# On the Thermal Stresses Due to Weathering in Natural Stones
## 2 Fundamentals and Methodology
### Solar Radiation  
The total solar radiation for a clear sky \((I_{T})\) can be estimated as:  
\[I_{T}=I_{t,D}+I_{t,d}+I_{t,R}, \tag{1}\]  
where \(I_{t,D}\) is the direct solar radiation, \(I_{t,d}\) is the diffuse solar radiation, and \(I_{t,R}\) is the reflected solar radiation.  
In this study, the reflected solar radiation flux \((I_{t,R})\) and the direct solar radiation \((I_{t,D})\) were considered to be equal to zero because the measurements on the marble slab were made in the shading area. For a vertical surface (wall) and clear-sky conditions, the diffuse solar radiation \((I_{t,d})\) can be calculated using Equations (2)-(5) [29; 30]:  
\[I_{t,d} = I_{d}\cdot Y \tag{2}\] \[Y = max(0.45,\ 0.55+0.437 cos(\theta)+0.313 cos^{2}(\theta)), \tag{3}\]  
where \(I_{d}\) is the diffuse horizontal irradiance, \(Y\) is the correction factor of the direct solar radiation, and \(\theta\) is the angle of incidence.  
The diffuse horizontal irradiance \((I_{d})\) is [31]:  
\[I_{d} = I_{0}\,e^{-\tau_{d}m^{d}} \tag{4}\] \[ad = 0.507+0.205\,\tau_{b}-0.080\,\tau_{d}-0.190\,\tau_{b}\,\tau_{d}, \tag{5}\]  
where \(I_{0}\) is the extraterrestrial radiant flux, \(\tau_{d}\) is the diffuse optical depth, \(m\) is the air mass, \(ad\) is the diffuse air mass, and \(\tau_{b}\) is the beam optical depth. The values of \(\tau_{b}\) and \(\tau_{d}\) for the month of August in 2009 in the city of Pescara were 0.494 and 1.935, respectively [32].  
According to the American Society of Heating, Refrigerating, and Air-Conditioning Engineers [31], the extraterrestrial radiant flux \((I_{0})\) can be approximated by:  
\[I_{0}=I_{sc}\bigg{[}1+0.033\ \cos\bigg{(}360\frac{n-3}{365}\bigg{)}\bigg{]}, \tag{6}\]  
where \(I_{sc}\) is the solar constant and \(n\) is the day of the year. As stated by [PERSON] (1983) [33], the solar constant \((I_{sc})\) assumes the value of 1367 W/m\({}^{2}\).  
Air mass \((m)\) and solar altitude \((\beta)\) can be expressed as [34]:  
\[m=\frac{1}{\sin(\beta)+0.50572(6.07995+\beta)^{-1.6364}} \tag{7}\]  
Figure 1: Daily variation of temperature in a ventilated structure.  
\[\sin(\beta)=\cos(l)\cos(h)\cos(d)+\sin(l)\sin(d), \tag{8}\]  
where \(l\) is the latitude, \(h\) is the hour angle, and \(d\) is the declination angle.  
The hour angle \((h)\) and declination angle \((d)\) are expressed as follows:  
\[h = 15(LST-12) \tag{9}\] \[d = 23.45\ \sin\biggl{(}360\cdot\frac{284+n}{365}\biggr{)}, \tag{10}\]  
where \(LST\) is the local solar time (h), and it is calculated according to Equation (11):  
\[LST = CT+\frac{1}{15}(L_{std}-L_{loc})+\frac{E_{t}}{60}-DT \tag{11}\] \[L_{std} = 15\cdot UTC, \tag{12}\]  
where \(CT\) is the clock time (h), \(L_{std}\) is the standard meridian for the local time zone, \(L_{loc}\) is the longitude, \(E_{t}\) is the equation of time, \(DT\) is the correction for daylight savings time, and \(UTC\) is the universal coordinated time.  
The equation of time \((E_{t})\), in minutes, can be approximated by [33]:  
\[E_{t} = 2.2918[0.0075+0.1868\ \cos(\Gamma)-3.2077\ \sin(\Gamma)-1.4615\ \cos(2\Gamma)-4.089\ \sin(2\Gamma)] \tag{13}\]  
\[\Gamma = 360\biggl{(}\frac{n-1}{365}\biggr{)}, \tag{14}\]  
where \(\Gamma\) is the angular correction.  
The angle of incidence \((\theta)\), the surface-solar azimuth angle \((\gamma)\), and the azimuth angle \((\phi)\) can be computed using the following equations:  
\[\cos(\theta) = \cos(\beta)\cos(\gamma)\cos(\Sigma)+\sin(\beta)\cos(\Sigma) \tag{15}\] \[\gamma = \phi-\psi\] (16) \[\cos(\phi) = \frac{\cos(d)\sin(l)\cos(h)-\sin(d)\cos(l)}{\cos(\beta)}, \tag{17}\]  
where \(\gamma\) is the surface-solar azimuth angle, \(\Sigma\) is the surface tilt angle (90\({}^{\circ}\) for vertical surfaces), and \(\psi\) is the surface azimuth.