# Investigation of the Pressure Coefficients Correlation Field for Low-Rise Building Roofs
## 3 Experimental Wind-Tunnel Results  
As discussed in previous papers by two of the authors [29; 30; 31; 32; 33; 34; 35; 36; 37; 38], the design of this kind of structure requires at a preliminary stage the knowledge of the mean values (sample shown in Figure 2) and of the peak values of the aerodynamic pressure coefficients \(c_{p}(t)\), whose instantaneous values are obtained by the recorded pressure time histories \(p(t)\) as follows:  
\[c_{p}(t)=\frac{p(t)-p_{0}}{\frac{1}{2}\rho V_{m}^{2}} \tag{1}\]  
where \(\rho\) denotes the air density, and \(V_{m}\) the mean speed at the roof level, and \(p_{0}\) the static pressure of undisturbed flow. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Models** & \(L_{1}\) & \(L_{2}\) & \(f_{1}\) & \(f_{2}\) & \(H\) \\ \hline
**RLF** & 400.0 & 800.0 & 26.7 & 53.3 & 213.3 \\
**RHF** & 400.0 & 800.0 & 26.7 & 53.3 & 346.3 \\
**RLC** & 400.0 & 800.0 & 44.4 & 88.9 & 266.6 \\
**RHC** & 400.0 & 800.0 & 44.4 & 88.9 & 399.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test model geometrical parameters, measurements in mm. Figure 1: Experimental setup: (**a**) RLF; (**b**) RHF; (**c**) RLC; (**d**) RHC (acronyms as in [30]). Depending on the dimensions and characteristics of the structure, aeroelastic effects, i.e., wind-structure interactions in the large displacement field, may become relevant depending on the flexibility of the structure. This may require ad hoc wind-tunnel tests and nonlinear numerical analyses, as widely discussed in the scientific literature [39]. Furthermore, the correlation between pressure-time histories at different points of the roof may also play a relevant role, because the resultant wind-induced loads on a given portion of the structure obviously depend on that correlation. Its underestimation or overestimation may lead to unsafe or redundant design of structural elements depending on their area of influence. However, the role of the correlation of the wind-induced pressure field on hyperbolic paraboloid roofs has so far been almost neglected. Based on some results of the experimental wind-tunnel study discussed in Section 2, this paper explores these aspects for four rectangular models with hyperbolic paraboloid roofs, discussing the dependence of the correlation functions on the height and the curvature of the scale models and on the mean direction of the wind. According to the scientific literature, the correlation coefficient between the pressure-time histories at a reference point \(P_{0}\) and another point \(P\) on the roof is defined as:  
\[r(P,P_{0})=\frac{cov(P,P_{0})}{\sigma_{P}\sigma_{P_{0}}} \tag{2}\]  
where \(\sigma_{P_{0}}\) and \(\sigma_{P}\) denote the standard deviations of the pressure coefficient time histories \(c_{p}(t)\) in \(P_{0}\) and \(P\), respectively, whose covariance \(cov(P,P_{0})\) is defined by:  
\[cov(P,P_{0})=E\big{\{}\big{[}c_{p}(P)-E\big{(}c_{p}(P)\big{)}\big{]}\big{[}c_{ p}(P_{0})-E\big{(}c_{p}(P_{0})\big{)}\big{]}\big{\}} \tag{3}\]  
where \(E\) denotes the expected value. In the following, the correlation functions relating a selected pressure tap (at point \(P_{0}\)) to all the other pressure taps \(P\) is discussed for the four models described in Section 2, for different positions \(P_{0}\) of the reference pressure tap. As a first check, the stationarity of the recorded pressure fields was verified for sample pressure taps, evaluating their correlation functions for a given length of the recorded pressure-time histories and for different starting points \(t_{0}\) of the wind-pressure acquisition. As reported in Figure 3, for the lower rectangular model with higher curvature (RLC), the level curves of the correlation function between the pressure-time history in the center of the roof (pressure tap #48) and the other points of the roof surface were almost independent on the initial time \(t_{0}\), confirming the stationarity of the experimental pressure field. Figure 2: Example of mean pressure coefficients maps for RLC at (**a**) 0\({}^{\circ}\) and (**b**) 90\({}^{\circ}\).