Engineering-Driven Approach for the Structural Design of Geometrically Complex Modular Artificial Reefs

Abstract

Over the years, artificial reefs with diverse forms and functions have become increasingly important in maritime engineering and ecosystem restoration plans, with socio-economic and ecological impacts on marine ecosystems. However, due to the increasing complexity of designs and durability requirements, the current structural design approaches lack adequate documentation and standardization. This work addresses this challenge by detailing an engineering-driven approach for the structural design of artificial reef structures made of reinforced concrete or composite steel and concrete. This study establishes the premises for the structural design, followed by the quantification of actions based on standards and recommendations for marine structures. Hydrodynamic and numerical models were utilized to assess the effects of these actions on the structure. A cross-section organic design was then implemented, followed by a parametric study exploring various structural and material combinations for optimization. This study validates the developed design methodology combining hydrodynamic actions and strength analysis for complex modular artificial reef structures of 3 to 15 m size, specifically tailored for open waters. The results confirm the applicability and reliability of the developed design methodology, demonstrating its potential for guiding future numerical and experimental studies on modular artificial reef structures in open waters.

1. Introduction

In recent years, there has been a growing interest in the development of marine infrastructures and technologies to harness maritime resources [1,2,3,4,5,6]. These efforts aim at supporting the expanding global population and promoting the sustainability of human activities [7,8,9]. Significant strides have been made, ranging from the creation of modular floating structures for energy generation and artificial energy islands, to the development of modular artificial reef structures dedicated to the protection and restoration of marine ecosystems [10,11,12,13,14]. In fact, artificial reefs (ARs) might contribute to multiple other purposes such as increasing fisheries yield [15], recreational dive tourism [16], improving surf conditions [17], coastal protection [2,18,19], and preventing bottom trawling [20]. Some authors have highlighted the importance of designing ARs specifically for the ecosystems projected to be enhanced [8,12]. However, according to the literature, the purpose for which artificial reefs have been created is seldom revealed [6].

Despite all the advantages, these innovative structures face numerous challenges characteristic of a new era of coastal and maritime engineering [21,22,23]. These challenges include structural complexity and durability requirements, complex hydrodynamic interactions with ocean currents, and suitability for sea operations. There is an urgent need for new approaches that not only address these problems effectively but also prioritize ecologically friendly solutions, bio-receptiveness, efficiency, and resilience. This is essential for fostering a sustainable and balanced relationship between human activities and the environment [4,21,24,25,26,27,28].

Over time, the design of ARs has predominantly relied on empirical methods, guided by subjective judgments and assessments. In some cases, this approach was limited to sinking structures without the adoption of a clear design process. However, growing interest in artificial reef technology led to the development of several guidelines and recommendations [29,30,31,32], as well as frameworks for the development of purpose-oriented ARs. In general, these frameworks address functional aspects and provide a global view of the entire process and, to some extent, also address the structural design, albeit superficially. Recently, Vogler proposed a conceptual framework to design artificial reefs, which includes a design component that focuses mainly on geometric design, materials, and manufacturing [33,34]. Carral et al. (2022) proposed a new general and integrated conceptual framework to guide the design of AR units by adopting an ecosystem ecology perspective [35]. The flowchart is composed of three main parts: (i) marine ecosystem ecology model, (ii) AR-ecosystem index transformation, and (iii) the stability analysis. Although there is no detailed framework for the structural design, the general methodology highlights this aspect. The authors presented also a methodology based on geographic information systems (GISs) to determine the best area for installing ARs in a specific estuary considering the geomorphological, hydrodynamic, and bathymetric characteristics of the area, as well as the distance to the harbour and economic factors [36]. ARs designed for surfing show a robust and heavy morphology, similar to breakwaters, following similar design principles [17]. A preliminary design achieved step by step, making use of the theory and state of the art of multifunctional artificial reefs for surfing, was proposed by Voorde [17,26,37,38]. On the other hand, the geometry complexity of ARs is also an important aspect in other cases, and some works have found that increasing reef complexity has a significant and positive impact on fish species flourishing, density, and increased biomass [39,40]. Therefore, in recent years there has been a demand for modular, slender, and complex geometries for ARs. As a result, more sophisticated structures need to be designed based on engineering methods using analytical, numerical, and experimental analysis to produce structures capable of withstanding the harsh conditions typically encountered in marine environments.

The development of artificial reefs from an engineering, reliability, and feasibility standpoint remains relatively unexplored in the literature. Given the growing importance of these structures in the rehabilitation of marine ecosystems, it is essential to integrate clear and technically sound engineering methodologies. While aspects related to geometric design are well established—covering conceptual requirements [30,41,42,43], as well as hydrodynamic and morphodynamic performance using numerical simulation [18,44,45,46,47,48,49,50,51,52] and physical modelling [26,53,54]—the structural design based on primary analytical and numerical methodologies is still insufficiently addressed. Existing studies that explore structural behaviour typically focus on advanced fluid–structure interaction (FSI) simulations [49,55,56]. This may hinder the replication of such structures in different geographies and scenarios, due to the lack of understanding of the overall relationships between general requirements including materials and action-effect relationships, and the design approach itself.

This work addresses this challenge by implementing an engineering-driven approach for the design of a geometrically complex and modular multifunctional artificial reef (MMFAR), allowing the safety verification and possibly the generalization of the methodology to other similar projects. The developed approach may also support subsequent stages of analysis based on CFD and FSI for detailed functional analysis.

2. Materials and Methods

2.1. Engineering-Driven Methodology

In this study, a systematic and engineering-driven approach was implemented having the structural design as the central process, based on analytical assumptions and simplified numerical models. The deployment site and the MMFAR requirements and functions were also considered as problem variables. Subsequent numerical analysis (CFD and FSI) and experimental characterization for detailed characterization of innovative materials and structural systems were also acknowledged. This approach is represented in Figure 1, and in this research the focus is mostly on processes 1 to 5, directly related to the structural design, summarized as follows:

• Concept and geometric design: in this stage, with the MMFAR objectives established, the process selects the site location and collects environmental data on physical and biological aspects. The design of shapes and dimensions is undertaken with the objective of meeting functional requirements. Stakeholders, including experts and the local community, participate actively.
• Constructive approach: the development of cross-sections and 3D models is initiated. It considers materials, structural systems, constructive processes, and durability specifications. These aspects must prevent marine pollution and establish a support that meets technical and functional requirements. Logistics and construction techniques are also considered.
• Site location and sea conditions: the analysis of the site location results in the identification of specific actions affecting the structure, such as dead, live, environmental, and accidental. Environmental actions are among the most important. The actions and load combinations follow the recommendations from ISO, API, DNV, and NORSOK [2,3,4,5]. Historical site data and prediction models are used to account for time and space variability.
• Structural model: the effects of the loads are obtained from a simplified numerical simulation. For that, the numerical model is discretized in bar elements, and a linear-elastic quasi-static analysis is performed. The materials, section properties, and supports are defined, along with load cases and combinations. The results are the axial (N), shear (V), flexural (M), and torsional (T) stresses for safety verifications.
• Cross-section design: determines the load capacity based on material strength and cross-section dimensions, checking limit states to assess performance and feasibility. A parametric study explores various structural combinations in order to identify the promising solutions. This approach optimizes the cross-section, eliminates less promising options, and encourages new structural systems.

2.2. Concept and Geometric Design

The concept of the MMFAR and the geometrical design were developed within the NEXTSEA project, as detailed by Maslov et al. (2024) [57]. The MMFAR was aimed at supporting marine restoration, enhancing fisheries, and promoting recreational diving [45,57]. In particular, the recovery of key species such as sea bass and seabream was targeted [57]. The geometric design considered also the marine protected area requirements, including site characteristics, bottom properties and habitats, and characteristics of the MMFAR resulting from its interaction with the flow, such as upwelling zones, velocities, and wake zones [45,57]. Figure 2 illustrates the evolution of the reef module geometry from module 1 to module 3. To ensure ease of transportation, the weight of each individual unit must not exceed 150 kg. The limb presents a variable cross-section, with a diameter of approximately 150 mm at the smallest section (at the connection end), increasing to around 196.5 mm at the junction with the central bulb. The modular units, united by connectors, allow various MMFAR configurations depending on the assembly scheme (niche, barrier, tunnel). The final assembled structures are reticulated, slender, and expandable, enabling the execution of various configurations from small niches to full reef barriers.

