# Analytical and Numerical Reliability Analysis of Certain Pratt Steel Truss
## 2 Structural Analysis
All the finite element method experiments were carried out in the civil engineering system Autodesk ROBOT, where linear truss elements were selected, one finite element was equivalent to a single structural element and each node had two degrees of freedom (vertical and horizontal displacements). This structure was simply supported at both ends in the lower chord, while the connections were all designed as perfectly non-deformable (due to the welds). The results of numerical modeling in both deterministic and stochastic contexts were contrasted with the analytical approach briefly presented below. It is known that a deflection of the periodic and regular truss having two parallel chords were determined using elementary knowledge following the rules of the strength of materials and, particularly, fourth-order ordinary differential equations relevant to the Euler-Bernoulli beam and its analogy to the truss structures with parallel upper and lower chords. One may demonstrate that deflection of such a truss can be determined using the following function \(u(x)\) including a distance \(0\leq x\leq l\) of the given point from its support  
\[u(x)=\frac{q\,l^{4}}{24 Ej}\left[\frac{x}{l}-2\left(\frac{x}{l}\right)^{3}+ \left(\frac{x}{l}\right)^{4}\right], \tag{1}\]  
where \(q\) is a characteristic value of the load, \(l\) is the theoretical span of the truss, \(E\) is the Young modulus, \(J\) is the inertia moment of a system of the upper and lower chords. It is very important that this model postpones a rigidity contribution of both columns and diagonals during bending analysis of the truss. The center of gravity of the truss chords is calculated as (see Figure 3)  
\[d=\frac{A_{1}\,h}{A_{1}+A_{2}}, \tag{2}\]  
where \(A_{1}\) is the total cross-sectional area of the upper chord, and \(A_{2}\) is a lower chord cross-sectional area. The resulting inertia moment of both truss chords according to the [PERSON] theorem equals  
\[I=I_{1}+A_{1}(h-d)^{2}+I_{2}+A_{2}\,d^{2}, \tag{3}\]  
where \(I_{1}\) is the moment of inertia of an upper chord, while \(I_{2}\)--inertia moment of the lower chord. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Upper Chord** & **Diagonals** & **Lower Chord** \\ \hline Member length \(l_{y}\) & 0.90 \(\times\)\(l\) & 1.00 \(\times\)\(l\) & 0.90 \(\times\)\(l\) \\ Member length \(l_{z}\) & 1.00 \(\times\)\(l\) & 1.00 \(\times\)\(l\) & 1.00 \(\times\)\(l\) \\ Buckling length coefficient \(y\) & 0.08 \(\times\)\(l\) & 0.90 \(\times\)\(l\) & 0.08 \(\times\)\(l\) \\ Buckling length coefficient z & 0.16 \(\times\)\(l\) & 1.00 \(\times\)\(l\) & 0.50 \(\times\)\(l\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of steel structural elements in the given truss. Having analytical formula for the deflection function and the given probability density function of the truss span, one may derive the basic four probabilistic characteristics of the deformation line \(u(x)\) of such a truss following classical definitions available from the theory of probability [1]. It yields  
\[E[u(x)]=0.0417\frac{qx}{EI}\left(E^{3}[l]+3E[l]\sigma^{2}(l)-2E[l]x^{2}+x^{3} \right), \tag{4}\]  
\[Var(u(x))= \tag{5}\]  
\[\mu_{3}(u(x))= 0.001302\frac{E[l][\sigma^{4}(l)q^{3}x^{3}}{E^{3}l^{3}} \tag{6}\] \[\left(45\sigma^{4}(l)+48E^{2}[l]\sigma^{2}(l)-24\sigma^{2}(l)x^{2 }+9E^{4}[l]-12E^{2}[l]x^{2}+4x^{4}\right)\,\] \[\mu_{4}(u(x))=9.04224\cdot 10^{-6}\frac{\sigma^{4}(l)q^{4}x^{4}}{E^{4} l^{4}}(3465\sigma^{8}(l)+17280E^{2}[l]\sigma^{6}(l)-2520x^{2}\sigma^{6}(l)+\] \[+11934E^{4}[l]\sigma^{4}(l)-8136E^{2}[l]\sigma^{4}(l)x^{2}+840 \sigma^{4}(l)x^{4}+2160E^{6}[l]\sigma^{2}(l)-3240E^{4}[l]\sigma^{2}(l)x^{2}+\] (7) \[+1440E^{2}[l]\sigma^{2}(l)x^{4}-160\sigma^{2}(l)x^{6}+81E^{8}[l]- 216E^{6}[l]x^{2}+216E^{4}[l]x^{4}-96E^{2}[l]x^{6}+16x^{8})\]  
where \(E[l]\) and \(\sigma(l)\) stand for the expectation and standard deviation of the truss girder length.