*5.4. Evaluation of Catenary and Flexural Mechanisms Under Di*ff*erent R<sup>i</sup>*

This section investigates the influence of different *R<sup>i</sup>* on flexural and catenary mechanisms. As discussed in Section 5.1, under the middle column removal condition, the vertical resistance of the beam-to-column connection is controlled by the contribution of flexural and catenary mechanisms. Figure 19 illustrates the gravity resistance development for the beam-column assembly of two specimens, having welded unreinforced flange-bolted web (WUF-BW) connections with the same beam section but different *R<sup>i</sup>* , as investigated by Li et al. [25]. The flexural (*f<sup>f</sup>* ) and the catenary (*fc*) mechanisms, recognized as two components of the gravity resistance, are separately plotted in Figure 20a respectively, and subsequently, their resultant is plotted in Figure 20b. Three distinctive phases are recognized, as shown in the line graphs, introduced by the flexure action-dominated phase I, the combination of flexure-catenary mechanism phase II, and finally, the catenary-dominated mechanism phase III. These three phases are normally separated from each other by the formation of a plastic hinge followed by an initial fracture in the connection's components.

**Figure 20.** Gravity resistance development for beam-to-column assembly: (**a**) flexural and catenary mechanism, and (**b**) overall gravity resistance.

It is evident that a specimen with larger *R<sup>i</sup>* (*R1*) can provide a higher vertical resistance as a result of higher catenary mechanism response. However, the specimen with lower *R<sup>i</sup>* (*R2*) results in a larger chord rotation ratio, leading to a more ductile response.

#### **6. Summary and Conclusions**

This paper presented the descriptions and experimental results of available full-scale double-span systems subjected to the middle column loss scenario. Several parameters and features including beam span-to-depth ratio, catenary mechanism, stiffness, and ductility have been investigated for fully rigid, semi-rigid, and flexible connections. The following conclusions can be drawn:

I. After middle column removal at the preliminary phases, the behavior of the beam is controlled by flexural resistance, and the tensile force is almost zero, recognized as a flexure action-dominated phase. With increased downward displacement, the axial tension also increases in the beams, developing a catenary mechanism recognized as a catenary-dominated mechanism phase. The results of this research show that the magnitude of axial force in the flexible connections, i.e., top and seat angle, is significantly small compared to fully rigid connections. This phenomenon can be justified by the failure mechanism that develops in the connection's components rather than the connected beam, preventing the catenary mechanism development.

II. The maximum rotation capacity versus connection depth for almost all beam-to-column connection categories significantly surpasses the DoD's recommended acceptance criterion. The suggested acceptance criteria are on the conservative side as it only considers pure flexural resistance. Therefore, connection depth alone is not a reliable indicator to predict the rotational capacity of beam-to-column connections.

III. The stiffness in fully rigid and semi-rigid connections generally experiences a decrease by increasing inter-storey drift angle. On the other hand, the flexible connections have a potential to develop the initial stiffness as the inter-storey drift angle increases. Such behavior can be explained by the geometry of these types of connections that allows rotation at preliminary steps, while the stiffness can be developed at higher drift angles depending on tensile capacity of connections' components and stiffness hardening.

**Author Contributions:** Original draft preparation, numerical models, validation, and methodology: I.F.; supervision, review, and editing: M.H.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by funding from the Federal State Autonomous Educational Institution of Higher Education, South Ural State University (National Research University).

**Conflicts of Interest:** The authors declare no conflict of interest.