2.3. Constructive Approach

Considering the requirements and complex geometry of the MMFAR, it was necessary to design an efficient structural system to (i) allow adequate adjustment of the reinforcing elements within the module and ensure uniform cross-sectional stress distribution as well as a minimum nominal reinforcement cover; (ii) ensure continuity of the internal metallic reinforcement elements and between elements, to facilitate electrical conduction for added functionalities (e.g., self-healing techniques); and (iii) the need to adopt a modular and adaptable construction system to allow the practical and simple assemblage, compatible to underwater operations.

Algorithmic thinking using parametric modelling tools as well as structural design approaches was used for the iterative design and analysis, as shown subsequently. Concrete was chosen for its versatility and cost-effectiveness, as well as its proneness to serve as a colonizing substrate [41,43]. Its brittle nature was countered by the incorporation of ductile reinforcements, such as hot-dip galvanised (HDG) steel which is technically and financially attractive while offering high corrosion resistance. Other options may include self-healing cementitious composites and fibre-reinforced polymers, although the design of these materials was not addressed in this study.

In this sense, different structural systems for the reef module were proposed, as shown in Figure 3: (i) a reinforced concrete structure approach (RC) based on the use of galvanized steel rebars along each limb of the unit and (ii) a composite steel and concrete structure (CS) where a HDG steel tube was adopted, aligned with the geometric centroid of each limb. The tube was perforated to allow the concrete to flow and fill the inner part of the tube, increasing the bond between the materials and providing a dead weight to counteract buoyancy.

The adoption of straight reinforcement elements was tried; however, the accommodation of the reinforcement elements in the geometry was found to be hard, in particular trying to guarantee the minimum concrete cover. Curved reinforcement elements, although requiring additional forming procedures, were easier to adapt to the complex geometry of the module, ensuring better distribution of the reinforcement area along the cross-section and providing adequate reinforcement cover.

The extension of the tube or rebar beyond the edges of each module was considered to support the connection system, particularly favourable in the case of the solution adopting the steel tubes as shown in Figure 4.

2.4. Site Location and Sea Conditions

As a case study, the developed MMFAR was assumed to be deployed in a Marine Protected Area (MAP), in Esposende—Portugal. This site represents a promising location for the implementation of MMFAR for ecosystems restoration and protection, and for recreational or scientific diving. Figure 5 shows the geographical location of the Littoral North Marine Park, as well as the bathymetry in the area. The ideal installation depth of the MMFAR was defined as 20 m, allowing recreational diving and avoiding the direct effect of high waves. The present study is related mainly to the structural design of an underwater structure; therefore in

Table 1. Wave height and period for different sea conditions [58,59,60,61].

Characteristic Event Ew	Typical Storm Condition Ew,storm	Extreme Storm Condition Ew,máxstorm	Abnormal Extreme Condition Ew,e
HS = 2.20 m	HS,storm = 4 m	HS,máxstorm = 9 m	HS,40 = 15.61 m
T = 7.89 s	T = 10 s	T = 14.6 s	Tp,40 = 20 s

the typical wave parameters in the area for different sea conditions of the study area are presented. Further details may be found in [58,59,60,61].

2.5. Structural and Hydrodynamic Model

The research conducted considered the development of a MMFAR niche, whose structure was composed of 16 units, each of which occupied a cube with an edge of 65 cm (Figure 6). With a width of approximately 2 by 2 m, the structure represents a bottom projection area of 4.807 m² and a total volume of 1.76 m³ [57]. The full assemblage was numerically studied based on a reticulated model of bars to evaluate the stresses for the structural mechanics analysis. For that, a finite element tool, with a 3D graphical user interface object-oriented, was used for the linear static analysis of the structure. In this process, the 3D elements were transformed into bar elements connected by nodes. The connection zone was not considered by simplification, and the structure was treated as monolithic with a perfect continuity between modules. The base was restricted with double supports, resembling real support conditions. Concrete and steel were considered for the sectional properties.

The structure was subjected to the self-weight and the hydrodynamic load (dF), which in turn was the result of the environmental actions (E) including waves (Ew), tidal currents (Ec), and marine growth (Em). In this process, the fluid velocities and accelerations were resolved into drag (dF_D) and inertial forces (dF_I) and applied as distributed loads on the members of the numerical model. The velocities (u_wave; u_current) and accelerations (∂u/∂t) of the fluid were calculated using the Stokes Wave Theory (

Table 2. 1st and 2nd order components of stokes wave theory [62,63].

Quantity	1st Order Component	2nd Order Component
Dispersion Relationship, $c^2$	${\displaystyle \frac{g}{k}}\tanh \hspace{0.17em}kd$	${\displaystyle \frac{g}{k}}\tanh \hspace{0.17em}kd$
Wave profile, η	${\displaystyle \frac{H}{2}} \mathrm{c}\mathrm{o}\mathrm{s}(kx-\omega t)$	${\displaystyle \frac{\pi H^2}{8L}}{\displaystyle \frac{\cosh \hspace{0.17em}kd}{{\sinh }^3\hspace{0.17em}kd}}\left[2+cosh 2kd\right]\cos \hspace{0.17em}2(kx-\omega t)$
Horizontal velocity, $u$	${\displaystyle \frac{\pi H}{T}}{\displaystyle \frac{\cosh \hspace{0.17em}ks}{\sinh \hspace{0.17em}kd}}\mathrm{c}\mathrm{o}\mathrm{s}(kx-\omega t)$	${\displaystyle \frac{3}{4c}}{\left({\displaystyle \frac{\pi H}{T}}\right)}^2{\displaystyle \frac{\cosh \hspace{0.17em}2ks}{{\sinh }^4\hspace{0.17em}kd}} \cos \hspace{0.17em}2(kx-\omega t)$
Vertical velocity, $v$	${\displaystyle \frac{\pi H}{T}}{\displaystyle \frac{\sinh \hspace{0.17em}ks}{\sinh \hspace{0.17em}kd}}sin(kx-\omega t)$	${\displaystyle \frac{3}{4c}}{\left({\displaystyle \frac{\pi H}{T}}\right)}^2{\displaystyle \frac{\sinh \hspace{0.17em}2ks}{{\sinh }^4\hspace{0.17em}kd}} \sin \hspace{0.17em}2(kx-\omega t)$
Horizontal acceleration, $\dot{u}$	${\displaystyle \frac{2{\pi }^{2}H}{T^2}}{\displaystyle \frac{\cosh \hspace{0.17em}ks}{\sinh \hspace{0.17em}kd}}sin(kx-\omega t)$	${\displaystyle \frac{3\pi }{2L}}{\left({\displaystyle \frac{\pi H}{T}}\right)}^2{\displaystyle \frac{\cosh \hspace{0.17em}2ks}{{\sinh }^4\hspace{0.17em}kd}} \sin \hspace{0.17em}2(kx-\omega t)$
Vertical acceleration, $\dot{v}$	${\displaystyle \frac{2{\pi }^{2}H}{T^2}}{\displaystyle \frac{\sinh \hspace{0.17em}ks}{\sinh \hspace{0.17em}kd}}cos(kx-\omega t)$	${\displaystyle \frac{3\pi }{4L}}{\left({\displaystyle \frac{\pi H}{T}}\right)}^2{\displaystyle \frac{\sinh \hspace{0.17em}2ks}{{\sinh }^4\hspace{0.17em}kd}} \cos \hspace{0.17em}2(kx-\omega t)$
Dynamic pressure, $p$	$pg{\displaystyle \frac{H}{2}}{\displaystyle \frac{\cosh \hspace{0.17em}ky}{\cosh \hspace{0.17em}kd}}\cos \hspace{0.17em}\left[k\left(x-ct\right)\right]$	$$\begin{gathered} {\displaystyle \frac{3}{4}}pg{\displaystyle \frac{\pi H^2}{L}}{\displaystyle \frac{1}{\sinh \hspace{0.17em}2kd}}\left[{\displaystyle \frac{\cosh \hspace{0.17em}2ks}{{\sinh }^2\hspace{0.17em}kd}}-{\displaystyle \frac{1}{3}}\right]\cos \hspace{0.17em}2\left(kx-\omega t\right) \\ -{\displaystyle \frac{1}{4}}pg{\displaystyle \frac{\pi H^2}{L}}{\displaystyle \frac{1}{\sinh \hspace{0.17em}kd}}\left[\cosh \hspace{0.17em}2ks-1\right] \end{gathered}$$

c: wave celerity, g: gravitational acceleration, k: wave number (k = 2π/L), L: wavelength, d: water depth, H_w: wave height, η: wave profile (surface elevation of the water), w: wave angular frequency (ω = 2πf, where f is the wave frequency), T: wave period, x: horizontal coordinate, t: time, u: horizontal velocity of the fluid, v: vertical velocity of the fluid, ρ: fluid density, p: dynamic pressure.

