# Optimal Design of Sustainable Reinforced Concrete Precast Hinged Frames

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emitted yearly [4]. Concrete is currently the most widely used material in construction. Therefore, a significant portion of the emissions is derived from its use [5,6,7].

_{2}emissions [11,12,13]. However, the real-world application of these ways of improvement is limited by the need to develop comprehensive experimental studies that establish conditions of use and appropriate practical recommendations [14]. Thus, the experimental nature of these studies means that the implementation period of these solutions is considerably affected.

_{2}emissions or the energy consumed in the construction of the designed structure [3]. These techniques have been highly improved during the last decade, mainly due to the simplicity of the algorithms and their high adaptability, in addition to the ability to avoid convergence to a local optimum. Due to the importance of improving the design process and the positive characteristics of the aforementioned methodologies, several studies have applied heuristic and metaheuristic optimization to design concrete structures, such as prestressed bridges [21], retaining walls [20,22,23], bridge piers [24] and building structures [25,26]. The results establish a direct relationship between the final cost of the structure and the CO

_{2}emissions associated with its construction. The optimization of precast girder bridges carried out by Yepes et al. [27] concludes that a reduction of EUR 1 in the final cost of the bridge equals the avoidance of emitting 1.74 kg of CO

_{2}.

## 2. Optimization Problem and Computational Model Definition

_{2}emissions associated with each design are evaluated by means of Equation (2). In this equation, similarly to the final cost, total CO

_{2}emissions are calculated as the product of the emissions associated with each material ${e}_{i}$ multiplied by the quantity used of each of those materials ${m}_{i}$. The values considered both in the evaluation of the objective function and the calculation of the associated CO

_{2}emissions are summarized in Table 1. These were obtained from the Construction Technology Institute of Catalonia by the BEDEC database [35]. The RCPHF design is subject to compliance with the requirements established by the standard regulations [36,37,38]. These, added to a series of technical considerations necessary for the complete verification of the RCPHF, are expressed in general terms through Equation (3).

#### 2.1. Parameters

#### 2.2. Variables

#### 2.3. Constraints

_{2}emissions, something relevant to the sustainability considerations presented in Section 1.

_{2}emissions and a series of structural verification coefficients. These represent the relation between the stress associated with the acting loads (${A}_{s}$) and the resistant limit of the structure (${R}_{s}$), which corresponds to Equation (4).

#### 2.4. Computational Model

## 3. Methodology

#### 3.1. Simulated Annealing Algorithm

#### 3.2. Threshold Accepting Algorithm

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

## 4. Results

_{2}emissions.

#### 4.1. SA Results

_{2}emissions and average and minimum values, obtained by each of the applied SA algorithms. In addition, the specific parameters of each of the metaheuristics are detailed.

_{2}that are emitted to the atmosphere as a result of its manufacture. With 5.24 tons of CO

_{2}, the RCPHF design obtained with the SA7 algorithm presents slightly lower emissions. Both this and the aforementioned design use C25/30 concrete and B 500 S steel. The main difference is that the SA7 design presents denser reinforcement while using smaller concrete amounts compared to the SA9 design with the lowest final cost. As stated in Section 1, this is somewhat to be expected since most of the associated CO

_{2}emissions originate in cement fabrication.

#### 4.2. TA Results

_{2}associated emissions values obtained by applying each of the TA algorithms in the economic optimization of the RCPHF. In addition, each of the heuristic parameters that configure the algorithms is detailed. The minimum cost values as a function of the computation cost are shown in Figure 9.

_{2}into the atmosphere. In this case, the RCPHF with the lowest final cost also happens to be the one that incurs the most negligible CO

_{2}emissions. This shows that in most cases minimal cost matches with the lowest CO

_{2}associated emission values.

#### 4.3. OBA Results

_{2}emission values obtained by applying each of the nine OBA algorithms. In this context, Figure 10 shows the evolution of the final cost obtained by each OBA algorithm as a function of the computational cost associated with obtaining each of these optimal RCPHF designs.

