# Response of Seismically Damaged Steel Reduced Beam Section Joints under Fire
## 3 Finite Element Models  
### Modelling Assumptions  
Finite element analyses (FEAs) were performed by means of Abaqus [28]. The adopted modelling assumptions are consistent with those presented by the authors of previous studies [29; 30]. Therefore, only the main hypotheses are described hereinafter.  
The FE models were discretized by means of structured meshes of C3 DSI solid elements (i.e., 8-node linear brick, incompatible mode), as shown in Figure 4; \"Tie constraints\"  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Beam–Column** & **RBS** & **Imposed Cyclic Damage** & **Fire Scenarios** \\
**Assembly** & **Resistance** & & \\ \hline External & 80\% M\({}_{\text{pl},\text{kd}}\) & No damage (ND) & Only one bay (I) \\ Internal & 65\% M\({}_{\text{pl},\text{kd}}\) & Significant damage (SD–0.04 rad) & Two lower bags (II) \\  & & Near collapse (NC–0.06 rad) & All bags (III) \\ \hline \hline \end{tabular} Total number of cases:  
\end{table}
Table 3: Investigated loadings and their combinations.  
Figure 3: Considered fire scenarios, and examples of the corresponding temperature distribution within the investigated joints: S1 (**a**), S2 (**b**) and S3 (**c**).  
were adopted to model the welds, while normal and tangential contacts were modelled using \"Hard contact\" and \"Penalty contact\" formulations.  
The geometrical imperfections of the members were accounted for by imposing the scaled shape of the imperfection-like engine modes obtained from buckling analysis as described by the authors of [31].  
European S355 steel grade was adopted for all the profiles and plates, while grade 10.9 was used for bolts. The expected mean yield stress of S355 was used through \(\gamma_{\text{ov}}\) factor equal to 1.25, as recommended by EN1998-1 [24], and combined hardening was modelled as in [3]. The non-linear behaviour of the bolts at ambient temperature was modelled as shown in [32; 33].  
The stress-strain curves at elevated temperatures were scaled through the reduction factors (k) for both structural steels and bolts as prescribed by EN 1993: 1-2 [34] (see Figure 5). The engineering curves in Figure 5 were converted into true stress-true strain curves and implemented in the FE models.  
Three analyses were performed per investigated joint, and each analysis comprised different intermediate steps as follows:  
* Cyclic loading at ambient temperature (i.e., 20 \({}^{\circ}\)C): Step 1, clamping the bolts; Step 2, cyclic displacements imposed at the tip of the sub-assembled beam (the AISC341 [27] loading protocol has been used to simulate the seismic damage);
* Fire scenario on undamaged joints: Step 1, clamping the bolts; Step 2, application of gravity loads; Step 3, application of ISO834 curve [35] on the surfaces of the structural elements depending on the simulated fire scenario (heating time equal to 60 min);
* Fire scenario on cyclically damaged joints: Step 1, clamping the bolts; Step 2, cyclic displacements imposed at the tip of the sub-assembled beam; Step 3, application of gravity loads; Step 4, application of the fire scenario (heating time equal to 60 min).  
Figure 4: Main features of the FE model: geometry (**a**) and mesh (**b**).  
Figure 5: Stress–strain curves (**a**) and relevant reduction factors (**b**) at elevated temperatures.  
In order to perform thermal analyses, a sequentially coupled thermal (transient) stress (implicit dynamic) analysis procedure was adopted; in particular, two sequential analyses were performed:  
* Heat transfer: this was carried out to simulate the heat transfer from the external surface of the heated elements through their cross section and along their length.
* Stress analysis: this was carried out to simulate the structural responses of heated elements that are exposed to fire or thermal loading resulting from the previous step.