) [62,63]. The local current profile was combined vectorially with the wave kinematics to determine local fluid velocities and accelerations adopting the Morison Equation (1).

dF = dF_D + dF_I
dF_D = C_d × ½ × ρ × D × (u_wave + u_current)² dz
dF_I = C_m × ρ × (π × D²)/4 × (∂u/∂t) dt

(1)

In Equation (1), ρ represents water density, dz the depth interval, and D the diameter of the bar element. The hydrodynamic drag coefficient (C_d) and hydrodynamic inertia coefficient (C_m) were adopted following the recommendation of API [63,64]. The water current velocity profile (u_current) assumed a constant velocity of 1.2 m/s between the water surface level and half of the depth (z/2) and a constant velocity of 0.3 m/s between z/2 and the seabed. According to DNV-RP-C205, a value of 0.1 m was assumed for the marine growth [65]. Buoyant loads were also taken into account. The resulting forces were applied in x direction (d_XX). The load combinations followed the recommendations of ISO 19903:2006 [66]. The crest wave phase was considered since it corresponds to the maximum positive horizontal velocity and to zero horizontal acceleration. The vertical velocity is zero, and the vertical acceleration is maximum (in negative value).

For the analysis the characteristic event, the typical storm condition and the extreme storm condition were considered. In addition, abnormal environmental extreme events were also considered as accidental loads.

2.6. Actions and Effects

Table 3. Results in terms of the maximum efforts obtained for different load combinations.

Load Combination	Ur	N−	N+	V2-2−	V2-2+	V3-3−	V3-3+	M2-2−	M2-2+	M3-3−	M3-3+	T−	T+
	[mm]	[kN]		[kN]		[kN]		[kNm]		[kNm]		[kNm]
Type of support: double support
LC1_ULS(A)_Ewcm_dXX	0.09	−26.89	-	−1.17	1.00	−0.29	0.44	−0.15	0.13	−0.24	0.37	−0.08	0.06
LC2_ULS(B)_Ewcm_dXX	0.14	−46.37	-	−1.62	1.38	−0.73	1.08	−0.37	0.25	−0.24	0.52	−0.11	0.08
LC3_ULS(A)_Ewcm,storm_dXX	0.12	−28.32	-	−1.96	1.97	−0.51	0.81	−0.27	0.21	−0.47	0.61	−0.13	0.08
LC4_ULS(B)_Ewcm,storm_dXX	0.22	−49.32	-	−3.22	3.33	−1.14	1.76	−0.59	0.38	−0.84	1.16	−0.20	0.15
LC5_ULS(A)_Ewcm,máxstorm_dXX	0.42	−38.23	-	−8.53	8.53	−2.32	3.03	−1.07	1.02	−2.75	2.93	−0.44	0.43
LC6_ULS(B)_Ewcm,máxstorm_dXX	0.79	−67.99	-	−15.43	15.51	−4.05	5.88	−2.00	1.89	−5.08	5.47	−0.81	0.81
LC7_ALS_Ewcme_dXX	1.79	−86.05	-	−36.43	36.43	−11.14	12.65	−4.70	4.61	−12.46	12.81	−1.94	1.94
Critical section	-	Support leg		Connection Leg zone		Support leg		Central node		Connection Leg zone		Connection leg zone

U_r—maximum global displacement; N—maximum axial stress; V_2-2—maximum shear stress in 2-2; V_3-3—maximum shear stress in 3-3; M_2-2—maximum flexural stress in 2-2; M_3-3—maximum flexural stress in 3-3; T—maximum torsional stress.

shows the results obtained for each of the load cases and load combinations, according to the limit states identified as the most severe, in terms of maximum axial (N), shear (V), flexural (M), and torsional (T) stresses. Figure 7 identifies the critical sections, i.e., the locations where the maximum stresses occur. As shown, the worst scenario is the abnormal extreme condition. In this context, the maximum deflection obtained was 1.789 mm, the maximum axial stress was −86.05 kN, the maximum flexural stresses were −4.70 kN·m and 12.81 kN·m, the maximum negative and positive shear stresses were 36.43 kN, and the maximum negative and positive torsional stresses were 1.94 kN·m.

2.7. Design Considerations for Durability

According to NP-EN 1992-1-1 [67], the exposure class to be considered is XS2 (parts of permanently submerged marine structures), and according to NP-EN 1990 [68] the structural class 4 (S4) was considered, corresponding to a service life of at least 50 years. The durability of concrete structures depends on its composition and performance [67], and Annex E of NP-EN 1992-1-1 suggests that concrete for corrosion protection should comply with higher strength classes, C35/45 for exposure condition XS2. However, the National Annex for Portugal specifies C30/37* (or *C40/50 for CEM I or CEM II/A). According to NP-EN 206-1 Annex F [69], the recommended concrete composition includes a maximum w/c ratio of 0.45, minimum strength class C35/45, and a minimum cement dose of 320 kg/m³. The concrete nominal cover (c_nom) of 50 mm was adopted, in accordance with the National Annex of NP-EN 1992-1-1 for Portugal.

2.8. Cross-Section Design

The cross-section design was conducted considering the two alternative structural systems studied, the RC and the CS, and adopting the organic sectional analysis based on the behaviour of the materials involved for ultimate limit states (ULSs).

Due to the geometric complexity and varying cross-sections of the members extending from the central core (sphere), the analysis focused on a typical cross-section that is geometrically representative (Figure 8). The critical cross-section, selected for its likely impact on design limitations, was analysed to determine its axial, flexural, and shear strengths, based on the area, geometry, and material properties of both structural systems. A parametric study was conducted varying the strength class of both materials and tube dimensions. NP-EN 1992-1-1 [67] was used for the analysis of RC cross-section, and NP-EN 1994-1-1 was applied for the CS cross-section [70]. Following, the analytical methodology is presented, and a parametric study developed.

2.8.1. Axial Strength

Considering that the most unfavourable scenario is the one where the entire section is under tension, the required amount of reinforcement in the cross section was obtained, ensuring that the stress in the reinforcement remains below the yield stress after concrete cracking, according to Equation (2).

Regarding the compressive strength (N_pl,Rd,comp), the contribution of both materials (concrete and steel) was considered. For the tensile strength (N_pl,Rd,trac), only the contribution of the steel was considered. N_pl,Rd,comp and N_pl,Rd,trac were obtained for smallest and largest cross-section areas, according to Equations (3) and (4) (

Table 4. Axial strength of the cross-section according to the structural system.

	RC	CS
As,min	$f_{ctm}\times \left[A_c+\left({\displaystyle \frac{E_s}{E_c}}-1\right)\times A_{s,min}\right]=A_{s,min}\times f_{yk}$		(2)
Npl,Rd,trac	${ f}_{yd}\times A_s$	$A_a{ f}_{yd}+A_s{ f}_{sd}$	(3)
Npl,Rd,comp	${{(E}_s\times {\mathcal{E}}_s)\times A}_s+\left(A_c\times f_{cd}\right)$	$A_a{ f}_{yd}+{{\alpha }_{cc}A}_c{f}_{cd}+A_s{ f}_{sd}$	(4)

A_a—steel profile area; A_c—concrete area; A_s—steel reinforcement area; E—modulus of elasticity (s—steel, c—concrete); ε—strain. f_ctm—concrete tensile strength; f_cd—compressive failure stress of the concrete; f_yk—characteristic yield strength; f_yd—design yield strength of the structural steel; f_sd—yield stress of the longitudinal reinforcement; α_cc was equal to 1.0 (tubular steel section covered and filled with concrete).

), respectively.

Given that the cross-sections were subjected to approximately centred stresses, the average compressive strain was limited to ${\mathcal{E}}_{\mathrm{c}2}$. Note that, in the case of class C45/55 concrete, then ${\mathcal{E}}_{\mathrm{c}2}=2\mathrm{‰}$. Thus, by strain compatibility, ${\mathcal{E}}_{\mathrm{s}}=2\mathrm{‰}$. Considering that the reinforcement exhibits elastic behaviour until yielding, the maximum tension to which the reinforcement can be subjected is given by σ_s = E_s × ε_s.

