# Analytical and Numerical Reliability Analysis of Certain Pratt Steel Truss
## 3 Numerical Results for Structural Elements and Their Discussion  
Numerical analysis was focused in turn on the following issues: (_i_) determination of polynomial response functions of the structural effort and of the extreme deformation as the functions of the truss span; (_ii_) determination of the basic probabilistic characteristics of structural responses with the use of FEM series and of the analytical formulas; (_iii_) determination of the reliability indices using both analytics and numerical analysis, including the new reliability index formula; and (_iv_) stochastic analysis of the most efforted structural joint in the truss. Three different probability distributions were selected for this analysis--triangular, uniform and Gaussian--to study the Monte Carlo statistical approach and the iterative generalized stochastic perturbation technique of the 10 th order. The detailed analytical derivations of the perturbation-based formulas for the first four central probabilistic moments could be found in [1] for the Gaussian PDF, whereas those relevant to triangular and uniform PDFs are presented in [29]. The results of stochastic analysis have all been collected for some range of the input uncertainty, i.e., \(\alpha(l)\in[0.00,0.15]\), which has been taken as relatively wide to study the convergence (and possible divergence) of all three tested probabilistic numerical techniques. The first results are presented in Figure 4 and they concern structural effort variations in all groups of the structural elements determined as the functions of the truss span. A numerical model was created in such a way that none of the designed steel profiles exceed the limit effort (100%), even for the extremely large span of this structure. Figure 4 clearly documents that all these efforts almost linearly depend upon this span--the largest effort was noticed within the welds, then, a slightly smaller one was observed for upper and lower chord members, while the smallest was in the truss diagonals. One may conclude that the numerical values of this effort ranged from about 65% up to 95% for the structural joints (whereas \(l\) varies from 15.0 up to 21.0 m), whereas the effort for lower and upper chord elements varied from 45% up to more than 85% at the same time. This enables us to conclude that the ULS-based reliability index should be based upon the joints' rational designing, while the SLS, traditionally, upon this truss midspan deflection. Having verified the deterministic safety of the designed Pratt truss, the polynomial response function was recovered via the series of FEM experiments with varying truss span. Discretization of the span computational domain, the corresponding series of the extreme truss deflections and third-order polynomial basis determined using the LSM procedure are shown in Figure 5. It can be seen that LSM fitting corresponds almost perfectly to the FEM data and that the approximating function is very smooth and regular. It should be mentioned that extreme deformation corresponds to the last equilibrium determined from the BFGS incremental path. Analytical representation was necessary for further perturbation-based derivations and it was obtained on the basis of numerical results delivered in Table 2 as  
\[u(l)=0.0001522l^{3}-0.003769l^{2}+0.04322l-0.17150. \tag{15}\]  
It should be mentioned that this relatively low order of polynomial basis causes, that higher than the fourth-order derivatives of the function \(u(l)\) simply vanish, so that the Taylor series expansions inherent in the stochastic perturbation technique are very short and contain a few terms only. Probabilistic results concerning extreme deflection of the truss were further collected (in Figures 6-9) and these are: expected value, coefficient of variation, skewness and kurtosis. They were collected for Monte Carlo analysis (abbreviated here as MC), perturbation method (PM), separately for the FEM approach (NC) and following analytical expressions (AC).