## 1 Introduction  
In current structural engineering practice, structures are typically designed in order to fulfil stability and serviceability requirements. Both requirements are related to a predefined period of time (durability and service life criteria) which should be achieved at minimum cost (economical and investment criteria). These requirements are typically formulated using a so-called limit state condition which associates the input parameters reflecting the resistance and the action sides of the structural evaluation. The input, which can be physical (dimensions, material strength, loads) or of dimensionless values, is then typically associated with a given level of uncertainty [1, 2]. Therefore, the input parameters are treated within a reliability framework as random variables. The limit state function (G) delineates the acceptable region of a system's performance with respect to a certain failure mode and typically represents a mathematical formulation in terms of the key input parameters. The failure is then attributed to specific combinations of the parameters and is conventionally defined by the condition G < 0. The failure probability of the considered state results from the system or element reliability. These notions determine whether a structural engineering design aims to ensure stability, serviceability or other limit states. The main focus of this investigation is the structural reliability assessment of designed slender concrete columns at the ultimate state. The general approach described above can be implemented by applying various formats of different intricacies together with their corresponding efficiencies. Recent experimental studies for slender reinforced concrete columns [3] have indicated that the classical design approaches for distinguished \(N\)-\(M\) combinations are too conservative and so cannot meet the reliability level that is required in current design standards and the desired level of structural safety required in modern societies [4; 5; 6]. Simultaneously, due to the rapidly developing computational technologies and formats, more sophisticated numerical probabilistic simulations and non-linear calculation formats are becoming more important and finding their way into engineering practice [7]. Such non-linear calculation formats require an extended understanding of a multitude of material specific properties, modeling procedures and analysis formats. Despite the explicit possibilities in EN 1992-1-1 [8] of using non-linear formats, there is still a need for the development of more consistent safety formats [9; 10; 11; 12; 13; 14; 15; 16] for predicting the stability failure of slender columns. A key aspect in deciding on the appropriate configuration of non-linear calculations is the consideration of possible variations in complex and interacting input parameters. To that end, [PERSON] et al. [4] indicated the reliability deficiencies in the above-mentioned design situation of slender columns by assessing to what extent a group of peer institutions can capture this deficiency with independent non-linear numerical analyses. Furthermore, this study allowed a quantification of the modeling and calculation uncertainties [15]. In this context, it should be mentioned that there are already several studies in which uncertainties are analyzed on the structural behavior and design of slender columns, like for example [17]. However, the uncertainties due to the variability of material properties, constitutive laws, geometries, loads and loading applications were not further investigated on the basis of a formal probabilistic analysis. Hence, the main objective of this paper is to discuss the probabilistic verification of the stability of slender columns. The associated research questions regarding the deterministic and probabilistic non-linear calculations for slender reinforced concrete columns, see also [18; 19; 20], provided the basis for the further studies presented in this paper. The studies gave rise to the discussion and analysis of the following novel elements:  
1.