Additionally, the steel contribution ratio (δ), which must lie between 0.2 and 0.9 in the case of the CS structures, was also a criterion. This is one of the aspects that differentiates the design of a RC structure from a CS structure.

2.8.2. Shear Strength

In the absence of specific transverse reinforcement in a given section, the effectiveness of the concrete’s contribution to shear strength diminishes under higher loads. Consequently, the longitudinal reinforcement assumes an important function in the transfer of shear forces. This scenario is particularly relevant in cases where the implementation of shear reinforcement is either impracticable due to spatial constraints or challenging to execute, such as in narrow elements or sections with intricate geometries, as is the case of the MMFAR structure under study.

Thus, the shear strength, V_Rd, of the cross-section was obtained by considering the contributions of concrete and longitudinal reinforcement, based on NP-EN 1992-1-1 [67], as detailed in

Table 5. Shear strength of the cross-section for both structural systems.

$V_{Rd}$	$V_{Rd,c}+V_{Rd,sl}$	(5)
$V_{Rd,c}$	$\left[C_{Rd,c}·k{\left(100·{\rho }_l·f_{ck}\right)}^{{\displaystyle \frac{1}{3}}}+k_1·{\sigma }_{cp}\right]·b_w·d$	(6)
$V_{Rd,sl}$	$A_{sl}·f_{yd}$	(7)

V_Rd,c—design shear resistance of the member without shear reinforcement; V_Rd,sl—design “shear” resistance of the longitudinal reinforcement, C_Rd,c—coefficient of shear strength of concrete (0.18/γ_c, with γ_c = 1.5), r_l—geometric ratio of longitudinal reinforcement (A_sl/b_wd), b_w—cross-section diameter, d—effective section height, k₁—shear behaviour coefficient, k = 1 + √(200/d) ≤ 2, σ_cp—compressive stress of concrete due to normal forces (N_Ed/A_c < 0.2f_cd), N_Ed—axial force in the cross-section due to loading, Ac—area of concrete cross section; A_sl—area of longitudinal steel reinforcement f_ck—characteristic compressive strength, f_cd—compressive failure stress of the concrete; f_yd—design yield strength of the structural steel.

by Equations (5)–(7). In this context, the smallest cross-section, located near the connection zone, was considered the critical cross-section, as it is where the maximum shear strength occurs, as indicated in Figure 7.

2.8.3. Flexural Strength

Considering the need to guarantee ductile failure, the minimum reinforcement A_s,min was obtained according to Equation (9). The corresponding cracking flexural stresses (M_cr) were computed by Equation (8).

Table 6. Flexural cracking strength (M_cr), flexural strength (M_Rd) and minimum reinforcement (A_s,min) of the cross-section for both RC and CS structural systems.

	RC Structure	CS Structure
Mcr	${\mathrm{M}}_{\mathrm{c}\mathrm{r}}={\displaystyle \frac{{\mathrm{f}}_{ctm}\times \overline{{\mathrm{I}}_{\mathrm{x}}}}{\overline{{\mathrm{y}}_{\mathrm{g}}}}}={\displaystyle \frac{{\mathrm{f}}_{ctm}\times \frac{\mathsf{\pi }{\mathrm{D}}^4}{64}}{\frac{{\mathrm{A}}_{\mathrm{c}}\times \frac{\mathrm{D}}{2}+\left(\mathsf{\alpha }-1\right){\mathrm{A}}_{\mathrm{s}}\times \mathrm{a}}{\mathsf{\pi }\times {\mathrm{r}}^2+\left(\mathsf{\alpha }-1\right){\mathrm{A}}_{\mathrm{s}}}}}$		(8)
As,min	${\mathrm{A}}_{\mathrm{s},\mathrm{m}\mathrm{i}\mathrm{n}}={\displaystyle \frac{{\mathrm{A}}_{\mathrm{c}\mathrm{c}}\times ⌊\frac{4\mathrm{r}}{3}\left(\frac{{\sin \hspace{0.17em}\left(\frac{\mathsf{\theta }}{2}\right)}^3}{\mathsf{\theta }-\sin \hspace{0.17em}\mathsf{\theta }}\right)-\left(\mathrm{d}-\mathrm{x}\right)⌋}{\mathsf{\alpha }\times \left(\mathrm{d}-\mathrm{x}\right)}}$		(9)
MRd	${\mathrm{F}}_{\mathrm{c}}\times \mathrm{z}$ ❖ ${\mathrm{F}}_{\mathrm{c}}=\left[{\mathrm{r}}^2{\cos }^{-1}\hspace{0.17em}\left(1-{\displaystyle \frac{\mathrm{h}}{\mathrm{r}}}\right)-\left(\mathrm{r}-\mathrm{h}\right)\sqrt{2\mathrm{r}\mathrm{h}-{\mathrm{h}}^2}\right]\mathsf{\eta }{\mathrm{f}}_{\mathrm{c}\mathrm{d}}$ ❖ $\mathrm{z}={\displaystyle \frac{4 r}{3}}\left({\displaystyle \frac{{\sin \hspace{0.17em}\left(\frac{\theta }{2}\right)}^3}{\theta -\sin \hspace{0.17em}\theta }}\right)$ ❖ $\theta =2{\cos }^{-1}\hspace{0.17em}\left({\displaystyle \frac{r-x}{r}}\right)$	${\mathrm{F}}_{\mathrm{c}}{\times {\mathrm{z}}_{\mathrm{c}}+\mathrm{F}}_{\mathrm{s}\mathrm{c}}\times {\mathrm{z}}_{\mathrm{s}\mathrm{c}}$ ❖ ${\mathrm{F}}_{\mathrm{c}}=\left(\left[{\mathrm{R}}^2{\cos }^{-1}\hspace{0.17em}\left(1-{\displaystyle \frac{\mathrm{h}}{\mathrm{R}}}\right)-\left(\mathrm{R}-\mathrm{h}\right)\sqrt{2\mathrm{r}\mathrm{h}-{\mathrm{h}}^2}\right]-\left[2{\cos }^{-1}\hspace{0.17em}\left({\displaystyle \frac{\mathrm{R}-\mathrm{h}}{\mathrm{r}}}\right)\mathrm{r}\mathrm{t}\right]\right)\mathsf{\eta }{\mathrm{f}}_{\mathrm{c}\mathrm{d}}$ ❖ ${\mathrm{F}}_{\mathrm{s}\mathrm{c}}=2{\cos }^{-1}\hspace{0.17em}\left({\displaystyle \frac{\mathrm{R}-\mathrm{h}}{\mathrm{r}}}\right)\mathrm{r}\mathrm{t}\times {\mathrm{f}}_{\mathrm{y}\mathrm{d}}$ ❖ ${\mathrm{z}}_{\mathrm{c}}={\displaystyle \frac{4\mathrm{R}}{3}}\left({\displaystyle \frac{{\sin \hspace{0.17em}\left(\frac{\mathsf{\theta }}{2}\right)}^3}{\mathsf{\theta }-\sin \hspace{0.17em}\mathsf{\theta }}}\right)+\left({\displaystyle \frac{{\mathrm{r}}^2(\sin \hspace{0.17em}\mathsf{\beta }-\sin \hspace{0.17em}\left(-\mathsf{\beta }\right)}{\mathrm{r}\left(2\mathsf{\pi }-\mathsf{\beta }\right)}}\right)$ ❖ ${\mathrm{z}}_{\mathrm{s}\mathrm{c}}={\displaystyle \frac{\mathrm{r}\times \sin \hspace{0.17em}\left(\frac{\mathsf{\beta }}{2}\right)}{\frac{\mathsf{\beta }}{2}}}+\left({\displaystyle \frac{{\mathrm{r}}^2(\sin \hspace{0.17em}\mathsf{\beta }-\sin \hspace{0.17em}\left(-\mathsf{\beta }\right)}{\mathrm{r}\left(2\mathsf{\pi }-\mathsf{\beta }\right)}}\right)$ ❖ $\mathsf{\theta }=2{\cos }^{-1}\hspace{0.17em}\left({\displaystyle \frac{\mathrm{R}-\mathrm{x}}{\mathrm{R}}}\right)$ ❖ $\mathsf{\beta }=2{\cos }^{-1}\hspace{0.17em}\left({\displaystyle \frac{\mathrm{R}-\mathrm{x}}{\mathrm{r}}}\right)$	(10)
As	${\mathrm{A}}_{\mathrm{s}}={\displaystyle \frac{{\mathrm{F}}_{\mathrm{c}}}{{\mathsf{\sigma }}_{\mathrm{s}}}}$	${\mathrm{A}}_{\mathrm{s}\mathrm{t}}={\displaystyle \frac{{\mathrm{F}}_{\mathrm{c}}+{\mathrm{F}}_{\mathrm{s}\mathrm{c}}}{{\mathsf{\sigma }}_{\mathrm{s}}}}$	(11)
Condition	${\epsilon }_{cu3}$$=3.5‰|{\epsilon }_s$$=“\infty ”$|λ = 0.8|η = 0.9	${\epsilon }_{cu3}$$=3.5‰|{\epsilon }_s$$=“\infty ”$|λ = 1.0|η = 0.85
Validation	${\epsilon }_c$$={\epsilon }_{cu3}$$\mathrm{and} {\epsilon }_{s,i}>{\epsilon }_{yd}$

