RC frame structure behavior should be in accord with the seismic principle of strong-column-weak-beam. Negative bending moment regions on both sides of a column that is to be removed are missing, however, these locations have maximum positive bending moment after removed column loss, and the spans of beams and slabs became larger, as shown in Fig. 11. This indicated that after a load-bearing column is removed, the beams that were connected to this column must transfer the load previously borne by the column and bridge over the damaged area. Thus, it was observed that columns were much stronger, and they may have a negligible effect on progressive collapse resistance compared to other elements in the system. The catenary mechanism of the beams and the tensile membrane mechanism of the slabs combine to resist vertical loads in the progressive collapse limit state.

### Catenary Mechanism of Beams

Figure 6 shows the failure pattern of the frame model. Due to the entire failure of the transverse frame beam and the transverse direction of the frame slabs in the failure zone, the effect of the transverse beam and the transverse direction of the slabs can be ignored, while calculating the progressive collapse resistance of the whole structure at a collapse limit state.

Based on the experimental observation, it can be found that the axes of the longitudinal frame beams are almost still straight at the collapse limit state. Figure 12 shows the vertical displacement of different positions on the longitudinal frame beams at the collapse limit state, which is obtained from the numerical results and is consistent with the experimental observation. Therefore, the model of the progressive collapse resistance of RC beams can be proposed as shown in Fig. 13. The progressive collapse resistance of frame beams was obtained as (Hou and Yang 2014)

$$ P_{\text{ub}} = \frac{{\left( {L_{1} + L_{2} } \right)v_{\text{u}} }}{{L_{1} L_{2} }} \cdot \left( {A_{\text{th}} } \right)f_{\text{y}} $$

(1)

where \( L_{1} \) and \( L_{2} \) are the spans of beam 1 and beam 2, respectively, \( v_{u} \) is the vertical displacement of the removed column, *A*
_{th} is the area of steel bars through whole span, and \( f_{y} \) is the yield stress of the steel bars in frame beams. The model was originally proposed by Li et al. (2011), and the reliability was verified by Hou and Yang (2014).

### Tensile Membrane of Slabs

Based on the failure phenomenon as shown in Fig. 6, the internal area of the frame slabs surrounded by the negative moment yield lines is viewed as an analysis object, and its boundaries are assumed to be rectangular. Thus, the model of the progressive collapse resistance of frame slabs is proposed as shown in Fig. 14. Based on the analysis of the progressive collapse process, just the effect of the longitudinal direction of the frame slabs on the progressive collapse resistance of the whole structure was considered, and the moments of resistance at the plastic hinge lines in the frame slabs can be ignored at the collapse limit state. Therefore, the boundaries of the analysis object can only bear a pulling force, as shown in Fig. 14. Let \( l_{y} \) (the side lengths of slabs ① and ② in the direction of the Y axis) be equal to the projection lengths of the positive moment yield lines in the corresponding position, it can be noted that the load-carrying capacity of the curved boundaries in Fig. 6 and the rectangular boundaries in Fig. 14 are equivalent.

In Fig. 6, for slab BOK, based on the vertical displacement of different positions on the longitudinal frame beams at the collapse limit state (Fig. 12) and the compatibility of deformation between frame beams and slabs, it can be found that the slab edge BK is still straight at the collapse limit state. Figure 15 displays the vertical displacement of different positions in the transverse span centers of slabs at the collapse limit state, which was obtained from the numerical results. It was observed that there was relatively little change in the vertical displacement in the Section OA1 range compared to Section OA2. Therefore, the line section OA1 can approximately be assumed to be straight. Thus, we assumed that the slabs GJK and HIK (Fig. 14), surrounded by positive and negative moment yield lines and outer edges of slabs, were still in two different planes at the collapse limit state, respectively.

For slab GJK, based on the vertical equilibrium condition, the progressive collapse resistance can be obtained as:

$$ R_{sGJK}^{tm} = F_{x1} l_{y} \frac{v}{{\sqrt {v^{2} + l_{x1}^{2} } }} $$

(2)

where \( F_{x1} \) is the yield-bearing capacities of steel bars in the range of unit width slab ①; \( l_{x1} \) and \( l_{y} \) are equal to the projection lengths of the positive moment yield lines in the corresponding position, respectively; and \( v \) is the vertical displacement of Point K.

In the same way, for slab HIK, the progressive collapse resistances can be given as:

$$ R_{sHIK}^{tm} = F_{x2} l_{y} \frac{v}{{\sqrt {v^{2} + l_{x2}^{2} } }} $$

(3)

The progressive collapse resistance of the whole frame slabs, based on the principle of superposition, as shown in Fig. 6, can be expressed as:

$$ P_{us} = R_{sGJK}^{tm} + R_{sHIK}^{tm} $$

(4)

### Validation

Based on analysis and experimental results, it is noted that the limit vertical displacement of the frame structure is controlled by frame beams on the A-axis. Based on the literature (Hou and Yang 2014), the calculated value of the limit vertical displacement of the removed column (\( v_{u} \)) is 356.7 mm. Substituting the value of \( v_{u} \), the geometric dimensions and the properties of steel bars of the frame beams on the A-axis into Eq. (1), the progressive collapse resistance of frame beams (\( P_{\text{ub}} \)) can be calculated 41.5 kN.

Based on the deformation compatibility condition of the frame beams and slabs, the limit vertical displacement of Point K (\( v \)) is equal to the limit vertical displacement of the removed column (\( v_{u} \)). Therefore, the value of \( v \) should be 356.7 mm. Substituting the value of \( v \), the geometric dimensions and the properties of steel bars of the frame slabs into Eqs. (2), (3) and (4), the progressive collapse resistance of frame slabs is 14.9 kN.

By using the principle of superposition, the progressive collapse resistance of the whole frame structure is 56.4 kN. The calculated value is 7.1 % smaller than the experimental result. Portions of the steel bars have entered the hardening stage in the collapse limit state, but steel hardening is not considered in the model. Thus, the progressive collapse resistance obtained from the model is somewhat conservative. Moreover, it can be found that the progressive collapse resistance of frame slabs is 26.4 % of the progressive collapse resistance of the whole frame structure.