# Tailored genetic algorithms for the detailed design optimization of reinforced concrete structures: case study on a flexural beam

Paul Quéva, Ludovic Jason, Gilles Arnaud, Gabriel Sarazin

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Paul Quéva, Ludovic Jason, Gilles Arnaud, Gabriel Sarazin. Tailored genetic algorithms for the detailed design optimization of reinforced concrete structures: case study on a flexural beam. Structural and Multidisciplinary Optimization, 2025, 68 (8), pp.161. 10.1007/s00158-025-04092-x. hal-05217969

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Submitted on 21 Aug 2025

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# Tailored genetic algorithms for the detailed design optimization of reinforced concrete structures: case study on a flexural beam

Paul Quéva<fnref n="1" />,<fnref n="2" /> ID · Ludovic Jason<fnref n="1" /> · Gilles Arnaud<fnref n="3" /> · Gabriel Sarazin<fnref n="3" />

Received: 24 April 2025 / Revised: 9 July 2025 / Accepted: 17 July 2025

© The Author(s) 2025

## Abstract

This contribution proposes an optimization method for the detailed design of reinforced concrete (RC) structures subject to NF EN 1992-1-1 requirements, integrating various criteria such as economic and sustainability. Traditional methods, focused on strength and durability, leave engineers with an infinite range of solutions to account for these criteria. To overcome these difficulties, genetic algorithm (GA)-based approaches have been used to automate the search for optimal trade-offs between competing objectives. However, when applied to the detailed design of RC structures, these approaches face challenges due to the combinatorial nature of the problem. In practice, simplifying assumptions—such as fixed bar diameters or reliance on predefined reinforcement arrangements—are often introduced to reduce the complexity of the search space and formulation. While effective computationally, these assumptions can limit the diversity and novelty of the resulting designs. To address these limitations, the proposed methodology introduces a GA-based optimization framework with a refined parametrization of the design space. Particular attention is given to the selection of variable inputs and the integration of tailored operators to enable efficient exploration of innovative designs. The performance of the proposed methodology is demonstrated on a constrained multiobjective optimization problem applied to the detailed design of an RC beam under three-point bending. The comparison against an exhaustive exploration of the search space reveals that the algorithm successfully converges to complex optimal designs while requiring at least 8 × 10⁷ times fewer computations than a complete enumeration of the search space. Finally, a discussion on the method's sensitivity to parameter calibration and its dependence on operator choices is also provided.

**Keywords** Reinforced concrete structure · Steel reinforcement design · Multiobjective optimization · Genetic algorithm · Carbon footprint minimization · Cost minimization

## 1 Introduction

Reinforced concrete (RC) structures are indispensable in modern construction, offering a balance of strength, durability, and cost-effectiveness across a wide range of applications. The traditional design process for RC members

typically begins with an initial pre-dimensioning of structural components, followed by a trial-and-error approach to determine the reinforcement layout. This iterative method involves specifying the diameters and placement of steel rebars within the structure. At each stage, structural engineers must ensure compliance with standards governing construction detailing, load resistance, and durability (AFNOR 2005), making the process time-consuming and reliant on expertise (Coello et al. 1997).

Moreover, RC structures are increasingly expected to address challenges beyond durability and safety. Since the construction industry is one of the largest consumers of energy and a major contributor to CO₂ emissions (Miller et al. 2016; IEA. 2024), sustainability has become a critical consideration (GCCA. 2023). In parallel, installation costs—encompassing both material and labor expenses—must also be accounted for in order to find a solution combining

Responsible Editor: Makoto Ohsaki.

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<footnote>Paul Quéva
paul.queva@cea.fr</footnote>

<footnote n="1">¹ Université Paris-Saclay, CEA, Service d'Etudes Mécaniques et Thermiques, 91191 Gif-Sur-Yvette, France</footnote>

<footnote n="2">² Laboratoire Energétique Mécanique Electromagnétisme, Université Paris Nanterre, 92410 Ville-d'Avray, France</footnote>

<footnote n="3">³ Université Paris-Saclay, CEA, Service de Génie Logiciel pour la Simulation, 91191 Gif-Sur-Yvette, France</footnote>
safety, cost, and constructability aspects. However, relying on a trial-and-error approach to address these considerations often leads to sub-optimal solutions.

In the scientific literature, optimization techniques have emerged as effective tools for enhancing designs that efficiently address cost, environmental impact, and other criteria. The method of Lagrange multipliers has been applied to the design optimization of various RC members (Haug and Kirmser 1967; Chou 1977; Prakash et al. 1988; Fadaee and Grierson 1998). However, this approach requires solving a complex system of equations and is limited to continuous optimization, making it unsuitable for detailed steel reinforcement design due to the discrete nature of rebar diameters available on the market. In (Chakrabarty 1992), the Geometric Programming (GP) method is applied to the cost-minimization problem of an RC beam. Nevertheless, this method is restricted to criteria and constraints that follow a specific functional form (Duffin et al. 1967).

Starting in the 1960s, metaheuristic algorithms began to emerge and are inspired by natural phenomena such as genetic algorithms (GA) (Goldberg 1989; Holland 1992), ant colonies (ACO) (Colorni et al. 1991), or particle swarms (PSO) (Kennedy et al. 2009), as well as physical processes like simulated annealing (SA) (Kirkpatrick et al. 1983) and social phenomena with Tabu Search (TS) (Glover and Laguna 2002). These methods have shown significant promise in RC design due to their ability to efficiently explore large and complex design spaces with minimal assumptions about the nature of design parameters, criteria, or constraints.

First application of GA to the optimization of RC structures proposed a cost-minimization design approach for RC beams using a continuous formulation (Coello et al. 1997). The methodology aimed to determine optimal cross-sectional dimensions and reinforcement areas while ensuring compliance with design code requirements. However, the choice of formulation does not support the explicit modeling of discrete reinforcement layouts, such as the individual diameters and positions of rebars. Thus, the generated solutions are not directly implementable, and additional post-processing is needed to obtain a constructible arrangement, accounting for the inherently combinatorial nature of reinforcement detailing.

As a result, subsequent studies reformulated this problem as a combinatorial optimization task, focusing initially on cost minimization for RC continuous beams (Leps and Sejnoha 2003; Govindaraj and Ramasamy 2005; Jahjouh et al. 2013) and frames (Lee and Ahn 2003; Kwak and Kim 2009; Akin and Saka 2015; Esfandiari et al. 2018). These efforts introduced discrete parameters, such as the number and diameter of tensile rebars (Leps and Sejnoha 2003; Jahjouh et al. 2013) but often relied on simplifying assumptions—e.g., using identical diameters

for all bars—to ease the formulation and reduce the search space. Other approaches incorporated databases of predefined reinforcement templates (Lee and Ahn 2003; Govindaraj and Ramasamy 2005; Kwak and Kim 2009; Akin and Saka 2015; Esfandiari et al. 2018), allowing practical considerations like constructability to be indirectly addressed. While these simplifications reduce the computational effort, they strongly limit the diversity of achievable solutions, as the search space is fully defined by the prior assumptions of the designers.

In both strategies, the optimization process is primarily tailored to automate conventional engineering practices or scale to larger problem instances, rather than to foster the emergence of innovative or unconventional reinforcement solutions. In essence, these methods aim to accelerate standard workflows rather than expand the design space.

The next phase of research in this field shifted toward using metaheuristics to solve multiobjective optimization problems for RC structure detailed design. These developments enabled consideration of additional criteria, which vary depending on the study, such as (1) uniformity of diameters (Koumousis and Arsenis 1998), (2) number of bars (Koumousis and Arsenis 1998; Paya et al. 2008), (3) sustainability (Paya et al. 2008; Eleftheriadis et al. 2018; Negrin et al. 2021; Zhang and Zhang 2021), (4) material waste (Zheng et al. 2019), (5) safety (Paya et al. 2008), (6) rebar clash avoidance (Li et al. 2021, 2023; Xu et al. 2022), and (7) steel labor costs (Eleftheriadis et al. 2018; Li et al. 2021). Such problems can be approached by either minimizing a weighted sum of multiple criteria, allowing the application of single-objective techniques (Koumousis and Arsenis 1998), or by searching for a set of solutions that achieve the best trade-offs among the various criteria. State-of-the-art GA is mostly used for the latter approach.

Despite significant progress in applying optimization techniques to the detailed design of RC structures, key challenges remain, as highlighted in the literature (Afzal et al. 2020):

* Performance-based detailed design optimization: Integrating mechanical quantities, such as flexural displacements and crack openings, into the optimization objectives and constraints, which often require evaluation through computationally expensive models (Hassanzadeh et al. 2024).

* Design of complex and irregular RC components: Optimization strategies allowing for a flexible and thorough exploration of the design space, unconstrained by prior assumptions.

