Since 1995, and after the Northridge and Kobe earthquakes, the Reduced Beam Section connection was introduced as a safety concept for structures in seismic zones. Extensive theoretical and experimental investigations lead to the prequalification of the radius-cut RBS connection in North America  , but regarding US and Canadian connection types of steel moment resisting frames. In Europe however, and although the introduction of the RBS was performed by Plumier  , and also patented in Asia  , there is poor existing data. Only recently    experiments and FEM analyses of RBS connections based on European Practice (beam-to-column welded or extended endplate bolted) were performed, while the only references of the Eurocodes regarding RBS are those given in EC8-Part 3  , as also mentioned in the companion paper . Therein, an optimization scheme under monotone loading was adopted, using European I-profiles for the beams and columns of the RBS connection and standard symmetrical extended endplate joints. This lead to eight optimum RBS connections, which in the present work are analyzed via the FEM under cyclic loading, in order to validate the whole design and prove that these connections meet to the maximum possible extent the performance criteria of seismic design   . After calibration of the proposed FE model with existing experimental data  and adopting a preliminary pushover analysis, the aforementioned optimum connections exhibited an excellent response as per every criterion. Hence, it is anticipated that both parts of the foregoing work will assist in incorporating the RBS concept in Eurocodes 3 and 8 as a new design methodology for both static and seismic applications.
Evidently, there are also other types of RBS connections that have been reported in the relevant literature, as for instance in the work by Rahnavard et al.  , where a comparison of the seismic response of these types with the one of the radius-cut RBS was investigated. Furthermore, fragility functions to estimate the probability of reaching or exceeding different damage states in reduced beam section (RBS) beam-to-column moment connections of steel moment resisting frames have been also reported . In the companion paper  , extended bibliographic information has been given, and it will not be repeated here for brevity. For a detailed overview of the parameters affecting the response of RBS connections, one may refer to an already cited work by theauthors .
2. Finite Element Modeling
2.1. Formulation of the Model
Let as considered a statically optimized connection  with European I-profiles for the beams and columns as well as with radius-cut RBS, as shown in Figure 1. The geometry and generalized forces of the optimum connection of the companion paper can be found in tabular form in Appendix, at the end of the manuscript. Such a connection is thereafter modeled in Abaqus Software  utilizing the C3D10M 10-node continuous solid element, suitable for large nonlinear deformations and contact problems. The parts-assembly concept was adopted and a denser mesh for the RBS and the beam-column interface was introduced. Moreover, fixed conditions were imposed at the ends of the column, while adequate constraints were also added to the beam, in order to minimize local buckling phenomena. The final FE model is illustrated in Figure 2, for which a mesh convergence was finally achieved.
Figure 1. Typical optimized RBS connection, as in  (for details see Appendix).
Figure 2. FE model used in the present analysis.
2.2. Material Model
Since the optimum solutions given in the companion paper  were made of S235 steel, an adequate material model was introduced in the FEM simulation, according to well accepted theoretical and experimental evidence of the relevant literature . The schematic picture of this trilinear kinematic hardening model is depicted in Figure 3, for which it is valid that
where εy and εu are the yield strain and ultimate strain respectively, εmax is the maximum strain and σy is the yield stress of S235 steel.
Figure 3. Material model used in the FE simulation (for S235 steel).
2.3. Model Calibration
Before attempting any cyclic analysis, the proposed model was calibrated with existing experimental data . After proper adjustments of this model to the experimental setup (as far as geometry, loading protocol and material properties were concerned), its cyclic response was found in very good agreement with the results of the experiments, as shown throughout Figure 4, in terms of hysteretic loops at two characteristic points of the connection.
2.4. Loading Protocol and Displacement-Based Cyclic Analysis
The cyclic analysis was based on initially evaluating the failure displacement Δf of each optimum connection, which was achieved via static pushover analysis. The goal was to impose the displacement at a proper point along the beam, such that the shear developing at the middle of the RBS was equal to VRBS, as evaluated during the optimization procedure. The whole scheme is shown in Figure 5, where the distance z is calculated and from its value, the failure displacement is appraised.
In the sequel, for each optimum solution, a well-established displacement protocol  was used, since both the values of z and Δf where found earlier. This protocol, in terms of cycle number vs. dimensionless amplitude imposed, is depicted in Figure 6.
