## 5 Structural Seismic Performance Analysis  
### Structural Collapse Performance Considering Record-to-Record Variability  
On the basis of the Chinese specifications [15], the maximum interstory drift ratio of 2% is used to determine the collapse limit state. The IDA method is used to obtain the seismic collapse fragility curves under 21 suites of ground motion records. The collapse vulnerability curves of the four structures with only record-to-record variability considered are illustrated in Figure 8a-d in red. The collapse data points are well fitted by the collapse vulnerability curves calculated with the lognormal distribution function. The corresponding collapse probabilities at MCE for NS-1, RS-1, RS-2, and S-1 are 25%, 5.4%, 2%, and 4.8%, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Structure** & \multirow{2}{*}{\(\sigma_{RTR}\)} & \multirow{2}{*}{\(\sigma_{MDL}\)} & \multirow{2}{*}{\(\sigma_{TD}\)} & \multirow{2}{*}{\(\sigma_{DR}\)} & \multirow{2}{*}{\(\sigma_{Sa,c}\)} \\
**Number** & & & & & \\ \hline NS-1 & 0.336 & 0.2 & 0.2 & 0.50 & 0.665 \\ RS-1 & 0.38 & 0.2 & 0.2 & 0.35 & 0.589 \\ RS-2 & 0.450 & 0.2 & 0.2 & 0.35 & 0.636 \\ S-1 & 0.364 & 0.2 & 0.2 & 0.1 & 0.472 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of various collapse uncertainties. The prestrengthened low-ductility structure NS-1 has the highest collapse probability, exceeding 10% (the acceptable collapse probability at MCE suggested by FEMA P695) and 5% (as stipulated by Chinese specification CECS 392). This condition implies the high collapse risk of the building, although the collapse probability remains smaller than 50%, and the associated CMR is larger than 1. The collapse probability of the steel brace-strengthened structure RS-2 is the lowest (only 1/12.5 of the collapse probability of NS-1). The additional steel braces provide a low-performance structure with an unsatisfactory collapse probability at MCE to meet the requirements of CECS 392. The enlarged column sections decrease the collapse probability from 25% to 5.4%, an improvement of 78.4% jump. This result accounts for the capability of the column section enlargement method in reducing the collapse probability at MCE of a structure. However, the collapse probability at MCE of RS-1 remains beyond the limit set by the Chinese code CECS 392. In a strict sense, the first strengthening method is not good enough but it is acceptable with the influences of other uncertainties except record-to-record variability. The structure designed with the current Chinese specifications meets the requirements of CECS 392 and FEMA P695 for collapse probability at MCE. Its collapse probability at MCE is only approximately 1/5 that of NS-1 but remains larger than that of the steel brace-strengthened structure. Additional steel braces can improve structural stiffness and reduce collapse probability at MCE more effectively than the two strengthening methods. The collapse probability at MCE of RS-1 is slightly larger than that of S-1. Although RS-1 has column sections that are 50 mm larger than those of S-1 at each side and a larger overall stiffness compared with that of the original structure, the reinforcement ratio of each column is decreased when the reinforcement remains the same. The improvement of stiffness and the decrease in reinforcement ratio show a counter-reaction with each other. The record-to-record variability of RS-1 is larger than that of S-1 because the structural period is smaller, leading to the slight inferior structural performance of RS-1. The section parameters, collapse probability at MCE, and record-to-record variability of RS-1 are similar to those of S-1. The calculation outcomes are similar to that considering record-to-record variability. As required by CECS 392, the acceptable collapse probability at MCE of Category-C buildings is 5% when record-to-record variability is considered.