Performance Assessment and Sustainable Applications of Steel Canopies with Saddle Modules

Abstract

Steel is an important construction material in civil engineering. In addition, the building industry is one of the global economy’s largest sectors, responsible for one-third of the energy consumption and significant CO₂ emissions. For this reason, there is a need to design effective structures that are characterized by the lowest possible steel consumption. This article presents an approach to sustainability considerations in steel structures, namely the approach of shaping efficient steel canopies with modular roofs using genetic algorithms. The shed structures, which were designed based on a regular polygonal plan, were constructed from grid modules that were formed on the basis of the hyperbolic paraboloid (HP) units arranged radially, supported by the columns, and covered by metal sheets. The algorithmic definitions allowed for the creation of numerous variants of the structures with the adopted preliminary criteria and for the performance of genetic optimization in order to select the best results. Twenty-four kinds of structures were analyzed and compared, differing in the quantity of modules, module shapes, arrangements, and dimensions. This made it possible to observe changes in the efficiency of the structures depending on the form of the roof applied. As a measure of structural efficiency, the coefficient representing the mass of the shed structure per square meter of the covered area was utilized. The presented design approach and optimal solutions can be helpful in shaping more complex sustainable structures, for which the analyzed sheds constitute modules.

1. Introduction

The shaping of steel bar structures can be defined as a gradual improvement of their forms and topologies based on initial criteria that guarantee reliability [1,2,3]. Steel structures should be optimized to withstand various loads without experiencing significant concentrated damage. To achieve adequate strength and stiffness and reduce susceptibility to severe damage under various load conditions, the structural elements must possess adequate ductility, stiffness, and durability. Over the years, numerous methodologies for shaping structures have been developed. The most common approaches include minimizing the energy that is needed to produce structural materials, minimizing the consumption of materials, and reusing materials. The performance of steel bar structures under loads is determined not only by the types of structural materials used, but also by their forms and topologies, the applied connections, and the supporting systems. For that reason, geometrical shaping criteria are crucial. Some of these criteria, like for modular hall frames, were determined based on many years of work practice [4]. However, for curvilinear steel bar structures shaped based on a specific curved surface with arranged structural nodes, the type of surface is critical, as it significantly influences the way that loads are transferred by the structure.

In reference to this, this study shows an optimization method for designing sustainable modular bar structures made from HP saddle modules that are arranged radially, supported by straight columns, and covered with metal sheets. In particular, it investigates the influence of the number of roof modules used, as well as their shapes, on the performance of shed structures under loads in order to obtain sustainable structures. These structures may, in turn, constitute modules of more complex structural forms.

2. Hyperbolic Paraboloid in Building Structures

The HP shape is widely used in the building industry. One of the first famous engineers who designed buildings using the shape of an HP was Felix Candela [5]. His projects mainly encompassed lightweight concrete structures that were reinforced with a grid of bars, like the Xochimilc restaurant in Mexico, Palmyra church, Chapel Lomas de Cuernavaca, and others. Many scientists discuss HP concrete shells in their publications [6,7,8,9,10,11]. However, a relatively small number of scientists deal with steel bar structures in HP shapes or wooden HP structures; some analyze aspects of the foundation of such structures or their coverings [12,13]. The mechanical static and dynamic properties of HP roofs, as well as their behavior under the influence of loads, also deserve attention, which was shown in [14,15]. Moreover, there are many well-known buildings based on the HP form but with other structural systems, like Qatar Russell Stadium with a cable net structure and Bird’s Nest Stadium in Beijing with a steel structure, which also constitute the basis of many studies and analyses [16].