2.5. Performance Criteria
For all eight optimum designs, strict seismic performance criteria were considered, in order to validate the whole analysis. These are the following:
Figure 4. Calibration of the proposed FE model, (a) at 3 cm away from the column face; and (b) at the middle of the RBS, in terms of M-δ curves.
Figure 5. Geometry (a) and static pushover curves (b) for establishing the point of imposing the cyclic displacement along the beam.
Figure 6. Loading protocol adopted (saw-tooth diagram).
1) Formation of the 1st plastic hinge in the Reduced Beam Section;
2) Development of a plastic rotation within the RBS of at least 0.03 rad and simultaneous minimal such rotations at the interface between beam and end-plate;
3) Large energy dissipation at the RBS, without any local buckling phenomena, that would reduce ductile behavior;
4) Avoidance of yield spreading in the bolts and minimum prying forces;
5) Steady hysteretic response of the panel zone with small energy dissipation; and
6) Elastic response of the column flange connected.
Hence, three specific and characteristic areas of the connections were examined, namely the middle of the RBS, the interface between beam and end-plate, and middle of the panel zone, as illustrated below, in Figure 7.
3. Numerical Results and Discussion
Numerical results, in terms of hysteretic loops―moment/rotation curves were obtained for all the eight statically optimized connections, at the three areas shown in the previous figure. For brevity, the outcome of the pushover and cyclic analysis will be presented 1) for the best optimum solution, i.e. connection 2 (see Appendix), and 2) as the overall results for the first four optimum connections. Similar response was obtained for the rest of the optimum connections.
More specifically, in Figure 8(a) and Figure 8(b) one may perceive the static pushover curve and the distribution of stresses of connection 2, while in Figures 9(a)-(c) the M-φ curves for this specific connection are shown. Moreover, Figures 10(a)-(c) depict the outcome of the FE cyclic analysis for the 1st four optimum solutions, in terms of M-φ curves, as previously stated.
Figure 7. The three characteristic areas of the connections, for which numerical results were obtained.
Figure 8. Static pushover curve (a); and distribution of stresses in the FE model of connection 2 (b), as in .
From these figures, it is proven that all the performance criteria were met in a very satisfactory manner. No local buckling was encountered (i.e. low cycle fatigue), prying forces did not seem to appear and additionally the panel zone was not affected by the increasing amplitude of the displacement until the plastic hinge was formed in the RBS. The bolts and the column flange connected performed as designed and expected, while the desired energy dissipation was concentrated within the RBS. The above results can be interpreted as a success of the whole analysis and optimum design methodology proposed in the foregoing as well as in the companion paper. This fact will hopefully assist in incorporating the RBS concept as a design alternative for moment resisting steel frames in both EC3 and EC8 as well as in everyday European engineering practice.
Future work requires the effect of a much larger axial force on the columns, in order to use alternative modeling of the RBS connections (details given in  ). The work is ongoing and will produce results that are more interesting in the near future, accompanied by experimental ones.
Figure 9. Results, in terms of moment-rotation curves, for connection 2 according to  , at the three characteristic areas described previously.
Figure 10. Overall results, in terms of moment-rotation curves, for the 1st four optimized connections, according to  (see also Appendix).
As declared also in the Abstract, the aim of this work was the validation of the results obtained via static optimization procedure  by a FE cyclic analysis. This was achieved in a very satisfactory extent, and hence one may draw the following conclusions:
1) The FE modeling of the RBS connections was adequate, since it was properly calibrated with existing relevant experimental result;
2) The displacement-based cyclic FE analysis adopted fully validated the optimization mehtodology under monote loading-outcome of the companion paper;
3) All oprimum designs showed excellent cyclic performance and met all strict performance criteria;
4) The goal of the whole scheme proposed will hopefully assist in incorporating the RBS concept for the seismic design of moment resting steel frames in Europe, both in Standards and practice.
The authors are indebted to Prof. E. S. Mistakidis at their home Department and to Associate Prof. K.-D. Tsavdaridis at the University of Leeds, UK, for their valuable suggestions and help.
Table A1. Geometry of optimized connections.
*if only one number appears, it means that e = e1.
Table A2. Additional geometry and generalized forces of the optimum connections.
*In all connections the value of the ratio VRBS/Vpl,RBS was found less than 42%.