Optimal design for continuous composite slabs via metaheuristic algorithms

Abstract

With the increase in greenhouse gas emissions from the construction sector, pursuing more sustainable solutions has become essential. The use of composite steel-concrete slabs can considerably reduce environmental impacts due to their rapid construction and the elimination of the need for shoring. This study aims to present the optimization problem formulation for composite steel-concrete slabs, considering continuity between slabs and the addition of positive and negative reinforcement. The design constraints adhered to the provisions of Brazilian standards for composite steel and concrete structures, while Particle Swarm Optimization and Grey Wolf Optimization algorithms were employed to solve the optimization problem. A comparative analysis was conducted with solutions provided by a formwork manufacturer to assess the effectiveness of the proposed solutions and the gains achieved through the application of the formulation. The optimization algorithms yielded similar optimal solutions in most analyses, with a slight advantage for Grey Wolf Optimization in cases where distinct solutions were obtained. The manufacturer's solutions exhibited a total CO₂ emission up to 54% higher than the optimal solutions. The constraints governing the optimization problem were the formation of a plastic hinge due to a negative bending moment and failure due to longitudinal shear.

Keywords:
continuous composite slabs; optimization; Particle Swarm Optimization; Grey Wolf Optimization; additional positive reinforcement.

1. Introduction

A slab is identified as a steel-concrete composite if there is shear transfer at the steel-concrete interface; only then are the forces resisted by the joint action of the two materials in the service phase. If there is no shear transfer between the steel formwork and the reinforced concrete, the two materials will resist the forces separately, resulting in slippage. Some researchers, such as Simões and Pereira (2020), proposed a new reinforcement system in the longitudinal shear strength of continuous and simply supported composite slabs after the insertion of additional bars in their ribs, aiming primarily to increase the longitudinal shear capacity and ductility of the composite slabs. Tian et al. (2021) studied continuous composite slabs with a bent plate connection, which exhibited better load-bearing capacity and performance. The studies conducted by Tian et al. (2021) concluded that the concrete strength and the thickness of the shear connection had little influence on the flexural resistance of continuous composite slabs. However, the shear connection can reinforce the connection between the slab and the steel beam, reducing the slippage of the slab over the steel profile. Bolina et al. (2021) evaluated the behavior of continuous composite slabs in a fire, concluding that negative reinforcements do not demonstrate effectiveness in maintaining the continuity of the slab in fire situations, and the steel profile loses strength and detaches from the concrete very quickly at the beginning of the fire. Therefore, Bolina et al. (2021) recommend not considering the elements as composite when designing the structure to resist fire. However, the authors highlighted that protecting the intermediate beam against fire can increase the resistance of this type of slab.

Several studies on composite slabs aim to improve the behavior of these structures and find the best relationship between the materials that constitute them, namely the steel formwork, concrete, and reinforcement. The performance of continuous steel fiber composite slabs in their negative moment region, demonstrating greater cracking control and bearing capacity, was studied by Abas et al. (2013). In their analyses, it was observed that slabs containing steel fibers above 20 kg/m³ provided significant improvements in resistance and cracking control. The shrinkage behavior of continuous composite slabs with recycled aggregates was studied by Zang et al. (2020). From the test results, the researchers concluded that using recycled aggregates in this structural system significantly increased mid-span deflections and crack widths in the negative moment region, with values exceeding the limits established by European and Chinese standards. The evaluation of additional bars in continuous composite slabs with two spans was studied by Grossi et al. (2020), highlighting that composite slabs with additional reinforcement bars exhibit more excellent ductility and load capacity. The authors demonstrated a direct relationship between stiffness and the increase in positive reinforcement. Furthermore, they also observed that composite slabs with additional reinforcement cracked less than those without this additional reinforcement.

