## 1 Introduction
Additionally, the parameters can be related with each other through convex models, such as a hyper-rectangle model or a hyper-ellipsoid model [6]. These methods are used in the analysis of structures with various uncertain parameters, such as uncertain geometry, material or load [7]. They are also used in multidisciplinary uncertainty problems, in which it is important to consider the coupling effect between uncertain variables [8]. The work of [9] shows the application of the method to a truss structure with an uncertain load, a Young's modulus and a cross-sectional area of bars. This approach can also be used for the distributed dynamic load identification [10]. However, in cases where insufficient data series is provided, a fuzzy description [11] is applicable, as exemplified in [12], where a structure in which information about uncertainty is limited and taken from measurements at single points of a model is analysed. It is also possible to use the fuzzy sets to reduce the influence of uncertainty on the obtained results. It is especially important when designing buildings for seismic loads, because such loads are unique in time and difficult to predict [13]. Mention should also be made of novel technologies, methodologies and processes which can help to all stages of the engineering structures by designing, testing, visualizing and analyzing such as Building Information Modeling (BIM) and Extended Reality (XR). The use of BIM and XR at all stages of planning, designing and construction [14] allows one to improve the control process and minimize uncertainty by more effective actions and information exchange [15]. The main motivation for the present author to take up this subject were noticeable gaps in the literature related to the uncertainty analysis. Most of the methods currently used lead to solution sets estimations [16; 17]. The present paper proposes an approach based on the use of convex sets properties. It allows one to find exact solution sets within structures, which can be described by algebraic equations of the form \(\mathbf{CEDx}=\mathbf{b}\) with diagonal elasticity matrix \(\mathbf{E}\) and with independent uncertain parameters. In this way, one can describe various problems of bar mechanics, from trusses, through plane and spatial structures made of thin bars, to theories of moderately thick bars [18; 19]. Many authors compare the solutions obtained by their algorithms with exact solutions (e.g., [20]) or close to exact ones [9]. However, exact solutions are usually only available for very small structures. In addition, often the exact solutions are accurate within one method, but they do not depict the true form of a solution set. In the paper [16], the authors present an exact interval solution for a tensile cantilever with varying stiffness. However, it should be noted that the solution set they provide is a box, i.e., an estimate of the exact solution set. The work [1] shows a comparison of this solution with the proposed method and Monte Carlo method. The proposed approach allows the extension of the set of available benchmark problems to structures with a greater number of unknowns and uncertain parameters. The postulated methodology is also insensitive to the level of uncertainty, while separate publications in the world literature are devoted to structures characterized by large uncertainties [21]. The techniques presented in the present paper enable the assessment of the impact of any uncertain parameters on the calculation results. However, it was decided to show and discuss solution sets depending on the uncertain [PERSON]'s module, which seems sufficiently general and very important from the point of view of engineering applications. It should be emphasized, however, that the methodology will not differ when one considers the uncertainty of geometric data of members, such as a cross-section area, moments of inertia and lengths. Assessment of the impact of uncertainty of other parameters, e.g., the location of structure nodes, remains outside the scope of this work.