The development of materials engineering and computer technology has led to considerable interest in buildings with saddle roofs [17,18]. Steel roofs based on the shape of an HP are composed of a lattice of bars that can be covered with different materials, which was presented in [13,19,20,21]. The double curvature of HP lattice structures provides them with a high degree of stiffness. Due to this fact, a lot of research is devoted to the analysis of the behavior of the HP-shaped structures under loads [21,22,23,24,25,26]. HP shapes are often used for tensile roofing systems [22]. The wind load for HP roofs is particularly specific, and many theoretical and experimental studies are devoted to this issue. In [22], experimental pressure coefficients for HP roofs with circular and elliptical plan, together with their simplified envelope maps, were presented.

The HP roof structures can be formed as modular structures consisting of repeating modules arranged in various ways, such as the approach presented in [20,26]. The characteristics and performance of these structures have been analyzed based on the arrangement of the modules within the structure [16,26]. This layout affects not only the overall geometry and form of the structures, but also the distribution of loads acting on them, and therefore also the values of stresses induced by them [27,28,29,30]. Due to the large number of variables that can define the geometry and topology of an HP steel bar structure, as well as its loading and supporting systems, it is beneficial to determine the structure parametrically. This approach helps identify the key parameters that describe the structure in an optimized manner. This topic has been explored in numerous studies, where the most effective shaping results were achieved through optimization with evolutionary algorithms, considering various optimization criteria [31,32,33]. Form and topology optimization is presented in [26,34], and shape optimization of the roof due to the amount of shadow it casts is shown in [35,36], whereas mass optimization is presented in [37]. However, in [26], photovoltaic modular HP roofs are analyzed, so one of the shape optimization criteria is the amount of solar radiation that reaches the roof. The presented research is a continuation of the authors’ previous research [26]. The previous article concerned the shaping of solar canopies with flat glass photovoltaic panels, which required an appropriate topology of the bar grids. In contrast, the current article concerns the shaping of roofs that are covered with metal sheets, where the topology of the bar grids is completely different (the grid cells are not flat). Unlike the previous article, the current research also considers structures that are supported by columns. Since metal sheets are used as roof coverings, solar energy reaching the roof surface is not an optimization criterion.

The main novelty and originality of the current research lie in the analysis of the efficiency of roof coverings with asymmetric HP modules. The efficiency of the structures is analyzed depending on the location of the external node of the structural frame in relation to the plane of symmetry of the module, followed by a comparison of the structural efficiency. In the previous article, only structures with symmetrical HP modules were analyzed.

3. The Method of Shaping Modular Canopies

The HP is a skewed ruled surface. It is composed of two families of rulings intersecting one another and creating a saddle HP surface. The rulings of the same family are mutually oblique, whereas the rulings belonging to various families intersect. Moreover, the rulings of the same family are included in the parallel planes, as shown in Figure 1. Therefore, the section of any HP surface, although doubly curved, can be formed from linear structural members. This is a great advantage of shaping HP grid shells. In this study, the horizontal projection of the HP section, which was to constitute a shaped roof module, was a rhombic grid of straight lines (Figure 1).

However, a single HP module could be arranged and combined in an infinite number of ways, giving various roof shapes. In the presented development, these modules were identical within the roof and were arranged radially. Due to this fact, shed structures formed over a plan of a regular polygon shape were considered. The fundamental stages of shaping these structures included the following:

• Defining the geometric model parametrically;
• Setting design constraints;
• Performing genetic optimization of the structures’ geometries;
• Initially selecting structures for further analysis;
• Conducting structural analysis;
• Optimizing cross-sections;
• Comparing the obtained results;
• Selecting the optimal solution.

The Rhinoceros 7 3D software, along with Grasshopper for parametric design, was used for shaping geometries, as well as the structures’ topologies, and for optimizing them [38]. Grasshopper, a visual programming tool for Rhinoceros 3D allows users to create intended geometries by connecting appropriate graphic components in specific relationships. These components are linked by cables that facilitate the flow of information between them. The geometry of any shed structure was composed of a modular HP roof that was supported by several columns. It was defined by block script using several variable parameters defining the roof’s form and topology. The bar grid of each module was obtained by dividing the sides of the spatial polygon which forms the module’s frame into equal parts. Four variants, 1, 2, 3, and 4, of the shed structures with modular roofs that were composed of several HP modules arranged radially over polygonal shapes like triangles, squares, pentagons, and regular hexagons were taken into consideration. Due to this fact, the quantity of modules applied for the roof matched the quantity of sides of the polygon, as shown in Figure 2.