* Enhancing the performance of optimization methodologies: Identifying appropriate hyperparameters for these algorithms and integrating

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them with heuristics or metaheuristics that are well suited to the specific nature of the problem.

Accordingly, to cope with the abovementioned challenges, this paper proposes a novel framework for GA-based multiobjective optimization, demonstrated through the sustainable, cost, and performance-based design of a flexural beam as a case study. The originality of the methodology lies in two key aspects:

*Design parametrization:* Treating individual rebar diameters as variable parameters, enabling the exploration of vast reinforcement configurations beyond conventional solutions;

*Algorithm adaptation:* Defining tailored operators to this formulation, allowing efficient exploration of the search space.

The remainder of this paper is structured as follows: Sect. 2 details the methodology, including problem formulation and the GA implementation. Section 3 provides a numerical example, applying the framework to the multiobjective design of flexural beam, comparing the proposed approach with a brute-force algorithm in terms of convergence and design optimality. Finally, Sect. 4 concludes with key insights and potential future research directions.

# 2 Methodology

## 2.1 General optimization framework

The proposed framework formulates the design of RC structures as a multiobjective constrained optimization problem. In this context, it aims to identify the set of optimal solutions, known as Pareto front (PF), which represents the best trade-offs between various objectives while satisfying all imposed constraints.

Formally, an optimization problem P is expressed as

$$P: \begin{cases} \min_{X \in S} \{ F(X, Y) = (f_i(X, Y))_{i \in \{1, \dots, p\}} \} \\ \text{subject to: } G(X, Y) \le 0 \quad \text{and} \quad H(X, Y) \le 0 \end{cases} \quad (1)$$

where $X = (x_i) \in S, i = 1, \dots, n$ are the n variable inputs; $Y = (y_i), i = 1, \dots, m$ represents the m known inputs (e.g., material properties); $F = (f_i), i = 1, \dots, p$ are the p functions to minimize called optimization criteria. The constraints are divided into two categories:

(1) Hard constraints $H = (h_i), i = 1, \dots, r$: Define the boundaries of the feasible domain $\mathcal{F} \subseteq S$, ensuring the designs are computationally valid and evaluable.

(2) Soft constraints $G = (g_i), i = 1, \dots, q$: Represent design code requirements and define the boundaries of the compliance domain $\mathcal{C} \subseteq \mathcal{F}$.

In this study, solutions within $\mathcal{F}$ and $S \setminus \mathcal{F}$ are, respectively, classified as feasible and infeasible, while those within $\mathcal{C}$ and $S \setminus \mathcal{C}$ are further categorized as compliant and non-compliant.

As illustrated in Fig. 1, the proposed framework solves P by coupling a computational model with an optimization algorithm. The computational model processes the input parameters to evaluate both the optimization criteria—e.g., installation cost, mechanical performances, and sustainability indicators—and soft constraints. This evaluation can leverage analytical expressions or numerical simulations, such as finite element method. The optimization algorithm iteratively navigates the search space $S$, invoking the computational model only for feasible solutions. For infeasible solutions, GA is tailored to modify the design, ensuring it becomes evaluable.

## 2.2 Formulation of the RC design problem of flexural beam

This section outlines the parametrization of the optimization problem for RC design to allow the framework to explore a wide range of designs. Additionally, the known and variable inputs as such as calculated parameters are summarized in Table 5, provided in the appendix.

To improve readability and manage the numerous dependencies between parameters induced by design calculations, the following convention is adopted throughout the paper: (1) in figures and equations, parameters derived from variable and known inputs are, respectively, expressed as functions of $X$ and $Y$; (2) in the main text, parameters are referred to by their designation, unless necessary for clarity. More information on functional dependence is found in Table 5.

### 2.2.1 Known and variable inputs

In traditional design methods, an initial pre-dimensioning step establishes the overall dimensions of the RC member, followed by a detailed design phase for a fixed section. In this phase, the minimum areas of longitudinal reinforcement—compressive and tensile bars—are determined based on the equilibrium of the cross-section under bending moment at Ultimate Limit State (ULS), while shear reinforcement is sized to resist shear forces. Transitioning from steel area to a detailed reinforcement layout is often left to the designer and involves an iterative trial-and-error process, particularly to ensure compliance with requirements at Serviceability Limit State (SLS).
<figure><img src="image_4.png" /><figcaption>Fig. 1 Optimization algorithm/Computational model coupling</figcaption></figure>

To address these limitations and automate the exploration of reinforcement configurations, this subsection presents a parametrization of a RC section. For the sake of clarity and demonstration, the methodology will be detailed for flexural beams. However, it could be adapted to other type of geometries or loadings by means of adaptations.

In the presented optimization framework, the known inputs of the formulation are categorized into four main groups:

(1) Geometric and loading condition: Span length L and boundary conditions at limit states.

(2) Concrete properties and reinforcement characteristics: Concrete mix formulation, the strength and ductility class of the reinforcing steel, and the diameter of the shear reinforcement.

(3) Cost, carbon footprint, and productivity indicators: Used to evaluate the installation costs and total carbon footprint of the RC member.

(4) Serviceability and durability criteria: Include the maximum allowable crack width $W_{perm}$ and maximum deflection $f_{perm}$, which are durability and serviceability limits prescribed by design standards.

While the serviceability and durability criteria are fixed according to NF EN 1992-1-1 (AFNOR 2005), the parameters in the first three categories are assumed to be determined during a preliminary design phase, incorporating cost and environmental studies. Fixing these parameters was chosen specifically to ensure that the search space S remains computationally manageable for brute-force algorithms, enabling direct comparison with the proposed optimization framework. However, the methodology could be applied without loss of generality by treating some of

these parameters as variables, particularly in scenarios where concrete mix formulation or strength class of steel is subject to optimization.

Building on these predefined parameters, Fig. 2 illustrates variable inputs and calculated parameters, associated with the reinforcement layout in a beam cross-section. As shown in Fig. 2a, the width B and the height H of the beam are considered inputs varying over the ranges $[B_{min}, B_{max}]$ and $[H_{min}, H_{max}]$, respectively, with increments of $\Delta B$ and $\Delta H$. This discretization is motivated by two key factors: (1) it ensures that the resulting dimensions are practical and achievable in real-world construction, and (2) it defines a finite search space, enabling the application of a brute-force algorithm for an exhaustive exploration.

For the parametrization of the tensile reinforcement (as shown in Fig. 2a), each bar is defined by its diameter $\phi_{i,j}$, where i is the row and j is the column index. Each diameter is selected from a discrete domain $D_{\phi}$, which comprises a predefined list of commercially available diameters and the value 0mm. 0mm characterizes an absence of bar, which therefore allows the framework to explore configurations with varying numbers of effective rows, columns, and bars. Finally, $\phi = (\phi_{i,j})_{(i,j) \in \{1, \dots, N_L\} \times \{1, \dots, N_{C,sym}\}}$ represents a reinforcement configuration organized in a grid with $N_{C,sym}$ columns and $N_L$ rows. The number of columns $N_{C,sym}$ is calculated with Eqs. (2) and (3), while the number of rows $N_L$ is chosen by the user.

$$ N_C(B_{max}, D_{\phi}, \mathbf{Y}) = \left[ \frac{B_{max} - 2c_h(\phi_{min}(D_{\phi}), \mathbf{Y}) + a_{min}(\phi_{min}(D_{\phi}), \mathbf{Y})}{a_{min}(\phi_{min}(D_{\phi}), \mathbf{Y}) + \phi_{min}(D_{\phi})} \right] \quad (2) $$
<figure><img src="image_5.png" /><figcaption>Fig. 2 Parametrization of structure and the reinforcement layout for the simplified RC flexural beam: (a) Cross section; (b) Longitudinal section</figcaption></figure>

$$N_{C,sym}(B_{max}, D_{\phi}, \mathbf{Y}) = \left[ \frac{N_C(B_{max}, \phi_{min}(D_{\phi}), \mathbf{Y})}{2} \right] + 1, \quad (3)$$

where $c_h$ is the horizontal cover for tensile rebars, which depends on concrete mix formulation, diameter of the tensile bar closest to the concrete surface and on the diameter of shear reinforcement $\phi_t$; $a_{min}$ denotes the minimum spacing between two adjacent bars which depends on the concrete mix formulation and on the diameters of the adjacent bars. $c_h$ and $a_{min}$ are, respectively, determined according to subsections 4.4.1 and 8.2 of NF EN 1992-1-1 considering $\phi_{min} = \min(D_{\phi} \setminus \{0mm\})$ as diameter of the tensile rebars (AFNOR 2005). This consideration allows $N_C$ to be equated to the maximum number of bars with diameter $\phi_{min}$ that can be arranged in a single row within a cross-section of width $B_{max}$, while complying with design code requirements for spacing between tensile bars and between the bars and the concrete.

