1. Introduction

2. Dam description

Figure 1. Downstream view of Alto Ceira dam.

Figure 1. Downstream view of Alto Ceira dam.

Figure 2. Downstream elevation view, geodetic surveying targets.

Figure 2. Downstream elevation view, geodetic surveying targets.

Figure 3. Horizontal displacements of the object points at level 663.0 m between 1950 and 1986.

Figure 3. Horizontal displacements of the object points at level 663.0 m between 1950 and 1986.

Figure 4. Vertical displacements of the object points on the crest between 1950 and 1986.

Figure 4. Vertical displacements of the object points on the crest between 1950 and 1986.

Figure 5. Crack mapping on the downstream face of the dam performed in 2001.

Figure 5. Crack mapping on the downstream face of the dam performed in 2001.

3. Finite Element Model

Figure 6. Finite element model of Alto Ceira dam.

Figure 6. Finite element model of Alto Ceira dam.

4. Thermal Analysis

4.5. Solar Radiation

4.5.1. The Path of the Sun across the Celestial Sphere

Solar radiation that reaches the boundaries of the Earth’s atmosphere is beam radiation, also known as direct radiation. This radiation travels directly from the sun to any point on the Earth along a straight line connecting the sun with the selected point on Earth. Upon passing through the Earth’s atmosphere, some radiation gets scattered. As a result, any point on the Earth’s surface is reached by both beam radiation and radiation scattered by the atmosphere. Beam radiation is strictly directional, while the scattered radiation reaches the Earth’s surface from all directions. Any tilted surface (in reference to a horizontal one) on Earth is reached by not only beam and scattered radiation, but radiation reflected by the Earth’s surface and various objects located nearby.

For a better understanding of the paths of the sun in the sky, one can imagine a celestial sphere with the Earth at its center and the sun revolving around it (geocentric system). Location of any surface in reference to the sun is determined with fundamental concepts of spherical geometry relating to the Earth’s movement in relation to the sun. The apparent motion of the sun is caused by the combined effects of the rotation of the Earth around its proper axis and the translation around the sun. Therefore, the angle at which the direct component of light will strike the Earth exhibits daily and yearly changes in east-west (E-W) and south-north (S-N) directions, respectively.

Error! Reference source not found. shows the apparent motion of the sun and its position at solar noon at the equinoxes and solstices, relative to the base of the Alto Ceira dam.

Figure 7. The sun’s paths across the sky as seen from the dam.

Figure 7. The sun’s paths across the sky as seen from the dam.

As can be seen in the figure the sun’s path reaches its extreme excursions at the winter and summer solstices, around December 22 and June 21, respectively, since Alto Ceira dam is located in the northern hemisphere. At the solstices, the altitude at solar noon is decreased or increased by the inclination of the earth’s axis (23°27’), respectively. At the equinoxes, around March 21 and September 23, the sun rises due east and sets due west, and at solar noon the altitude equals 90° minus the latitude.

4.5.2. Solar Position

Before describing the equations that govern the solar flux, it is important to note that the time used in these equations is the apparent solar time AST, which directly tracks the diurnal motion of the sun. Meanwhile, the chronological time that clocks use, called local standard time $LST$, corresponds to a certain time zone, in which noon does not necessarily correspond to the time when the sun is highest in the sky. Additionally, the Earth does not rotate perfectly, rather, it wobbles on its axis creating what is known as an analemma. Therefore, to convert between local standard time and apparent solar time involves the equation of time $ET$ and a longitude correction, which corresponds to four minutes of time per degree difference between the local longitude $LON$ and the longitude of the local standard meridian $LSM$ for the time zone [17]:

$$AST=LST+\frac{ET}{60}+\frac{LON-LSM}{15}+Day saving$$

(16)

with

$$ET=0.0075+0.1868\cos \gamma -3.2077\sin \mathsf{\gamma }-1.4615\cos 2\mathsf{\gamma }-4.089\sin 2\mathsf{\gamma }$$

(17)

where $\gamma = \frac{2\pi }{365}\left(N-1\right)$, where $N$ is the day of the year considering a year with 365 days, and $Day saving$ is either 0 or 1 hour, depending on whether daylight saving time is applied.