Mcr—cracking flexural stress; f_ctm—mean concrete axial tensile strength; $\overline{{\mathbf{I}}_{\mathbf{x}}}$—equivalent moment of inertia of the section; $\overline{{\mathbf{y}}_{\mathbf{g}}}$—distance from the centroid of the section to the outermost fibre; Ac—cross sectional area of concrete; D—diameter of the circular section, r or R—radius of the concrete circular section; t—wall tube thickness; Ei—materials Young’s Modulus; α—Es/Ec, a—distance from the centroid of the reinforcement to the geometric centre of the section; As—cross sectional area of reinforcement, Acc—compressive area of the concrete section; Ast—tensile area of the steel section; θ—angle of the compressed concrete section; β—angle of the compressed steel section; r—tube radius; d—effective depth of a cross-section; x—neutral axis depth; h and l—effective height of the compression zone; Fc—concrete compressive force; Fsc—steel compressive force; fcd—design compressive strength of concrete; η—effective strength; fyd—design tensile strength of steel; z—lever arm of internal forces; σ_s—steel stress; ε_cu3—ultimate compressive strain in the concrete; ε_s—strain in the steel; ε_c—strain in the concrete.

shows the main equations and validation criteria related to the flexural behaviour of the cross-section for each structural system. Figure 9 shows the strains and stresses developed in both cross-section types assuming ULS and the simplified Bernoulli conditions for simple flexure.

The approach consisted of determining the equilibrium of the cross-section stresses that maximizes the flexural strength to verify the safety condition, M_Rd > M_Ed. The problem was solved through the implementation of a flexural inverse analysis, which identified the critical cross-section as the one located in the proximity of the connection zone, since it shows simultaneously the minimum area and the maximum stress, as illustrated in Figure 7.

The value of ${\mathcal{E}}_{\mathrm{s}}$ was adopted to ensure the plastic regime and the optimal position of the neutral axis could be calculated. For that, the rectangular stress block for the distribution of stresses in the concrete and an elastic-perfectly plastic diagram for steel (${\mathcal{E}}_{\mathrm{c}\mathrm{u}3}, {\mathcal{E}}_{\mathrm{s}=\infty }$, l and h) were assumed. Subsequently, the required reinforcement to balance the tensile and compression forces in the cross-section was analysed, and the M_Rd was calculated according to Equations (10) and (11). The phenomenon of cross-section steel saturation was considered, whereby an increase in A_s does not necessarily result in a proportional increase in flexural strength due to cross-section limitations. The binary arms z, z_c, and z_sc and the parameter that translates the geometric relationship between the concrete and steel cross sections and the position of the neutral axis θ and β are shown in Table 6. Design assuming reinforcement in the elastic regime should be avoided, as it risks uneconomical solutions and brittle failure.

3. Results and Discussion

3.1. Parametric Study of RC and CS Design Without Cross-Sectional Diameters Modifications

The results of the parametric study conducted in terms of axial and flexural strength for both structural systems, where concrete and steel classes were varied, are shown in

Table A1. Results of the parametric analysis of RC alternatives.

Materials (Concrete + Steel)	As,min			Tensile Strength	Compressive Strength	Flexural Strength			MEd
	As,req	Øeff	As,eff	Npl,Rd	Npl,Rd	As,satur	Mpl,Rd	Øsatur	ULS(B) Ewcm	ULS(B) Ewcm,storm	ULS(B) Ewcm,maxstorm	ALS Ewcme
	cm2	mm	cm2	kN	kN	cm2	kNm	mm	kNm	kNm	kNm	kNm
C30/37 + S400	2.28	1Ø20	3.14	109.22	462.65	1.75	2.77	1Ø16	0.52	1.16	5.47	12.81
C30/37 + S500	1.81	1Ø16	2.01	87.39	440.82	1.25	2.60	1Ø16	0.52	1.16	5.47	12.81
C30/37 + DIN 4.8	2.88	1M22	2.99	83.20	436.63	2.40	2.92	1M20	0.52	1.16	5.47	12.81
C30/37 + DIN 5.8	2.28	1M20	2.41	83.83	437.26	1.75	2.77	1M18	0.52	1.16	5.47	12.81
C30/37 + DIN 8.8	1.41	1M16	1.54	85.70	439.13	0.83	2.37	1M12	0.52	1.16	5.47	12.81
C35/45 + S400	2.52	1Ø20	3.14	109.22	521.55	2.01	3.23	1Ø16	0.52	1.16	5.47	12.81
C35/45 + S500	2.00	1Ø16	2.01	87.39	499.73	1.46	3.23	1Ø16	0.52	1.16	5.47	12.81
C35/45 + DIN 4.8	3.19	1M24	3.46	96.28	508.61	2.80	3.40	1M22	0.52	1.16	5.47	12.81
C35/45 + DIN 5.8	2.52	1M22	2.99	104.00	516.33	2.01	3.23	1M20	0.52	1.16	5.47	12.81
C35/45 + DIN 8.8	1.55	1M18	1.89	105.18	517.52	0.98	2.76	1M14	0.52	1.16	5.47	12.81
C45/55 + S400	3.01	1Ø20	3.14	109.22	639.36	2.62	4.16	1Ø20	0.52	1.16	5.47	12.81
C45/55 + S500	2.39	1Ø20	3.14	136.52	666.67	1.87	3.89	1Ø16	0.52	1.16	5.47	12.81
C45/55 + DIN 4.8	3.81	1M27	4.52	125.77	655.92	3.60	4.37	1M27	0.52	1.16	5.47	12.81
C45/55 + DIN 5.8	3.01	1M24	3.46	120.35	650.49	2.62	4.16	1M22	0.52	1.16	5.47	12.81
C45/55 + DIN 8.8	1.85	1M18	2.41	134.12	664.27	1.27	3.55	1M16	0.52	1.16	5.47	12.81
C50/60 + S400	3.26	1Ø25	4.91	170.78	759.83	2.91	4.62	1Ø20	0.52	1.16	5.47	12.81
C50/60 + S500	2.58	1Ø20	3.14	136.52	725.57	2.08	4.33	1Ø20	0.52	1.16	5.47	12.81
C50/60 + DIN 4.8	4.12	1M27	4.52	125.77	714.82	4.00	4.86	1M27	0.52	1.16	5.47	12.81
C50/60 + DIN 5.8	3.26	1M24	3.46	120.35	709.40	2.91	4.62	1M22	0.52	1.16	5.47	12.81
C50/60 + DIN 8.8	2.00	1M20	2.41	134.12	723.17	1.41	3.95	1M16	0.52	1.16	5.47	12.81
C55/67 + S400	3.33	1Ø25	4.91	170.78	818.74	3.02	4.91	1Ø20	0.52	1.16	5.47	12.81
C55/67 + S500	2.64	1Ø20	3.14	136.52	784.48	2.14	4.56	1Ø20	0.52	1.16	5.47	12.81
C55/67 + DIN 4.8	4.22	1M27	4.52	125.77	773.73	4.19	5.51	1M27	0.52	1.16	5.47	12.81
C55/67 + DIN 5.8	3.33	1M24	3.46	120.35	768.30	3.02	4.91	1M24	0.52	1.16	5.47	12.81
C55/67 + DIN 8.8	2.05	1M20	2.41	134.12	782.08	1.43	4.11	1Ø16	0.52	1.16	5.47	12.81
C60/75 + S400	3.49	1Ø25	4.91	170.78	877.64	3.18	5.25	1Ø25	0.52	1.16	5.47	12.81
C60/75 + S500	2.77	1Ø20	3.14	136.52	843.38	2.24	4.85	1Ø20	0.52	1.16	5.47	12.81
C60/75 + DIN 4.8	4.42	1M27	4.52	125.77	832.63	4.43	5.59	1M27	0.52	1.16	5.47	12.81
C60/75 + DIN 5.8	3.46	1M24	3.46	120.35	827.21	3.18	5.25	1M24	0.52	1.16	5.47	12.81
C60/75 + DIN 8.8	2.15	1M20	2.41	134.12	840.98	1.49	4.35	1M16	0.52	1.16	5.47	12.81
C70/85 + S400	3.65	1Ø25	4.91	170.78	994.45	3.57	5.98	1Ø25	0.52	1.16	5.47	12.81
C70/85 + S500	2.89	1Ø20	3.14	136.52	961.19	2.50	5.47	1Ø20	0.52	1.16	5.47	12.81
C70/85 + DIN 4.8	4.62	1M30	5.52	153.60	978.27	5.00	6.40	1M30	0.52	1.16	5.47	12.81
C70/85 + DIN 5.8	3.65	1M27	4.52	157.22	981.89	3.57	5.98	1M27	0.52	1.16	5.47	12.81
C70/85 + DIN 8.8	2.24	1M20	2.41	134.12	958.79	1.65	4.89	1M18	0.52	1.16	5.47	12.81
C80/95 + S400	3.81	1Ø25	4.91	170.78	1113.26	3.99	6.75	1Ø25	0.52	1.16	5.47	12.81
C80/95 + S500	3.02	1Ø20	3.14	136.52	1079.00	2.79	6.18	1Ø20	0.52	1.16	5.47	12.81
C80/95 + DIN 4.8	4.82	1M30	5.52	153.60	1096.08	5.62	7.24	1M33	0.52	1.16	5.47	12.81
C80/95 + DIN 5.8	3.81	1M27	4.52	157.22	1099.70	3.99	6.75	1M27	0.52	1.16	5.47	12.81
C80/95 + DIN 8.8	2.34	1M20	2.41	134.12	1076.60	1.84	5.49	1M18	0.52	1.16	5.47	12.81
C90/105 + S400	3.97	1Ø25	4.91	170.78	1231.07	4.49	7.59	1Ø25	0.52	1.16	5.47	12.81
C90/105 + S500	3.14	1Ø20	3.14	136.52	1196.81	3.14	6.95	1Ø20	0.52	1.16	5.47	12.81
C90/105 + DIN 4.8	5.02	1M30	5.52	153.60	1213.89	6.32	8.15	1M33	0.52	1.16	5.47	12.81
C90/105 + DIN 5.8	3.97	1M27	4.52	157.22	1217.50	4.49	7.59	1M27	0.52	1.16	5.47	12.81
C90/105 + DIN 8.8	2.44	1M22	2.99	166.40	1226.69	2.07	6.16	1M20	0.52	1.16	5.47	12.81