This parametric representation incorporates a positioning function to determine the location of each reinforcement bar within the beam cross-section, considering its diameter, row,

and column. The horizontal positioning of tensile bars is carried out iteratively based on the following steps:

(1) The total available horizontal space is calculated, taking into account $B$, $c_h$, and tensile rebar diameters of the first row;

(2) The available horizontal space is uniformly distributed among all the spacings between the bars;

(3) If any spacing falls below the minimum specified by design codes, it is constrained to the minimum;

(4) The remaining space is computed taking into account the constrained spacings, and then it is redistributed uniformly among the unconstrained ones;

(5) Steps 3 and 4 are repeated until all minimum spacings are satisfied.

Figure 3 illustrates the vertical positioning process for tensile bars. The first row of reinforcement bars is placed according to the vertical concrete cover $c_v$ which depends on the concrete mix formulation, the diameter of the tensile bar closest to the concrete surface, and on the diameter of shear reinforcement $\phi_t$. For the upper rows, NF EN

### Vertical positioning algorithm

1. **Row n°1:** Placement based on $c_v$

2. **Row n°2:** Bar stacking if permitted by NF EN 1992-1-1

3. **Row n°3:** Minimum space $a_{min}$ (stacking with row n°2 is not permitted)

<figure><img src="image_6.png" /><figcaption>Fig. 3 Vertical positioning of tensile bars illustrated with a 3-row steel arrangement</figcaption></figure>
1992–1-1 permits vertical stacking of tensile bars with those in the lower row, as shown in Fig. 3 with rows 1 and 2. The arrangement must satisfy a maximum allowable diameter ratio between the largest and smallest bars, and vertical alignment is permitted only for a maximum of two adjacent rows (AFNOR 2005). If these conditions are not met, the bars are placed at the minimum spacing $a_{min}$.

Finally, the design and positioning of the compression and shear reinforcement follow a set of assumptions that simplify the optimization problem. For the compression reinforcement, the bars are arranged in a single row and horizontally aligned with the tensile reinforcement (as shown in Fig. 2a). All bars have the same diameter $\phi_c \in D_\phi$. This diameter is calculated to ensure that the total section is sufficient to maintain equilibrium of cross-section under bending moment. For the shear reinforcement (see Fig. 2b), the diameter $\phi_t$ is fixed. Number of shear reinforcement sets $N_{st}$ and then the transverse spacings ($s_i$) ($i \in \{1, ..., N_{st} - 1\}$) are determined using the Strut-and-Tie method outlined in Sect. 6.2 of NF EN 1992-1-1 (AFNOR 2005). Regarding the placement of a shear reinforcement set in the cross-section, it is arranged as shown in Fig. 2a and depends on the number of tensile rebars with nonzero diameter.

## 2.2.2 Soft constraints

Outline in design codes, design constraints are a fundamental aspect of civil engineering as they ensure that structures can withstand expected loads, maintain structural performance over their intended lifespan, and fulfill serviceability requirements under ULS and SLS.

Within the proposed optimization framework, soft constraints incorporate the requirements of NF EN 1992-1-1 for the tensile reinforcements. The computational model used for the evaluations is based on analytical equations provided in the same standard. For clarity, these equations are not included in the present document; however, readers are referred to the corresponding references indicated alongside each constraint below.

$$A_{min}(\mathbf{X}, \mathbf{Y}) \le A_S(\mathbf{X}) \le A_{max}(\mathbf{X}, \mathbf{Y}) \text{(art.9.2.1.1 of NF EN 1992-1-1)} \quad (4)$$

$$A_S(\mathbf{X}) \ge A_{min,ULS}(\mathbf{X}, \mathbf{Y}) \text{(art.6.1 of NF EN 1992-1-1)} \quad (5)$$

$$(\varepsilon_{sm} - \varepsilon_{cm})(\mathbf{X}, \mathbf{Y}) \ge 0.6\varepsilon_S(\mathbf{X}, \mathbf{Y}) \text{(art.7.3.4 of NF EN 1992-1-1)} \quad (6)$$

$$W_k(\mathbf{X}, \mathbf{Y}) \le W_{perm} \text{(art.7.3.4 of NF EN 1992-1-1)} \quad (7)$$

$$f_{max}(\mathbf{X}, \mathbf{Y}) \le f_{perm} \text{(art.7.4.3 of NF EN 1992-1-1)} \quad (8)$$

Equation (4) defines the permissible range for the reinforcement area, where $A_S$ represents the total tensile reinforcement area, and $A_{min}$ and $A_{max}$ denote the boundaries based on

the steel ratios specified in design codes to ensure the ductility of the RC member. In Eq. (5), $A_S$ should exceed the required reinforcement area $A_{min,ULS}$ to adequately resist the bending moment at ULS. Equations (6), (7), and (8) establish the serviceability and durability requirements at SLS. Specifically, in Eq. (6), the mean strain difference between the tensile reinforcement and the concrete in the tensile zone, $(\varepsilon_{sm} - \varepsilon_{cm})$, must exceed 60% of the reinforcement strain $\varepsilon_S$. Finally, Eqs. (7) and (8) define constraints to limit crack width $W_k$ and maximum deflection $f_{max}$ to maximum allowable values (respectively $W_{perm}$ and $f_{perm}$).

## 2.2.3 Hard constraints

In the context of the optimization framework, the choice of parametrization presented in 2.2.1 offers the advantage of generating a high level of complexity in the reinforcement arrangements. This search space $S$ can be particularly useful for optimizing the design of complex structures for which it is not feasible to establish an exhaustive list of predefined solutions. However, as illustrated in Fig. 4, there are solutions within $S$ that are not feasible.

For the application of the framework to the optimization of a simplified RC flexural beam design, the hard constraints are used to determine the feasibility of the reinforcement configuration with respect to (1) horizontal spacing between reinforcement bars and between the bars and the concrete surface, and (2) mechanical considerations.

$$B_{req}(\mathbf{X}, \mathbf{Y}) \le B \quad (9)$$

$$\text{with } B_{req}(\mathbf{X}, \mathbf{Y}) = 2c_h(\mathbf{X}, \mathbf{Y}) + \sum_{i=1}^{N_B(\mathbf{X})} \phi_{1,i} + \sum_{i=1}^{N_B(\mathbf{X})-1} a_{min,i}(\mathbf{X}, \mathbf{Y}). \\ \forall i \in \{1, ..., N_L - 1\}, \forall j \in \{1, ..., N_{C,sym}(B_{max}, D_\phi, \mathbf{Y})\}, \phi_{i+1,j} \le \phi_{i,j} \quad (10)$$

To address this, the minimum required section width, $B_{req}$, necessary to accommodate the first row of $N_B$ tensile bars of nonzero-diameter ($\phi_{1,i}$) (where $i \in \{1, ..., N_B(\mathbf{X})\}$), is determined. This calculation considers the cover $c_h$ and minimum horizontal spacings between tensile bars with nonzero-diameter ($a_{min,i}$) (where $i \in \{1, ..., N_B(\mathbf{X}) - 1\}$), as shown in Fig. 4a. The first constraint in Eq. (9) ensures that the section width $B$ is greater than or equal to the minimum required width $B_{req}$. Finally, Eq. (10) verifies that for each tensile rebar in row $i$ and column $j$, its diameter $\phi_{i,j}$ is greater of the rebar directly above it. As illustrated in Fig. 4b, positioning larger-diameter tensile bars above smaller ones reduces the contribution of the reinforcement to resisting the forces induced by the bending moment.

## 2.2.4 Optimization criteria

In this study, the objective is to find the designs making the best trade-offs between installation cost $f_c$, carbone footprint
<figure><img src="image_7.png" /><figcaption>Fig. 4 Illustrative example of infeasible tensile reinforcement arrangements: a Unsatisfied minimum required section width; b Limitation of the diameters in the upper rows</figcaption></figure>

$CO_2$, and crack width $W_k$. They are formulated, respectively, in Eqs. (11), (12) and (13):

$$f_{\mathcal{C}}(\mathbf{X}, \mathbf{Y}) = V_c(\mathbf{X}, \mathbf{Y})c_c + \frac{V_L(\mathbf{X}, \mathbf{Y}) + V_T(\mathbf{X}, \mathbf{Y})}{\rho_s} c_s + \frac{T_p(\mathbf{X}, \mathbf{Y}) + T_T(\mathbf{X}, \mathbf{Y})}{\tau} c_{\text{labor}} \quad (11)$$

where $V_c$, $V_L$, and $V_T$ represent the volume of concrete, longitudinal—compressive and tensile bars—and shear reinforcement, respectively (in $m^3$), $c_c$ and $c_s$ are the unitary cost of concrete (in €/$m^3$) and steel (in €/kg), and $\rho_s$ is the density of reinforcing steel (in kg/$m^3$). $T_p$ and $T_T$ refer to the time of rebar placing and tying (in min). Depending on productivity of bar placing and bar tying, they are expressed in the following references (Eleftheriadis et al. 2018; Li et al. 2021). $c_{\text{labor}}$ represents the labor cost per man-day (in €/man-day) and $\tau$ is a constant 480min/man-day for unit conversion assuming an 8-h workday.

$$CO_2(\mathbf{X}, \mathbf{Y}) = V_c(\mathbf{X}, \mathbf{Y})c_{CO_2,c} + \frac{V_L(\mathbf{X}, \mathbf{Y}) + V_T(\mathbf{X}, \mathbf{Y})}{\rho_s} c_{CO_2,s} \quad (12)$$

where $c_{CO_2,c}$ and $c_{CO_2,s}$ refer to the unit carbon footprint of concrete (in kg$CO_2e/m^3$) and steel reinforcement (in kg$CO_2e/kg$).