_{2}emission into the atmosphere. The optimal design reached by the OBA algorithm presents a slightly higher final cost than those obtained by the SA and TA algorithms. Although the OBA design might indeed be able to improve traditional designs used for the considered structural typology, it is interesting to evaluate which of the metaheuristics applied achieves the best results. This, together with other considerations, is detailed in the following section.

## 5. Discussion

_{2}emissions obtained as a result of the RCPHF economic optimization developed in the present study. In addition, some of the most relevant characteristics of the RCPHF, such as the upper slab and lateral wall depths and the upper slab flexural reinforcement area are presented.

_{2}to the atmosphere, a considerably higher value when compared with the proposed modular structure.

_{2}emissions. In this context, reducing EUR 1 in the final cost of the RCPHF using the SA algorithm allows for avoiding the emission of 1.94 kg of CO

_{2}. This corresponds with the avoidance of a 1.96 kg CO

_{2}per euro reduction for the TA and 1.72 CO

_{2}kg per euro in the case of the OBA algorithm. This is in line with values presented in studies of similar characteristics [20].

^{3}in the case of the SA algorithm result, 89.12 kg/m

^{3}for the TA and 86.01 kg/m

^{3}in the case of the OBA. Some optimal designs present localized reductions in shear reinforcement, a lack solved by local increases in flexural reinforcement. The passive reinforcement density of the OBA algorithm result is slightly lower than those obtained by applying the rest of the considered metaheuristics. This reduction in the general passive reinforcement of the RCPHF is solved by increasing the upper slab and lateral wall depth. This results in using 6.75% and 5.68% more concrete when compared to the optimal RCPHF obtained by the SA and TA algorithms, respectively.

## 6. Conclusions

_{2}emissions associated with the optimal designs obtained under restrictive budgets. In this context, the authors consider it appropriate to draw the following conclusions:

- The TA and SA algorithms generate designs with very similar characteristics, achieving the lowest costs and associated CO
_{2}emissions. Reductions of EUR 1 in the final cost for a reference cast-in-place frame are equivalent to the avoidance of emitting 1.94, 1.96 and 1.72 kg of CO_{2}in the case of SA, TA and OBA algorithms, respectively; - Optimizing the final cost of a precast structure conforms to a suitable methodology aiming to reduce the use of materials. Thus, economic optimization allows the reduction of associated CO
_{2}emissions. Economic interest drives the industry, making this an especially interesting approach to optimal design; - Optimal RCPHF designs present thin sections with reduced depths compared to traditional designs. This is solved with particularly dense passive reinforcement designs, which reach up to 91 kg/m
^{3}, 89.12 kg/m^{3}and 86.01 kg/m^{3}for the SA, TA and OBA algorithms, respectively; - The optimal design with the lowest final cost obtained by the OBA algorithm presents a slightly lower passive reinforcement density than those obtained by applying the SA and TA. Upper slab and lateral wall sections with greater depths compensate for these reductions. Consequently, the final cost of the frame and the associated CO
_{2}emissions are higher than other optimal designs; - The reduction in the use of materials from restricting the RCPHF final cost results in designs with lower self-weight. This also reduces the associated loads, limiting the structural requirements, which allows optimal use of the concrete and steel;
- The SA and TA algorithms found optimal design solutions where the values of the lateral walls and bottom slab take the minimum bound of the established range which may denote interest in slender designs. This is limited by the constructability conditions imposed. However, the study of this event is an avenue for improvement that will be considered in subsequent studies;
- The prefabrication of the considered structural typology conforms to a way of improvement in the use of natural resources when compared to the cast-in-place alternative. A complete study of the Life Cycle Assessment (LCA) of both structures and a parametric study conforms to a line of research under current development.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RCPHF | Reinforced Concrete Precast Hinged Frame |

