## 5 Influence of Stud Behavior on the Beam Response
### Global Behavior of the Deck  
Analyses were developed for the three cases of stud behavior, but the same number of connectors, N\({}_{f}\), designed for the full interaction of the reference case were considered. Furthermore, a case of partial interaction with half the number of reference cases (\(\eta=N/N_{f}=0.5\)) was added. The plastic moment of the partially connected beam, \(M_{pl,\eta}=0.5\), is simply calculated assuming the linear trend between the plastic moment of the steel section (\(M_{pl,\alpha}\)) and the plastic moment of the composite section (\(M_{pl}\)) when the ratio \(N/N_{f}\) varies between 0 and 1. Therefore, it results in:  
\[\frac{M_{pl,\eta=0.5}}{M_{pl}}=\frac{(M_{pl}-M_{pl,\alpha})}{M_{pl}}\cdot\left( \frac{N}{N_{f}}\right)=0.57 \tag{10}\]  
It means that the partially connected beam with a grade \(\eta=0.5\) has a flexural resistance equal to 57% of the fully connected beam.  
According to the modeling of a simple supported deck under a uniform load, the maximum moment and deflection occurred at the mid-span of the bridge. The load-deflection relationship was elaborated considering the load is non-dimensional with respect to the strength capacity of the deck, i.e., the load \(q_{p}\) corresponds to the plastic moment in the middle span (\(q_{p}=8\cdot M_{p}/l^{2}\)). The non-dimensional load-deflection curves are depicted in Figure 4; the value 1 on the vertical axis corresponds to load \(q_{p}\), and the horizontal and vertical dashed lines correspond to the yielding moment (according to the analytical behavior of the composite section) and the corresponding deflection given by the model, respectively. (B) The vertical dotted line indicates the limit elastic behavior of the case with partial interaction \(N/N_{f}=0.5\) (A).  
Load q\({}_{p}\) is related to the plastic moment evaluated by stress blocks, while the elastic-plastic behavior is assumed for concrete and steel; therefore, it can only be asymptotically reached by the models. The results show that 90% of q\({}_{p}\) is attained in case RS because the demand of the end slip overcomes 6 mm and softening of the stud behavior occurs, i.e., the connection fails before the complete plastification of the most-stressed section of thedeck, reducing the global ductility of the structure. Conversely, a higher global capacity in strength and ductility is realized with EP or OS studs, which indicates better performance with greater ductility and strength of the deck. Case OS also shows an increase in stiffness due to the higher stiffness of the studs.  
The case designed with N/Nf = 0.5 evidences a brittle failure of the deck; furthermore, the non-dimensional load 0.5-\(q_{p}\) is reached, which is lower than the designed one.  
\[\frac{M_{pl,q=0.5}}{M_{pl}}=0.57 \tag{11}\]  
This means that the design of ductile connections in partial interaction results in brittle global failure of the structure without assuring the design resistance.  
The results underline that a ductile stud defined by Eurocode 4 [10] with a slip capacity of 6 mm would not be safe. The effect of the partial safety factors of materials and connection strength could provide a final safe design, but these factors would not cover the uncertainties of the materials and connector behaviors, nor necessarily the uncertainty of the global behavior of the beam that is examined in this work.