## 4 Structural Analysis
### Linear Elastic and Nonlinear Elastic-Plastic Analysis
#### 4.3.2 Condition of Nonlinear Elastic-Plastic Analysis  
The bending moments and compression on bridge elements occur as whole but not as separate elements. However, Eurocode 3 recommends verifying structures in SLS and ULS as constituent members [35]. Conversely, in-situ structures interact as a whole, and their interaction should be analyzed similarly. Bailey Bridges have numerous critical points of tension (bottom chord) and compression (top chord) members. They have pronounced stresses at the center span chord, as presented in Figures 7 and 8 from the field experimental data. During loading and unloading, Bailey Bridges have random dynamic and static behavior and are vulnerable to instability. The Newton-Raphson method does not account for nonlinearity for structures exhibiting buckling and material nonlinearity. Therefore, the reliability of the structure was assessed using the Arc length method. This method can check material and geometric nonlinearity [46]. In this study, it was first used to identify the linear response about the base state before the affected member fails--buckling. Moreover, the nonlinearity of systems when they exhibit numerous critical points were investigated, i.e., a body that cannot support an increase in load. Procedures of linear and nonlinear buckling analysis were detailed after this. It is worth noting that the used structural model, boundary conditions, and the material property were the same as aforementioned in Figures 10 and 11.  
Linear Buckling Analysis--Linear Perturbation  
A four-point bending of \(-\)1 N on each loading point was applied on the deck section, coinciding with the position of the load cases used in the linear buckling analysis as the initial load. Then, the linear perturbation procedure was selected in Step and buckle as the option and subspace eigensolver. Though the number of the eigenvalue(s) required is only one--Mode 1 Eigenvalue, \(n=10\) was input as the minimum number of eigenvalues requested where (\(2n\), \(n+8\)) is the minimum number of vectors. Hence, 18 vectors were used per iteration of a maximum of \(\approx\)300. A lower number of iterations was avoided to ensure convergence. The program applies increments to the load on the load control.  
Non-Linear Buckling Analysis--Static Riks  
First, geometrical nonlinearity and residual stresses were introduced through equivalent geometric imperfections, as recommended in EN 1993-1-1 (EC-3), as a prerequisite for designing slender structures. The imperfection was imported from the linear perturbation step. Secondly, the step was created using the general procedure Static Riks method that exploited the arc length method to account for the nonlinearity using Nonlinear geometry (Nlgeom). The maximum number of increments was set to 1.0 \(\times\) 10\({}^{6}\) with an arc length of approximately 1. The load was concentrated force set as the First Eigenvalue from Linear Perturbation Analysis with similar boundary conditions, loading point positions,and material properties. Because the method does not terminate, it was terminated when the arc length is approximately less than or equal to 1. It was assumed that the structure fails when the stress values on critical members (chords) surpass \(f_{\text{p}}\), corresponding to span/800 of deflection value greater than or equal to 39.763 mm. Finally, the Load Proportionality Factor (LPF) was extracted, and the product of the values by the first eigenvalue from Linear perturbation [46] was determined and plotted against the deflection and the load-displacement curve.