**5. Discussions of Test Results**

## *5.1. Catenary Mechanism*

A steel beam is mainly subjected to the combined effect of shear force and bending moment under the gravity loading condition, whereas there is no considerable axial force. In the case of sudden column removal, the bending moment at both ends and mid-span of the beam will increase accordingly, resulting in large deflection, developing plastic hinges in these particular locations. In this situation, the beam will experience significant axial force. As the axial force increases gradually, the whole cross-section will experience yielding, resulting in reducing bending moment at plastic hinge locations. At the final stage, the resistance to the external loading only depends on the axial capacity of the beam. This process is generally recognized in the engineering community as tensile catenary action.

The pushdown force versus vertical deflection for a beam with axially restrained support is plotted in Figure 7 [44]. Figure 7a is representative of the behavior of all tested specimens in which failure takes place at a different stage of their response depending on rotation and ductility capacity. In the elastic range, the behavior is generally controlled by connection stiffness, while at the post-elastic stage (beyond point B), the behavior largely depends on geometric and material nonlinearity. Depending on connection categories, different behavior is observed during the different stages, as shown in the typical graph of Figure 7b. Generally, the connected beams and connections are subjected to relatively high axial compression and bending moment at the compressive arching stage, leading to premature instability in the vicinity of compressive components. After point C, the compressive arching impact steadily declines, and the beam axial compression force is converted to tensile (after point D). By increasing the axial tension, the connection's bending moment becomes less significant and they experience severe tensile deformation. Finally, following the point E, the catenary tensile action becomes the governing force-carrying mechanism.

Considering a tensile catenary stage in the beam in the case of sudden column removal (point D onward in Figure 7b), a simplified model for plastic interaction between axial force and bending moment is developed according to the following assumptions:


**Figure 7.** Beam and connection performance following column removal. (**a**) beam nonlinear load-deflection response, (**b**) beam axial load - connection moment interaction

Assuming the entire cross-section yield, the bending moment and tensile force correlation is given by the following equation in the case where the neutral axis is in the web:

$$\frac{M}{M\_Y} + \frac{\left(1+\alpha\right)^2}{\alpha[2(1+\beta)+\alpha]} \left(\frac{p}{p\_y}\right)^2 = 1\tag{5}$$

where *M<sup>y</sup>* and *p<sup>y</sup>* are the plastic moment capacity and ultimate plastic axial force of the beam respectively, calculated by:

$$M\_y = Zf\_{y\varepsilon}p\_y = \left(2A\_f + A\_w\right)f\_{y\varepsilon}$$

where *fye* is the expected material yield strength, *Z* is the plastic section modulus, *A<sup>f</sup>* is the beam's flange area, and *A<sup>w</sup>* is the beam's web plate area. β is the ratio of flange thickness to web height, and α is also defined by the following equation:

$$\alpha = A\_w / \left(2A\_f\right)$$

 (1) ଶ ( ) <sup>ଶ</sup> = 1 In the case that the plastic neutral axis appears in the beam's flange, the moment and tensile correlation is given by the following equation:

$$\frac{1-\gamma}{1-\frac{(1+\alpha)^2\gamma^2}{a[2(1+\beta)+\alpha]}}\frac{M}{M\_y} + \frac{P}{P\_y} = 1\tag{6}$$

$$\gamma \quad f\_Y$$

$$\gamma = A\_w / \left(2\mathcal{A}\_f^\sharp + \mathcal{A}\_w^\sharp\right) f$$

where

<sup>௬</sup> <sup>௬</sup>

$$A \qquad A$$

௪

Since the web height is much bigger than flange thickness, it is acceptable to assume β = 0. Accordingly, Equations (5) and (6) can be rewritten as: = ௪/(2 ௪) =0

ଶ

 <sup>௬</sup> ௬ = 1

1− 1 − (1 )ଶ

ሾ2(1) ሿ

$$\frac{M}{M\_y} + \zeta \left(\frac{p}{p\_y}\right)^2 = 1\tag{7}$$

$$
\lambda \frac{M}{M\_y} + \frac{P}{P\_y} = 1\tag{8}
$$

ଶ

where

$$\zeta = \frac{\left(1+\alpha\right)^2}{\left[\alpha\left(2+\alpha\right)\right]}; \ \lambda = \frac{1-\gamma}{1-\zeta\gamma^2}\_{\zeta}$$

Figure 8a presents the interaction of Equations (7) and (8), indicating that the curves of Equations (7) and (8) intersect at point 1 − ζγ<sup>2</sup> .γ . For simplicity, the nonlinear domain of Equation (7) can be replaced by a straight line, using the following equation: (1 − <sup>ଶ</sup> . )

*M My* <sup>+</sup> ζγ *<sup>P</sup> Py* = 1 (9) <sup>௬</sup> ௬ = 1

**Figure 8.** (**a**) Plastic interaction between bending moment and tensile force, and (**b**) beam axial force versus mid-span deflection.

