For this study 5-storey,10- storey and 15 storey structures, three different cases of column removal are considered as GIVEN in Fig. 4. The adjacent column and beam numbers with respect to actual column removal C1, C22 and C9 which are used to investigate the effect of demand capacity ratio (DCR). And to present the comparison results between with and without removal of column on neighbouring column and beam elements with and without the removal of column.
4.1 Linear Static Analysis (LSA)
Following GSA, 2013 guidelines LSA are carried
out in several steps. Initially, the structure is analysed for the load combinations given by GSA 2013. Extra loads calculated from equation-1 are applied on a slab for corner column removal, double slab on peripheral internal column removal (C22) and four slabs for internal column removal (C9) to get the effect of sudden column removal. Figure 5 shows the deformed shapes obtained for 5- storey structures for C1, C22 and C9 columns removal respectively. The location of column removal largely effects the joint displacement and deformation behaviour.
Table 3
Displacement of 5,10 and 15 storey(*all the units are in ‘mm’)
Removal location | 5-Storey | 10-Storey | 15-Storey |
---|
Corner Removal (C1) | -115.7 | -109.8 | -105.5 |
Interior Removal (C9) | -139.5 | -89.4 | -100.7 |
Middle Removal (C22) | -111.7 | -104.2 | -103.5 |
For 5-storey structure, at core location column removal (C9) is observed to have maximum displacement compared to other two locations, which means the effect of the extra load acting on floor slabs is directly seen. Comparing between C1 removal and C22 removal, it is found there is the reason for more displacement in former case due to larger cantilever effect. For 10-storey, 15-storey structures it is observed that corner column removal has led to maximum vertical deflection compared to other two locations due to the sudden decrease in stiffness and two-way cantilever effect. Whereas for C22 location column removal it has one-way cantilever effect and for C9 location column removal no cantilever effect is observe.
Comparing the results between different storey levels, it is very clear that height of the structure will also largely affect the progressive collapse behaviour. When column ‘C9’ is removed in three cases, maximum displacement is seen in 5-storey and minimum is observed for 15-storey structures. It indicates as height increases the load transfer elements would be more and, larger deflection would decrease for the same scenario.
4.2 Non-linear Static Analysis (NLSA)
In general, NLSA is considered to be the more sophisticated process and nearer to the real behaviour when compared to LSA,especially when larger loads and larger deflections are considered. As progressive collapse behaviour involves both conditions, the NLSA would be more preferable. Similar to LSA, the load combinations applied on the structure are based on GSA 2013 guidelines. The only difference in load calculation between linear and non-linear is the value of ‘load increase factor’ which is calculated based on ASCE 41 − 06. It can be observed that this load increase factor for e.g. the value is calculated to be 1.13 which is less for NLSA and the same has resulted in fewer displacement values compared to LSA.
Figure 6 shows the formation of hinges in all the floors of 15-storey structure. For beam and column joints immediately above the column removal the displacement behaviour obtained is explained below. Similar behaviour can be observed in 5 and 10-storey structures.To evaluate the potential of failure, the performance of the hinges is investigated. Three performance levels are explained in ASCE-41-10 the green colour indicates immediate occupancy, sky blue colour indicates life safety, pink colour indicates collapse prevention and red colour indicates failure of the structure.
4.3 Linear Dynamic Analysis (LDA)
In LSA and NLSA cases, where extra loads are calculate to create the effect of sudden column removal. In LDA, these extra loads are not applied to the structure, as the dynamic loading itself would create the effect. So, to simulate the phenomenon of removing the column abruptly, the axial forces (P) acting on the C1, C9 and C22 of the ground floor are calculated for GSA 2013 load combination given in equation-4. Then the column is replaced by equivalent point load of its member force. Now to impart the actual behaviour the forces and loads (W) acting on the structure are increased linearly for first five seconds until they reach their actual values. To obtain the position of equilibrium, two more seconds dynamic analysis is carried out without changing any parameter. Once it reaches a stable position, the column is removed abruptly at 7 seconds. The behaviour of loading applied and removed is shown in the Fig. 7.
Deflection behaviour is similar to LSA. But in terms of vibrations observed from the results obtained (Fig. 8 to Fig. 10), it can be concluded that vibrations depend on the number of storeys as well as on the location of column removal. Vibrations are almost negligible if the internal column is removed from the structure.
