## 3 Experimental Results of Static Collapse Tests
### Equivalent Dynamic Increase Factors of Collapse Load  
Structural collapse actually is a dynamic process. Therefore, dynamic analysis is the most accurate method to assess the collapse responses of structures in the scenario of load-bearing members loss. However, researchers are most familiar with the static collapse analysis on the collapse behavior of structures. Dynamic increase factor (DIF) is usually used to take into account the dynamic effect of collapse load when static collapse analysis is conducted. As a result, taking proper DIF values in static collapse analysis becomes very important. A method based on energy conservation provided an effective method to determine DIF values of collapse load [24]. As the method above is only suitable for the non-prestressed frame structures, the following is mainly to discuss the method to determine DIF values for prestressed frame structures.  
As the prestressing strand in the frame beams was arranged in a parabolic shape, the pretension of the strand can be converted into segmentally distributed uniform loads along the beams by the load balance method proposed by [PERSON] [25]. It is assumed that the vertical displacement at any section of the  
Figure 10: Overall deformation of the frame beams: (**a**) S1, (**b**) S2, (**c**) S3, (**d**) S4, (**e**) S5, (**f**) S6.  
frame beams was proportional to that of the failed middle column, \(\delta\). Then, the work done by the equivalent uniform loads corresponding to \(\delta\) can be given by Equation (1):  
\[W_{p}(\delta)=\sum_{i=1}^{n}\int_{0}^{l_{i}}q_{i}\delta(x)dx \tag{1}\]  
where, \(q_{i}\) and \(l_{i}\) are the equivalent uniform loads of the initial pretension and the corresponding distribution lengths on the beam, \(n\) is the total number of the equivalent uniform load \(q_{i}\), \(\delta(x)\) is the vertical displacement at any section of the frame beams and proportionate to \(\delta\) according to the assumption, \(x\) is the coordinate of the beam section on the axis.  
The initial prestressing is regarded as an external force here, so the total work, \(W_{ps}\), done by the external forces can be obtained by Equation (2):  
\[W_{ps}=\int_{0}^{\delta_{t}}P_{s}(\delta)\delta d\delta+W_{p}(\delta_{t}) \tag{2}\]  
where, \(\delta_{t}\) is a certain vertical displacement at the failed middle column under static load \(P_{s}(\delta_{t})\), \(P_{s}(\delta)\) is the load applied on the top of the middle column corresponding to \(\delta\). Then, the corresponding equivalent dynamic load,\(P_{dq}\), at the vertical displacement of \(\delta_{t}\) can be got:  
\[P_{dq}=W_{ps}/\delta_{t} \tag{3}\]  
and the DIF value corresponding to \(\delta_{t}\) is  
\[DIF(\delta_{t})=\frac{P_{s}(\delta_{t})+P_{p}(\delta_{t})}{P_{dq}} \tag{4}\]  
where, \(P_{p}(\delta_{t})=W_{p}(\delta_{t})/\delta_{t}\) is the equivalent load of initial pretension of the strand corresponding to \(\delta_{t}\).  
The curves of DIF values versus vertical displacement in the middle column are shown in Figure 11. It can be seen that (1) the maximum DIF values of the sub-assemblages reinforced with BFRP bars in frame beams were close to 2.0 in the initial loading stage, while those for S2 and S3 were 1.38 and 1.44, respectively, due to the action of prestress. (2) In accordance with Figure 11b, the specimen with greater span to depth ratio possessed higher DIF values. (3) From Figure 11c, it can be seen that DIF values of the sub-assemblage were little affected during collapse when steel stirrups of the frame beams were replaced by BFRP bars. Based on the analysis above, it is suggested that the DIF values for the non-prestressed sub-assemblages should be conservatively taken as 2.0, while DIF values for the prestressed sub-assemblages could be taken between 1.44 and 2.0.  
Figure 11: _Cont._