RCCPF | Reinforced Concrete Cast in Place Frame |

ULS | Ultimate Limit State |

SLS | Service Limit State |

LCA | Life Cycle Assessment |

SA | Simulated Annealing |

TA | Threshold Accepting |

OBA | Old Bachelor’s Acceptance |

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Unit | Material | Unit Cost (€) | Associated CO_{2} Emissions (kg) |
---|---|---|---|

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

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

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

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

kg | B 400 S | 1.40 | 0.70 |

kg | B 500 S | 1.42 | 0.70 |

Geometrical parameters | ||

Free height (m) | H | 5 |

Horizontal span (m) | L | 10 |

Hinge height (m) | $HH$ | 3 |

Earth cover (m) | $HE$ | 1.50 |

Lower corner reinforcement length (m) | $CR1$ | 2 |

$CR3$ | 2 | |

Upper corner reinforcement length (m) | $CR2$ | 1.5 |

$CR4$ | 3 | |

Upper slab shear reinforcement length (m) | $SR1$ | 3 |

Lower slab shear reinforcement length (m) | $SR2$ | 2 |

Upper slab flexural reinforcement length (m) | $BR1$ | 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 |

Coefficient of active earth pressure | ${K}_{A}$ | 0.33 |

Coefficient of resting earth pressure | ${K}_{R}$ | 0.50 |

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

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

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

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

Economic and CO_{2} emissions parameters | ||

Unit costs (€) | c_{i} | Table 1 |

Exposure-related parameters | ||

Exposure class | XC2 | |

Legislative-related parameters | ||

Standard regulations | CEN [36,37]/MFOM [38] | |

Code considerations | MFOM [28] |

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

Upper slab depth | (m) | ${D}_{US}$ | 41 | 0.40 to 1.20 |

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

Lateral walls depth | (m) | ${D}_{LW}$ | 41 | 0.40 to 1.20 |

Materials variables | ||||

Concrete grade | (MPa) | C | 4 | 25 to 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 | |

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

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

Flexural reinforcement ${R}_{11}$ | (mm) | ${\varphi}_{{R}_{11}}$ | 6 | 10 to 32 |

(bars) | ${n}_{{R}_{11}}$ | 10 | 3 to 12 | |

Shear reinforcement ${R}_{12}$ | (mm) | ${\varphi}_{{R}_{12}}$ | 7 | 8 to 32 |

(m) | ${s}_{{R}_{12}}$ | 7 | 0.10 to 0.40 | |

Shear reinforcement ${R}_{13}$ | (mm) | ${\varphi}_{{R}_{13}}$ | 7 | 8 to 32 |

(m) | ${s}_{{R}_{13}}$ | 7 | 0.10 to 0.40 |

**Table 4.**Algorithm parameters in addition to the minimum and mean final cost and associated CO

_{2}emissions results obtained.

SA Algorithm | Markov’s Chain | TemperatureCoefficient | Iterations | MinimumCost (€) | Mean Cost (€) | MinimumCO_{2} (kg) | Mean CO_{2} (kg) |

SA1 | 500 | 0.80 | 8158 | 3821.57 | 4003.31 | 5831.25 | 6200.67 |

SA2 | 500 | 0.90 | 15,451 | 3755.27 | 3828.82 | 5274.81 | 5596.65 |

SA3 | 500 | 0.95 | 26,497 | 3713.99 | 3804.87 | 5463.01 | 5611.49 |

SA4 | 1000 | 0.80 | 15,338 | 3709.59 | 3837.97 | 5316.74 | 5667.25 |

SA5 | 1000 | 0.90 | 31,686 | 3761.14 | 3825.08 | 5320.67 | 5660.71 |

SA 6 | 1000 | 0.95 | 46,246 | 3708.99 | 3810.27 | 5314.10 | 5725.23 |

SA7 | 5000 | 0.80 | 84,257 | 3691.30 | 3787.28 | 5241.13 | 5455.81 |

SA8 | 5000 | 0.90 | 138,728 | 3707.36 | 3769.21 | 5458.70 | 5585.77 |

SA9 | 5000 | 0.95 | 266,363 | 3683.84 | 3749.43 | 5311.64 | 5569.85 |

TA Algorithm | Chain | Thresholdcoefficient | Iterations | Minimumcost (€) | Mean cost (€) | MinimumCO_{2} (kg) | Mean CO_{2} (kg) |

