quadratic and linear relationships with the span and burial depth, respectively.
- Previous designs do not condition the structures conceived using the proposed methodology. Thus, any configuration that verifies the requirements is considered a feasible solution. In this context, the 8 and 10 m span RMHF buried one meter deep presented specific characteristics that differ from the general. With considerable reductions in shear reinforcement, these structures have upper slabs with greater depth and mid-span reinforcement areas.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SA | Simulated Annealing |

TA | Threshold Accepting |

OBA | Old Bachelor’s Acceptance |

MO | Mutation Operator |

GA | Genetic Algorithm |

RMHF | Road Modular Hinged Frame |

CPRF | Cast in Place Road Frame |

DoE | Design of Experiments |

SAMO | Simulated Annealing with Mutation Operator |

TAMO | Threshold Accepting with Mutation Operator |

OBMO | Old Bachelor’s Acceptance with Mutation Operator |

ULS | Ultimate Limit State |

SLS | Service Limit State |

MCL | Markov Chain Length |

CC | Cooling Coefficient |

SC | Stopping Criterion |

VN | Variable Number |

SD | Standard Deviation |

CL | Chain Length |

RC | Reduction Coefficient |

TC | Termination Criterion |

IC | Increase Coefficient |

DC | Decrease Coefficient |

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**Figure 5.**Associated CO

_{2}emissions analysis as a function of: (

**a**) horizontal span; (

**b**) earth cover.

Unit | Material | Unit Cost (EUR) | CO_{2} Emissions (kg) | Energy (kWh) |
---|---|---|---|---|

