# Real Geometrical Imperfection of Bow-String Arches--Measurement and Global Analysis
## Pilot Study
### 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. Figure 4: Derived horizontal deviation of the right arch from the vertical plane. Figure 3: A point cloud as a result of the 3D laser scanning: (**a**) the global view; (**b**) in detail. 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 \(\alpha_{\text{cr}}\) to reach the global instability, by which the design load has to be increased to cause the loss of stability. Figure 5: Finite element method (FEM) model in ADINA with the shell finite elements applied to most parts. Figure 6: Finite element method (FEM) model in SCIA Engineering with most parts as beam finite elements. 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\({}_{\text{RG}}\)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\({}_{\text{RG}}\)A.