The last optimization criterion is the crack width, calculated using Eq. (13) provided the subsection 7.3.4 of NF EN 1992-1-1 (AFNOR 2005):

$$W_k(\mathbf{X}, \mathbf{Y}) = S_{r,max}(\mathbf{X}, \mathbf{Y}) \times (\varepsilon_{sm} - \varepsilon_{cm})(\mathbf{X}, \mathbf{Y}) \quad (13)$$

where $S_{r,max}$ represents to the maximal crack spacing (in mm), $\varepsilon_{sm}$ is the mean strain in the reinforcement including the effect of imposed deformations and the effect of tension stiffening, and $\varepsilon_{cm}$ is the mean strain in the concrete between cracks.

## 2.3 Tailored optimization technique based on GA

The design optimization of RC members is an NP-hard problem due to its high space complexity. In scenarios where classical methods fail to efficiently address such challenges, metaheuristic approaches, particularly GA, have proven effective in approximating optimal solutions (Collette and Siarry 2002). However, these methods often require a significant number of computational model evaluations to achieve convergence, which can be a limitation.

Section 2.3.1 provides an overview of the fundamental principles of GA. Sections 2.3.2 to 2.3.5 deal with specific adaptations made to customize the algorithms for the problem at hand and the chosen representation, aiming to enhance convergence efficiency. Finally, Sect. 2.3.6 focuses on performance indicator for optimization algorithm.

### 2.3.1 General principles of GA and tailored operators

As illustrated in Fig. 5, GA is a family of stochastic algorithms based on the evolution of a population of $N_{pop}$ individuals, each representing a candidate solution $\mathbf{X} = (B, H, \phi)$, toward an approximated Pareto front $PF_{GA}$. Quality of $PF_{GA}$ and convergence and stability of GA depend on the operators shown in Fig. 5 and hyperparameters presented in Table 1.

The algorithm begins with an initialization phase, generating $N_{pop}$ solutions within the feasible domain $\mathcal{F}$. At each iteration, the computational model evaluates optimization criteria $\mathcal{F}$ and soft constraints $\mathcal{G}$ for each individual.
<figure><img src="image_8.png" /><figcaption>**Fig. 5** Flowchart of the GA tailored for RC Member Design Optimization Framework</figcaption></figure>

<table><caption>**Table 1** Hyperparameters of the optimization framework and their role</caption><thead><tr><td>Hyperparameter</td><td>Definition</td></tr></thead><tbody><tr><td>N<sub>pop</sub></td><td>Population size: number of candidate solutions at each iteration</td></tr><tr><td>λ</td><td>Elitism rate: proportion of the best-performing solutions retained in the next iteration</td></tr><tr><td>ρ<sub>mut</sub></td><td>Mutation rate: probability of an individual to be mutated by the mutation operator</td></tr><tr><td>N<sub>eval</sub></td><td>Number of calls to the computational model: stopping criteria</td></tr></tbody></table>

These evaluations guide the ranking and selection processes, which are based on Pareto-dominance and diversity strategy.

In a multiobjective context, the ranking methodology follows an approach proposed by D. Goldberg, as employed in widely adopted algorithms such as the *Multiple Objective Genetic Algorithm* (Fonseca and Fleming 1993). This approach involves sorting individuals based on

their rank, determined by the number of times an individual is Pareto-dominated by others within the population. This dominance count is computed through pairwise comparisons across all possible pairs in the population. In the presence of soft constraints, constraint-handling approach must be defined to compare compliant and non-compliant solutions. In this matter, the presented framework uses the following rules:
(1) if both solutions are compliant, apply Pareto-dominance on optimization criteria;

(2) if both solutions are non-compliant, apply Pareto-dominance on soft constraints;

(3) Otherwise, if the non-compliant solution is not Pareto-dominated by the compliant one, the solutions are considered non-dominated.

The effect of the aforementioned principle is that it retains the best non-compliant solutions to explore better the boundaries of C. In contrast to the constraint-handling approach proposed by (Deb et al. 2002), this approach is particularly suitable for problems where the compliance domain C is significantly narrower compared to the feasible F or search space S ($\frac{|C|}{|F|} \ll 1$ or $\frac{|C|}{|S|} \ll 1$) (Siarry 2014).

In order to maintain the diversity of the non-dominated set throughout the iterations, this framework employs the *crowded-distance* metric from *Non-dominated Sorting Genetic Algorithm—II* (Deb et al. 2002). This method, free of additional hyperparameters, outperforms other techniques in preserving both convergence and diversity of the Pareto front (Deb et al. 2002).

The *environmental selection* operator retains a portion of the ranked individuals for the subsequent population. In this study, an elitist selection strategy is employed, preserving the $\lambda N_{pop}$ top-performing individuals in each iteration. This approach offers two key advantages: (1) it "protects" the best solutions at every iteration, and (2) it enhances the convergence rate of the algorithm (Rani et al. 2019).

The *crossover*, *mutation*, and *repair* operators are responsible for generating new individuals from the ranked population to form the next one. The two former operators enable

both exploration of S and exploitation of regions where local optimal solutions have been identified. The latter adjusts the individuals so that the population remains within F.

Finally, the operators described above are iteratively applied until a stopping criterion is met. In this study, the loop halts after performing $N_{eval}$ calculations. This stopping criterion allows for the evaluation of the quality of $PF_{GA}$ by controlling the computational effort, which facilitates parametric studies when the other hyperparameters vary.

### 2.3.2 Tailored initialization

The initialization phase plays a crucial role in the ability of GA to converge toward global optima, as it establishes the diversity of the initial population. In the presented problem, evaluation of $|F|$ and $|S|$ with a brute-force approach on study cases shows that F is significantly constrained compared to the overall search space ($\frac{|F|}{|S|} = \frac{10^9}{10^{12}} \ll 1$). Thus, instead of performing a random sampling in S as shown in Fig. 6a, the framework randomly generates individuals within F as illustrated in Fig. 6b.

In this context, the latter approach follows the procedure described in Algorithm 1. The cross-sectional dimensions (width and height) are sampled uniformly within their respective discrete domains. For the tensile reinforcement layout, a random number of columns $n_c$ is first selected. Then, for each selected column, its position $i_c$ and a list of bar diameters in descending order are chosen, with a progressively decreasing maximum diameter to help ensure the feasibility of the resulting reinforcement configuration. Unlike rejection-based methods, Algorithm 1 directly generates $N_{pop}$ feasible individuals without requiring post-generation filtering.

<figure><img src="image_9.png" /><figcaption>**Fig. 6** Illustrative examples of initialization phases: a Simple random sampling in the search space; b Presented initialization method where individuals are exclusively created in the feasible space</figcaption></figure>
**Algorithm 1 Random generation of a feasible individual**

**Input:**

- Parameter ranges: $[B_{min}, B_{max}]$, $[H_{min}, H_{max}]$, $D_{\phi}$ and steps $\Delta B, \Delta H$

- Dimensions of the tensile reinforcement grid: $N_L$ (rows) and $N_{C,sym}$ (columns)

**Output:** Individual $\mathbf{X} = (B, H, \phi) \in \mathcal{F}$, $\phi = (\phi_{i,j})$, $i \in \{1, ..., N_L\}, j \in \{1, ..., N_{C,sym}\}$

**Procedure:**

1. Sample width and height

Select $B \in \{B_{min}, B_{min} + \Delta B, B_{min} + 2\Delta B, ..., B_{max}\}$ uniformly at random

Select $H \in \{H_{min}, H_{min} + \Delta H, H_{min} + 2\Delta H, ..., H_{max}\}$ uniformly at random

2. Sample of feasible tensile arrangement $\phi$

$D_{\phi}^* \leftarrow D_{\phi} \setminus \{0\}$

$\phi_{i,j} \leftarrow 0, \forall i \in \{1, ..., N_L\}, j \in \{1, ..., N_{C,sym}\}$

$l_c \leftarrow \emptyset$

Select $n_c \in \{1, ..., N_{C,sym}\}$ uniformly at random

Repeat $n_c$ times:

