**1. Introduction**

In general, steel bar structures are determined as spatial structures made of slender members which are directly connected in order to carry loads. Historically, curvilinear steel bar structures, mostly in the form of cylindrical lattice structures, began to be created in the mid-nineteenth century. However, due to serious difficulties in both calculating and constructing from repeatable elements, they began to be used on a larger scale only in the 1940s. During this period, the beginning of steel mass production and the invention of many devices influenced the great development of various manufacturing technologies of steel roof structures. The most popular were layered geodesic domes, which were shaped using the procedures of sphere division into triangles elaborated by Buckminster Fuller [1,2]. Due to this fact, the problem of the most regular subdivision of the spherical surface was one of the major challenges for scientists in the steel structures field. Various ways of dividing a sphere have been developed over the years, in order to achieve different types of grid, like Lamell's lattice and Schwedler's lattice [3]. However, combining different parts of the sphere into larger forms was one of the ways of obtaining new shapes of grid shells [3]. The broad review of various types of spatial grid structures and their development is described in [4], whereas broad analytical approaches concerning plane bar grids and double layer trusses are given in [5]. The method of forming steel bar structures, placing their vertices on the so-called base surfaces which are Catalan surfaces—has been presented in [6]. On the other hand, the shaping of bar structures based on minimal surfaces, especially the Enneper surfaces, is presented in [7,8].

A hyperbolic paraboloid constitutes an especially interesting and important basic shape for various single or complex architectural roof forms. The use of a hyperbolic paraboloid shape for constructing thin shells was pioneered in the post-war era as the result of the combination of modern architecture with structural engineering. The hyperbolic paraboloid as the element for creating complex forms was used widely by F. Candela for the implementation of lightweight shell concrete structures, constituting coverings that are free of intermediate supports [9]. The great interest in this shape was caused by its positive static properties, allowing the creation of shells with a large span, as well as a great possibility of various arrangements of single shells in compound ones.

Hyperbolic paraboloids are exceptionally stiff, due to their double curvature [10]. They exhibit membrane action, wherein internal forces are efficiently transmitted through the surface, which is the subject of various publications [11,12]. Most of the research concerns theoretical, experimental and constructional problems related to hyperbolic paraboloid concrete or reinforced concrete shells [9,13]. The method of shaping freeform buildings, roofed with profiled steel sheets effectively transformed into strips of screw ruled surfaces, is presented in [11,14]. However, the behaviour of gabled hyperbolic paraboloid shells is studied in [15]. Although a hyperbolic paraboloid as a ruled surface constitutes a good basis for creating lattice grids, there is little research into the effect of the division of this surface, as well as received grid pattern, on the bearing capacity of the bar structure created. However, the variety of compound roof structures that can be obtained by combining several hyperbolic paraboloid grid modules is presented in [12]. The canopy roofs of hyperbolic paraboloid shape are worth considering due to both their interesting form and relative simplicity of construction.

At present, the use of curvilinear steel bar structures is increasing thanks to advanced technology in the field of steel bar structures. Thus, more and more curvilinear steel bar structures are created, with a great variety of geometric forms and technical solutions. Therefore, a rational and effective attempt to shaping of this type of structures is important. The shaping phase is the design phase preceding all subsequent stages of the design process, which is why it is the most creative phase, as well as having a significant impact on the final form of the structure [9,16]. Due to this fact, it is very important to consider as many aspects of the future project as possible in the early stage. The principles of the rational shaping of steel bar structures are presented in many publications [17,18]. However, shaping the curvilinear steel bar structures can sometimes be a challenging task.

The possibilities of rational shaping depend not only on creativity and practical skills, but also on the design tools used. In the last twenty years, the progress of digital technologies has affected the entire field of both architectural design and structural engineering. That is due to the fact that digital tools greatly facilitate the creation of complex geometry, as well as performing advanced structural calculations [19]. The practical application of the digital design tools by European design studios is presented in the research reported in [20]. Various computer-aided design (CAD) tools enable both the creation of two-dimensional documentation and the creation of three-dimensional models based on two-dimensional drawings [21–23]. CAD technology enabled the generation of digital models, their geometry visualization and, finally, analysis of their structural behavior. Moreover, the progress in design caused by development of computer technology and integration of digital modeling systems has facilitated cooperation in various design areas, such as architecture and structural engineering [24,25]. Especially, in the field of steel bar structures, whose shaping is accompanied by a number of issues, the interdisciplinary approach is often required.

Recently, development of the modeling process based on Non-Uniform Rational B-Splines (NURBS) has had a great impact on forming the structures' shapes. NURBS can be controlled during modeling, and therefore they can constitute a base for the generation of various digital changeable forms with diverse topologies. Moreover, digital environment, especially algorithmic-aided shaping structures, has created new possibilities for performing various simulations, which further enable structural optimization [26]. In 1977, the idea of solving evolutionary optimization problems was introduced by means of a computer simulation of evolutionary transformations [27]. The evolutionary algorithms for such a simulation are the stochastic search methods that mimic natural biological evolution. These algorithms have been developed in order to arrive at near-optimum solutions to large-scale optimization problems for which traditional mathematical techniques might fail [28]. A comparison of the formulation and results of five recent evolutionary-based algorithms—genetic algorithms, particle swarm algorithms, ant-colony systems, and shuffled frog leaping—is presented in [28].

