## 2 Stiffness of the Stringer-to-Cross-Girder Connection
### Theoretical Study  
The analysis of the bending stiffness of the stringer-to-cross-girder connection was the subject of a parametric study, which we have processed in our department. In this study, we observed the influence of different configurations of a stringer-to-cross-girder connection corresponding to those in Figures 2 and 3 on the bending moment in the stringer at the place of its connection to the cross girder. Three types of joint configurations were considered in this study:  
* Type 1--stringers are connected to the cross girder by their webs only using connecting angles;
* Type 2--in addition to the web connection, the upper flanges of the adjacent stringers are also interconnected using splice plates. The lower flanges are not connected directly, but the bottom part of the joint is stiffened by haunches;
* Type 3--in addition to the web connection, both stringer flanges are interconnected using splice plates. Theoretical analysis of the behaviour of these three types of connections was carried out using three different types of finite element models (FEM models) of a fictitious superstructure of a plate girder bridge with a lower member deck. The numerical models were processed with SCIA Engineer software [15]. The two models represented a common way of modelling this type of bridge when all the structural members of the superstructure were modelled with beam elements (Figure 4a,c). The stringer-to-cross-girder connection was considered once as fully hinged and once as a perfectly rigid joint. In the third model, the whole superstructure was modelled using shell elements for all load-carrying structural members (Figure 4b). Such a model allowed for a better approximation of the behaviour of the analysed type of connection at the cost of a more demanding modelling process as well as the necessary computing time. Visualizations of the shell models of the three considered types of stringer-to-cross-girder connections are illustrated in Figure 5. The bolts joining the connecting angles and webs of the stringers and cross girders as well as the splice plates and stringer flanges were modelled by means of beam elements rigidly connected with the shell elements  
Figure 4: Three types of numerical models of the bridge superstructure: (**a**) beam FEM model with hinged stringers, (**b**) shell FEM model, (**c**) beam FEM model with continuous stringers. modelling webs, flanges and angles. The shell elements modelling the cross girders were rigidly connected to the shell elements modelling the main girders including vertical stiffeners. Standard rectangular 2D finite elements were used in the mesh generation by the computational program. The average size of the 2D elements was considered to be 150 mm, with a refinement to the value of 20 mm in the case of the examined stringers and 10 mm in the case of the connecting angles. Two different spans of stringers, given by the cross girder distances D, have been considered for each investigated case in this study, namely D = 2.0 m and D = 3.0 m. Thus, 18 numerical models were processed for the needs of this study in total. In order to eliminate the global effects of the considered loads, the main girders were supported in the vertical direction at the locations of the cross-girder connections. The results of the theoretical study are presented in Figure 6. The bending stiffness of the observed three types of stringer-to-cross-girder connections is compared indirectly by means of the courses of bending moments from the uniformly distributed load on an intermediate stringer calculated with respect to the different boundary conditions as they are presented hereinbefore. The bending moments are given in a relative scale related to the minimum calculated moment corresponding to the beam FEM model with a perfectly rigid stringer-to-cross-girder connection (M\({}_{\text{ymin,RC}}\)). The results of the numerical analysis confirmed the expected behaviour of the three investigated types of stringer-to-cross-girder connections.