$l_{\phi} \leftarrow \emptyset$

Select $i_c \in \{1, ..., N_{C,sym}\} \setminus l_c$ uniformly at random

$l_c \leftarrow l_c \cup \{i_c\}$

Do:

Select $n_{\phi} \in \{1, ..., N_L\}$ uniformly at random

Select $(\phi_{1,i_c}, ..., \phi_{n_{\phi},i_c}) \in (D_{\phi}^* \setminus l_{\phi})^{n_{\phi}}$ uniformly at random such that $\phi_{1,i_c} \ge \dots \ge \phi_{n_{\phi},i_c}$

$(\phi_{n_{\phi}+1,i_c}, ..., \phi_{N_L,i_c}) \leftarrow (0, ..., 0)$

$l_{\phi} \leftarrow l_{\phi} \cup \{\phi \in D_{\phi}^*, \phi > \phi_{1,i_c}\}$

$\mathbf{X} \leftarrow (B, H, \phi)$

While $\mathbf{X} \notin \mathcal{F}$ and $\phi_{1,i_c} \ne \min D_{\phi}^*$

Return $\mathbf{X}$

### 2.3.3 Tailored crossover

Generally, the crossover operator uses two randomly selected individuals from the population to create one or two offspring. These offspring inherit characteristics from the parents to varying degrees, depending on the implemented operator. While there are no strict rules for designing crossover operators, classical methods in the literature often exhibit the following properties: (1) stochasticity, and

(2) crossing two identical individuals results in an identical offspring. Beyond these considerations, the effectiveness of the crossover operator relies on the data representation and the specific problem being addressed (Siarry 2014).

In the presented framework, the crossover operator follows the Algorithm 2. It is an adaptation of the uniform operator used in binary-coded GA to the discrete parametrization of the problem at hand.
**Algorithm 2 Discrete Uniform Crossover**

**Input:**

- Dimensions of the tensile reinforcement grid: $N_L$ (rows) and $N_{c,sym}$ (columns)

- Parent individuals: $X_1 = (B_1, H_1, \phi^1), X_2 = (B_2, H_2, \phi^2) \in \mathcal{F}$
  where $\phi^1 = (\phi_{i,j}^1), \phi^2 = (\phi_{i,j}^2), \forall i \in \{1, ..., N_L\}, j \in \{1, ..., N_{c,sym}\}$

**Output:** Offspring $X = (B, H, \phi) \in \mathcal{S}, \phi = (\phi_{i,j}), \forall i \in \{1, ..., N_L\}, j \in \{1, ..., N_{c,sym}\}$

**Procedure:**

Select $B \in \{B_1, B_2\}$ uniformly at random

Select $H \in \{H_1, H_2\}$ uniformly at random

Select $\phi_{i,j} \in \{\phi_{i,j}^1, \phi_{i,j}^2\}$ uniformly at random, $\forall i \in \{1, ..., N_L\}, j \in \{1, ..., N_{c,sym}\}$

$X \leftarrow (B, H, \phi)$

Return $X$

**2.3.4 Tailored mutation**

Initially, the mutation operator involves selecting $\rho_{mut}\%$ of the population and randomly altering one or more characteristics of the individuals. Alternatively, a local exploration operator can replace the classical mutation. In both approaches, the mutation operator prevents premature convergence to sub-optimal solutions by ensuring population diversity throughout the iterations.

In the presented framework, the mutation operator based on local approaches is performed following the Algorithm 3.

For each individual to be mutated, the structural dimensions are randomly selected within a range centered on the initial dimensions. For the arrangement of tensile steel bars, $\phi$ is altered by targeting either a column (variable $mut = 1$) or a specific tensile bar (variable $mut = 2$). The operator explores options such as (1) increasing or decreasing steel diameters, (2) replacing a column by a closer one, and (3) add new rebars on void positions.
**Algorithm 3 Tailored Mutation**

**Input:**

- Parameter ranges: $[B_{min}, B_{max}]$, $[H_{min}, H_{max}]$, $D_{\phi}$ and steps $\Delta B, \Delta H$

- Dimensions of the tensile reinforcement grid: $N_L$ (rows) and $N_{C,sym}$ (columns)

- Feasible Individual: $\mathbf{X} = (B, H, \mathbf{\phi}) \in \mathcal{F}$ where $\mathbf{\phi} = (\phi_{i,j}), \forall i \in \{1, ..., N_L\}, j \in \{1, ..., N_{C,sym}\}$

- Set of all possible $N_L$-diameters combinations with repetition from $D_{\phi}: \text{Columns}$

**Function:** *SteelQuantity(C) → total steel quantity of a N<sub>L</sub>-tuple of diameters C*

**Output:** Mutated individual $\mathbf{X}^{mut} = (B^{mut}, H^{mut}, \mathbf{\phi}^{mut}) \in \mathcal{S}, \mathbf{\phi}^{mut} = (\phi_{i,j}^{mut}), \forall i \in \{1, ..., N_L\}, j \in \{1, ..., N_{C,sym}\}$

**Procedure:**

1. Mutation of width and height

Select $B^{mut} \in \{\max(\{B_{min}, B - \Delta B\}), B, \min(\{B_{max}, B + \Delta B\})\}$ uniformly at random

Select $H^{mut} \in \{\max(\{H_{min}, H - \Delta H\}), H, \min(\{H_{max}, H + \Delta H\})\}$ uniformly at random

2. Mutation of $\mathbf{\phi}$

$$D_{\phi}^{*} \leftarrow D_{\phi} \setminus \{0\}$$

$$\phi_{i,j}^{mut} \leftarrow \phi_{i,j}, \forall i \in \{1, ..., N_L\}, j \in \{1, ..., N_{C,sym}\}$$

Select $mut \in \{1,2\}$ uniformly at random

Select $i_C \in \{1, ..., N_{C,sym}\}$ uniformly at random

If $mut = 1$ Then:

If $\phi_{1,i_C} = 0$ Then:

Select $(\phi_{1,i_C}^{mut}, ..., \phi_{N_L,i_C}^{mut}) \in (D_{\phi}^{*})^{N_L}$ uniformly at random such that $\phi_{1,i_C}^{mut} \ge \dots \ge \phi_{N_L,i_C}^{mut}$

Else:

$C_{ref} \leftarrow (\phi_{1,i_C}, ..., \phi_{N_L,i_C})$

$C_{min} \leftarrow \max(\{C \in \text{Columns}, \text{SteelQuantity}(C) < \text{SteelQuantity}(C_{ref})\})$

$C_{max} \leftarrow \min(\{C \in \text{Columns}, \text{SteelQuantity}(C) > \text{SteelQuantity}(C_{ref})\})$

Select $(\phi_{1,i_C}^{mut}, ..., \phi_{N_L,i_C}^{mut}) \in \{C_{min}, C_{ref}, C_{max}\}$ uniformly at random

End If

Else:

Select $i_L \in \{1, ..., N_L\}$ uniformly at random

If $\phi_{i_L,i_C} = 0$ Then:

Select $r \in [0,1]$ uniformly at random

If $r < 0.8$ Then

$$\phi_{i_L, i_C}^{mut} \leftarrow 8mm$$

End If

Else:

$\phi_{min} \leftarrow \max(\{\phi \in D_{\phi}^{*}, \phi < \phi_{i_L, i_C}\})$

$\phi_{max} \leftarrow \min(\{\phi \in D_{\phi}^{*}, \phi > \phi_{i_L, i_C}\})$

Select $\phi_{i_L, i_C}^{mut} \in \{\phi_{min}, \phi_{max}\}$ uniformly at random

End If

End If

$$\mathbf{X}^{mut} \leftarrow (B^{mut}, H^{mut}, \mathbf{\phi}^{mut})$$

Return $\mathbf{X}^{mut}$
### 2.3.5 Tailored repair

The two previous operators generate new individuals randomly from the population but do not guarantee that the hard constraints are verified. Consequently, a repair operator is essential for correcting infeasible individuals while limiting alterations to characteristics that heavily influence their performance.

In the presented problem, structural dimensions and the tensile steel area significantly influence installation costs, carbon footprint, and structural performance. To address this,

a repair strategy based on Algorithm 4 has been developed to ensure compliance with spacing constraint defined in (9). The core principle involves replacing columns of bars with the smallest steel quantity by columns with even less steel, iteratively, until an arrangement satisfies the constraint. Although not detailed in Algorithm 4, compliance with hard constraint defined in (11) is straightforward and involves rearranging the bars within each column in descending order of their diameter.

