# Seismic Behavior and Modeling of Ductile Composite Steel-Trussed Concrete Beam to Column Joints
## 3 Numerical Analysis  
With the aim of validating and interpreting the experimental results, two different modeling approaches were considered using Sap2000 software [21], based respectively on lumped plasticity or fiber modeling of the most stressed sections. The first modeling approach considers a lumped plasticity model [20; 21; 22]. In the case of the hinged base column (specimen 1), the model is composed of two elastic elements connected by a plastic hinge inserted at the end of the beam (Figure 11a). In the case of the fixed base column (specimen 2), the column is divided into eight elastic segments 50 cm in length connected by plastic hinges (Figure 11b). The segmental modeling allows for better distribution of the elastic deformation along the column height; the location of displacement transducers installed at the base of the CFT column (Figure 6b) has been defined coherently with the segment length. Figure 10: Comparison of experimental results: (**a**) moment-rotation envelope and (**b**) force-drift envelope. Figure 9: Detected crack pattern: (**a**) joint of hinged column model; (**b**) joint of fixed column model; and (**c**) base of fixed column model. The overall dimensions and external restraints adopted for the models are reported in Figure 11a for specimen 1 and Figure 11b for specimen 2. In the lumped plasticity model, the inelastic behavior is concentrated in a zero-length plastic hinge composed of a nonlinear rotational spring calibrated on the basis of the experimental responses of both tests. The reinforced concrete connection elements have the same geometric characteristics in both specimens, which, however, are made of different materials that affect the moment-rotation relationship. The overstrength capacity of the panel zone is defined in the model by means of a rigid end offset. For the fixed column model, a rigid zone was also defined at the base of the column. In the second modeling approach, numerical fiber element plasticity models [23] able to take into account the yield of steel, including strain hardening, are defined to validate the nonlinear behavior of both experimental tests (Figure 12a,b). In fiber modeling, the inelastic behavior is distributed in the plasticity model within the length of the deformable region, taking into account the spread of inelastic behavior both over the cross-sections and along the member length. The proposed models may be easily implemented in a finite element software program adopting elastic and nonlinear mono-dimensional elements. With particular regard to the current case, column and beam elements were modeled with frame elements divided into a number of segments in which the section properties are assumed constant; the cross-section of each segment is divided into a number of fibers (Figure 12c) having a nonlinear stress-strain relationship with both steel and concrete materials (Figure 12d). Fiber sections are defined only based on geometrical and mechanical properties without applying any empirical calibration procedure. With the aim of accurately reproducing the distribution of the plastic strains in the first part of the beam close to the column edge, having the RC cross-sections A-A represented in Figure 2, a dense discretization has been employed with a segment of size 50 mm for an extension equal to \(2h\), where \(h\) is the section element depth, while the remaining elements of the prefabricated beam with dimension 205 mm, having the cross-sections B-B represented in Figure 2, have been considered elastic. The cross-section of the beams has been discretized into 20 \(\times\) 20 fibers of unconfined concrete and steel reinforcement rebars. A linear behavior has been assumed for the tensile model of the concrete, considering an elasticity modulus of \(E_{ct}=E_{c}/3\) and a tensile strength of \(f_{ct}=f_{cu}/10\). The contributions of stiffness and strength of the beam provided by the reinforcement of the truss and of the bottom steel plate in the proximity of the joint are neglected.