TA1 | 500 | 0.80 | 7416 | 3805.03 | 3984.38 | 5663.23 | 6080.31 |

TA2 | 500 | 0.90 | 14,517 | 3675.73 | 3873.48 | 5268.30 | 5747.97 |

TA3 | 500 | 0.95 | 25,376 | 3719.86 | 3797.42 | 5366.11 | 5665.19 |

TA4 | 1000 | 0.80 | 17,857 | 3759.21 | 3854.84 | 5437.94 | 5776.78 |

TA5 | 1000 | 0.90 | 27,859 | 3766.73 | 3819.90 | 5476.89 | 5732.29 |

TA6 | 1000 | 0.95 | 27,913 | 3705.83 | 3771.47 | 5290.85 | 5672.44 |

TA7 | 5000 | 0.80 | 79,219 | 3682.57 | 3741.59 | 5290.80 | 5586.77 |

TA8 | 5000 | 0.90 | 146,111 | 3697.23 | 3762.97 | 5292.91 | 5648.35 |

TA9 | 5000 | 0.95 | 253,145 | 3678.59 | 3711.77 | 5372.34 | 5488.02 |

OBA Algorithm | Iteration limit | Iterations | Minimumcost (€) | Mean cost () | MinimumCO_{2} (kg) | Mean CO_{2} (kg) | |

OBA | 500,000 | 48,195 | 3804.83 | 3855.26 | 5714.00 | 5637.64 | |

OBA1 | 500,000 | 32,028 | 3909.94 | - | 5430.45 | - | |

OBA2 | 500,000 | 48,195 | 3804.82 | - | 5714.00 | - | |

OBA3 | 500,000 | 49,409 | 3808.72 | - | 5785.66 | - | |

OBA4 | 500,000 | 41,295 | 3847.44 | - | 5472.21 | - | |

OBA5 | 500,000 | 48,878 | 3819.64 | - | 5551.71 | - | |

OBA6 | 500,000 | 15,698 | 3874.68 | - | 6000.80 | - | |

OBA7 | 500,000 | 13,336 | 3830.57 | - | 5927.97 | - | |

OBA8 | 500,000 | 14,118 | 3866.86 | - | 5522.67 | - | |

OBA9 | 500,000 | 47,809 | 3934.59 | - | 5333.26 | - |

Reference | SA | TA | OBA | |
---|---|---|---|---|

Final cost (€) | 4867.64 | 3683.84 | 3675.73 | 3804.83 |

Associated CO_{2} emissions (kg) | 7608.97 | 5311.64 | 5268.30 | 5714.00 |

Upper slab depth (m) | 0.75 | 0.82 | 0.82 | 0.86 |

Lateral walls depth (m) | 0.44 | 0.40 | 0.40 | 0.46 |

Upper slab flexural reinforcement (mm^{2}) | 4785.50 | 2827.43 | 2827.43 | 2412.74 |