Colour legend: ${\mathrm{A}}_{\mathrm{s},\mathrm{satur}} \ge {\mathrm{A}}_{\mathrm{s},\mathrm{eff}}; {\mathrm{Ø}}_{\mathrm{satur}} \ge {\mathrm{Ø}}_{\mathrm{eff}}$; ${\mathrm{A}}_{\mathrm{s},\mathrm{satur}} << {\mathrm{A}}_{\mathrm{s},\mathrm{eff}}; {\mathrm{Ø}}_{\mathrm{eff}} = {\mathrm{Ø}}_{\mathrm{satur}}$; M_Rd > M_{Ed,ALS_Ewcme}; M_Rd < M_{Ed,ALS_Ewcme}.

(Appendix A) and

Table A2. Results of the parametric analysis of CS alternatives.

Materials (Concrete + Steel)	Steel Tube		Steel Ratio 1	Compressive Strength	Tensile Strength	Flexural Strength	MEd	Reef Unit Weight	Reef Unit Cost
	Dext	tw	0.2 < d < 0.9	Npl,Rd,comp	Npl,Rd,trac	MRd	ALS_Ewcme
	[mm]	[mm]	[-]	[kN]	[kN]	[kN.m]	[kN.m]	[kg]	[€]
C30/37 S235	88.9	6	0.20	493.24	210.32	13.30	12.81	167.76	52.24
C30/37 S275	76.1	6.3	0.21	500.79	215.71	12.71	12.81	153.79	48.38
	88.9	6	0.23	529.04	230.19	14.77	12.81	167.76	54.47
C30/37 S355	76.1	5	0.22	515.20	227.30	13.28	12.81	153.93	42.55
	88.9	4	0.21	507.89	219.57	13.90	12.81	168.03	40.85
C30/37 S420	60.3	5	0.20	497.91	207.13	11.57	12.81	139.72	37.81
	76.1	4	0.21	509.91	219.73	13.09	12.81	154.05	39.23
	88.9	3.2	0.20	501.36	210.76	13.59	12.81	168.14	35.77
C35/45 S275	88.9	6	0.21	576.19	246.12	15.54	12.81	167.76	55.07
C35/45 S355	76.1	6	0.22	600.28	266.95	15.45	12.81	153.82	50.19
	88.9	5	0.22	600.78	269.61	16.66	12.81	167.89	49.76
C35/45 S420	76.1	5	0.22	604.80	268.92	15.64	12.81	153.93	47.91
	88.9	4	0.21	596.15	259.77	16.37	12.81	168.03	46.02
C45/55 S355	88.9	6	0.21	742.10	317.72	20.04	12.81	167.76	58.56
C45/55 S420	76.1	6	0.20	744.39	315.82	19.32	12.81	153.82	56.00
	88.9	5	0.21	747.33	318.98	20.24	12.81	167.89	55.51

¹ steel ratio for perforated section; colour legend: M_Rd > M_{Ed,ALS_Ewcme}; M_Rd ≈ M_{Ed,ALS_Ewcme}; M_Rd < M_{Ed,ALS_Ewcme}.

(Appendix B). Figure 10 and Figure 11 illustrate the correlation between the concrete compressive strength (f_c) and steel class and the obtained flexural (M_pl,Rd) and normal (N_pl,Rd) strengths based on Table A1 and Table A2. The plotting colour-code in the graphs is combined with the reinforcement configuration according to steel class.

• RC cross-section

The reinforcement rebar (1ØD) was of conventional steel classes S400 and S500 used in civil construction, and threaded rods (1MD) of DIN 4.8, 5.8, and 8.8 steel class. The minimum concrete strength class was C30/37.

In terms of axial strength, it appears that the most unfavourable combination of materials for determining A_s,min was concrete C90/105 and DIN 4.8 steel rebar. These require a minimum reinforcement of 5.02 cm², which can be fulfilled by a 30 mm threaded steel rod. As demonstrated, for all scenarios, the minimum reinforcement guaranteed that the compressive and tensile strengths (N_pl,Rd) (Figure 11) clearly exceed the maximum axial stress N_Ed (−86.05 kN, refer to Table 3) obtained from the numerical model.

In terms of flexural strength, the minimum reinforcement could be fulfilled using 1Ø8 mm (M_cr = 1.65 kN·m). The obtained A_s,min was in all cases below the A_s required for ULS. Figure 10 presents the flexural strength versus concrete strength for various structural combinations of RC cross-section. In general, the reinforced concrete critical cross-section showed limitations in terms of flexural strength, since it did not achieve safety for abnormal extreme conditions and, in some cases, extreme storm action (Figure 10). The lowest flexural strength, M_pl,Rd = 2.37 kN, was obtained for the combination of concrete class C30/37 and DIN 8.8 steel rebar, ensuring strength capacity only for the load combinations E_wcm and E_wcm,storm. The highest M_pl,Rd, 6.32 kN, was achieved for the concrete class C90/105 and DIN 4.8 steel rebar, meeting all requirements except the ALS_E_wcme load combination. As shown in Figure 10a, the M_pl,Rd decreases as the steel strength increases for the same class of concrete, as higher-strength steel reduces the saturation reinforcement area (A_s,satur) of the cross-section, reducing the lever arm of internal forces and decreasing the section’s flexural strength. This trend is attributed to the restriction of the cross-section diameter to a minimum of 150 mm, imposed by the geometric design. This highlights the importance of balancing steel reinforcement layout with steel strength in small sections to optimize structural performance. The most promising combinations of materials were those that resulted in A_s,satur (flexural) ≥ A_s,eff (axial) and Ø_satur (flexural) ≥ Ø_eff (axial), identified in green shading in Table A1. While the combinations identified in yellow coincide in diameter (Ø_eff = Ø_satur), these exhibit A_s,eff >> A_s,satur, disrupting the cross-section equilibrium and preventing reinforcement yielding. In this case, it would be necessary to reconfigure and optimize the cross-section to accommodate the reinforcement imposed by A_s,min, as discussed in Section 3.2 and Section 3.3.

