# Wind-Induced Dynamic Performance Evaluation of Tall Buildings Considering Future Wind Climate
## 3 Fragility Analysis for Human Occupant Comfort
### Probabilistic Assessment of Vibration Analysis  
This section presents a comprehensive fragility analysis of the tall building against human occupant comfort criteria, focusing on its response to historical, mid-future, and late-future wind conditions. The analysis aims to evaluate the probability of exceeding given performance thresholds, particularly regarding human occupant comfort. The modal peak acceleration is regarded as the perception threshold for the occupant comfort assessment. The flowchart for the dynamic performance assessment and failure probability evaluation is shown in Figure 2. The mean wind speeds are considered to evaluate the probability of failure using the Monte Carlo simulation. A performance function for the probabilistic assessment of vibration exceedance of a wind-induced tall building could be defined as follows:  
\[G(U_{r},\theta)=a^{U}-max[a(U_{r},\tau,\theta)] \tag{26}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Level** & **Human Subjective Response** & **Peak Acceleration Limit (milli-g)** \\ \hline I & No sense & \(\leq\)5 \\ II & Low vibration sense & 5–20 \\ III & Medium vibration sense & 20–35 \\ IV & Trouble & 35–50 \\ V & Very Trouble & 50–150 \\ VI & Intolerable & \(\geq\)150 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Occupant comfort levels.  
where \(U_{r}\) represents the design wind speed and \(\theta\) represents the modal property vector. \(a^{U}\) is the peak acceleration criterion, and \(a(U_{r},\tau,\theta)\) refers to the wind-induced vibration of the building over a specified time duration \(\tau\). The time duration \(\tau\) can be set to the values such as 10 min, 1 h, or more, and assuming that the wind loads and vibrations induced by storms or typhoons can be modeled as a stationary process. The design wind speed is averaged over the same time duration \(\tau\) used for wind-induced vibration. Based on the performance function defined in Equation (23), the probability of failure due to the vibration exceedance is defined by the limit state \(G(U_{r},\theta)=0\), where \(G\leq 0\) indicates failure. Theoretically, the failure probability can be assessed through a multidimensional integral, as shown in Equation (27):  
\[P_{f_{p}}=P\big{(}G_{p}(\mathrm{x},\mathrm{d})\leq 0\big{)}=\int G_{p}\leq 0 \cdots\int f_{x}(x_{1},\mathrm{x}_{2},\ldots)dx_{1}dx_{1}\cdots dx_{n} \tag{27}\]  
where \(G_{p}\) represents directional performance function \(G\) in the direction \(p\). The vector \(\mathrm{x}\) consists of random variables that quantify uncertainties in wind loads and structural properties for probabilistic exceedance evaluation, while \(\mathrm{d}\) denotes the deterministic building parameters. \(f_{x}(x_{1},x_{2},\ldots)\) represents the joint PDF of the random variables. The performance function in Equation (27) determines whether vibration exceedance leads to failure (\(G_{p}\leq 0\)) or not (\(G_{p}>0\)). By conducting a sufficiently large number of simulations, the failure probability due to vibration exceedance can be approximated using Equation (28):  
\[P_{f_{p}}=P(G_{p}(\mathrm{x},\mathrm{d})\leq 0)\approx\widetilde{P}_{f_{p}}= \frac{1}{N}\sum_{k=1}^{N}\Pi_{G_{p}}(x) \tag{28}\]  
where \(\Pi_{G_{p}}\) is the so-called indicator function, which is equal to one when \(G_{p}\leq 0\) and zero otherwise.