<caption>**Algorithm 4 Tailored Repair**</caption>

**Input:**
- Diameter set: $D_{\phi}$
- Dimensions of the tensile reinforcement grid: $N_L$ (rows) and $N_{C,sym}$ (columns)
- Individual: $\mathbf{X} = (B, H, \mathbf{\phi})$ where $\mathbf{\phi} = (\phi_{i,j}), \forall i \in \{1, ..., N_L\}, j \in \{1, ..., N_{C,sym}\}$
- Set of all possible $N_L$-diameters combinations with repetition from $D_{\phi}$: Columns

**Function:**
- $\text{SteelQuantity}(C) \rightarrow \text{Total steel quantity of a } N_L\text{-tuple of diameters } C$
- $\text{SortIndex}(lC) \rightarrow \text{Sorted indices of a list of } N_L\text{-tuples of diameters } lC \text{ based on their total steel quantity, in ascending order}$

**Output:** Repaired individual $\mathbf{X}^{\text{rep}} = (B, H, \mathbf{\phi}^{\text{rep}}) \in \mathcal{F}, \mathbf{\phi}^{\text{rep}} = (\phi_{i,j}^{\text{rep}}), \forall i \in \{1, ..., N_L\}, j \in \{1, ..., N_{C,sym}\}$

**Procedure:**
$\phi_{i,j}^{\text{rep}} \leftarrow \phi_{i,j}, \forall i \in \{1, ..., N_L\}, j \in \{1, ..., N_{C,sym}\}$
If $\mathbf{X} \notin \mathcal{F}$ Then:
$C_i \leftarrow (\phi_{1,i}, ..., \phi_{N_L,i}), \forall i \in \{1, ..., N_{C,sym}\}$
$\text{SortedColumnsIndex} = \text{SortIndex}((C_1, ..., C_{N_{C,sym}}))$
For each $j$ in $\text{SortedColumnsIndex}$:
$l_{\phi} \leftarrow \{C \in \text{Columns}, \text{SteelQuantity}(C) < \text{SteelQuantity}(C_j)\}$
Repeat:
$C_{\text{trial}} \leftarrow \text{argmax}_{C \in l_{\phi}}(\text{SteelQuantity}(C))$
$l_{\phi} \leftarrow l_{\phi} \setminus C_{\text{trial}}$
$(\phi_{1,j}^{\text{rep}}, ..., \phi_{N_L,j}^{\text{rep}}) \leftarrow C_{\text{trial}}$
$\mathbf{X}^{\text{rep}} \leftarrow (B, H, \mathbf{\phi}^{\text{rep}})$
Until $\mathbf{X}^{\text{rep}} \in \mathcal{F}$ or $l_{\phi} = \{(0, ..., 0)\}$
If $\mathbf{X}^{\text{rep}} \in \mathcal{F}$ Then:
Exit For
End If
End For
End If
Return $\mathbf{X}^{\text{rep}}$
### 2.3.6 Performance indicator for optimization algorithm

In this section, we introduce the performance indicators used to assess the quality and diversity of the set of optimal solutions, referred as Pareto front $PF$.

The literature provides various metrics to quantitatively assess (1) the proximity of the obtained optimal solutions, referred as $PF_{GA}$, to the true Pareto front, noticed $PF_{true}$, (2) the dominance over other Pareto fronts, and (3) their distribution across the objective space (Van Veldhuisen 1999; Knowles and Corne 2002; Zitzler et al. 2003; Ishibuchi et al. 2015). Among the various performance indicators, the hypervolume HV is widely employed (Zitzler and Thiele 1998). As illustrated in Fig. 7, it measures the hypervolume in the objective space dominated by the obtained Pareto front $PF$ and bounded by a reference point $\rho$. This indicator effectively captures both the proximity of the obtained front to $PF_{true}$ and its distribution across the objective space.

In order to enhance the interpretability of the HV, the hypervolume ratio HVR is used in this study (Knowles and Corne 2002). As expressed in Eq. (14), this indicator equals 1 when $PF_{true}$ is identical to $PF_{GA}$. The use of HVR is particularly suited to this study because the search space $S$ is finite. This property allows for an exhaustive exploration of $S$, enabling the exact determination of $PF_{true}$. For problems where $S$ cannot be exhaustively explored within a reasonable time, HV can still be employed as a reliable metric (Shang et al. 2021).

$$HVR(PF_{true}, PF_{GA}, \rho) = \frac{HV(PF_{GA}, \rho)}{HV(PF_{true}, \rho)} \quad (14)$$

However, HV and HVR are not suitable for evaluating performance across all types of multiobjective problems. These indicators exhibit significant variability when small Pareto fronts are involved, particularly in cases where slight

<figure><img src="image_10.png" /><figcaption>Fig. 7. Illustrative example of hypervolume indicator $HV(PF, \rho)$ for a bi-objective minimization problem</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Illustrative example of IGD calculation for a bi-objective minimization problem</figcaption></figure>

changes in the positions of solutions are observed. This sensitivity makes them unsuitable for analyzing algorithm performance in non-conflicting multiobjective problems.

In such cases, the Inverted Generational Distance IGD (Coello Coello and Reyes Sierra 2004) with a modified distance calculation is adopted in this study, noticed as $IGD^+$. As illustrated in Fig. 8, IGD measures the average distance from each solution in $PF_{true}$ to its nearest element in $PF_{GA}$. This metric is therefore only suitable if $PF_{true}$ is known and finite. Early applications of this metric relied on Euclidean distance. However, studies have demonstrated that this approach can produce inconsistent results and have proposed an alternative distance calculation, as expressed in Eq. (15) (Ishibuchi et al. 2015).

$$d^+(z, a) = \sqrt{\sum_{i=1}^{p} (\max\{a_i - z_i, 0\})^2} \quad (15)$$

where $z = (z_i)$ and $a = (a_i)$, $i \in \{1, ..., p\}$ are, respectively, elements of $PF_{true}$ and $PF_{GA}$ for a p-objective optimization problem.

To mitigate the sensitivity of the indicator to scale differences among the optimization criteria, HVR and $IGD^+$ are evaluated within a normalized space.

# 3 Illustrative examples

## 3.1 Basic information

The presented optimization framework is applied to the detailed design of a RC beam subjected to three-point bending, as illustrated in Fig. 9. For this structure, two
<figure><img src="image_12.png" /><figcaption>Fig. 9 Beam subjected to three-point bending</figcaption></figure>

bi-objective optimization problems are considered, each associated with different actual load amplitudes, as summarized in Table 2. In accordance with NF EN 1990, these loads are factored to account for uncertainties in real operating conditions, leading to the corresponding limit states—ULS and SLS—as shown in Table 2 (AFNOR 2003). The self-weight of the beam is neglected in this case, as it is relatively small compared to the actual load amplitudes. However, it could be easily integrated into the loading scheme without compromising the methodology presented in Sect. 2. The first optimization problem focuses on balancing installation cost and carbon footprint, while the second addresses the trade-off between installation cost and crack width. In both scenarios, the proposed beam designs must comply with the requirements on structural performance at ULS and SLS, as detailed in 2.2.2, and feasibility aspects outlined in 2.2.3.

For the parametrization (summarized in Table 3), the problems consist of 14 discrete variable inputs, including the width and height of the beam section, as well as 12 diameters of tensile bars. The latter are arranged in a grid of 2 rows and 6 columns ($N_L = 2$ is fixed, while $N_{C,sym} = 6$ is calculated following (3)). In total, $S$ contains approximately $4.23 \times 10^{12}$ beam designs.

For the known parameters, the beam has a fixed span of 4 m. The ready-mix concrete used is specified as C30/37—CEM I—XC1—Dmax 20 mm—S3. Its mechanical behavior is modeled using the parabola—rectangle stress—strain relationship, as prescribed by NF EN 1992-1-1 (AFNOR 2005). The reinforcing steel is classified as B500B and

<table><caption>**Table 3** Discrete variable inputs and characteristics of the search space for the case study</caption><thead><tr><th>Variable inputs X</th><th>Search space</th><th>Size</th></tr></thead><tbody><tr><td>B</td><td>{20,30,40cm}</td><td>3</td></tr><tr><td>H</td><td>{30,40,50,60,70cm}</td><td>5</td></tr><tr><td>φ = (φ<sub>i,j</sub>)<br>where i ∈ {1,2}<br>,j ∈ {1,...,6}</td><td>D<sub>φ</sub><sup>12</sup> = { 0,8, 10,12,14,16<br>, 20,25,32mm }<sup>12</sup></td><td>9<sup>12</sup></td></tr><tr><td>X = (B, H, φ)</td><td>S</td><td>≈ 4.23 × 10<sup>12</sup></td></tr></tbody></table>

<table><caption>**Table 4** Hyperparameters of the optimization framework for $P_{\epsilon/CO_2}$ and $P_{\epsilon/w_k}$</caption><thead><tr><th>Problem</th><th>$N_{pop}$</th><th>$\lambda$</th><th>$\rho_{mut}$</th><th>$N_{eval}$</th></tr></thead><tbody><tr><td>$P_{\epsilon/CO_2}$</td><td>{300,500,700}</td><td>20%</td><td>1%</td><td>[1000,50000]</td></tr><tr><td>$P_{\epsilon/w_k}$</td><td>{300,500,700}</td><td>20%</td><td>1%</td><td>[1000,150000]</td></tr></tbody></table>

follows an elastoplastic behavior with linear hardening, also compliant with NF EN 1992-1-1 (AFNOR 2005). The diameter of the stirrups and ties for shear reinforcement is set at 6 mm. The unit cost for the supply and placement of concrete is €130/m³, with an associated carbon footprint of 200kgCO₂e/m³. For reinforcing steel, the unit cost is €1.50/kg, and its carbon footprint is 1.04 kgCO₂e/kg. The labor cost is assumed to be €65 per man-day, with productivity rates for bar placement and tying set at 2.5 bars per minute and 10 ties per minute, respectively. Finally, the crack width $W_k$ and maximum deflection $f_{max}$ are verified to ensure they do not exceed 0.4 and 16 mm, respectively. These thresholds are recommended by the NF EN 1992-1-1 (AFNOR 2005).

