## 1 Introduction  
The present article is focused on the optimal design of a plane steel frame structure in a fire design situation with the use of system reliability analysis. Previous research [1, 2] has proven that system reliability analysis may be a valuable tool that allows anyone to design truss structures in a more effective way. However, the structural behavior in terms of reliability may be completely different for structures with pinned and rigid connections [3]. Therefore, the reliability analysis of the frame structures under fire influence has to be conducted separately. The analysis, including reliability methods, enables us to design a safety structure with less material compared to the traditional method based on partial safety factors. Therefore, it appears that similar groups of methods can be used to reduce the mass and final cost of frame steel structures. The present research initially characterizes the issue as the considered problem is complex and has many limitations, which will be described in the following sections. To make this and further research easier to compare with other results, the analyzed structure was inspired by publications [4, 5]. The design of a structure in a deterministic way for fire conditions is well established not only by normative documents [6; 7] but also in handbooks [8; 9]. The present article combines a static analysis of a steel frame during fire with a reliability analysis. The basics of the reliability theory are well known; the real flowering of this field of science appeared after introduction of the Hasofer-Lind index [10], which, to this day, is the most popular measure of the safety of a structure. In the following years, many \"classical\" publications were published [11; 12; 13; 14]. The present research focuses on the analysis of system reliability. For the analyzed frame structures, three typical mechanisms of failure (collapse modes) are known: sway (outward), beam (inward), and combined (Figure 1). Fire analysis of a frame structure is not an easy task, as the fire process is very unpredictable, and even small details in construction may determine the structural response. Therefore, the best way to design the structure is to perform a parallel numerical and experimental analysis. A very interesting publication that analyzes a one-story single-nave steel frame was presented in [4]. The partial results of this analysis, i.e., the inward collapse mode, are presented in Figure 2.  
Another detailed analysis of the structural behavior of a steel frame under fire influence was described in [5]. The structures analyzed in this article were inspired by the last  
Figure 1: The typical collapse modes for frame structures [4]. Figure 2: Inward collapse mode—experimental research [4]. two mentioned publications. The details will be described in the following part of this article, although the presented analyses will be developed upon in the future, so the author would like to compare some results. Therefore, it should be underlined that the simplifications made in the present research may have a significant influence on the results. This subject in the context of steel frames was thoroughly analyzed by the authors of [15]. Nevertheless, at the present stage of research, simplified assumptions are essential to narrow down the analyzed tasks and obtain some initial results. Reliability methods in the last years have been applied more and more willingly in engineering analyses, and also in civil engineering problems [16; 17]. Furthermore, a reliability analysis enables the optimization of the structure, not only in the persistent design situation [18] but also under fire influence [19]. As the reliability analysis may be quite a complex task, dedicated software is still being developed not only in Poland [20] but also abroad [21]. The reliability analysis of a steel frame system has been the subject of two publications [22; 23]. The use of this method for the structure under fire load is also discussed in [24; 25].