Regardless of the concrete class, it seems that the structural combinations showing the best compromise between the strength of the materials, the position of the neutral axis, and the normal and flexural strength, include the use of DIN 4.8, as evidenced by the highest M_pl,Rd values shown in Table A1 and Figure 10a. In fact, the threaded rebar DIN 4.8 steel resulted in the most significant overlap between steel reinforcement diameters that satisfy both axial and flexural reinforcement requirements. For S500 steel, there is a central range in the graph where the effective steel area (A_s) remained constant, irrespective of the concrete strength (f_c), specifically within the range of f_c = 50 MPa to 90 MPa. This occurred due to a significant gap between A_s,pl and A_s,eff, making the choice of S500 inefficient for lower concrete strength classes where A_s,pl is significantly higher than A_s,eff. Furthermore, a higher concrete strength class results in the raise of the position of the neutral axis, increasing the A_s,pl and respective M_Rd.

Summarizing, the results showed that the tensile behaviour of the cross-section limits the amount of reinforcement that must be provided in the cross-section. It was observed that the minimum reinforcement required for a RC element in tension with a variable cross-section (largest section being the most critical) is higher than the one obtained for flexure (A_s,satur) (smallest section being the most critical). However, the flexural strength of the section, although optimised internally, appears to be the limiting strength, primarily due to the reduced cross-sectional dimension. The results indicate that it may be more advantageous to optimize the reinforcement area rather than merely increasing the steel strength, particularly in the case of sections with reduced diameter. This analysis shows that stronger materials do not always lead to proportional structural gains, especially when geometric constraints and internal force distributions are significant.

• CS cross-section

Tube diameters 60.3 mm, 76.1 mm, and 88.9 mm were selected to comply with the concrete cover and to offer a range of possible solutions for adapting the steel section to specific strength requirements. The minimum thickness of the tube wall was chosen to comply with the A_s, as well as to verify the requirement of a minimum contribution of steel (δ) for the non-perforated and perforated variants (two opposite holes with D_hole = D_ext/2). The minimum strength class for concrete was set to C30/37. In this case it was deemed unnecessary to expand the study to include high-strength concrete classes, as the combinations achieved with lower-strength concrete classes were found to be favourable. The minimum reinforcement was not a limitation when compared to the ULS analysis since A_s,min (0.41 cm²) was substantially lower when compared to A_s,pl.

In general, this structural solution easily achieved flexural strengths exceeding 12.81 kNm, ensuring compliance with the M_Rd > M_Ed criterion, even for the lower-strength concretes and steel classes as shown in Table A2 and Figure 10. As shown, for a given steel strength class and tube diameter, an increase in the concrete class resulted in a design that required a thicker tube, yielding an increase in M_pl,Rd. However, this increase was less relevant, since the material combinations analysed for lower-class concretes (e.g., C30/37) already met the safety criteria.

The geometric configuration that ensured the greatest nominal cover were associated to the tube with a diameter of 60.3 mm, which is important to prevent the corrosion caused by continuous exposure to seawater. However, in most cases, the adoption of this diameter and the maximum available thickness of 5 mm did not meet the required steel contribution ratio (δ). This criterion was only fulfilled when C30/37 concrete and S420 steel were combined. The parametric study showed that tube configurations with larger diameters (especially 88.9 mm) provided a greater number of efficient combinations, which is crucial for ensuring structural stability when the structure is subjected to significant horizontal forces. However, concrete cover in the connection region was small, and steel protection was reduced. Implementing additional protective measures or adjusting the section in this area should be considered.

The variety of structural solutions available for CS configuration resulted in a wider multitude of alternatives, withstanding the most extreme actions and ensuring optimal structural efficiency. A significant factor in this type of system is the D/t ratio, which directly impacts the M_pl,Rd. Additionally, the increase of fc was linked to the need for increased t_w, thus ensuring harmonious integration of materials and sufficient sectional resilience. Nevertheless, to have the section classified as mixed concrete–steel according to NP-EN 1994-1-1 [67], for increased fc the use of lower strength steels, such as S235, is not feasible.

The lowest M_pl,Rd values were observed in sections comprising lower strength steels, such as S235, while the highest values were observed in sections comprising high-strength steels, such as S420. The highest steel classes (S355 and S420) resulted in the most efficient use of the full cross-section, providing greater strength and better efficiency in materials usage. Moreover, higher compressive strengths of concrete (f_c) resulted in higher normal plastic strengths (N_pl,Rd), with a nearly linear relationship (Figure 11b).

Figure 12 presents the shear strength of both structural systems obtained for the most critical scenarios. As observed, the shear strength clearly exceeded the applied shear stress, therefore ensuring shear safety.

3.2. Optimization of the RC Cross-Section

In the previous sections it has been shown that the majority of combinations designed to optimise the structural efficiency of the original geometry invariably result in a RC cross-section imbalance. This, in turn, compromises the structure’s ability to withstand applied loads and increases the risk of obtaining failure modes where the reinforcement is in the elastic regime. In order to address this challenge, two optimisation approaches were explored, where the variation of the initial cross-section size was allowed. The first was aimed at minimising modifications to the cross-section, albeit with certain trade-offs. The second approach allowed the free resizing of the cross-section to meet the most demanding scenario. Considering the variable geometry of the cross-section, a minimum and maximum diameter will be defined for the smallest and largest cross-sections of the reef branch, respectively.

In approach 1, a diameter of 196.5 mm was considered to the cross-section, corresponding to the largest cross-section found the initial design. The A_s,min was then calculated, and an initial solution was determined (1ØD). This resulted in a real steel area (A_s,real), which served as a reference for establishing a maximum limit for the section diameter (D_max). For the flexural analysis, which was critical when the smaller cross-section was considered, the minimum viable diameter (D_min) was calculated to ensure that A_s,pl (flexural) matched A_s,real (axial). For each configuration, the flexural (M_Pl,Rd) and axial (N_pl,Rd) strengths were obtained.

Figure 13 presents the concrete and reinforcement combinations that meet the constraints of this approach for the RC cross-section. The upper graph illustrates the changes in minimum (D_min) and maximum (D_max) cross-sectional diameters correlated with different reinforcement configurations and concrete strengths. The lower graph displays the corresponding M_Pl,Rd and M_Ed. The plotting colour-code in the graphs distinguished the steel class.

The results show that the alteration of the cross-section dimensions resulted in a significant improvement of M_pl,Rd. Most of the material combinations exceeded the requirements of the load combination ULS(B)_E_{wcm,máxstorm}, but none met the abnormal combination requirements, ALS_E_wcme.

A common trend across most combinations is the need to increase the diameter of the initial cross-section to optimize performance. The analysis identifies combinations that result in minimal changes (−7% to 7%) in the cross-sectional diameters, maintaining the original design’s geometric effect. This reduction in material contributes to a more efficient and economical structural solution.

The maximum value found for D_min was 208 mm, occurring in the C35/45 + DIN 8.8 combination. This represents a 38.7% increase compared to the initial section (150 mm). On the other hand, the minimum value found for D_min was 140.19 mm, obtained for the C90/105 + 4.8 combination. D_max varied between 196.5 mm, for the C60/75 + DIN 5.8 combination, and 241.44 mm, for the C50/60 + S400 combination. The latter combination showed a 22.8% increase compared to the baseline design.

In approach 2, a free optimization was carried out to find the smallest cross-section and structural combination that guarantees the most challenging scenario.

For each combination of concrete and steel types, the reinforcement solution (1ØD) was imposed, and the minimum cross-section diameter (D_min) was found, ensuring that A_s,pl matches A_s,real, while satisfying both axial and flexural strength criteria. Subsequently, the maximum diameter (D_max) was calculated considering that the minimum reinforcement area (A_s,min) determined by axial stresses did not exceed the real reinforcement area (A_s,real).

