## 1 Introduction  
### Motivation and Literature Overview  
Steel structures have become integral in the construction industry, celebrated for their versatility, strength, and efficiency. Unlike concrete, steel allows for the design and implementation of complex geometries that can substantially reduce material consumption and, in turn, lower the overall cost of construction projects. This advantage is particularly important in an era where sustainable practices and cost-effective solutions are tremendously sought after. The push for sustainability has made resource efficiency a critical factor in civil engineering, and steel's ability to minimise material utilisation plays a crucial role in meeting these demands. The ability to create sophisticated shapes with steel offers not only economic benefits, but also opens up new possibilities in civil engineering from structural and architectural points of view. [PERSON] et al. described in [1] selected methods for finding patterns for shell structures, highlighting their potential to enhance both the aesthetic appeal and structural efficiency of engineered systems. Their research emphasises how innovative geometric configurations can lead to a more effective load distribution and improved performance in complex structural designs. Complex steel structures can be prefabricated with immense precision, reducing waste and enhancing construction speed and safety. Prefabrication also ensures better quality control, as components are manufactured in a controlled environment and then assembled on-site, leading to fewer defects and a more streamlined construction process. Moreover, steel's inherent properties, such as a high tensile strength and ductility, make it an ideal material for constructing buildings that need to withstand dynamic loads, such as those from wind or seismic activity. Although the understanding of steel structures has greatly evolved in recent years, there are still some particular areas that have significant development potential. Among them are bidirectionally sinusoidal corrugated steel shells, which find practical applications in several areas within the construction industry. In structural engineering, various types of steel webs are utilised in welded plate girders to enhance their performance and efficiency. Traditional flat webs provide straightforward load transfer, but can be prone to a local loss of stability under high loads. To address this issue, corrugated webs have been introduced, offering an increased stability and resistance to buckling without significantly increasing material consumption. Trapezoidal and sinusoidal corrugations are common examples. [PERSON] and [PERSON] (2015) explored the reliability issues associated with corrugated I-beams, emphasising their improved stability compared to flat webs [2]. [PERSON] and [PERSON] (2016) applied the stochastic finite element method to further analyse the performance of these corrugated web steel plate girders, demonstrating enhanced resistance under various conditions [3]. [PERSON] and [PERSON] (2017) extended this work to investigate the impact of corrosion and fire on the reliability of corrugated web girders, revealing additional benefits of corrugated designs in maintaining structural integrity under adverse conditions [4]. Research by [PERSON] et al. (2010) focused on the transverse load-carrying capacity of sinusoidally corrugated steel web beams with web openings, showcasing the advantages of sinusoidal corrugations in load distribution [5]. [PERSON] et al. (2021) examined methods for determining the minimum number of stiffeners required for stiffened corrugated steel walls, further highlighting the structural benefits of corrugation patterns [6]. [PERSON] (2009) provided insights into the shear buckling strength of curved corrugated steel webs for bridges, demonstrating the improved performance of these advanced geometries in practical applications [7].