m${}^{3}$ | C25/30 Concrete | 88.86 | 256.66 | 402.44 |

m${}^{3}$ | C30/37 Concrete | 97.80 | 277.72 | 428.29 |

m${}^{3}$ | C35/45 Concrete | 101.83 | 278.04 | 429.95 |

m${}^{3}$ | C40/50 Concrete | 104.83 | 278.04 | 429.95 |

kg | B 400 S | 1.40 | 0.70 | 3.38 |

kg | B 500 S | 1.42 | 0.70 | 3.38 |

Geometrical Variables | Num. Values | Range Values | ||
---|---|---|---|---|

Upper slab depth | (m) | ${D}_{US}$ | 46 | 0.30 to 1.20 |

Lower slab depth | (m) | ${D}_{LS}$ | 41 | 0.40 to 1.20 |

Lateral walls depth | (m) | ${D}_{LW}$ | 46 | 0.30 to 1.20 |

Materials variables | ||||

Concrete grade | (MPa) | C | 4 | 25, 30, 35 or 40 |

Steel grade | (MPa) | S | 2 | 400 or 500 |

Passive reinforcement variables | ||||

Flexural reinforcement ${R}_{1}$ | (mm) | ${\varphi}_{{R}_{1}}$ | 6 | 10 to 32 |

(bars) | ${n}_{{R}_{1}}$ | 9 | 4 to 12 | |

Flexural reinforcement ${R}_{2}$ | (mm) | ${\varphi}_{{R}_{2}}$ | 6 | 10 to 32 |

(bars) | ${n}_{{R}_{2}}$ | 9 | 4 to 12 | |

Flexural reinforcement ${R}_{3}$ | (mm) | ${\varphi}_{{R}_{3}}$ | 6 | 10 to 32 |

(bars) | ${n}_{{R}_{3}}$ | 9 | 4 to 12 | |

Flexural reinforcement ${R}_{4}$ | (mm) | ${\varphi}_{{R}_{4}}$ | 6 | 10 to 32 |

(bars) | ${n}_{{R}_{4}}$ | 9 | 4 to 12 | |

Flexural reinforcement ${R}_{5}$ | (mm) | ${\varphi}_{{R}_{5}}$ | 6 | 10 to 32 |

(bars) | ${n}_{{R}_{5}}$ | 9 | 4 to 12 | |

Flexural reinforcement ${R}_{6}$ | (mm) | ${\varphi}_{{R}_{6}}$ | 6 | 10 to 32 |

(bars) | ${n}_{{R}_{6}}$ | 10 | 3 to 12 | |

Flexural reinforcement ${R}_{7}$ | (mm) | ${\varphi}_{{R}_{7}}$ | 6 | 10 to 32 |

(bars) | ${n}_{{R}_{7}}$ | 9 | 4 to 12 | |

Flexural reinforcement ${R}_{8}$ | (mm) | ${\varphi}_{{R}_{8}}$ | 6 | 10 to 32 |

(bars) | ${n}_{{R}_{8}}$ | 9 | 4 to 12 | |

Flexural reinforcement ${R}_{9}$ | (mm) | ${\varphi}_{{R}_{9}}$ | 6 | 10 to 32 |

(bars) | ${n}_{{R}_{9}}$ | 9 | 4 to 12 | |

(m) | ${L}_{{R}_{9}}$ | 151 to 451 | 3 to 0.75 $\xb7L$ | |

Corner reinforcement $C{R}_{1}$ | (mm) | ${\varphi}_{C{R}_{1}}$ | 6 | 10 to 32 |

(bars) | ${n}_{C{R}_{1}}$ | 10 | 3 to 12 | |

(m) | ${H}_{C{R}_{1}}$ | 76 to 226 | 1.5 to 0.375 $\xb7L$ | |

(m) | ${V}_{C{R}_{1}}$ | 76 | 1 to 2.5 | |

Corner reinforcement $C{R}_{2}$ | (mm) | ${\varphi}_{C{R}_{2}}$ | 6 | 10 to 32 |

(bars) | ${n}_{C{R}_{2}}$ | 10 | 3 to 12 | |

(m) | ${H}_{C{R}_{2}}$ | 101 to 251 | 1 to 0.375 $\xb7L$ | |

(m) | ${V}_{C{R}_{2}}$ | 26 | 1 to 1.5 | |

Shear reinforcement $S{R}_{1}$ | (mm) | ${\varphi}_{S{R}_{1}}$ | 7 | 8 to 32 |

(m) | ${s}_{S{R}_{1}}$ | 7 | 0.10 to 0.40 | |

(m) | $L{D}_{S{R}_{1}}$ | 76 to 226 | 1.5 to 0.375 $\xb7L$ | |

Shear reinforcement $S{R}_{2}$ | (mm) | ${\varphi}_{S{R}_{2}}$ | 7 | 8 to 32 |

(m) | ${s}_{S{R}_{2}}$ | 7 | 0.10 to 0.40 | |

(m) | $L{D}_{S{R}_{2}}$ | 101 to 251 | 1 to 0.375 $\xb7L$ |

Geometrical Parameters | ||
---|---|---|

Free height (m) | H | 5 |

Horizontal span (m) | L | 8 to 16 |

Hinge height (m) | $HH$ | (3/5) · H |

Earth cover (m) | $HE$ | 1 to 5 |

Loading parameters | ||

Earth specific weight (kN/m${}^{3}$) | ${\gamma}_{E}$ | 20 |

Reinforced concrete specific weight (kN/m${}^{3}$) | ${\gamma}_{C}$ | 25 |

Earth internal friction angle (${}^{\circ}$) | $IF$ | 30 |

Active earth pressure coefficient | ${K}_{A}$ | 0.33 |

Resting earth pressure coefficient | ${K}_{R}$ | 0.50 |

Heavy traffic vehicle load (kN/m${}^{3}$) | $TL$ | 150 |

Heavy traffic vehicle load lenght (m) | $TLL$ | 1.20 |

Uniform overload (kN/m${}^{3}$) | $UO$ | 10 |

Ballast coefficient (MN/m${}^{3}$) | ${B}_{E}$ | 10 |

Economic and sustainability parameters | ||

Unit costs (EUR) | ${c}_{i}$ | Table 1 |

Unit CO_{2} emissions (CO_{2} kg) | $a{e}_{i}$ | Table 1 |

Unit embodied energy (kWh) | $e{e}_{i}$ | Table 1 |

Exposure related parameters | ||

Exposure class | XC2 | |

Legislative related parameters | ||

Standard regulations | CEN [30,31]/MFOM [32] | |

Code considerations | MFOM [33] |

SAMO | P1 | P2 | P3 | P4 | P5 |
---|---|---|---|---|---|

Parameter | MCL | SD | VN | CC | SC |

Lower bound (−) | 1000 | 0.1 | 1 | 0.8 | 1 |

Upper bound (+) | 5000 | 0.3 | 5 | 0.9 | 5 |

TAMO | P1 | P2 | P3 | P4 | P5 |

Parameter | CL | SD | VN | RC | SC |

Lower bound (−) | 1000 | 0.1 | 1 | 0.8 | 1 |

Upper bound (+) | 5000 | 0.3 | 5 | 0.9 | 5 |

OBAMO | P1 | P2 | P3 | P4 | P5 |

Parameter | TC | SD | VN | IC | DC |

Lower bound (−) | 10,000 | 0.1 | 1 | 1 | 1 |

Upper bound (+) | 50,000 | 0.3 | 5 | 5 | 5 |

SAMO | TAMO | OBAMO | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

P1 | P2 | P3 | P4 | P5 | Cost (EUR) | Iter. | % Min. | Cost (EUR) | Iter. | % Min. | Cost (EUR) | Iter. | % Min. |