4.4 Non-linear Dynamic Analysis (NLDA)
The procedure carried out for LDAis also followed for NLDA as given by GSA, 2013. And for the inclusion of non-linear behaviour, hinges have been defined for all the structural elements. It can be observed that all the structures considered for this study have not shown any effect of non-linearity, but there a minor difference in values was obtained. One of the major reasons for the difference in obtaining the values between linear and non-linear analysis is the procedure followed by the software. All the simulations for non-linear analysis have taken a huge CPU time and hard disk space for every single computation. It directly indicates that several number of steps have been carried out when compared to linear analysis.
Another major observation made is the formation of hinges in NLSA whereas the absence of formation of hinges in the NLDA is due to the load factor considered.
Figure 8 shows the displacement versus time at C1, C9 and C22 column removal in all four analyses of 5-storey. LSA results show maximum joint displacement when compared with all cases due to large dynamic amplification factor considered as per GSA 2013. In all the above cases of LDA and NLDA joint displacement is almost equivalentbecause same load combinations were considered, which is derived from LSA of the un-factored area as per GSA 2013. And the structure is more ductile as it does not show the effect of non-linearity. The joint displacement obtained for LDA and NLDA are nearly half as that of LSA. Similar, trends can be observed in other the two cases. For 10-storey and 15-storey structures similar pattern of joint displacement results can be observed as shown in Fig. 9–10.
Figure 11 shows the effect of column removal location for 5-storey and 10-storey structures. In case of the 15-storey structure due to more alternate path available similar trends are not observed and for all the locations, joint displacement is nearly same. Also, the effect of height of the structure can be clearly understood. Figure 12 shows the corner column removal resulting in maximum joint displacement in the 5-storey structure. This trend is not seen in the case of middle column removal due to alternate load path availability. In case of internal column removal scenario as the height of the structure increases joint displacement decreases.
4.5 Demand capacity ratio (DCR) of Special Moment Resisting Frames
Demand capacity ratio (DCR) is defined as, the ratio of member force and member strength. It is an important parameter for the structural assessment. As per GSA-2013 and UFC-2013 guidelines if DCR exceeds 2.0, then the linear static procedure is not recommended. For all the four analyses at three different column removal locations, the effect of DCR values is shown in Fig. 13–15 and compared with the results before and after removal scenarios.
The most important aspect of any design is to maintain the proportion of demand to capacity ratio for rational distribution of loads to the foundation. In this study, demand capacity ratios (DCR) of elements adjacent to removal column are considered to understand the sudden effect on DCR. In case of symmetric columns or beams, only one of them is considered as the obtained results are similar. Example, for C1 column removal the adjacent members to be studied are C2, C8, B1 and B43. Since the DCR values obtained for column C2 and column C8 are similar, so only the values of column C2 are considered. In similar way beam B1 and beam B43 DCR values are observed to be similar so only B1 values are considered and plotted as shown in the Fig. 13. The same phenomenon has been used for all column removal cases and for all three structures. When C1 column is removed from the ground floor, the DCR value of C1 column in the first floor has increased and then it has decreased for all other floors. Whereas column C2 has little change in DCR and same is continued for all other floors. Similar to C1 of other floors B1 has also shown the increment of load transfer.
From the results obtained, it is observed that for all three 5-storey, 10-storey and 15-storey structures having column C1, C22 and C9 removal the effect of DCR values can be clearly seen on the adjoining beams having their DCR values 2. It indicates that the beams nearer to the removal location have failed according to guidelines. But the same effect is not seen on adjoining columns indicating that the column has failed at some particular location and clearly has susceptibility to progressive collapse.
As the height of the structure increases, adjoining elements of removal column show higher DCR values for 15 storeys, as compared to 5-storey and 10-storey. This is due to increase in the floor weight of the structure. DCR values of load transferring elements is abruptly changes near to the column removal location. This effect is reducing as height of structure increases. Similarly, for all three corner column removal cases the DCR values for connecting beams and column are larger as compared to other two locations. Since large partial collapse may occur in this case, so the results obtained for DCR show similar trends which can be seen in the joint displacement scenario.