#### 3.3. Finite Element Method Analyses

Numerical global analyses were executed by using of finite element method (FEM). Two programs and two different approaches concerning the mesh of the main superstructure were applied.

The first spatial numerical model was created in ADINA software (ver. 9.4.2, Adina R&D, Inc., Watertown, MA, USA, 2019). Almost all the parts of the bridge superstructure including the orthotropic deck were modelled within this model by shell finite elements (

Figure 5). The beam elements were assumed only for upper bracings and for the diagonals between the hangers as well. The changes in thicknesses of each modelled 2D member were carefully taken into account in accordance with the bridge documentation and in situ verification. Special attention was paid to the approximation of the arch-to-girder connection, where all the relevant eccentricities, stiffeners and additional ribs were precisely considered.

The second 3D model was developed in the SCIA Engineer software (ver. 19.1, SCIA nv, Hasselt, Belgium, 2019) widely used in civil engineering practice. Contrary to the first one, almost all the parts of the bridge superstructure were modelled by 1D finite elements (

Figure 6) in this model. Shell elements were applied only for the plate of the orthotropic deck, but the longitudinal and transversal stiffeners of the deck were modelled as ribs of the shell elements again. Within this model, all the eccentricities of the modelled members and all the changes in the thicknesses of the plates were taken into account as well.

The FEM models were loaded by the same dead loads and also by Load Model 71 (LM71) including the dynamic effects in accordance with EN 1991-2. In addition to the consideration of the actual position of the railway track, the code-based eccentricity of vertical loads produced by the unequal load of train wheels was thought as well. Corresponding partial safety factors and combination rules were applied to create a design combination of loads. The longitudinal position of the LM71 came from the checking process as a crucial element for the normal stresses in the outer arch, which is affected by the compression combined with biaxial bending. Basis static values such as the sum of the mass, the reactions, deformations, etc. were compared between the models, to make sure that the static behaviour of both models was similar.

Naturally, the linear analyses (LA) were done first in both software. The linear stability analysis (LSA) were also ran in order to get the first eigenmode of the loss of the arches’ structural stability, see

Figure 7. This structural buckling mode is quantified by the minimum load factor α

_{cr} to reach the global instability, by which the design load has to be increased to cause the loss of stability.

In order to see the influence of real imperfections on the internal forces and stresses in linear approach, a real shape of the arches was implemented into the geometry of the models. Therefore, the analysis executed on such imperfect models can be designated as a linear analysis with the real geometric imperfections included (LI_{RG}A).

The geometrically nonlinear analyses with the imperfections included (GNIA) of both spatial models of the bridge superstructure was the next step. By introducing the “real geometric imperfections” into the nonlinear analysis, taking into account second-order effects, an influence of the deformed shape on the static behaviour of the structure were correctly estimated. As imperfection was taken from the laser scanning, the results of this analysis will be designated as GNI

_{RG}A. The scheme of axis of arches with real geometrical imperfections are in

Figure 8a.

The last analysis came from the discussion on the comparison of the effect of the measured real geometric imperfections with the results taking into account the geometric imperfections reflecting the normatively determined tolerances. Such an analysis can be then designated as geometrically nonlinear analyses with theoretical geometric imperfections were included (GNI

_{TG}A). With reference to the statements given in the introduction of this paper, for the shape of such theoretical imperfections, the first mode of the loss of stability can be utilised. Thus, the supposed shape of the global and local geometric imperfections were considered in the form of the first elastic buckling mode of the arch structure. The key problem is to estimate the amplitude of this kind of unique global and local imperfection (see designation “ugli” imperfection in [

5]). As the presented research deals with geometrical imperfections only, the so-defined imperfection in this paper does not represent an equivalent geometrical imperfection, which should also include structural imperfections. Therefore, it will subsequently be referred to as a “theoretical geometrical imperfection”. Hence, the amplitude scaling of the bucking shape was based on the allowances in tolerances for bridges given in EN 1090-2.

Simplistically, the value of the maximum horizontal displacement of the arch from its designed position was estimated as a sum of the global and local deviation from straightness. For the local deviation, the value of L

_{loc}/750 was considered, but not less than 6 mm. Similarly, for the global tolerance, L

_{glob}/500 was assumed, with a limit value of at least 12 mm. The first mode of out-of-plane stability reaches its maximum between the 11

^{th} and 12

^{th} hangers, therefore the local length L

_{loc} = 8.96 m was taken, while for the “global” scale, the buckling length of the arch calculated by LSA was adopted, i.e., L

_{glob} = 13.28 m. Thus, the amplitude of theoretical geometrical imperfection in the shape of the first out-of-plane buckling mode in this GNI

_{TG}A analysis was the sum of the local and global tolerances 12.0 + 26.5 = 38.5 mm. The scheme of the axis of arches with theoretical geometrical imperfections are presented in

Figure 8b

#### 3.4. Comparison of Results

Some results from the presented study are compared in

Table 1. Maximum stresses in four chosen cross-sections of the right arch are compared. The values in

Table 1 were found as the maximum stresses picked from the FEM analyses at the same corner point of each arch cross-section marked in

Figure 9 from A1 to A4.

From the comparison it can be stated that geometric imperfections had a smaller influence on the stresses than expected. The consideration of the measured imperfection produced stresses not far from analyses, where the theoretical geometrical imperfections derived from the allowed tolerances were taken into account.

The graph in

Figure 10 shows a percentage increase in stresses in the analyses with imperfections included in comparison with linear analyses without any imperfections (LA). Except for the differences between both software analyses, only a few percent of effect is evident. Surprisingly, the linear analyses with the real geometrical imperfections included (LI

_{RG}A) were very close to their more complex geometrically nonlinear alternative (GNI

_{RG}A).

The absolute values of differences are small, but relative variation between the four chosen arch cross-sections is evident. The real deformation can be sometimes very different from the theoretical shape of buckling, thus it could probably better reflect the increase in stresses in the most affected cross-sections. However, if the values are compared, it can be clearly seen that the geometrical imperfection influenced stresses only slightly in the case of this bridge.

At the same time, it has to be pointed that almost three times higher differences were produced by the application of the different FEM model than by the different type of analysis. Such a comparison is given in

Figure 11, where the percentage differences in the stress obtained by the SCIA Engineering model are presented in comparison to the stresses produced by the analyses in the ADINA software.

In the cross-sections near the arch foot and in the middle of the span the difference are smaller. The influence of using 2D versus 1D finite elements, respectively, is evident even though both the models acted statically very similarly in global comparison. For instance, the differences in the calculated deformations (deflections) between the models were only about 1%.