1 − <sup>ଶ</sup> <sup>௬</sup> <sup>௬</sup> Figure 8b shows the correlation between the beam's axial force and vertical displacement in the case of middle column removal. According to the experimental results, two main different failure mechanisms in the beam are recognized, as presented in Figure 8b. These two failure mechanisms are introduced as the beam mechanism (phase 1 to phase 2) and the catenary mechanism (phase 2 to phase 4), respectively. Experimental results indicate that the tensile force in the beam is smaller than the ultimate plastic axial force before the failure. Accordingly, it is reasonable to assume that the moment of the beam is equal to 1 − ζγ<sup>2</sup> *M<sup>y</sup>* and the axial force is equal to γ *P<sup>y</sup>* before the beam experiences failure (refer to practice failure point 3 in Figure 8b).

ோ Figure 9 shows the vertical reaction, *VR*, of connection, which can be calculated by Equation (10).

$$V\_R = V\_{\text{i}} \cos \varphi\_{\text{i}} + P\_{\text{i}} \sin \varphi\_{\text{i}} = F\_{\text{f}} + F\_{\text{c}} \tag{10}$$

where ϕ*<sup>i</sup>* , *V<sup>i</sup>* , and *P<sup>i</sup>* are rotation of deflected beam, transverse shear, and axial force in the case of middle column removal, respectively. The internal transverse shear and axial force are measured through installed strain gauges distributed on a connected beam. *F<sup>f</sup>* in Equation (10) represents flexural mechanism resistance while *F<sup>c</sup>* is the resistance component due to the catenary mechanism.

**Figure 9.** Development of internal forces in the beam-to-column assembly in the case of middle column removal. ,

ோ = cos sin = , Figure 10 shows the development of axial force and bending moment in the case of middle column removal for three tested specimens in reference [43]. According to Figure 10a, at the preliminary phases, the behavior of the beam is controlled by flexural resistance, and the tensile force is almost zero. With increased downward displacement, the axial tension also increases in the beams, developing catenary mechanism until the beam or connection can no longer bear the combined flexural stresses and tensile force and fails (see Figure 10b). 

 axial force and a bending moment of connected beam compared to the I-W specimen. In other words, the type of beam-to-column connection plays an important role in developing the catenary mechanism. This issue is also acknowledged in Figure 11, where the flexible connections, i.e., top and seat angle and TSWA = 8 mm, fail to develop the catenary mechanism of the connected beam. This issue confirms the fact that the failure is mainly concentrated at the connection components in a flexible and semi-rigid connection, and therefore, these types of connections have limited capacity to develop the catenary mechanism.

#### *5.2. Maximum Rotation and Ductility Capacity*

In steel structures subjected to sudden column removal, the maximum deformation and rotation capacities of connections within the affected areas have a major contribution to the ultimate load-carrying capacities of the system. The main progressive collapse guidelines, i.e., the Unified Facilities Criteria (UFC) 4 023 03 [45], specify a series of plastic rotation angles for several beam-to-column connection categories based on a nonlinear modeling simulation, as shown in Table 4.

**Figure 11.** Correlation between axial force and bending moment in studied specimens at the final stage of the pushdown test.



dbg = depth of bolt group, inch; d = depth of beam, inch.

0.036 0.048 0.2 0.03

0.2 0.0502 − 0.0015

0.016 0.024 0.8 0.013 0.1125 − 0.0027 0.150 − 0.0036 0.4 0.112 − 0.0027 0.0502 0.072 Figure 12 shows the nonlinear force–vertical displacement curve of several fully rigid tested specimens along with idealized multi-linear curves. For a welded unreinforced flange WUF connection, represented by I-WB and ST-WB in this research, the results indicate large ductility when subjected to mid-column removal. The plastic rotation angle *b* for I-WB and ST-WB is around 0.16 rad, well beyond the recommended acceptance criteria by the UFC presented in Table 4. This indicates that tested specimens can resist large plastic rotation without a significant decline in resisting vertical force.

