# Inter-Story Drift Ratio Detection of High-Rise Buildings Based on Ambient Noise Recordings
## 3 Data and Processing Methods
### Deflection Function Fitting  
Generally, the wind load on high-rise buildings is the uniformly distributed load of an inverted triangle, and the shear force is the uniformly distributed load of a triangle [28]. If a high-rise building is regarded as a cantilever beam fixed at one end under a triangularly distributed load, its stress condition can be shown, as indicated in Figure 3.  
In Figure 3, H is 98 m, the total height of the aboveground part of the building, X is the distance from the ground to the calculation point, and q\({}_{0}\) is the maximum value of the uniformly distributed load of the inverted triangle. As shown in Figure 3, the bending moment function for the structure is a quadratic function of distance X. The second derivation of the deflection line allows for computing the curvature, depending on the stiffness and the bending moment of the considered static system, and their relationship is as follows [29]:  
\[\frac{M}{\varphi}=EI \tag{2}\]  
where \(EI\) is the stiffness of the component. According to [PERSON] [30], the idealized \(\frac{M}{\varphi}\) curve of a reinforced concrete component can be represented by three straight lines, which means that if the concrete can be sustained in one condition, \(M\) and \(\varphi\) can be considered as homogeneous functions. For any normal building, its concrete components always align in one condition, without changing drastically. Therefore, the bending moment function shall be the second derivative of the building deflection function. Based on this, it can be concluded that (1) the deflection function of the building structure should be a quartic function of the distance X, and (2) the derivative of the deflection curve function is a cubic function of the distance X. As shown in Figure 4, the parameter \(\theta_{1}\) data do not correspond to IDRs but rather to the tangential slope of the deflection curve calculated indirectly at specific heights; if there is no initial inter-story drift, the change value of \(\theta_{1}\) (\(\Delta\theta_{1}\)) can be considered to be the tangent angle of the deflection curve at each floor's height, and \(\tan\Delta\theta_{1}\) can be considered as the tangent slope of the deflection curve at each floor's height. Because \(\tan\Delta\theta_{1}\) approximately equals the tangent slope of the deflection curve and the data set formed by the tangent slope at specific heights and the corresponding heights can be used to fit the derivative function of the deflection curve, the data set formed by \(\tan\Delta\theta_{1}\) and its corresponding heights can be used to approximately fit the derivative function of the deflection curve. The following steps can be used to fit the Zhonghe Building's deflection function:  
1. Use the data set formed by \(\tan\Delta\theta_{1}\) and its corresponding heights to fit the derivative function of the deflection curve \(f(x)\). 2. The deflection function \(\rho(x)\) can be calculated via the following formula: \[\rho(x)=\int_{x_{min}}^{x_{max}}f(x)\] (3) where \(x\) is the distance from the ground to the calculation point and \(x_{max}\) is 98 m, the maximum height of the aboveground part of the building. \(x_{min}\) is 0 m.  
Figure 3: Stress model diagram under wind load. In Figure 4, the deflection curve of the building's structure is in the shape of a quartic function. Represented by \(\alpha\), IDRs are the secant angle of the deflection curve. Ignoring a tiny initial drift, \(\Delta\theta_{1}\) can be considered the tangent angle of the deflection curve. Calculation of the IDRs  
The IDR is the ratio between the relative translational drift between two consecutive floors and the story height. Therefore, the IDR \(\alpha\) can be calculated via the following formulas:  
\[\left\{ \begin{matrix}\alpha_{i}=\frac{\Delta\rho(x)_{i}}{\Delta x_{i}}\\ \Delta\rho(x)_{i}=\rho(x)_{i+1}-\rho(x)_{i}\\ \Delta x_{i}=x_{i+1}-x_{i}\end{matrix} \right. \tag{4}\]  
where \(i\) and \(i+1\) are the floor numbers.