## 4 Structural Analysis
### Determination of the Bearing Capacity by Nonlinear Elastic-Plastic Analysis  
Figure 14a,b present the results of linear and nonlinear buckling analysis. They show the location of the local buckling mode one shape identified from the first out-of-plane eigenvalue and the location of maximum mises stress generation, respectively. The linear buckling process accounted for the linear response before the initial imperfection was regarded as a failure mode, one that is buckling. Hence, first, the analysis applied the weight of the concrete and superstructure preceded by the live load. Initially, the superstructure (critical members--top-chords) was linear. The loads were incremented until failure mode 1 of the superstructure corresponds to a magnitude of 1. The phase is regarded as the initial imperfection caused by an incrementing load of 1 N, as described in Figure 14a. The resultant eigenvalue mode 1 was 1.06 \(\times\) 10\({}^{6}\). The value was then used as the initial load on the subsequent nonlinear buckling analysis procedure [46]. The process is like linear buckling analysis; however, the structure experiences bifurcation when the compression chord starts to buckle on stresses beyond the yield strength of 325 MPa. With the onset of buckling, the structure begins behaving nonlinearly, indicated by stresses beyond the yield strength. The superstructure deforms with no significant load varying from a deflection more than span/800 at 353.936 MPa, as illustrated in Figure 14b. Due to the consequent geometric and material nonlinearity, the superstructure releases energy related to strains. The strains are associated with the LPF extracted from the failed structure presented in Figure 15a. The proportionality factor also corresponds to the displacement of the structure with respect to the applied load that was also extracted from the nonlinear buckling analysis, as illustrated in Figure 15b. \begin{table}
\begin{tabular}{l l c c c c} \hline \hline \multirow{2}{*}{\(\sigma\)**Distribution**} & \multirow{2}{*}{**Standard**} & \multirow{2}{*}{\(\lambda_{\text{p}}\)} & \multirow{2}{*}{\(\lambda_{\text{r}}\)} & \multicolumn{2}{c}{**Slenderness Ratio**} \\ \cline{5-6}  & & & & **Chord** & **I-Section** \\ \hline \multirow{4}{*}{FLB} & Japanese & 56.76 & - & & \\  & Eurocode & 7.65 & 8.50 & & \\  & British & 8.28 & 9.20 & & \\  & Philippines & 9.43 & 24.81 & & \\ \hline \multirow{4}{*}{WLB} & Japanese & 56.76 & - & & \\  & Eurocode & 61.22 & 70.58 & & \\  & British & 25.76 & 29.44 & & \\  & Philippines & 93.27 & 141.40 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Cross-sectional classification of critical members. Figure 14: Results of linear and nonlinear buckling analysis (**a**) Location of local buckling mode 1 (**b**) Location of maximum stress generation. Figure 15: Relationship between Arc length and (**a**) LPF. (**b**) Displacement. Figure 16 illustrates the location of the onset of yielding and eventual failure. Though failure occurs randomly, from the field experiment it was observed that the center span and top chord are the most vulnerable to stresses due to loading, similarly to dynamic loading [30]. In essence, the section can be used to determine the bearing capacity of the structure. A product of the LPF values by the eigenvalue mode one was considered the load or Force in Newtons (\(F_{n}=LPF_{n}\times\mathbf{X}\)). Where \(F_{n}\) Force, \(LPF_{n}\)th Load proportionality factor and \(\mathbf{X}\) eigenvalue mode 1 (constant). On the other hand, displacement was considered as the structural deformation. The onset of nonlinearity from 26.16 Tons was observed as the SLS with a material yield stress of 325 MPa. Thus, the capacity was regarded as the minimum load that caused the initial failure of the critical member. From the bilinear model, necking denotes the ULS of 47 Tons with an ultimate strength of 353 MPa, as described in Figure 16. The estimates agreed with the suggested hearsay of 25 Tons carrying capacity.