materials Article Modeling of Curvilinear Steel Rod Structures Based on Minimal Surfaces Jolanta Dzwierzynska * and Igor Labuda   Citation: Dzwierzynska, J.; Labuda, I. Modeling of Curvilinear Steel Rod Structures Based on Minimal Surfaces. Materials 2021, 14, 6826. https://doi.org/10.3390/ma14226826 Academic Editor: Georgios Savaidis Received: 13 September 2021 Accepted: 8 November 2021 Published: 12 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Architectural Design and Engineering Graphics, Faculty of Civil and Environmental Engineering and Architecture, Rzeszow University of Technology, Al. Powstancow Warszawy 12, 35-959 Rzeszow, Poland; labuda@prz.edu.pl * Correspondence: joladz@prz.edu.pl Abstract: The article deals with shaping effective curvilinear steel rod roof structures using genetic algorithms by implementing them for the analysis of various case studies in order to find new and efficient structures with positive characteristics. The structures considered in this article are created on the basis of the Enneper surface and minimal surfaces stretched on four arcs. On the Enneper surface, a single layer grid is used, while on the other surfaces, two-layer ones. The Enneper form structure with four supports and the division into an even number of parts along the perimeter of the covered place proved to be the most efficient, and the research showed that small modifications of the initial base surface in order to adapt the structure to the roof function did not significantly affect its effectiveness. However, the analysis and comparison of single and double-shell rod structures based on minimal surfaces stretched on four arcs have shown that a single-shell structure is much more effective than a double one. The paper considers the theoretical aspects of shaping effective structures, taking their masses as the optimization criterion. The optimization helped to choose the best solutions due to structures’ shapes and topologies. However, the obtained, optimized results can find practical applications after conducting physical tests. Keywords: engineering materials; engineering structures; optimization; genetic algorithms; para- metric design; shaping structures; curvilinear steel rod structure; modeling structures; Enneper surface; Grasshopper 1. Introduction The paper deals with the use of steel as the structural material for modeling curvilinear rod structures. The systematics of steel rod structures is quite complex as they can be differentiated due to their function, shape, kind of material used, as well as prefabrication methods. However, due to their overall shape, that is, the shape of the surface they form, the following types of steel rod structures can be distinguished: flat, single-curved as well as double-curved. On the other hand, in terms of their grid systems, the steel rod structures can be divided into single-layer and multi-layer structures. Geometric grids are characterized by a large topological diversity. However, single-layer coverings usually occur in a curved form, whereas multi-layer coverings are more often applied as flat ones [1]. Modeling/shaping of any steel rod structure can be defined as the continuous optimization of its form and shape due to assumed criteria and constraints. From the architectural point of view, during shaping, the main emphasis is put on functionality and aesthetics, whereas from a structural aspect, shaping can be treated as determination of the system that meets the assumed criteria for strength, which are specified by standards [2–5]. However, architectural and structural shaping criteria are interdependent [6]. Moreover, as far as curvilinear steel rod roof structures are concerned, they are subjected to the specific geometric shaping criteria for this kind of structure. These criteria relate mainly to geometric parameters of the structure’s module, its height, span, and topology [7]. Materials 2021, 14, 6826. https://doi.org/10.3390/ma14226826 https://www.mdpi.com/journal/materials