The design variable parameters adopted in the script describing the shed forms and topologies were as follows:

h—roof structure height;

r—radius of the circle that circumscribes the polygon;

n—number of the polygon edges;

d—distance of the outer corner of the module from the level of column heads;

s—distance from the polygon edge to the rectangular projection of the module’s outer corner;

m—number of parts of the side division;

h_c—column height, as shown in Figure 2.

While modeling a shed structure of one of the four variants, genetic optimization was carried out. It consisted in minimizing the surface area of the roof in relation to the surface area of the covered place with the assumed design criteria. It guaranteed obtaining a minimized amount of structural material in relation to the covered area. Subsequently, for each structure, three modifications were generated, which differed in module geometry and height. Finally, 24 different structures with various roof shapes were subjected to static analysis by Autodesk Robot Structural Analysis Professional 2023 (ARSA) software, taking into account both dead and environmental loads [39]. The structural components were optimized based on their masses. Subsequently, an evaluation based on the adopted parameters and a comparative analysis of the resulting models were conducted.

4. Results

4.1. General Rules of Generating the Geometries of the Structures

To define the shape of each bar structure made up of several HP modules, a block script was created using Grasshopper, as illustrated in Figure 3.

However, the same radially arranged modules with the same bar grids were used for each structure. The same column heights were applied too. In this way, the geometrical models of shed structures of the primary type A were characterized by the roofs with the symmetrical modules. The variable number n of modules in the script made it possible to obtain shed structures of various forms. However, this research considered the structures with three-, four-, five-, and six-module roofs. These were called the structures of variants 1, 2, 3, and 4, respectively. Their front and top views are presented in Figure 4.

For further research, it was assumed that each polygonal plan was inscribed in a circle with a radius of r = 6.0 m. The column heights were established as h_c = 2.0 m, and each module edge was divided into m = 5 parts (see Figure 2 and Figure 4). Subsequently, genetic optimization was conducted for each variant structure (three-, four-, five-, and six-module) to minimize the roof area and maximize the covered area. This approach ensured that the roof used the least amount of steel per unit area of the covered area. Particularly, single-criteria optimizations were performed. The objective function was to maximize the parameter that was the quotient of the roof area and the area of the covered square. Simulation I and simulation II were performed assuming the same radius of the circle circumscribing the polygonal place, as well as the same column height, for functional reasons. The structures varied in the range and size of the parameters defining roof geometries.

The range of the parameter s, which determines the horizontal dimensions, was set identically for all structures. However, different ranges of variables h and d, which determine the vertical dimensions of the roofs, were adopted. The ranges of variable parameters were determined based on functional considerations. The adopted difference between the lower and upper values of the h and d parameters was identical for both simulations. However, the lower range of h and d variables in the second simulation was assumed to be half of that in the first simulation (rounded to 0.5 m). The values of the fixed parameters and the ranges of variable parameters adopted during simulations I and II are presented in

Table 1. The values of fixed and ranges of variable parameters adopted during simulations.

Kind of Parameter	Parameter Value of Simulation I [m]	Parameter Value of Simulation II [m]
r	6.0	6.0
hc	2.0	2.0
h	3.5–5.5	2.0–4.0
d	3.0–4.0	1.5–2.5
s	0.0–4.0	0.0–4.0

.

As a result of both simulation I and simulation II, four basic structures of variants 1, 2, 3, and 4 and of type A were obtained, Figure 4. These structures were characterized by their parameter values, as presented in

Table 2. The values of fixed and ranges of variable parameters adopted during simulations.

