IDA is performed to produce the collapse fragility curves of the steel MRFs and with reference the collapse resistance at the 50% probability of the steel MRFs without dampers, the allowable values of coefficient θ for the steel MRFs with dampers are established. For a pair of steel MRF and ground motion, until the drifts increase without bound given a very small increment of Sa(T1) and the MRF becomes globally unstable, the Sa(T1) is systematically scaled up in increments. The aforementioned procedure has been repeated for all steel MRFs and 44 ground motions; resulting in the IDA curves provided in Appendix E. Sources of collapse uncertainty (e.g. record-to-record, design requirement, test data and modeling) are not considered in this study.
Figure 5.1 and Figure 5.2 show the collapse fragility curves of the 5-storey steel MRFs, Figure 5.2 and Figure 5.3 show the collapse fragility curves of the 10-storey steel MRFs and Figure 5.4 and Figure 5.5 show the collapse fragility curves of the 20- storey steel MRFs. Sa(T1) is normalised by Sa,MCE(T1) in order to enable the comparison of steel MRFs having different fundamental periods.
The Sa(T1) at 50% probability of collapse for the 5-storey MRF is 3.74·Sa,MCE, for the 10-storey MRF is 2.42·Sa,MCE and for the 20-storey MRF is 3.61·Sa,MCE. The Sa(T1) at 50% probability of collapse for the 5-storey MRFs with dampers (ξtot=10%) and θ equal to 0.154, 0.112, 0.111 are 3.25·Sa,MCE,3.89·Sa,MCE, 4.15·Sa,MCE respectively, for the 5-storey MRFs with dampers (ξtot=15%) and θ equal to 0.175, 0.165, 0.152, 0.144 are 3.11·Sa,MCE, 3.17·Sa,MCE, 3.50·Sa,MCE, 3.71·Sa,MCE respectively, for the 5-storey MRFs with dampers (ξtot=20%) and θ equal to 0.325, 0.188, 0.137, 0.084 are 2.95·Sa,MCE,3.34·Sa,MCE, 4.12·Sa,MCE, 5.81·Sa,MCE respectively.
The Sa(T1) at 50% probability of collapse for the 10-storey MRFs with dampers (ξtot=10%) and θ equal to 0.330, 0.188 are 2.38·Sa,MCE,2.96·Sa,MCE respectively, for the 10-storey MRFs with dampers (ξtot=15%) and θ equal to 0.270, 0.215, 0.177 are 2.90·Sa,MCE, 3.11·Sa,MCE, 3.50·Sa,MCE respectively, for the 10-storey MRFs with dampers (ξtot=20%) and θ equal to 0.229, 0.181 are 3.49·Sa,MCE, 3.69·Sa,MCE respectively. The Sa(T1) at 50% probability of collapse for the 20-storey MRFs with dampers (ξtot=10%) and θ equal to 0.197, 0.141, 0.110 are 4.17·Sa,MCE,4.36·Sa,MCE,
Figure 5.1 Collapse fragility curves of the steel MRF and the MRFs with viscous dampers (solid line indicates median)
Figure 5.2 Collapse fragility curves of the steel MRFs and the MRFs with viscous dampers
Figure 5.3 Collapse fragility curves of the steel MRF and the MRFs with viscous dampers
Figure 5.4 Collapse fragility curves of the steel MRF and the MRFs with viscous dampers
Figure 5.5 Collapse fragility curves of the steel MRF and the MRF with viscous dampers
4.43·Sa,MCE respectively, for the 20-storey MRF with dampers (ξtot=15%) and θ equal to 0.174 is 4.98·Sa,MCE, for the 20-storey MRF with dampers (ξtot=20%) and θ equal to 0.171 is 5.81·Sa,MCE.
The 5-storey MRFs with dampers (ξtot=10%) and θ equal to 0.112, 0.111 show higher performance and with θ equal to 0.154 shows lower performance compared to the 5-storey MRF without dampers. Linear interpolation can be approximately adopted to calculate the allowable values of coefficient θ, using the Sa(T1) at 50% probability of collapse of the MRFs with θ equal to 0.154, 0.112 and the conventional MRF. The allowable value of coefficient θ for a 5-storey MRF with dampers (ξtot=10%) is equal to 0.120. The 5-storey MRFs with dampers (ξtot=15%) and θ equal to 0.175, 0.165, 0.152 show lower performance and with θ equal to 0.144 shows marginally lower performance compared to the 5-storey MRF without dampers. The allowable value of coefficient θ for a 5-storey MRF with dampers (ξtot=15%) is equal to 0.142 after performing linear interpolation on the Sa(T1) at 50% probability of collapse of the MRFs with θ equal to 0.152 and 0.144 and the conventional MRF. The
5-storey MRFs with dampers (ξtot=20%) and θ equal to 0.137, 0.084 show higher performance and with θ equal to 0.325, 0.188 show lower performance compared to the 5-storey MRF without dampers. The allowable value of coefficient θ for a 5- storey MRF with dampers (ξtot=20%) is equal to 0.158 after performing linear interpolation on the Sa(T1) at 50% probability of collapse of the MRFs with θ equal to 0.188 and 0.137 and the conventional MRF.
The 10-storey MRF with dampers (ξtot=10%) and θ equal to 0.188 shows higher performance and with θ equal to 0.330 shows lower performance compared to the 5- storey MRF without dampers. The allowable value of coefficient θ for a 10-storey MRF with dampers (ξtot=10%) is equal to 0.317 after performing linear interpolation on the Sa(T1) at 50% probability of collapse of the MRFs. The 10-storey MRFs with dampers (ξtot=15%, 20%) and the 20-storey MRFs with dampers (ξtot=10%, 15%, 20%) show higher performance compared to the corresponding MRFs without dampers. In particular, the more flexible MRFs with dampers show lower performance compared to stiffer MRFs with dampers but still higher performance than the conventional MRFs. In these cases of MRFs with dampers, limit of coefficient θ can be neglected in the design procedure and other loading conditions, such as wind, is expected to result in designs of similar or less flexible MRFs that are considered in this study.
The allowable values of coefficient θ for the 5-storey and the 10-storey MRFs with dampers are shown in Figure 5.6. Linear interpolation is suggested to approximately calculate the allowable values of coefficient θ for total viscous damping ratio between 3-10%, 10-15% and 15-20% for 5-storey MRFs with dampers and between 3-10% for 10-storey MRFs with dampers.