As highlighted, civil construction must use structural elements with less environmental impact to reduce the consequences of its structures. Many researchers use structural optimization to find the best engineering solutions to reduce environmental impacts with different optimization algorithms. The optimization of weight or cost of composite floors using Particle Swarm Optimization (PSO), damage verification in composite beams, and multiparametric optimizations performed on composite floor systems were studied (Poitras et al. 2011; Vosoughi and Gerist 2014). The PSO in the new statistical approach to analyzing earthquake engineering problems is applied by Xiao et al. (2016). An Advanced Particle Swarm Optimization (APSO) to study the flutter critical velocity of bridges is proposed by Yu et al. (2022).

Similarly, the efficiency of Grey Wolf Optimizer (GWO) in the study of damage detection of trusses and frame structures under vibration is analyzed by Ghannadi et al. (2020). An Improved Grey Wolf Optimizer (IGWO) and the analysis of the algorithm’s efficiency in the optimal design of pipe racks, comparing the results with those of PSO and GWO, are proposed by Zakian et al. (2021).

Civil construction has sought structural elements with good bearing capacity, lower costs, and reduced environmental impact. Research evaluating the environmental impact generated by raw materials, such as reinforced concrete and steel during production and transportation, is becoming common. Santoro and Kripka (2020) evaluated the CO₂ emissions of reinforced concrete raw materials, aiming to optimize these elements to minimize environmental impact and cost. The study of composite steel and concrete slabs has gained traction, as they can be constructed quickly, minimize the need for tensile reinforcement, and eliminate the requirement for shoring. Therefore, using composite steel and concrete slabs can reduce CO₂ emissions in the construction sector, which typically arise from using steel and concrete. This reduction is achieved by ensuring continuity between slabs and utilizing additional reinforcement to resist positive and negative moments. As noted in the article published by Teixeira et al. (2023) for simply supported slabs, this approach helps to reduce the thickness of the steel forms and the concrete volume used in the slabs, a point further explored in this research.

An optimization study of composite tubular columns was conducted by Guimarães et al. (2022). The main objective of this study was to present a formulation for optimizing the design of composite columns, focusing on reducing financial costs and CO₂ emissions during manufacturing. In all analyzed cases, it was observed that steel is the most expensive and least environmentally friendly material, contributing to more than 80% of the cost and emissions of unreinforced columns.

Fiorotti et al. (2023) proposed an optimization formulation for steel beams using monosymmetric or double symmetric profiles with external pretension using straight or polygonal tracing cables, aiming to analyze these structures' environmental and economic impacts. From the results obtained, the authors found that monosymmetric profiles emit less CO₂ and are more economical compared to double-symmetric profiles. Additionally, it was observed that straight cables result in better CO₂ emission values and cost reduction compared to polygonal cables.

The composite steel and concrete floor system, consisting of composite beams and slabs, was optimized by Arpini et al. (2022) to reduce these systems' economic and environmental impacts. The optimization process used the Genetic Algorithm toolbox available on the MATLAB^® platform. The optimization program's solutions reduced the structure's financial cost by around 17%. The steel deck form generated the highest percentage of environmental impact, while the beams had the highest financial cost. This indicates that the best environmental solution does not always provide the lowest financial cost. Silva et al. (2024) analyzed the composite floor system composed by cellular beams. The results show that cellular beams give the best solutions when compared with the floor with full web beams.

Loureiro et al. (2023) implemented the optimization problem of steel formwork for steel-concrete composite slabs using Particle Swarm Optimization, the Direct Strength Method, and the traditional Effective Width Method. Numerical analyses were carried out on four different steel geometries in the Brazilian market. The optimization reduced steel consumption by an average of 21.6%. Thickness had a greater effect on the steel savings, and this variable decreased in the optimal solution while the height and web bend angle increased.