Kind of Parameter	Parameter Value of Simulation I [m]	Parameter Value of Simulation II [m]
r	6.0	6.0
hc	2.0	2.0
h	3.5	2.0
d	3.0	1.5
s	4.0	4.0

.

The structures that were characterized by the parameters obtained during the first simulation were called the structures of version H (with high roofs); however, the structures that were characterized by the parameters obtained during the second simulation were called the structures of version L (with low roofs). As a result, four variants, 1, 2, 3, and 4, of A-type structures of version H and four variants, 1, 2, 3, and 4, of A-type structures of version L were obtained.

4.2. Generating the Structures with Assymetrical Modules

Based on the basic type A, the additional types B, C, and D were generated for each variant of the structure. The new types B, C, and D were characterized by different locations of the external nodes of the module frames. These locations were established by the division of the length a of the plan edge into two, three, or four parts according to Figure 5. The type A structure was a basic structure with symmetrical modules and external node locations, characterized by parameter 1/2a.

In this way, for each structural variant (1, 2, 3, and 4), four types (A, B, C, and D) were generated in two versions (H and L), as shown in Figure 5. These shed structures were characterized by roofs with asymmetrical modules. However, due to the unfavorable shape in terms of roof covering, as shown in Figure 6, the D-type structures were omitted from further considerations and analyses.

4.3. Structural Analysis

4.3.1. Definition of the Computational Model

Each structural model of the shed was exported to ARSA software for static analysis [39]. Consequently, the supports and roof coverings were introduced for each previously obtained model to ensure proper load transfer. The number of supports in each structure corresponded to the number of modules in the roof structure. Structural bars were assigned properties based on their function within the structure. All models featured rigid supports at the column bases and rigid connections. Steel of grade S235 was used as the structural material.

The dead/permanent loads were modeled, taking into account both the mass of the structure and the mass of the applied roofing material, i.e., metal sheets (0.1 kN/m²). However, variable environmental loads were introduced for the location of the city of Rzeszow (Poland), which is situated in snow load zone III and wind load zone I [31,32]. Two versions of snow loads were assumed (distributed both evenly and unevenly), considering characteristic values of snow load on the ground equal to 1.2 kN/m². The automatic simulation of wind loads was performed with the same assumptions for all structures, i.e., the wind velocity pressure of the location in Rzeszow [32]. The cases of analyzed snow and wind loads for variants 1, 2, and 3 of the four-module structure types A, B, and C are presented in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. In the case of wind load, the number of wind directions that were assumed for simulation depended on the roof shape. In the case of a type A structure with two planes of symmetry, only three wind action directions were used, while in the case of type B and C structures, four were used, as shown in Figure 8, Figure 10, and Figure 12.

Subsequently, static and strength calculations were performed, and structural members were optimized in terms of minimum mass. To standardize the cross-sections, the structural members were categorized into three groups based on their function within the structure. The first group comprised the internal members of each frame, while the second group included the external members. The third group, forming a network of roof structural bars, consisted of the bars connecting the previously established frames. A first-order analysis was performed, and both the ultimate limit states (ULSs) and the serviceability limit states (SLSs) were verified.

4.3.2. Structural Analysis Results of the Considered Structures

The individual structures were designed to maximize bar utilization, with cross-sections chosen from a range of commercially available options. However, the structural analyses enabled the evaluation of the shed structures’ performance under various loads and allowed for the comparison of their key parameters.

The analyses indicated that the stress distribution for each assessed structure was similar. Specifically, the highest normal stresses were found in the columns, while the lowest stress values were observed in the lattice bars inside the module frames. However, the maximum deflections occurred in the outer frame members. In the case of the asymmetric modules, they were located in the longer outer frame member. However, the maximum stresses in the columns varied significantly depending on the number of roof modules used.

Table 3. The results of the structural analyses of various types of the structure L (A, B, and C).

