## 1 Introduction
It is also worth mentioning the studies that have contributed to the development of methodologies and technologies for evaluating the structural condition of bridges [16], monitoring their health [17], understanding their responses to environmental factors like temperature variations [18; 19], and identifying potential damage using advanced optimization techniques [20]. Throughout the staged construction of a cable-stayed bridge, cable tensioning significantly influences the structure's development [21]. Depending on the adopted method for assembling the bridge span structure, the most delicate operation is the activation, or tensioning, of the stay cables. This process involves phased changes in the static system of the bridge structure until the final system configuration is achieved. The tensioning of the stay cables simultaneously induces prestressing in the span structure, impacting the stress and strain states of all active elements of the system at any given moment. During this operation, tensioning individual cables must not result in a loss of tensile force in other cables, while ensuring that the deformations of the span structure and pylons remain within permissible limits. The tensioning protocol, or program, specifies the sequence, required number of phases, and the magnitude of the applied tensile force for each cable in each phase. In defining the tensioning program, besides the aforementioned considerations, the applied cable system and the equipment available to the contractor are highly relevant. Given the time and labor intensiveness of the procedures of optimizing cable tensioning during installation and construction, control becomes very important [22]. The primary objective is to minimize repetitive cable tensioning and achieve the targeted cable force according to the design specifications, thereby eliminating the necessity for final cable force adjustments [23]. Generally, the tensioning of stay cables can be performed either by the simultaneous tensioning of all strands or by individually tensioning each strand. For simultaneous tensioning, powerful hydraulic jacks, known as multi-strand jacks, are used. If these large, heavy jacks can be accommodated on the construction site, despite their size and weight, it is possible to stress the cables in one pull. This process requires careful planning, precise execution, and monitoring to prevent overloading and structural imbalance, and ensuring that the bridge structure is not subjected to excessive forces too quickly. Instead of applying the full required tension in one pull, the tensioning process can be divided into smaller increments. Each increment involves applying a portion of the designed cable force, followed by monitoring and adjustments before proceeding to the next increment. Another method of performing cable tensioning is the individual tensioning of each strand of the cable, also known as strand-by-strand tensioning. As the concept for calculating the tensioning of a stay cable strand-by-strand presented in this paper originated during the construction of a pedestrian bridge over the Ni\(\check{\text{s}}\)ava River in Ni\(\check{\text{s}}\)[24; 25], in 2003 (Figure 1), this method will be further explained. The tensioning of cables composed of parallel strands was performed using a lightweight hydraulic jack, also known as a mono-strand jack, or a single-strand post-tensioning jack (Figure 2), which is a crucial tool utilized in the construction of stayed engineering structures. Its primary function is to apply controlled tension to a single steel strand, typically consisting of numerous high-strength steel wires encased within a protective sheath. This specialized device plays a pivotal role in the stay cable tensioning process, ensuring the success and safety of such operations. Strand-by-strand application of tension force to a cable can be performed in a single cycle or in several cycles. Single-cycle strand tensioning, known as \"Isotension,\" was developed and patented by the Freyssinet Institute [27].