# Analytical and Numerical Reliability Analysis of Certain Pratt Steel Truss
## 2 Structural Analysis  
The numerical experiments in this work were entirely focused on the simple Pratt truss (Figure 1) schematically presented in Figure 2, whose height was adopted as \(h=1.50\) m, whose number of segments equaled 12, and whose span \(l\) was treated as the input random variable, having in turn triangular, uniform and also Gaussian probability distributions. The random length was initially represented by the discrete set of values ranging from 15.00 up to 21.00 m every 0.60 m to recover polynomial response functions via the series of traditional FEM experiments. They connectextreme deformations and reduced stresses with input uncertainty sources (e.g., parameter \(l\) in this study). Each truss FEM model in this series was uploaded with the same constant load \(q\) redistributed throughout the upper chord nodal points as the concentrated forces. Its characteristic value was set as \(q_{k}=15\) kN/m, while the design load was proposed for an illustration as \(q_{d}=20\) kN/m--the adoption and combination of the safety indices had a marginal influence on this study so it was postponed here by arbitrary load values. All the cross-sections of structural components were assumed as hot-finished square hollow steel profiles made of structural steel S235, whose material parameters were assumed according to the design code Eurocode 3 [26]; structural parameters assigned to the individual elements have been presented in Table 1 below. The buckling lengths of the members making up the girder were assumed on the basis of annex BB.1 to EN-1993-1-1: 2005 [26]. In-plane buckling of the chords was assumed to be 1/12 of the length of the truss. Diagonals reduce the buckling length to 0.08 \(\times\)\(l\). In the case of out-of-plane buckling of the upper chord, the buckling length was reduced by using every second purlin to support the transverse roof bracing to 0.16 \(\times\)\(l\). The out-of-plane buckling coefficient for the lower chord was assumed with the assumption of vertical bracing in the center of the spar. The buckling length of the diagonals was reduced to 0.90 \(\times\)\(l\) due to the degree of fastening of the bars in the chords, i.e., welded joints. Let us note that the proposed cross-sections of structural elements were determined based on initial FEM calculations, and designing procedures provided for the mean value of the truss span \(l\). This structure was initially designed by verification of both ultimate and serviceability limit states (ULS and SLS) in all truss elements and also the bearing capacity of the truss connections. Finally, the following profiles were proposed: (_i_) upper chord--SHS 140 \(\times\) 140 \(\times\) 8, (_ii_) diagonals--SHS 90 \(\times\) 90 \(\times\) 8, and (_iii_) lower chord--SHS 120 \(\times\) 120 \(\times\) 8. The numerical experiments were completed assuming geometrical nonlinearities and P-delta effect in the truss structural behavior, so that incremental Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm was engaged to determine the resulting deformations and stresses. The FEM discretization was completed with the use of 49 linear two-noded truss finite elements. The following additional parameters were used in the FEM analysis: (_i_) load increment number equaled 5, (_ii_) maximum iteration number for one increment equaled 40, (_iii_) increment length reduction number was set as 3, (_iv_) increment length reduction factor was equal to 0.5, (_v_) the maximum number of line searches equaled 0, (_vi_) control parameter for the line search method was as 0.5, (_vii_) the maximum number of the BFGS corrections was automatically set as equal to 10, while (_viii_) relative tolerance for the residual forces and displacements was predefined as 0.0001. Of course, all system matrices were updated after each subdivision.