Structure Type	Member	Cross-Section [mm]	Bar Utilization [%]	Structure’s Mass [kg]	Max Fx/Ax [MPa]	Maximum Deflection [cm]
AL-1	Inner frame Outer frame Lattice Column	140 × 140 × 5 120 × 120 × 5 80 × 80 × 4 180 × 180 × 5	91 88 60 88	2595	23.41	1.4
AL-2	Inner frame Outer frame Lattice Column	120 × 120 × 5 100 × 100 × 5 80 × 80 × 4 150 × 150 × 5	90 75 54 88	2975	22.81	1.4
AL-3	Inner frame Outer frame Lattice Column	140 × 140 × 5 90 × 90 × 4 60 × 60 × 4 140 × 140 × 5	80 73 58 92	2878	18.21	1.0
AL-4	Inner frame Outer frame Lattice Column	120 × 120 × 6 60 × 60 × 4 60 × 60 × 4 140 × 140 × 5	90 94 59 75	3142	19.22	0.9
BL-1	Inner frame Outer frame Lattice Column	140 × 140 × 5 140 × 140 × 5 100 × 100 × 4 180 × 180 × 5	96 94 50 87	3112	25.79	2.5
BL-2	Inner frame Outer frame Lattice Column	140 × 140 × 5 120 × 120 × 5 90 × 90 × 4 150 × 150 × 5	95 84 57 88	3661	23.06	2.0
BL-3	Inner frame Outer frame Lattice Column	120 × 120 × 6.3 90 × 90 × 5 80 × 80 × 4 140 × 140 × 5	99 85 75 98	3770	22.00	1.6
BL-4	Inner frame Outer frame Lattice Column	120 × 120 × 6.3 80 × 80 × 4 80 × 80 × 4 120 × 120 × 6.3	89 96 74 89	4250	22.63	1.3
CL-1	Inner frame Outer frame Lattice Column	140 × 140 × 5 140 × 140 × 5 100 × 100 × 4 180 × 180 × 5	94 75 50 84	3269	27.19	2.6
CL-2	Inner frame Outer frame Lattice Column	140 × 140 × 5 120 × 120 × 5 90 × 90 × 4 150 × 150 × 5	97 81 71 88	3710	24.79	2.7
CL-3	Inner frame Outer frame Lattice Column	140 × 140 × 5 100 × 100 × 4 90 × 90 × 4 140 × 140 × 5	85 88 72 94	4073	22.51	2.2
CL-4	Inner frame Outer frame Lattice Column	120 × 120 × 6 90 × 90 × 4 90 × 90 × 4 120 × 120 × 6	87 76 60 89	4674	23.91	1.7

presents the results of the structural analysis for various types of structure L (A, B, and C), including bar cross-sections, maximum normal stresses (Fx/Ax), maximum deflections, and total structural masses. Similarly,

Table 4. The structural analysis results for types A, B, and C of structure H.

