## 1 Introduction
The first examples are the Hearst Tower in New York (Figure 2a) and the 30 St. Mary Axe (also known as Swiss Re Tower or The Gherkin) in London (Figure 2b), both by Sir [PERSON]. These buildings reached 180 m and provided the first references for the suitability of diagrid systems in tall building design. Thanks to the stiff diagrid facades, which create a pleasant diamond-like pattern, the Hearst Tower was realized using 20% less steel than an equivalent conventional moment frame structure [4]. The aerodynamic form of the Swiss Re Tower, obtained through an external free-form diagrid envelope, allowed the reduction of wind action on the building, and led to column-free flexible internal spaces [5]. These two examples already  
Figure 1: Different structural systems for 20 th century tall buildings: (**a**) Moment resisting frame: Empire State Building (New York, NY, USA); (**b**) Braced tube: John Hancock Center (Chicago, IL, USA). Pictures taken by [PERSON]. showed the valuable features of diagrids for tall buildings: enhanced structural performance, saving of material consumption compared to traditional solutions, and significant aesthetic potential. Many diagrid structures were realized worldwide in the following years, where various forms and shapes were adopted for the external diagrid facades. Among others, examples worthy of note are the Guangzhou Financial Center, the CCTV Tower and the Poly International Plaza in China, the Tornado Tower (Figure 2c) in Qatar, the Capital Gate in Arab Emirates, and the Bow Tower in Canada [3]. Nowadays, most of the built diagrid structures are made of steel, mostly due to the easier and faster construction, simpler joints and less expensive formworks [3]. However, concrete and composite diagrids are also experiencing an increasing popularity, as they allow the realization of even more complex-shaped diagrid patterns, e.g., the O-14 Building in Dubai [6]. The significant use of diagrid systems in recent tall buildings is mainly due to the following reasons:  
1. high lateral stiffness (thus low lateral deformability), which allowed the reaching of the lateral deflection limit target using a lower amount of structural material compared to other conventional systems;
2. architectural flexibility, allowing a more rational use of the interior space with fewer columns;
3. modularity, which led to the realization of complex-shaped structures of various forms. These three points arise from the successful use of the triangular module coupled with the inherent structural performance of the tubular structure [7; 8]. The triangular element, which is made up of two inclined diagonals and a ring beam, is the basic component of the diagrid facade. The diagonals carry the vertical and lateral loads mostly by axial forces (compression or tension). For this reason, they are usually pinned at the panel nodes, as reflected in Figure 3a. Since the inclined diagonals often extend over multiple stories, the external floor beams of intermediate stories are often supported by the diagonals, which consequently induces slight shear and bending stresses on them. However, in preliminary design stages, these are usually neglected when compared to the high axial stresses arising from the vertical and lateral loads on the building. Note that in Figure 3a, a 3-story module is reported, as the height of each diagrid module has the same height as a triangular element. In other research works, as will be shown in further figures in the remainder of the paper, the diagrid module is defined in such a way that it covers two triangular elements. In Figure 3b, the three-dimensional view of the tubular diagrid structure is shown, as reported in [9]. Usually, the tube-in-tube configuration is found in real diagrid buildings, where an internal (concrete or steel braced) core is coupled with the external diagrid tube. In preliminary design stages, the diagrid is usually designed to carry the lateral actions alone, while the internal core is designed only for gravity loads.