The algorithms presented in the paper are genetic algorithms inspired by Darwin's theory of evolution, which mimic natural selection and gene mutation. In each genetic algorithm, the optimization process of the given problem begins with creating a set of random solutions called individuals. The set of variables is treated as chromosomes. As in nature, the whole set of possible solutions is considered a population. The most efficient and strong chromosomes are selected in order to create the next population. The evolutionary principles of genetic algorithms allow the generation of well-performing instances and search for the solution closest to the optimal one in the given space [29]. In order to solve the optimization problem, the parameters and constraints of the problem should be identified. Depending on the nature of the objective function applied, the optimization problem can be classified into either single objective or multi objective. In the literature of the subject, the term multi-objective optimization refers to problems with up to four objectives [27].

The optimization of bar structures can deal with many aspects: the weight of the structure, appropriate support method, or topology related to both ultimate limit states (ULS) and serviceability limit states (SLS) [6,8,30,31]. In the case of a steel structure analyzed in the paper, the ultimate limit state referring to internal failure involves the resistance of cross sections and the resistance of the structure and its members. If the design value of effect of actions *E*<sup>d</sup> does not exceed design value of corresponding resistance *R*d, then this should be verified. The design value of the effects of actions *E*<sup>d</sup> is determined by combining the various values of actions that are considered to occur simultaneously.

However, verification of SLS primarily aims at preventing excessive movements or vibrations of structures [31]. Whether the design value of the effects of actions specified in the serviceability criterion *E*<sup>d</sup> does not exceed limiting design value of the relevant serviceability criterion (e.g., design value of displacement) should be verified.

In the paper, the effects of displacements and deformations are mostly taken into account, assuming deflection limits equal to

$$f = \text{L/250} \tag{1}$$

where *L*–span of the structure [31].

The application of evolutionary structural optimization (ESO) for the shaping of steel bar structures is a new field of research, which can lead to obtaining effective structures.

Referring to the above conditions, the article attempts the multi-objective optimization of curvilinear steel bar structures forming roofs of hyperbolic paraboloid shape. Although the hyperbolic paraboloid as a ruled surface may be a convenient base for forming bar grids, there are few studies on the effect of its division and the topology of the obtained bar structures on their load-bearing capacity. In order to fill this gap, the aim of the presented research is the comparison of the effectiveness of canopies—curvilinear steel bar structures formed based on hyperbolic paraboloids covering the same plane. The research goal is to determine the most effective pattern of grids and the optimal supporting system, as well as the mass of the structure.

### **2. Materials and Methods**

The research was conducted with the application of modern digital tools working in Rhinoceros 3D software developed by Robert McNeel and Associates [32]. These tools are: Grasshopper plug-in for parametric modeling and Karamba 3D plug-in developed to predict the behaviour of structures under external loads [33]. The active use of Rhinoceros 3D/Grasshopper software in the architectural design process is becoming increasingly popular in the world, mainly as a tool for generating models with complex geometry. Moreover, interactive structural evolutionary optimization has recently gained some popularity for optimization in structural design [6,8,9,34]. New methods of design solutions based on genetic optimization are analyzed in [35]. However, algorithmic structural shaping, which is the process in which both the geometric model and structural analysis are carried out using multi-objective interactive structural evolutionary optimization algorithms, is a new field of research. Therefore, the approach presented in the paper to shape curvilinear steel bar structures of hyperbolic paraboloid canopy roofs is innovative.

During the tests, in order to generate geometric models, Rhinoceros 5.0 version was used in combination with Grasshopper. This enabled the creation of complex generative algorithms and the parallel exploration of the shaped geometric models in the Rhinoceros 3D viewport. The shaping strategy presented in the paper consisted of forming of curvilinear steel bar structures by placing their structural nodes on the so-called base surfaces, which were hyperbolic paraboloids. However, the structures' geometries were generated algorithmically using a set of various specified input parameters.

Then, on the basis of the created geometric models of the analyzed structures, as well as the adopted boundary conditions concerning the supporting systems (loads), as well as the type of joins and material properties, the structural models were established. The integration of geometrical shaping and structural analysis took place by the Karamba 3D. The topology and cross-sections of the structures' bars were optimized taking into account the minimal structure's self-weight, as well as minimal deflection, as the optimisation criteria.

The general scheme of the conducted analysis dealing with shaping hyperbolic paraboloid canopy roof is presented in Figure 1. However, a more detailed description of the individual steps is provided in the following sections.

**Figure 1.** The general scheme of the conducted analysis.