Regarding the implementation details, the presented optimization framework was entirely developed on the uncertainty quantification and optimization platform Uranie (Blanchard et al. 2019). Ranges of hyperparameters for the application of the optimization framework are presented in Table 4. A preliminary scatter search indicates that $N_{pop}$ and $N_{eval}$ play a significant role in quality, convergence, and stability of the optimization framework. Consequently, performances of the algorithm will be evaluated based on the variations in these hyperparameters.

<table><caption>**Table 2** Optimization problems addressed by the presented optimization framework</caption><thead><tr><th>Optimization problem</th><th>Actual load</th><th>Load at SLS</th><th>Load at ULS</th><th>Optimization criteria</th></tr></thead><tbody><tr><td>Installation Cost/Carbon footprint minimization $P_{\epsilon/CO_2}$</td><td>$F_1 = 200kN$<br>$F_{1,SLS} = 200kN$<br>($F_{1,SLS} = 1 \times F_1$)</td><td>$F_{1,ULS} = 270kN$<br>($F_{1,ULS} = 1.35 \times F_1$)</td><td>$F = (f_{\epsilon}, CO_2)$</td></tr><tr><td>Installation cost/crack width minimization $P_{\epsilon/w_k}$</td><td>$F_2 = 300kN$<br>$F_{2,SLS} = 300kN$<br>($F_{2,SLS} = 1 \times F_2$)</td><td>$F_{2,ULS} = 405kN$<br>($F_{2,ULS} = 1.35 \times F_2$)</td><td>$F = (f_{\epsilon}, W_k)$</td></tr></tbody></table>
## 3.2 Validation methodology

The validation of the optimization framework for this illustrative case involves assessing (1) the quality of $\mathcal{PF}_{GA}$, (2) convergence, and (3) stability. The quality is measured with the $IGD^+$ for $P_{\epsilon/CO_2}$ and HVR for $P_{\epsilon/W_k}$, with a reference point set to (1,1). $\mathcal{PF}_{true}$ is determined using a brute-force algorithm that exhaustively explores $\mathcal{F}$. The stability is assessed by running the optimization algorithm 30 times in order to obtain a distribution of HVR values. Convergence is analyzed by studying the evolution of HVR and its range depending on $N_{pop}$ and $N_{eval}$. The variation ranges of the two hyperparameters are specified in Table 4.

<figure><img src="image_13.png" /><figcaption>**Fig. 10** Optimal beam cross-section for $P_{\epsilon/CO_2}$ produced by the optimization framework (Hyperparameters for the optimization framework: $N_{pop} = 700, N_{eval} = 50000$)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 11** Convergence and stability of the optimization framework $N_{pop} = 300, 500, 700$ for $P_{\epsilon/CO_2}$ (solid lines: mean $IGD^+$ on 30 runs of GA)</figcaption></figure>

## 3.3 Results for the installation cost and carbon footprint minimization

Figure 10 presents the optimal solutions obtained by the proposed optimization framework. Due to the limited size of the Pareto front (only two solutions), the key characteristics of these optimal designs (including reinforcement configurations, beam cross-sectional dimensions, and compliance with soft constraints) are detailed. Notably, these solutions align with those identified using the brute-force method. The small size of the Pareto front is attributed to the low degree of conflict between the optimization criteria in this case: reducing both concrete and steel volumes simultaneously decreases $f_{\epsilon}$ and $CO_2$. Among the optimal solutions, the design with a lower steel-to-concrete ratio achieves the

<figure><img src="image_15.png" /></figure>
minimum installation cost, while a higher steel-to-concrete ratio results in a more favorable carbon footprint.

Regarding convergence, Fig. 11 shows that the average IGD⁺ decreases significantly with $N_{eval}$. For $N_{pop} = 300$, the average IGD⁺ stabilizes at approximately $4.78 \times 10^{-3}$ beyond $N_{eval} = 4 \times 10^3$. For $N_{pop} = 500$ and $700$, IGD⁺ converges to $0$ beyond $N_{eval} = 1 \times 10^4$. In terms of performance stability, significant IGD⁺ variations are observed for $N_{eval} < 10^4$ with a decrease of the min-max band widths as $N_{eval}$ decreases. Beyond $N_{eval} = 10^4$, this band width stabilizes at varying levels depending on $N_{pop}$. Stability appears to improve with larger population sizes, with $N_{pop} = 700$ achieving the most stable results.

In terms of computational efficiency, the algorithm successfully converges to $\mathcal{PF}_{true}$ with sufficient stability for $10^4$ evaluations and $N_{pop} = 500$. This represents a computational effort reduction by a factor of $4.23 \times 10^8$ compared to exhaustive exploration of $S$.

## 3.4 Results for the installation cost and crack width minimization

An example of $\mathcal{PF}_{GA}$ and $\mathcal{PF}_{true}$ for $P_{\epsilon/w_k}$ is presented in Fig. 12. With an HVR of 99.983%, the solutions in $\mathcal{PF}_{GA}$ are

overall very close to those in $\mathcal{PF}_{true}$ and are well distributed across the objective space. Examples of beam cross-sections shown in Fig. 12 highlight the efficiency of the optimization framework in exploring detailed beam designs in compliance with design code requirements with more than three different diameters of bars.

Figure 13 shows the variations in the stability and convergence of the algorithm as a function of $N_{pop}$ and $N_{eval}$. For the lowest values of $N_{eval}$, the performance is comparable to that of random search since the generated population has not undergone enough iterations to identify and exploit interesting regions of $S$. For $N_{eval} \in [10^3, 10^4]$, the HVR increases significantly surpassing the 97.5% threshold on average. Over this range, the HVR is greater when $N_{pop}$ is lower. Pour $N_{eval} > 5 \times 10^4$, the average HVR stabilizes to 99.983% without any significant difference between curves.

In terms of computational efficiency, the algorithm achieves convergence to $\mathcal{PF}_{true}$ with good stability after $5 \times 10^4$ evaluations for $N_{pop} = 500$, yielding a computational effort reduction by a factor of $8.5 \times 10^7$.

<figure><img src="image_16.png" /><figcaption>**Fig. 12** Comparison of $\mathcal{PF}_{GA}$ and $\mathcal{PF}_{true}$ and examples of optimal beam cross-section for $P_{\epsilon/w_k}$ (Hyperparameters for the optimization framework: $N_{pop} = 500$, $N_{eval} = 50000$)</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Fig. 13 Convergence and stability of the optimization framework $N_{pop} = 300, 500, 700$ for $P_{E/W_k}$ (solid lines: mean HVR on 30 runs of GA)</figcaption></figure>

# 4 Conclusion

This study presents a multiobjective constrained optimization methodology for the detailed design of RC structures with a formulation adapted to the combinatorial nature of reinforcement layout. It is based on the coupling of a calculation code that evaluates the compliance of the structure with design standards and a GA incorporating problem-specific operators. The design problem is specifically parametrized to enable the exploration of a wide range of reinforcement configurations by treating each diameter of the tensile bars as a variable. The methodology takes advantage of the high modularity of GA by introducing tailored operators designed to explore a search space encompassing complex reinforcement layouts, while ensuring compliance with both structural performance and constructability requirements.

Applied to the optimization problems of the detailed design of an RC beam subjected to a three-point bending load, the proposed framework demonstrates promising results. It achieves convergence and stability toward the reference Pareto front identified through a brute-force algorithm, while reducing the computational effort by a factor exceeding $8 \times 10^7$ compared with a complete enumeration of the search space. The diversity of the resulting designs highlights the strong exploratory capability of the algorithm, which converges toward a wide range of reinforcement layouts—including configurations with more than three different bar diameters—while still ensuring compliance with feasibility constraints.