Structure Type	Member	Cross-Section [mm]	Bar Utilization [%]	Structure’s Mass [kg]	Max Fx/Ax [MPa]	Maximum Deflection [cm]
AH-1	Inner frame Outer frame Lattice Column	140 × 140 × 5 120 × 120 × 5 80 × 80 × 4 180 × 180 × 5	95 93 71 90	2867	25.14	1.9
AH-2	Inner frame Outer frame Lattice Column	120 × 120 × 6 100 × 100 × 5 80 × 80 × 4 150 × 150 × 5	97 84 61 95	3432	24.39	1.8
AH-3	Inner frame Outer frame Lattice Column	140 × 140 × 5 100 × 100 × 4 60 × 60 × 4 140 × 140 × 5	89 78 63 92	3648	22.21	1.4
AH-4	Inner frame Outer frame Lattice Column	120 × 120 × 6 100 × 100 × 4 60 × 60 × 4 140 × 140 × 5	95 85 68 82	4035	23.98	1.2
BH-1	Inner frame Outer frame Lattice Column	140 × 140 × 5 140 × 140 × 5 80 × 80 × 4 160 × 160 × 5	84 77 64 99	3938	27.63	1.9
BH-2	Inner frame Outer frame Lattice Column	120 × 120 × 6 120 × 120 × 5 100 × 100 × 4 140 × 140 × 5	94 67 57 91	4054	26.52	1.8
BH-3	Inner frame Outer frame Lattice Column	120 × 120 × 5 120 × 120 × 5 80 × 80 × 4 120 × 120 × 5	90 89 76 93	4320	24.82	1.3
BH-4	Inner frame Outer frame Lattice Column	120 × 120 × 5 120 × 120 × 5 80 × 80 × 4 120 × 120 × 5	86 85 63 91	4829	25.67	1.3
CH-1	Inner frame Outer frame Lattice Column	120 × 120 × 6.3 140 × 140 × 5 70 × 70 × 4 160 × 160 × 5	89 64 46 91	4386	28.41	1.9
CH-2	Inner frame Outer frame Lattice Column	120 × 120 × 6.3 120 × 120 × 5 80 × 80 × 4 140 × 140 × 5	96 61 74 93	5139	27.58	1.9
CH-3	Inner frame Outer frame Lattice Column	120 × 120 × 5 120 × 120 × 5 90 × 90 × 4 120 × 120 × 6	94 64 53 97	4503	25.00	1.5
CH-4	Inner frame Outer frame Lattice Column	120 × 120 × 5 100 × 100 × 5 90 × 90 × 4 120 × 120 × 5	85 53 56 92	5011	25.86	1.5

shows the structural analysis results for various types of structure H (A, B, and C).

However, the values of the maximum bending moments, M_y,max, occurring in the structures L and H of various types (A, B, and C and versions 1, 2, 3, and 4) are presented in

Table 5. The values of the maximum bending moments occurring in structures L and H.

Type of the Structure L	Value of My,max [kNm]	Type of Structure H	Value of My,max [kNm]
AL-1	51.93	AH-1	53.48
AL-2	39.36	AH-2	41.56
AL-3	26.12	AH-3	16.83
AL-4	29.10	AH-4	22.89
BL-1	50.15	BH-1	42.33
BL-2	37.32	BH-2	31.22
BL-3	22.82	BH-3	17.38
BL-4	28.52	BH-4	21.70
CL-1	45.26	CH-1	37.93
CL-2	36.87	CH-2	30.29
CL-3	22.23	CH-3	17.75
CL-4	26.02	CH-4	21.69

. The maximum M_y,max bending moments relative to local coordinate systems occurred in columns for each structure. It is worth mentioning that the values of bending moments occurring in the roof grid bars are very small, in the range of 0–0.56 kNm. This is due to the fact that the hyperbolic paraboloid has properties similar to those of a minimal surface.

5. Discussion—Assessment of the Considered Structures

Analyzing the results presented in Table 3 and Table 4, it can be stated that the greatest normal stresses in columns occur in the structures of variant 1 (three-module) and decrease with the increase in the number of modules in the structure. The same relationship applies to bending moments, which have the highest values in the case of variant 1 structures. As a rule, the values of the maximum normal stresses in the bars decrease with the increase in the number of roof modules. However, the structures of variant 3 (five-module) deserve special attention, as they are characterized by the lowest values of normal stresses and bending moments occurring in their members.

In turn, for a given variant of structures 1, 2, 3, and 4, the change of the basic type A to the type B or to the type C causes an increase in normal stresses in the bars and a decrease in bending moments. However, regardless of the types of structures analyzed (A, B, or C), the five-module structures (variant 3) are characterized by the lowest normal stresses and bending moments and the smallest masses.

In order to determine the efficiency of the structures by showing the amount of structural material used per unit of the covered area, the coefficient v was introduced. The quantities required to calculate the coefficients v, along with the values of these coefficients for both structures L and H, are shown in

Table 6. The values of the coefficient v for structures L.

