# Analytical and Numerical Reliability Analysis of Certain Pratt Steel Truss
## 1 Introduction  
Stochastic computational mechanics is still a very extensively explored research area with many concurrent methods. However, many of them do not have widely accessible computational implementations, especially in the context of the finite element method [1, 2]. One could recall here not only the classical Monte Carlo simulation [1, 3] or the stochastic perturbation technique [1, 4, 5, 6] in its various order implementations. Moreover, still very popular is the Bayesian approach [7], Karhunen-Loeve decomposition and polynomial chaos expansion [8], stochastic kriging models [9], different analytical and semi-analytical techniques [1, 10] and fuzzy stochastic analysis [11]. It should be mentioned that various computational studies concerning numerical error for stochastic methods are available [1, 12]. Stochastic analysis has been implemented together with acceleration [13] and reduction algorithms [14], and is recommended for the homogenization approach in micromechanics [1]. The aforementioned numerical approaches may be used even for a solution to the insufficient material data problem in solid mechanics [15]. Unfortunately, practicing engineers must use some additional software or must create new small programs to determine the reliability indices mandatory for modern engineering structures. A very specific role is in the distribution and usage of the computer algebra systems such as MAPLE, MATLAB or MATHEMATICA, where many stochastic methods can be and areimplemented. Let us note that the finite element method (FEM) modeling in the presence of any uncertainty still remains a separate part of the engineer's work. It is well known that a huge portion of mathematical effort in the development of analytical formulas for structural engineering analyses has been lost in the last few years thanks to a common and fast methodology of \"computing everything\". However, many relatively simple structures may be quite efficiently designed using classical formulas, extended relatively easily with mathematical software towards reliability assessment. The guidelines for structural safety are based upon the very simplified reliability index formulated by Cornell many years ago. It belongs to the first-order reliability method, so that even if the engineers follow Eurocode 0 statements [16], their analyses have rather limited importance. This index is based upon the assumption that structural resistance \(R\) and the effect of external loadings \(E\) on the given structure both have Gaussian distributions, and that the limit function has linear character. Furthermore, such a designing code does not offer any algorithm for how to efficiently compute the first two probabilistic moments for the functions \(R\) and \(E\). On the other hand, an application of the second-order reliability method (SORM) for practicing engineers is usually impossible because it demands knowledge about a curvature of the limit state surface; this curvature cannot be directly determined using the FEM software. This gave the authors a motivation to look over mathematical works concerning probabilistic distance (divergence), which could be applied to measure structural safety using some relative entropy in between the distributions of \(R\) and \(E\). It should be mentioned here that statistical divergence (or diversity) has been studied in different areas of the applied sciences, particularly in biology [17], econometrics [18], statistical physics [19] and even accidentally in civil engineering, where it served in some seismic analyses [20]. Therefore, this paper aims to present an application of analytical and numerical probabilistic analysis in a reliability assessment of exemplary steels and popular civil engineering structures. The Pratt steel truss girder has been selected to demonstrate such an approach due to its popularity in academic analyses during civil engineering academic courses and a huge number of the existing old steel bridges all around the world.