In contrast with previous studies, which primarily aimed to extend the applicability of generic GAs to the optimization of large-scale RC structures by simplifying the problem

formulation, the presented methodology leverages the inherent modularity of GAs to efficiently navigate richer search spaces arising from a finer parametrization of the design problem. This approach enables adaptation to complex structures and provides enhanced robustness and flexibility to address diverse perspectives.

However, the optimization framework has three main limitations that require further study:

1. The case study demonstrates the feasibility of the methodology on a structure with one identified tensile reinforcement area. This approach could be extended to more complex problems.

2. The evaluation of optimization criteria can be improved. For installation costs, it is easy to include material and labor costs, such as for formwork. For environmental performance, the integration of a life cycle analysis software would be worth exploring.

3. The impact of hyperparameters and operators on the performance of optimization algorithms is not fully understood, especially for solving specific problems like the one studied in this paper. Sensitivity analysis would help to better understand the importance of hyperparameters, operators, and their interactions, specifically in the application of RC design.

# Appendix

This section provides an overview of the parameters involved in the formulation of the optimization problem. For clarity and quick reference, these parameters and their respective descriptions their variability are summarized in Table 5.
<table><caption>**Table 5** Parameters used for the formulation of the design optimization of RC member</caption><thead><tr><th>Category</th><th>Designation</th><th>Definition</th></tr></thead><tbody><tr><td colspan="3"><strong>Known inputs</strong></td></tr><tr><td>Geometric and loading conditions</td><td>L<br>F</td><td>Span length of the RC member (mm)<br>Actual load amplitude (kN)</td></tr><tr><td>Material properties</td><td>CM = (f<sub>ck</sub>, CEM, E, D<sub>max</sub>, C<sub>c</sub>)</td><td>Concrete mix formulation: Specifies the compressive strength f<sub>c</sub> (MPa), class of cement CEM, environmental class E, largest aggregate size D<sub>max</sub> (mm), consistency class C<sub>c</sub></td></tr><tr><td rowspan="4">Characteristics of steel bars</td><td>f<sub>yk</sub></td><td>Yield strength of tensile reinforcement (MPa)</td></tr><tr><td>D<sub>c</sub></td><td>Ductility class D<sub>c</sub> of the steel reinforcement</td></tr><tr><td>D<sub>φ</sub></td><td>Finite set of rebar diameter (mm)</td></tr><tr><td>φ<sub>t</sub></td><td>Diameter of shear reinforcement (mm)</td></tr><tr><td rowspan="4">Unitary cost and carbon footprints and productivity indicators</td><td>N<sub>L</sub></td><td>Number of rows of the tensile reinforcement configuration</td></tr><tr><td>c<sub>c</sub>, c<sub>CO<sub>2</sub>c</sub></td><td>Unitary cost (€/m<sup>3</sup>) and carbon footprint of concrete (kgCO<sub>2</sub>e/m<sup>3</sup>)</td></tr><tr><td>c<sub>s</sub>, c<sub>CO<sub>2</sub>s</sub></td><td>Unitary cost (€/kg) and carbon footprint of steel (kgCO<sub>2</sub>e/kg)</td></tr><tr><td>c<sub>labor</sub></td><td>Labor cost per man-day (€/man-day)</td></tr><tr><td rowspan="2">Serviceability and durability criteria</td><td>p<sub>p</sub>, p<sub>t</sub></td><td>Productivity for rebar placing and tying (min<sup>-1</sup>)</td></tr><tr><td>W<sub>perm</sub></td><td>Maximum allowable crack width (mm)</td></tr><tr><td></td><td>f<sub>perm</sub></td><td>Maximum allowable deflection (mm)</td></tr><tr><td colspan="3"><strong>Variable inputs</strong></td></tr><tr><td>Dimensions of the RC cross-section</td><td>B ∈ {B<sub>min</sub>, B<sub>min</sub> + ΔB, ..., B<sub>max</sub>}</td><td>Width of the RC member (mm)</td></tr><tr><td rowspan="2">Tensile reinforcement</td><td>H ∈ {H<sub>min</sub>, H<sub>min</sub> + ΔH, ..., H<sub>max</sub>}</td><td>Height of the RC member (mm)</td></tr><tr><td>φ = (φ<sub>i,j</sub>) where<br>i ∈ {1, ..., N<sub>L</sub>}</td><td>Diameters of the tensile reinforcement configuration, organized in a grid of N<sub>L</sub> rows and N<sub>C,sym</sub> columns (mm)</td></tr><tr><td></td><td>j ∈ {1, ..., N<sub>C,sym</sub>}</td><td></td></tr><tr><td colspan="3"><strong>Calculated parameters</strong></td></tr><tr><td rowspan="4">Positioning of tensile reinforcement</td><td>c<sub>h</sub>(φ, CM, φ<sub>t</sub>)</td><td>Horizontal and vertical cover for tensile rebars (mm)</td></tr><tr><td>c<sub>v</sub>(φ, CM, φ<sub>t</sub>)</td><td>Minimum allowable spaces between tensile rebars (mm)</td></tr><tr><td>a<sub>min</sub>(φ, CM)</td><td>Number of columns of the tensile reinforcement configuration</td></tr><tr><td>N<sub>C,sym</sub>(B<sub>max</sub>, D<sub>φ</sub>, CM, φ<sub>T</sub>)</td><td>Diameter of compressive rebars (mm)</td></tr><tr><td rowspan="3">Quantification of compressive and shear reinforcement</td><td>φ<sub>c</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)</td><td>Number of shear reinforcement sets</td></tr><tr><td>N<sub>st</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)</td><td>i<sup>th</sup> transverse spacing (mm)</td></tr><tr><td>s<sub>i</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)<br>with i ∈ {1, ..., N<sub>st</sub>}</td><td></td></tr><tr><td rowspan="7">Soft constraints parameters</td><td>A<sub>S</sub>(φ)</td><td>Total tensile reinforcement area (mm<sup>2</sup>)</td></tr><tr><td>A<sub>min</sub>(CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)</td><td>Boundaries of tensile steel ratios (mm<sup>2</sup>)</td></tr><tr><td>A<sub>max</sub>(CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)</td><td>Required tensile reinforcement area (mm<sup>2</sup>)</td></tr><tr><td>A<sub>min,ULS</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)</td><td>Mean strain difference between the tensile reinforcement and the concrete in the tensile zone</td></tr><tr><td>ε<sub>sm</sub> - ε<sub>cm</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)</td><td>Tensile reinforcement strain</td></tr><tr><td>ε<sub>s</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)</td><td>Crack width at loading condition (mm)</td></tr><tr><td>W<sub>k</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)</td><td>Maximal deflection (mm)</td></tr><tr><td>Hard constraints parameters</td><td>f<sub>max</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)</td><td>Minimum required section width for a given tensile reinforcement configuration (mm)</td></tr><tr><td rowspan="2">Construction times</td><td>B<sub>req</sub>(φ)</td><td>Time of rebar placing (min)</td></tr><tr><td>T<sub>p</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, p<sub>p</sub>, p<sub>t</sub>, B, H, φ)</td><td>Time of rebar tying (min)</td></tr><tr><td></td><td>T<sub>t</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, p<sub>p</sub>, p<sub>t</sub>, B, H, φ)</td><td></td></tr></tbody></table>
<table><caption>**Table 5 (continued)**</caption><thead><tr><th>Category</th><th>Designation</th><th>Definition</th></tr></thead><tbody><tr><td rowspan="3">Optimization criteria</td><td>f<sub>E</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, c<sub>c</sub>, c<sub>s</sub>, c<sub>labor</sub>, p<sub>p</sub>, p<sub>t</sub>, B, H, φ)</td><td>Installation cost of the RC member (€)</td></tr><tr><td>CO<sub>2</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, c<sub>CO<sub>2,c</sub></sub>, c<sub>CO<sub>2,s</sub></sub>, p<sub>p</sub>, p<sub>t</sub>, B, H, φ)</td><td>Carbone footprint of the RC member (kgCO<sub>2</sub>e)</td></tr><tr><td>W<sub>k</sub>(L, F, CM, f<sub>yk</sub>, D<sub>c</sub>, φ<sub>t</sub>, B, H, φ)</td><td>Crack width at loading condition (mm)</td></tr></tbody></table>

**Author contributions** All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Paul Quéva, Ludovic Jason, and Gilles Arnaud. The first draft of the manuscript was written by Paul Quéva, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

**Funding** Open access funding provided by Commissariat à l'Énergie Atomique et aux Énergies Alternatives. The authors did not receive support from any organization for the submitted work.

**Data availability** Datasets generated in the presented study are available from the corresponding author on reasonable request.

## Declarations

**Conflicts of interest** The authors have no relevant financial or non-financial interests to disclose.

**Replication of results** The original contributions presented on this study are included in the article, further inquires can be directed to the corresponding author.

**Consent to participate** Not applicable.

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