Variant of the Structure	Structure’s Mass [kg]	Plan Area [m2]	Value of v [kg/m2]
AL-1	2595	93.53	27.75
AL-2	2975	122.91	24.20
AL-3	2878	138.50	20.78
AL-4	3142	147.53	21.30
BL-1	3112	100.74	30.89
BL-2	3661	136.39	26.84
BL-3	3770	154.14	24.46
BL-4	4250	164.23	25.80
CL-1	3269	93.90	34.81
CL-2	3710	132.72	27.95
CL-3	4073	151.89	26.82
CL-4	4674	162.72	28.72

and

Table 7. The v coefficient values for H structures.

Variant of the Structure	Structure’s Mass [kg]	Plan Area [m2]	Value of v [kg/m2]
AH-1	2867	93.59	30.65
AH-2	3432	122.91	27.92
AH-3	3648	138.50	26.34
AH-4	4035	147.53	27.35
BH-1	3938	100.74	39.09
BH-2	4054	136.39	29.72
BH-3	4320	154.14	28.03
BH-4	4829	164.23	29.40
CH-1	4386	93.90	46.71
CH-2	5139	132.72	38.72
CH-3	4503	151.89	29.65
CH-4	5011	162.72	30.80

, respectively.

The data presented in Table 6 and Table 7 for the structures of various types and variants are compared in the charts presented in Figure 13 and Figure 14 for structures L and H, respectively. This allowed for the assessment of the structure’s effectiveness due to the amount of structural material used.

Assessing the coefficient v across various structure types reveals that type A structures are the most efficient, whereas type C structures are the least efficient, as shown in Figure 13 and Figure 14. The above properties are characteristic for both the L and H versions of structures. It is worth emphasizing that in the case of L structures with low roofs, the difference in steel consumption between the individual variants of structures 1, 2, 3, and 4 is smaller than in the case of H structures with high roofs. However, in the case of the H structure, the largest difference in the value of the v coefficient occurs in the case of the variant 1 structure. To sum up, the five-module L structures of type A, characterized by a symmetrical external frame for each module, are characterized by the lowest v coefficient.

In general, these analyses have shown that type A structures with symmetrical modules are the most efficient in terms of material consumption. The greater the deviation of the module shape from the symmetrical form (the structures of type C) is, the lower the efficiency of the modular shed is. Moreover, the symmetry of the modules makes the number of structural elements used more repeatable, which is also beneficial for technological reasons. Roof structures were compared in terms of the amount of structural material used. The overall cost of a single structure is also influenced by technological aspects related to, among others, the design of connections, which will be the subject of further research. On the other hand, the adopted criterion for comparing structures allows for their preliminary assessment at an early stage of design.

Each modular structure resulting from optimization can be a module of a more complex structure. Individual modular structures can be combined with each other in a variety of ways, which makes it possible to obtain complex roofs that are spread out along a straight line or along an arc. Modular roof structures arranged linearly, and the modular roof structures arranged along an arc are shown in Figure 15 and Figure 16, respectively.

6. Conclusions

This research demonstrated a method for designing efficient steel structures with minimal steel usage, providing tangible sustainable benefits. Scripts developed for the parametric shaping of shed structures with modular roofs based on an HP shape enabled the creation of various shed models and an initial comparative analysis based on different criteria. Twenty-four structures were examined, which varied in the number of modules, as well as in their shapes and dimensions.

Their efficiency was compared, considering the amount of structural material per square meter of the covered surface. This research highlighted how the shapes and number of modules of the structures influenced their efficiency. The most effective versions of the structures with symmetrical roof modules have been determined. However, the observed properties and behavior of the structures under loads may be an indication for shaping more complex sheds using a given modular shed structure as a single module. Additionally, the research confirmed that genetic algorithms are highly effective in identifying effective, unconventional steel structures, and their efficiency can be optimized in the initial design phase by appropriately shaping their forms.