## 4 Considerations for the Reliability Analyses of the Structural System
### Stochastic Models for the Basic Variables  
The stochastic models for the basic variables that affect the design equations described in Section 2 were defined based on the provisions of the JCSS Probabilistic Model Code [38]. These models are presented and described throughout Tables 1-3. For the sake of this investigation, all the basic variables are considered uncorrelated. The stochastic models for the concrete compressive strength \(f_{\rm c}\) described in Table 1 are particularly relevant. As a material property, concrete compressive strength is typically described using a logarithmic normal distribution (i.e., lognormal distribution), as it does not allow for negative values (e.g., [39]). Since concrete compressive strength is affected by multiple factors, such as limited sample sizes, the production site and the time of year, it is generally accepted that its mean and its standard deviation are also considered random variables. In statistics, the conjugate prior of a normal or lognormal distribution with an unknown mean and standard deviation is a normal-gamma distribution (e.g., [39]), which is a bivariate four-parameter family of a continuous probability distribution. These four parameters are commonly denoted as \(m,\ n,\ s\) and \(\
u\), with \(m\) referring to the mean value of a sample with size \(n\) and \(s\) being the empirical standard deviation of a sample of size \(\
u+1\). To describe the stochastic model for the concrete compressive strength, a four-parameter distribution (log-Student's-\(t\) distribution) was used, whose parameters are given in the JCSS Probabilistic Model Code [38] for different concrete strength classes (Table 1). Note that a log-Student's-\(t\) distribution arises as the marginal distribution, which results from the integration of a normal-gamma posterior distribution and is the required distribution density for concrete compressive strength (e.g., [39]). Another relevant basic variable is the model uncertainty. The model uncertainty generally accounts for effects that are neglected in model simplifications and mathematical relationships (e.g., [40,41]). As previous studies already demonstrated (e.g., [18,40,41]), uncertainties in resistance models can play a significant role in structural safety evaluations and the calibration of partial factors in semi-probabilistic designs and, therefore, their thorough consideration is critical for the feasibility of any reliability-based investigation. According to the JCSS Probabilistic Model Code [38], the model uncertainty variable is typically characterised by a lognormal distribution. Nonetheless, in the context of this investigation, this assumption needed to be properly evaluated. Thus, the statistical parameters of the distribution function describing the model uncertainty were estimated from the data stored in the experimental database described in Section 4.1. To this end, the theoretical (i.e., predicted) punching shear resistance (according to DIN EN 1992-1-1+NA(D) [4, 5, 6, 7]) was compared with the experimental failure loads. Note that for the comparison, the punching shear \(V_{\text{R},c}\) was used, all the partial factors in the design equations were taken as unity and the mean value of concrete compressive strength (for cylinders) was adopted. Then, the statistical parameters of the related resistances were determined under the assumption of both the normal and lognormal distributions. By doing this, the quotients of the experimental failure loads and the theoretical punching shear resistances were determined. In addition to the statistical characterisation of the model uncertainty, three consecutive analyses were carried out. First, a goodness-of-fit test-in this case, the Kolmogorov-Smirnov test (e.g., [42, 43, 44, 45])--was adopted to evaluate whether the data followed the assumed distribution function (i.e., a lognormal distribution).