Teixeira et al. (2023) conducted an optimization study of simply supported composite steel and concrete slabs to achieve solutions that promote more sustainable development in civil construction. The analyses adopted two optimization processes: Particle Swarm Optimization (PSO) and Gray Wolf Optimizer (GWO). The element that contributes most to CO₂ emissions, around 59%, is the steel formwork. Both algorithms selected the steel form with the smallest thickness, the least concrete cover, the lowest characteristic strength of the concrete, and the highest rate of additional positive reinforcement. This occurred because the weight of the steel form and the concrete are the variables that most influence CO₂ emissions. Positive reinforcements contribute to CO₂ reduction as they increase the slab's strength without significantly impacting the total CO₂ emissions.

As highlighted, few studies in literature address the optimization problem of continuous composite steel-concrete slabs, considering the design criteria required in the service phase. So, the main objective of this research is to minimize the CO₂ emissions of continuous composite steel-concrete slabs and compare the optimum solutions against those from the formwork manufacturer. For this purpose, a computational code was developed on the MATLAB (MATrix LABoratory) platform, in which the PSO and GWO algorithms. The implemented PSO was proposed by Kennedy and Eberhart (1995), combined with the Adaptive Penalty Method proposed by Lemonge and Barbosa (2004). For GWO, implemented was the algorithm proposed by Mirjalili et al. (2014) and modified by Kohli and Arora (2018), who proposed the Chaotic Gray Wolf Algorithm using chaotic maps to search for the global minimum. This article is structured as follows. After this introduction, the optimization formulation of the optimal design is presented in Section 2. The numerical examples and results are studied in Section 3. Finally, the conclusions are presented in Section 4.

2. Optimization problem formulation

2.1 Design variables

The design variables in the optimization problem are the concrete thickness above the formwork (x₁ = t_c), the characteristic compressive strength of the concrete (x₂ = f_ck), the thickness of the steel formwork (x₃ = t_f), the additional positive reinforcement ratio (x₄ = 〉_R⁺), the type of steel formwork according to the manufacturer (x₅), the negative reinforcement ratio (x₆ = 〉_R^-) and the diameter of the negative reinforcement (x₇ = φ_R^-). The composite slab with an indication of the design variables is shown in Figure 1 and the analyzed Polydeck 59S formwork, in Figure 2.

Figure 1
Continuous composite slab and indication of the variables.

Figure 2
Slab with the Polydeck 59S steel formwork. Search: Adapted of ArcelorMittal Perfilor (2022).

2.2 Objective function

The objective function

1 $$C{O}_2=E{m}_{\mathrm{F}}+E{m}_{\mathrm{C}}+E{m}_{\mathrm{R}}+E{m}_{\mathrm{w}}+E{m}_{\mathrm{NR}}+E{m}_{\mathrm{CR}}$$

was used to minimize the total CO₂ emission associated with the slab. This equation estimates the CO₂ emission value considering the contributions from the steel formwork, Em_F, concrete, Em_C, additional reinforcement for positive moment, Em_R welded mesh, Em_W, negative reinforcement, Em_NR and crack control reinforcement, Em_Cr. The analysis was conducted considering a unit slab width (B_s = 1). The emission of the steel formwork is given by:

2 $$E{m}_F=p_F\cdot A_F\cdot C{O}_{2F}\cdot N_S$$

where p_F is the mass of the steel formwork per unit area, A_F is the area of the steel formwork, CO_2F is the CO₂ emission of the steel formwork per kilogram, and N_S is the number of slab spans. The concrete emission is determined by:

3 $$E{m}_C=V_c\cdot C{O}_{2C}\cdot N_s$$

where V_c is the volume of concrete of one slab, and CO_2C is the CO₂ emission of the concrete per unit volume. The emission of the additional positive reinforcement is represented by:

4 $$E{m}_R=L\cdot {\gamma }_S\cdot C{O}_{2R}\cdot A_R^{+}\cdot N_S$$

where L is the span of the composite slab (m), γ_S is the specific mass of the steel, and CO_2R is the CO₂ emission of the steel bars per kilogram. For the additional reinforcement, 5 mm diameter steel bars are considered, with only the usage rate varying. The area of the additional reinforcement A_R⁺ is defined by:

5 $$A_R^{+}={\rho }_R^{+}\cdot t_c\cdot B_s$$

where 〉_R⁺ is the positive reinforcement rate determined at each step of the optimization process, t_c is the thickness of the concrete layer (m), and B_s is the width of the composite slab (m), taken as 1 m. The emission of the welded mesh is represented by:

6 $$E{m}_{\mathrm{w}}=p_w\cdot A_L\cdot C{O}_{2R}\cdot N_S$$

where p_w is the mass of the mesh per unit area (kg/m²) and A_L is the required steel area of cracking mesh. The emission of the negative reinforcement is represented by:

7 $$E{m}_{NR}=0.5L\cdot {\gamma }_s\cdot C{O}_{2R}\cdot A_R\cdot N_{ls}$$

where N_Is is the number of internal supports of the continuous slab, and A_R^- is the negative steel area given by:

8 $$A_R^{-}={\rho }_R\cdot t_c\cdot B_s$$

where 〉_R^- is the negative reinforcement rate that is optimized at each step of the optimization process. The number of negative reinforcement bars is given by Eq. (9), where A_ϕ is the area of a steel bar. The steel area of the negative reinforcement A_R^-* is rounded according to the number of bars given by:

9 $$N_{\varphi }=\left(A_R^{-}/A_{\varphi }\right)+1$$

10 $$A_R^{-\ast }=N_{\varphi }\cdot A_{\varphi }$$

The emission of the cracking reinforcement is represented by:

11 $$E{m}_{CR}=0.5L\cdot {\gamma }_s\cdot C{O}_{2R}\cdot A_R^C\cdot N_{ls}$$

where A_R^C is the area of the cracking reinforcement.

The CO₂ emission values for steel were based on the Life Cycle Inventory (LCI) study published by the World Steel Association (2020). This assessment considered the "Cradle to Gate" approach, excluding recycling, for 1 kg of steel. The data does not account for any burden from scrap input or credits for recycling. On the other hand, CO₂ emissions from concrete were established according to the research conducted by Santoro and Kripka (2020). In their study, the authors calculated the values by adding the CO₂ emissions from the raw materials for various characteristic strengths of concrete to those generated by the batcher during the production and transport of the concrete. This analysis considered the extraction and production phases, as well as transportation over a radius of 100 km. Table 1 shows the CO₂ emissions for concrete and steel.

Table 1
CO₂ emissions in concrete and steel production.

Component	Strength MPa	Symbol Units	CO2 Emission	Observations
Concrete	20	CO2C kgCO2/m3	129.85	Obtained values by adding the CO2 emissions from the raw materials to those generated by the batcher during the production and transport of the concrete. Search: Santoro and Kripka (2020)
	25		142.71
	30		153.68
	35		163.25
	40		171.73
	45		189.60
	50		199.72
Steel formwork	280	CO2F kgCO2/kg	2.6380	Hot-dip Galvanized Steel is obtained by passing a cold-rolled coil through a molten zinc bath, coating the steel with a thin layer of zinc to provide corrosion resistance. Search: World Steel Association (2020)
Wire mesh Positive reinforcement Negative reinforcement Cracking reinforcement	600	CO2R kgCO2/kg	1.9240	Steel reinforcing bar rolled on a hot rolling mill. This product strengthens concrete and is also the primary product for the wire rod process. Search: Worldsteel Association (2020)

2.3 Constraints

The optimization algorithm considers five constraints, C(1) to C(5), related to the limit states design methodology, in which a building's structure is evaluated under various extreme conditions that define ultimate limit states (ULS) and serviceability limit states (SLS). As shown in Chart 1, constraint C(1) pertains to the verification of ULS concerning the positive bending moment; C(2) addresses vertical shear, and C(3) addresses longitudinal shear. Constraint C(4) checks the SLS for excessive deflection, while C(5) relates to the verification of ULS for the negative bending moment. The other constraints, C(6), C(7), and C(8), pertain to construction requirements, specifically the maximum and minimum spacing for negative reinforcement and the maximum diameter for negative reinforcement.

