**2.** *R<sup>i</sup>* **E**ff**ects on Progressive Collapse**

To evaluate the effects of the *R<sup>i</sup>* ratio, one set of pushdown analyses was performed in this research using structural analysis program (SAP) 2000 academic version 21. In this case study, three steel double-span assemblies with *R<sup>i</sup>* ratios of 5, 10, and 15 were considered. All specimens possess fully rigid connections which were designed according to the strong column–weak beam theory in compliance with The American. Institute of Steel Construction (AISC) seismic design [40]. Figure 1 shows the topology of one studied frame with double-span assemblies.

**Figure 1.** Double-span assembly.

In Figure 1, the pushdown force, *F,* represents the concentrated load in the pushdown analysis applied at the top of the removed column. This concentrated load is equivalent to the progressive collapse resistance of the double-span assembly. Table 1 shows the beam and column section properties used in progressive collapse analysis.

The plastic hinge capacity of connected beams, *Fp*, of double-span assembly can be calculated from the following equation:

$$F\_p = \frac{2\ M\_p}{\frac{L\_0}{2}} = \frac{4\ W\_p\ f\_y}{L\_0} \tag{1}$$

where *M<sup>p</sup>* is the plastic moment of the connected beam, *L*<sup>0</sup> is the double-span assembly length, *W<sup>p</sup>* is the plastic modulus of the connected beam, and *f<sup>y</sup>* is the yield stress.


**Table 1.** Beam and Column Sections of the Steel Double-Span Assemblies (AISC Database).

To facilitate the comparison of different types of double-span assemblies against each other independently and regardless of connected beam plastic capacity, the pushdown force, *F,* should be normalized against the plastic hinge capacity of connected beams, *Fp*, as in the following equation: 

$$\frac{F}{F\_p} = \frac{F}{4} \frac{f\_0}{\mathcal{W}\_p} \tag{2}$$

Besides, to highlight the differences of different beam sections during the development of the catenary action, it is necessary to normalize the chord rotation, θ, over the plastic rotation, θ*p*, based on the following equations: *θ θ* 

$$
\theta = \frac{\delta}{L\_{\theta}} \tag{3}
$$

$$\theta\_p = \frac{\delta\_p}{\frac{L\_0}{2}} = \frac{F\_p \ 2}{K\_e L\_0} = \frac{\frac{4 \text{ W}\_p \ f y}{L\_0}}{\frac{48 \text{ E } I\_b}{L\_0^3}} \frac{2}{L\_0} = \frac{\mathcal{W}\_p \ f\_y L\_0}{6 \text{ E } I\_b} \tag{4}$$

where *I<sup>b</sup>* is the moment of inertia of a connected beam, and *K<sup>e</sup>* is the elastic stiffness of a simply supported beam subjected to pushdown force. Figure 2 shows the plots of the pushdown analysis results of double-span assemblies with different *R<sup>i</sup>* ratio, in which the vertical and horizontal axes are the normalized force, *<sup>F</sup> Fp* , and the normalized rotation, θ/θ*p*, respectively. Figure 2 shows that the frame with the larger *R<sup>i</sup>* has a higher capacity to develop progressive collapse resistance. Also, the double-span assembly with the larger *R<sup>i</sup>* possesses relatively high initial stiffness. ி ி *θ θ*

**Figure 2.** Normalized resistance against normalized rotation.