− 0.0022

− 0.0015

Rotation (rad)

0

0.05

0.1

0.15

Rotation (rad)

0.2

0.25

TSWA-L8mm Top&Seat Angle

Web Cleat Fine Plate DWA

0.3

**Figure 12.** Ductility assessment of fully rigid tested specimens.

In this section, the plastic rotation angle of all tested specimens is compared to the acceptance criteria, estimated based on the depth of the connection, recommended by the UFC. Figures 13–15 show the maximum rotational capacity of studied connections versus connection depth along with the acceptance criteria line recommended by the UFC, as presented in Table 4.

**Figure 13.** Maximum rotation capacities versus connection depth for fully rigid connections.

**Figure 14.** Maximum rotation capacities versus connection depth for semi-rigid connections.

Web Cleat, Fine Plate, DWA (Shear in Bolts, Simple Shear Tab)

Top&Seat Angle, TSWA-L 8mm (Flexure in Angles)

*ϴ −*

*ϴ −*

*ϴ −*

0 2 4 6 810 12 14 16 18 20

Connection depth (inch)

0

0.05

0.1

0.15

Rotation (rad)

0.2

0.25

0 5 10 15 20

*ϴ ϴ*

Extended End-Plate, Flush End-Plate (Tension in bolt, Tension in Tee)

*ϴ −*

TSWA (Shear in Bolt, Flexure in Tee)

*ϴ ϴ*

TSWA (L=90mm) - EXP TSWA (L=12mm), EXP Extended End-Plate - EXP

Flush End-Plate - Exp

Connection depth (inch)

**Figure 15.** Maximum rotation capacities versus connection depth for flexible connections.

Overall, Figures 13–15 indicate that the average plastic rotation angle varies from 0.1 to 0.2 rad, which significantly surpasses the UFC recommended acceptance criterion. This indicates that all three beam-to-column connection categories can address adequate plastic rotation during sudden column removal. In addition, the results reveal that the suggested ductility acceptance criteria are on the conservative side for all three beam-to-column connection categories. The current in-practice acceptance criteria employed in progressive collapse analysis are based on cyclic simulations proposed by the American Society of Civil Engineers (ASCE/SEI) 41 13 [46] that do not consider large axial demands imposed over sudden column removal. Actually, the results show that the connection depth alone is not a reliable indicator to predict the rotational capacity of a connection, where different connection types with the same *R<sup>i</sup>* result in totally different maximum rotation capacity. Moreover, the results show that although the stiffness of fully rigid connections is higher than flexible connections, both categories result in almost the same rotation capacity, by around 0.15 rad.

The previous literature indicates that the rotational ductility of connections is significantly affected by bolt rupture and brittle failure of welding [47]. In other words, a variation in failure mode will affect the maximum rotational ductility of connections. To meet the criterion for rotation capacity, in this research, an acceptance criterion is proposed by considering initial stiffness and the design flexural resistance. To this end, the maximum rotation capacity of the connection, ϕ*u*, is defined by the rotation at the point where either (i) the connection resistance has dropped to 0.8 *Mn*, or (ii) the deformation is more than 0.03 rad. The first yielding rotation ϕ*<sup>y</sup>* is also defined as follows:

$$
\rho\_{\mathcal{Y}} = \frac{\frac{2}{5} \, M\_{\mathcal{Y}}}{\mathcal{S}\_{j, \text{ini}}} \tag{11}
$$

The above-mentioned parameters have been used to determine the ductility of studied connections. The ability to sustain large inelastic deformations without significant loss in strength is referred to as ductility, µ, and is defined as follows:

$$
\mu = \frac{\varphi\_u}{\varphi\_y} \tag{12}
$$

Figure 16 illustrates the nonlinear idealization of the moment-rotation curve of a typical steel beam-to-column connection.

<sup>௬</sup> =

 = <sup>௨</sup> <sup>௬</sup>

<sup>௨</sup>

<sup>௬</sup>

**Figure 16.** Typical moment-rotation response.

Table 5 illustrates the yielding rotation, maximum rotation capacity, and ductility of studied specimens. Table 5 indicates that fully rigid and semi-rigid connections result in bigger ductility compared to flexible connections. Generally, in a situation where the connection strength exceeds the beam strength, the ductility of the whole system is controlled by the beam, and the connection remains elastic. For instance, the behavior and overall ductility of the SidePlate moment connection system are defined by the plastic rotational capacity of the beam, where the limit state is ultimately the failure of the beam flange, away from the connection. On the other hand, if the beam capacity exceeds the connection capacity, then deformations only take place in the connection itself.


**Table 5.** Yielding Rotation, Maximum Rotation Capacity, and Ductility of Studied Specimens.