Chart 1
Optimization problem design constraint.

The design efforts (M_Sd⁺,M_Sd^- and V_Sd) and design deflection (δ) are determined through elastic analysis with the non-cracked section, if all loads act on the composite section and considering the worst scenarios of distributed loading. The 30% redistribution in negative moments is considered.

The design resistances (M_Rd⁺, M_Rd^- and V_Rd) are determined according to Brazilian Standard (2008) in the slab sections without additional positive reinforcement and according to Grossi et al. (2020) in the slab sections with additional positive reinforcement. Thus, Chart 2 and Chart 3 give the design moment resistance, considering the plastic neutral axis in the concrete cover.

Chart 2
Positive resistant moment equations.

Chart 3
Negative resistant moment equations.

Chart 4
Resistant longitudinal and vertical shear equations.

In Chart 3, N_c is the compressive design force of concrete due to negative moment, A_c is the concrete area above the top face of the formwork, z_c is the distance from the center of gravity of the compressed concrete area to the top of the slab, z_s is the distance from the negative reinforcement to the top of the slab, z is the lever arm of the negative bending moment, h_t is the total height of the composite slab, and b₀ is the average width of the ribs. The longitudinal and vertical design shear forces are determined according to Chart 4.

In Chart 2, N_pa is the resistant tensile design force of the steel formwork, A_F,ef is the effective section area of the formwork, N_s is the resistant tensile design force of the additional reinforcement, A_s is the area of longitudinal tensile reinforcement, f_yFd is the calculated yield strength for tension, f_sd is the design yield strength of the reinforcement, f_cd is the design compressive strength of the concrete, a is the height of the concrete compression block, d_F is the distance from the upper face of the concrete slab to the centroid of the effective formwork section, and d_s is the distance from the upper face of the concrete slab to the center of the longitudinal tensile reinforcement.

In Chart 4, A_v is the resisting area of concrete for vertical shear, f_ck is the characteristic compressive strength of concrete, m and k are empirical constants obtained through testing, b is the unit width of the slab, b_n is the width between two consecutive ribs, L_s is the shear span, γ_sl is the weighting coefficient of longitudinal shear strength, and τ_Rd is the design resistant shear stress.

2.4 Implementation of optimization routines

Optimization routines were implemented using the following parameters for each algorithm:

• • Grey Wolf Optimizer (GWO): Maximum number of iterations: 50; Population size: 150 individuals; 10 runs;

• • Particle Swarm Optimization (PSO): Adaptive Penalties method studied by Lemonge and Barbosa (2004); Maximum number of iterations: 50; Population size: 150 individuals; 10 runs.

For both algorithms, the convergence criterion was defined as either reaching the maximum number of iterations or verifying that, over 10 consecutive iterations, the relative variation of the objective function between successive iterations remains below 10^-6. Figure 3 shows a flow chart of the optimization procedure.

Figure 3
Optimization procedure.

3. Numerical results

This study analyzed continuous Polydeck 59S steel formwork composite slabs from the Perfilor catalog with the geometry in Fig. 2. It analyzed slabs with 2.6, 3.4, and 4 m with three spans. It also considered the manufacturer’s values for all the cases in the catalog as overlapping loads. It should be noted that the manufacturer’s catalog has no load prediction for all analyzed spans. Thus, this research used the span load immediately below the table or that corresponding to the span immediately below the research. The highlighted items in the load tables represent the values adopted for the analysis, since the manufacturer’s catalog provides no values for these situations.

This research also considered a steel elasticity modulus (E_a) of 200 GPa, a 7850 kg/m³ specific mass of steel, a strength weighting factor for concrete (γ_c), the steel of the formwork and reinforcement (γ_a) and longitudinal shear strength (γ_sl ) equal to 1.4, 1.15 and 1.43, respectively.