− | − | − | − | + | 3722.99 | 11,472 | 0.0067 | 4484.78 | 7790 | 0.2127 | 4266.87 | 5621 | 0.1538 |

+ | − | − | − | − | 3700.97 | 56,759 | 0.0008 | 4206.81 | 28,405 | 0.1376 | 3935.18 | 15,901 | 0.0641 |

− | + | − | − | − | 3809.90 | 10,551 | 0.0302 | 4005.26 | 6823 | 0.0831 | 3939.44 | 6688 | 0.0653 |

+ | + | − | − | + | 3723.28 | 44,999 | 0.0068 | 3933.48 | 19,124 | 0.0636 | 4108.21 | 30,886 | 0.1109 |

− | − | + | − | − | 3893.24 | 13,428 | 0.0528 | 3844.05 | 14,008 | 0.0395 | 4131.20 | 6607 | 0.1171 |

+ | − | + | − | + | 3764.81 | 75,442 | 0.0180 | 3764.61 | 64,237 | 0.0180 | 4420.60 | 31,150 | 0.1954 |

− | + | + | − | + | 3854.21 | 14,101 | 0.0422 | 3917.74 | 12,470 | 0.0594 | 4470.99 | 3043 | 0.2090 |

+ | + | + | − | − | 3787.60 | 76,644 | 0.0242 | 3764.19 | 62,072 | 0.0179 | 3978.78 | 23,037 | 0.0759 |

− | − | − | + | − | 3759.89 | 32,726 | 0.0167 | 3900.58 | 20,817 | 0.0548 | 4233.84 | 5175 | 0.1449 |

+ | − | − | + | + | 3735.80 | 137,994 | 0.0102 | 3990.50 | 61,719 | 0.0791 | 3808.35 | 30,193 | 0.0298 |

− | + | − | + | + | 3745.25 | 38,517 | 0.0127 | 3953.74 | 22,026 | 0.0691 | 3878.68 | 7326 | 0.0488 |

+ | + | − | + | − | 3717.33 | 135,874 | 0.0052 | 4376.86 | 41,030 | 0.1835 | 4103.01 | 28,609 | 0.1095 |

− | − | + | + | + | 3810.35 | 53,194 | 0.0304 | 3759.25 | 53,703 | 0.0165 | 4135.06 | 8395 | 0.1182 |

+ | − | + | + | − | 3740.23 | 254,796 | 0.0114 | 3711.42 | 227,138 | 0.0036 | 3788.71 | 30,451 | 0.0245 |

− | + | + | + | − | 3808.14 | 52,211 | 0.0298 | 3755.12 | 52,050 | 0.0154 | 4188.70 | 4486 | 0.1327 |

+ | + | + | + | + | 3730.33 | 266,381 | 0.0087 | 3698.10 | 258,351 | 0.0000 | 3838.12 | 43,960 | 0.0379 |

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**MDPI and ACS Style**

Ruiz-Vélez, A.; Alcalá, J.; Yepes, V.
A Parametric Study of Optimum Road Modular Hinged Frames by Hybrid Metaheuristics. *Materials* **2023**, *16*, 931.
https://doi.org/10.3390/ma16030931

**AMA Style**

Ruiz-Vélez A, Alcalá J, Yepes V.
A Parametric Study of Optimum Road Modular Hinged Frames by Hybrid Metaheuristics. *Materials*. 2023; 16(3):931.
https://doi.org/10.3390/ma16030931

**Chicago/Turabian Style**

Ruiz-Vélez, Andrés, Julián Alcalá, and Víctor Yepes.
2023. "A Parametric Study of Optimum Road Modular Hinged Frames by Hybrid Metaheuristics" *Materials* 16, no. 3: 931.
https://doi.org/10.3390/ma16030931