## 3 Bridges Analyzed and Method of FE Analysis
### FE Analysis  
#### 3.2.1 FE Modeling  
The finite element models were developed using MSC. Marc. The mesh size will affect the accuracy and calculation time of the FE analysis. Since the number of elements and nodes in a single FE model in this paper is not large, after ensuring that nodes are set at special positions such as springing, vault, and intersection point of arch rib and suspender, arch rib and transverse brace are divided by elements with small length and approximately equal length, each element is about 1.2 m long. As the longitudinal beam, cross beam, and suspender are not the focus of this paper, their elements are not subdivided. The nodes and elements of the FE model are divided, as shown in Figure 8a. The beam elements of type 14 and 79 were adopted to develop the members of box sections and I-shaped sections, respectively. The deck of the bridge was ignored. To simplify calculation according to the codes in the following sections, the arch ribs were simplified to have a constant section. An initial out-of-plane deflection of \(\sqrt{20L}/300\) in the shape of the first-order out-of-plane buckling mode was assumed. Three degrees of translation freedom at the springing of two arch ribs were fixed. The dead load of the bridge was automatically determined by the software according to the inputted material density and element volume. According to the influence line of springing axial force, the live load was applied as a full-span uniform load, as shown in Figure 8b. The established FE models of through-type, half-through type, and deck-type steel arch bridges are shown in Figure 8c-e.  
There are two kinds of arch instability: branch point instability and extreme point instability. Extreme point instability often occurs in practical steel arch bridges. To obtain a relatively accurate evaluation of out-of-plane critical axial force, it is advisable to adopt the analytical method considering the material and geometric nonlinearity [13,24]. The nonlinear equilibrium equation was solved by the Newton-Ra  
Figure 8: FE Model of Steel Arch Bridge: (**a**) FE Model Meshing; (**b**) Full-span Uniform Loading; (**c**) Through-Type Bridge; (**d**) Half-through Type Bridge; (**e**) Deck-Type Bridge.  
Figure 7: Geometric Notation for Coefficient \(\gamma\): (**a**) General View; and (**b**) Elevation View.  
Large deformation effects were considered by using the updated Lagrangian formulation to conduct geometrical nonlinear analysis. The nonlinearity of the material is considered in the establishment of the FE model. For the box sections and I-shaped sections, 28 and 24 integration points were specified, respectively. The constitutive relation of steel proposed by [PERSON] et al. [30] was employed, and the stress-strain curve is shown in Figure 9. Here, \(\sigma_{y}\) represents the yield stress and the \(\varepsilon_{y}\) is the yield strain, \(\varepsilon_{st}\) and \(E_{st}\) represent the strain at the onset of strain hardening and initial strain-hardening modulus, respectively. Equation (9) shows the calculation of strain hardening modulus \(E^{\prime}\):  
\[E^{\prime}=E_{st}\exp(-\xi\frac{\varepsilon-\varepsilon_{st}}{\varepsilon_{y} })\hskip 28.452756 pt(\varepsilon_{st}\leq\varepsilon) \tag{9}\]  
where \(\xi\) is a material coefficient. Japanese Industrial Standards (JIS) SM490Y steel is assumed in this study, with \(E\) = 2.06 \(\times\) 10\({}^{5}\) MPa, \(\sigma_{y}\) = 355 MPa, Poisson's Ratio \(\
u\) = 0.3, \(\varepsilon_{st}\) = 7\(\varepsilon_{y}\), \(E_{st}\) = \(E/30\) and \(\xi\) = 0.06.