3.1 Continuous composite slab with three spans

This study compared PSO and GWO to evaluate the efficiency of the values found. Table 2 shows the load cases defined by the manufacturer that optimized the composite slabs according to the lowest CO₂ emission.

Table 2
Analyzed load cases on the optimization problem.

Load Case	2.6 m kN/m2	Load Case	2.6 m kN/m2	Load Case	3.4 m kN/m2	Load Case	3.4 m kN/m2	Load Case	4 m kN/m2	Load Case	4 m kN/m2
1	7.19	24	19.96	1	4.79	24	8.17	1	4.79	24	4.93
2	8.37	25	9.31	2	4.94	25	3.98	2	4.94	25	3.98
3	10.73	26	15.18	3	6.86	26	5.20	3	6.86	26	2.76
4	7.92	27	21.16	4	2.56	27	8.65	4	2.56	27	5.21
5	9.21	28	9.82	5	3.3	28	4.18	5	3.3	28	4.18
6	12.12	29	16.04	6	7.55	29	5.48	6	3.28	29	2.90
7	8.65	30	22.36	7	2.76	30	9.12	7	2.76	30	5.49
8	10.06	31	10.32	8	3.57	31	4.39	8	3.57	31	4.39
9	13.52	32	16.90	9	8.25	32	5.76	9	3.55	32	3.04
10	9.37	33	23.56	10	2.96	33	9.60	10	2.96	33	5.78
11	10.91	34	10.83	11	3.84	34	4.60	11	3.84	34	4.60
12	14.92	35	17.76	12	8.95	35	6.03	12	3.82	35	3.18
13	10.1	36	24.77	13	3.16	36	10.08	13	3.16	36	6.06
14	11.76	37	11.34	14	4.11	37	4.81	14	4.11	37	4.81
15	16.32	38	13.96	15	6.76	38	6.31	15	4.10	38	3.32
16	10.84	39	25.98	16	3.36	39	10.56	16	3.36	39	6.34
17	12.61	40	11.85	17	4.38	40	5.01	17	4.38	40	5.01
18	17.56	41	14.59	18	7.23	41	6.59	18	4.37	41	3.46
19	11.57	42	27.19	19	3.57	42	11.04	19	3.57	42	6.63
20	13.47	43	12.63	20	4.65	43	5.22	20	4.65	43	5.22
21	18.76	44	15.23	21	7.7	44	6.87	21	4.65	44	3.60
22	12.3	45	28.41	22	3.77	45	11.53	22	3.77	45	6.92
23	14.32	-	-	23	4.93	-	-	23	2.62	-	-

The CO₂ emission results for each algorithm varied by 8.8% between algorithms. Figure 4 shows the ratio between the CO₂ emissions of the manufacturer’s solutions and the optimization problem for the 2.6, 3.4, and 4 m spans, respectively. In this graph, the horizontal axis indicates the height of the concrete thickness (t_c) and the thickness of the steel decking (t_f) for each slab case analyzed.

Figure 4
CO₂ emission ratio - Manufacturer’s solution/optimized solution for slabs with three spans.

Optimal solutions performed better than the manufacturer’s solutions in 93% of cases. The maximum ratio reaches 1.51 for the 4m span and high load value, indicating that the manufacturer's solution is approximately 51% more polluting. The first 3.4m span case obtained a higher CO₂ emission than the manufacturer, since the catalog provides no load value for this situation. Thus, this study used a load value corresponding to the 3.2m span on the catalog, showing that it can find composite slab solutions for this load for 3.4m spans. For 4m spans, some cases had no manufacturer’s values (Table 3), and this research used 3.4 m span values.

Table 3
Analysis of the design constraints.