Final cost reduction (%) | - | 24.32 | 24.49 | 21.75 |

Associated CO_{2} emissions reduction (%) | - | 30.19 | 30.76 | 24.90 |

Geometrical Variables | SA | TA | OBA | ||
---|---|---|---|---|---|

Upper slab depth | (m) | ${D}_{US}$ | 0.82 | 0.82 | 0.86 |

Lower slab depth | (m) | ${D}_{LS}$ | 0.40 | 0.42 | 0.46 |

Lateral walls depth | (m) | ${D}_{LW}$ | 0.40 | 0.40 | 0.46 |

Materials variables | |||||

Concrete grade | (MPa) | C | 25 | 25 | 25 |

Steel grade | (MPa) | S | 500 | 500 | 500 |

Passive reinforcement variables | |||||

Flexural reinforcement ${R}_{1}$ | (mm) | ${\varphi}_{{R}_{1}}$ | 16 | 12 | 16 |

(bars) | ${n}_{{R}_{1}}$ | 7 | 12 | 6 | |

Flexural reinforcement ${R}_{2}$ | (mm) | ${\varphi}_{{R}_{2}}$ | 12 | 10 | 16 |

(bars) | ${n}_{{R}_{2}}$ | 7 | 10 | 4 | |

Flexural reinforcement ${R}_{3}$ | (mm) | ${\varphi}_{{R}_{3}}$ | 10 | 10 | 12 |

(bars) | ${n}_{{R}_{3}}$ | 8 | 8 | 7 | |

Flexural reinforcement ${R}_{4}$ | (mm) | ${\varphi}_{{R}_{4}}$ | 16 | 16 | 20 |

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

Flexural reinforcement ${R}_{5}$ | (mm) | ${\varphi}_{{R}_{5}}$ | 20 | 20 | 25 |

(bars) | ${n}_{{R}_{5}}$ | 9 | 9 | 6 | |

Flexural reinforcement ${R}_{6}$ | (mm) | ${\varphi}_{{R}_{6}}$ | 20 | 20 | 16 |

(bars) | ${n}_{{R}_{6}}$ | 9 | 9 | 12 | |

Flexural reinforcement ${R}_{7}$ | (mm) | ${\varphi}_{{R}_{7}}$ | 12 | 16 | 12 |

(bars) | ${n}_{{R}_{7}}$ | 11 | 6 | 7 | |

Flexural reinforcement ${R}_{8}$ | (mm) | ${\varphi}_{{R}_{8}}$ | 20 | 20 | 16 |

(bars) | ${n}_{{R}_{8}}$ | 5 | 5 | 8 | |

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

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

Flexural reinforcement ${R}_{10}$ | (mm) | ${\varphi}_{{R}_{10}}$ | 12 | 12 | 12 |

(bars) | ${n}_{{R}_{10}}$ | 8 | 8 | 7 | |

Flexural reinforcement ${R}_{11}$ | (mm) | ${\varphi}_{{R}_{11}}$ | 10 | 10 | 12 |

(bars) | ${n}_{{R}_{11}}$ | 4 | 4 | 7 | |

Shear reinforcement ${R}_{12}$ | (mm) | ${\varphi}_{{R}_{12}}$ | 32 | 20 | 32 |

(m) | ${s}_{{R}_{12}}$ | 0.35 | 0.15 | 0.40 | |

Shear reinforcement ${R}_{13}$ | (mm) | ${\varphi}_{{R}_{13}}$ | 10 | 10 | 8 |

(m) | ${s}_{{R}_{13}}$ | 0.35 | 0.35 | 0.40 |

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## Share and Cite

**MDPI and ACS Style**

Ruiz-Vélez, A.; Alcalá, J.; Yepes, V.
Optimal Design of Sustainable Reinforced Concrete Precast Hinged Frames. *Materials* **2023**, *16*, 204.
https://doi.org/10.3390/ma16010204

**AMA Style**

Ruiz-Vélez A, Alcalá J, Yepes V.
Optimal Design of Sustainable Reinforced Concrete Precast Hinged Frames. *Materials*. 2023; 16(1):204.
https://doi.org/10.3390/ma16010204

**Chicago/Turabian Style**

Ruiz-Vélez, Andrés, Julián Alcalá, and Víctor Yepes.
2023. "Optimal Design of Sustainable Reinforced Concrete Precast Hinged Frames" *Materials* 16, no. 1: 204.
https://doi.org/10.3390/ma16010204