Figure 14 illustrates the RC cross-section diameter obtained for each concrete strength class adopting the aforementioned procedure, showing the minimum and maximum section diameters and flexural strengths for different material combinations.

The minimum diameter (D_min) required decreases with the increase in the concrete strength class. The influence of the concrete strength on the section size is significant, with increments ranging between 23% (for material combinations including high fc) and 99% (for material combinations with lower fc). The smallest admissible diameter was 185 mm, corresponding to the materials combination C90/105 + DIN 5.8. The largest diameter was 298 mm for combination C35/45 + S400 and C30/37 + S500. Moreover, the obtained steel reinforcement in the new cross-section leads to a new maximum diameter for the element in order to comply with A_s,min requirement, when subjected to a normal tensile stress. In this context, a minimum diameter of 234.32 mm (+19%) was found for the C90/105 + DIN 8.8 combination and 350.76 mm (+79%) for the C35/45 + A400 combination. The combinations that best preserve the original dimensions, i.e., those requiring the smallest average diameter increase, correspond to higher concrete classes, namely C90/105 (S500, DIN 5.8, and 8.8), with average increases of 25%, 27%, and 24%, respectively.

3.3. RC Versus CS Cross-Section

The RC unit was in general found to be easier to build and adapt, especially regarding rebar shaping. However, creating an effective and simple connection system between modules to ensure structural continuity can be challenging, as documented by [71]. While using a single rebar offers benefits such as better reinforcement cover and consistent mechanical strength across all loading directions, there are disadvantages that cannot be neglected. The small cross-sectional dimensions limit optimization, especially under high flexural stresses, such as extreme storm or abnormal conditions. The close proximity of compression and tension forces reduces the torque arm and, consequently, flexural strength. Additionally, the original RC cross-section faces saturation of the steel, where increasing reinforcement does not significantly improve flexural strength. To avoid this, reinforcement ratios must be respected, steel distribution optimized, or section dimensions increased to ensure safety and ductility. In this context, using compression reinforcement in a polar array and a tubular profile with the appropriate diameter and thickness, as shown in Figure 15, is advantageous.

Although the CS solution is more complex to construct, it offers significant advantages in terms of structural behaviour due to its geometric flexibility and material strength. Additionally, the incorporation of bidirectional holes along the reinforcing tube improves material adherence and facilitates concrete filling, as suggested in [28]. The combination of materials allows for better optimization of the cross-section, increasing structural efficiency, especially under bending conditions. This system also provides a practical solution for connecting modules as stated by [71]. However, the need for specific equipment to bend the steel tubes is a disadvantage in the construction process.

3.4. Cost-Effectiveness

Figure 16 illustrates the correlation between the weight and cost of the reef module, based on the CS cross-section geometry and the optimized RC cross-section that ensures structural safety under the most demanding loading scenario. The analysis was conducted to assess the cost implications of strengthening the RC modules in comparison to the CS module, addressing concerns related to cost and construction feasibility. In the case of RC solution, only configurations resulting in a sectional increase of less than 35% in the critical area (D_min) were considered, and these were identified as the green zone in Figure 16. In addition, the maximum dimension of the limb near the spherical zone was constrained to maintain the original geometric ratio between D_max and D_min, thereby ensuring consistency with the initial design morphology. This aimed to preserve the slenderness of the geometric form, as a bulkier structure could significantly alter the local hydrodynamic behaviour, potentially requiring further studies in that domain.

The analysis of the reinforced concrete (RC) and composite steel (CS) cross-sections reveals distinct trade-offs between weight, cost, and structural efficiency (Figure 16). Most RC solutions have a weight of 160 kg, exceeding in 33% the initially desired 120 kg, which requires the use of medium equipment for handling the units. The cost of these RC solutions ranges from 18.7€ to 68.8€ per module. Conventional reinforcement results in lower costs, while the use of threaded bars significantly increases the unit cost. The RC combination with the best cost-construction feasibility ratio is the one using C90/105, S500, and 1Ø25, which has a weight of 159.8 kg and costs 18.7€ per unit. This solution offers an optimal balance between concrete usage and cost-efficiency, making it a more sustainable choice in terms of material optimization.

In comparison, CS solutions, which utilize a composite concrete and steel structure, present a lower increase in unit weight but come at a significantly higher cost (Figure 16). The lowest cost for CS solutions is approximately double that of the RC unit. Several CS solutions, with weights around 150 kg, have costs ranging from 39.23€ per unit to 56€ per unit. A notable example is the combination of C30/37, S420, Ø76.1, and t_w 4, which weighs 154 kg and costs 39.23€ per unit. Whilst these CS solutions may exhibit a reduced weight relative to RC solutions, they necessitate a higher financial investment due to the materials employed. However, they can utilise lower-strength, more readily available concretes. The solution that results in the lowest possible weight for the CS cross-section uses the combination of C30/37, S420, Ø60.3, with a 5 mm thick wall, which achieves a weight of 139.72 kg (+16%) and costs 37.81€. However, it is important to note that this solution has a slightly lower M_pl,Rd than required, meaning additional care is needed during application.

The cost-effectiveness analysis reveals that the optimized RC solution offers the most economical and structurally adequate configuration for the reef module. However, this comes at the expense of increased volumetry, which may alter local hydrodynamic behaviour and potentially impact biological colonization. If maintaining the original geometry is a priority—whether for ecological performance, fluid-structure interaction, or design constraints—the composite (CS) solution becomes the preferable alternative. Despite its higher unit cost, it provides a structurally efficient design while using lower-strength, cement-reduced concretes, contributing to lower carbon emissions and greater sustainability. Thus, the choice between RC and CS depends on project priorities: cost and simplicity favour RC, while environmental performance and geometric fidelity favour CS.

4. Conclusions

This work introduces the concept of different structural systems and presents a design methodology for developing and analysing modular and geometrically complex artificial reef structures. This study is based in a cross-sectional analytical approach based in the principles of structural mechanics supported by numerical analyses to evaluate actions and their effects on the structure.

By detailing the geometric design, structural conception, and relevant environmental actions, an approach was proposed for the evaluation of possible structural configurations. The analytical methodology, along with the parametric study, provided crucial insights into the performance and feasibility of different design configurations. In particular, and for the specific case of the cross-sections developed, the main findings were as follows:

• Reduced fixed diameters establish an important geometric limit for optimizing the cross-section of the RC type.
• Tensile strength emerged as the primary limiting factor, particularly in the RC type with variable cross-sections, where the minimum reinforcement was constrained by the larger section.
• Flexural strength was not a limiting factor for the CS structural solution, as the variety of structural combinations indicates sufficient robustness under flexural stresses.
• In terms of axial and flexural strength, the CS system provided significantly higher design strengths when compared to RC sections. Additionally, the CS structure did not require the use of high-strength concrete, simplifying construction and potentially reducing costs and environmental impact.
• Shear strength was not a limiting factor for either structural solution.
• Full optimization of the cross-section to withstand the most demanding loading scenario would require a minimum size increase of 23%, making it bulky, heavier, and less attractive.
• If cost is not a limiting factor and preserving the original geometry is a priority, the CS solution stands out for its high structural efficiency, reduced carbon footprint, and minimal impact on hydrodynamic and ecological performance.
• The methodology needs to include factors such as fatigue or long-term plastic deformation, which are crucial for structures exposed to high load cycles, like artificial reefs.

These findings emphasize the importance of a systematic engineering-driven analysis to the design and construction of artificial reefs, as it allows a comprehensive assessment of structural viability before advancing to prototyping and more complex numerical analyses. This stage enhances efficiency and ensures the most promising designs are pursued, optimizing resources and efforts. The design methodology has limitations due to the incomplete consideration of axial and flexural stresses interactions, which may affect performance and lead to an underestimation of the forces acting on the structure. This highlights the need for further analysis in future work to ensure robustness under real exposure conditions. Additionally, in situ monitoring strategies, such as wave height measurements, pressure sensors, strain gauges, accelerometers, or acoustic emission, need to be explored to assess real-time structural performance and detect signs of damage or degradation [72].

Given that marine structures are exposed to dynamic ocean loads over extended periods, future research will address the cumulative fatigue effects. This will require dedicated studies to characterize fatigue resistance as a function of the number of cycles and stress levels in critical sections, based either on the Model Code 90 framework or on experimental data. These investigations will build upon the structural insights developed in this initial study and aim to enhance the durability and reliability of the proposed modular reef system.