Solution	Span 2.6 m		Span 3.4 m		Span 4 m
Constraint	VSd⁄ Vl,Rd	MSd- ⁄ MRd	VSd ⁄ Vl,Rd	MSd- ⁄ MRd	VSd⁄ Vl,Rd	MSd- ⁄ MRd
Percentage of cases	9%	91%	0%	100%	22%	78%

Figure 5 shows the CO₂ emission plots of each composite slab component for each span size. The order of contribution to the total CO₂ emission of the structure was maintained: steel formwork, concrete, cracking mesh, negative reinforcement, and additional positive reinforcement.

Figure 5
CO₂ emission per component for slab with three spans.

Figure 6 shows the frequency optimal slab solutions are chosen, concrete layer height, steel formwork thickness, additional reinforcement, f_ck, and negative reinforcement ratio.

Figure 6
Values more selected for the slab properties with three spans.

Figure 7 shows the analysis of the constraints that control the optimization process of the slabs with a Polydeck 59S steel formwork and 2.6, 3.4, and 4 m spans to load cases 01 and 30.

Figure 7
Analysis of the constraints for Polydeck 59S slabs and spans of 2.6 m, 3.4 m, and 4 m for a continuous composite slab with three spans

As shown in Figure 7, and similarly in the analysis for slabs with 2 spans, longitudinal shear force and negative bending moment established the criteria that governed the designs for the different analyzed spans. This behavior was consistent across all 45 analyzed load cases. On the other hand, the constraints related to transverse shear, positive moment, and maximum displacement had little impact on the final design of the slabs.

Table 3 shows the percentage of cases governed by longitudinal shear stress and negative bending moments for three-span continuous slabs. The negative bending moment ruled most cases.

4. Conclusions

This research solved the problem of optimizing continuous steel and concrete composite slabs to minimize CO₂ emissions. Regarding the optimization algorithms, PSO and GWO efficiently searched for solutions, as there was a high percentage of equal solutions.

The manufacturer’s solutions demonstrated CO₂ emissions that were 44% and 51% higher than those calculated by the algorithms for two and three spans of continuous slab, respectively. The slab's formwork contributes the most to the structure's total CO₂ emissions, followed by concrete, the cracking mesh, negative reinforcement, and additional positive reinforcement.

Notably, all cases adopted a 0.8 mm thick steel formwork as their optimal solution, which is lighter and results in lower CO₂ emissions. The constraint ruling the design is often referred to as plasticization by a negative bending moment, reducing the use of additional reinforcement. On the other hand, the optimal solutions for continuous composite slabs adopted a f _ck of 30, 35, and 40 MPa, respectively. Continuous composite slabs used lower negative reinforcement rates, avoiding impacts to total CO₂ emissions.

In addition, the results obtained indicate that the use of additional positive reinforcement is promising. In some analyses, the inclusion of reinforcement bars reduced the total CO₂ emissions of the slabs by decreasing the steel area in the formwork. The use of high-strength concrete, despite having higher CO₂ emissions, is advantageous because it reduces the volume of concrete required, and consequently lowers the total CO₂ emissions of the slab.

In this study, the constraints considered in the optimization process relate to the checks of the slab in its composite phase after the curing of the concrete. Future research suggests that checks be included for the steel formwork during the construction phase before the concrete curing. Additionally, it is possible to incorporate other objective functions, such as minimizing costs, maximizing loads, or optimizing the span of the slab, and even to integrate these functions into a multi-objective analysis of the problem.

Acknowledgements

The authors acknowledge the Brazilian Federal Government Agency CAPES and State Government Agency FAPES for the financial support provided during the development of this study. The second author (Process: 2023-PXGBRF) and fourth author(Process: 2024-DS87Z) thank FAPES for the research productivity grant.

• Funding information
  Brazilian Federal Government Agency CAPES and State Government Agency FAPES for the financial support provided during the development of this study.

Data availability

Datasets related to this article will be available upon request to the corresponding author.

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• Associate Editor
  Herlander Mata Fernandes Lima

Publication Dates

• Publication in this collection
  26 Sept 2025
• Date of issue
  2025

History

• Received
  25 Feb 2025
• Accepted
  22 June 2025