The solar hour angle converts the solar time $AST$ into the number of degrees that the sun moves across the sky. By definition, the solar hour angle is 0° at solar noon. Since the Earth rotates 15° per hour, each hour away from solar noon corresponds to an angular motion of the sun in the sky of 15°. In the morning, the solar hour angle is negative, in the afternoon the solar hour angle is positive:

$$\omega =15\left(AST-12\right)$$

(18)

while the Earth orbits the sun, its axis is constantly tilted at 66°33’ to the orbit plane. Seasonal changes in the mutual location of the Earth and the sun are described by the parameter known as the sun’s declination $\delta$. The declination is defined as the angle between the line connecting the Earth and the sun, and the equatorial plane at the solar noon. It can be represented by Spencer’s formula, given by:

$$\delta =0.006918-0.399912\cos \gamma +0.070257\sin \gamma -0.006758\cos 2\gamma +0.000907\sin 2\gamma -0.002697\cos 3\gamma +0.00148\sin 3\gamma$$

(19)

The sun position is described by two angles: the solar altitude above the horizontal ${\alpha }_s$ and the solar azimuth measured from the south ${\psi }_s$. Both angles are a function of the sun’s declination $\delta$, the earth’s latitude $\varphi$ and the solar hour angle $\omega$:

$$\sin {\alpha }_s=\cos Z=\sin \varphi \sin \delta + \cos \varphi \cos \delta \cos \omega$$

(20)

$${\psi }_s=\mathrm{sign}\left(\omega \right) \left[{\cos }^{-1}\left(\frac{\cos \delta \cos \omega -\cos Z\cos \varphi }{\sin Z\sin \varphi }\right)\right]$$

(21)

4.5.3. Angle of Incidence

Usually, the infrastructure boundaries are tilted against the horizontal surface. Tilt is described by the tilt angle $Y$ against the horizontal plane, which varies in the range of $0\le Y\le 180°$.

The incident angle $\alpha$ of beam radiation on a tilted surface of any orientation is expressed in terms of the surface angle $Y$, the Sun’s declination $\delta$, the Earth’s latitude $\varphi$, the solar hour angle $\omega$ and the azimuth $\psi$ [18]:

$$\cos \alpha =A\sin \delta +B\cos \omega \cos \delta +C\sin \omega \cos \delta$$

(22)

with

$$\begin{array}{c}A=\cos Y\sin \varphi -\sin Y\cos \varphi \cos \psi \\ B=\cos Y\cos \varphi +\sin Y\sin \varphi \cos \psi \\ C=\sin Y\sin \psi \end{array}$$

(23)

If the right-hand side of Eq. (22) is negative, the sun’s rays will not strike the front side of the surface and therefore it will be shaded.

3.5.4. Solar Radiation Components

According to the Liu-Jordan theory [19], the solar radiation $I_T$ that reaches a surface tilted at an angle $Y$ (in reference to a horizontal plane) is described by the equation:

$$I_T\left(t\right)= I_b\left(t\right)R_b\left(t\right)+ I_d\left(t\right)R_d+\left[I_b\left(t\right)+I_d\left(t\right)\right]{\rho }_o{R}_r$$

(24)

where $I_b$ and $I_d$ are, respectively, the beam and the diffuse radiation intercepting on an horizontal surface, ${\rho }_o$ is the surface (ground) reflectance and $R_b$, $R_d$ and $R_r$ are, respectively, the correction factors for beam, diffuse and reflected radiation

$$R_b=\frac{\cos \alpha }{\cos Z}$$

(25)

$$R_d=\frac{1+\cos Y}{2}$$

(26)

$$R_r=\frac{1-\cos Y}{2}$$

(27)

4.5.5. Solar Irradiance

The average solar irradiance in the site where the dam was located can be represented by the average solar radiation measured at Penhas Douradas station given by [20,21]:

$$I_b=I_o\exp (-0.968+0.760\cos Z)\cos Z$$

(28)

where $I_o$ is the solar constant (1367 W/m²). In fact, the solar irradiance measured in the solar radiation station corresponds to the global horizontal irradiation, which is the sum of the beam and diffuse radiations. However, as no data is available to decompose the radiation, all measured irradiance was considered as beam radiation and the diffuse radiation was neglected.

The absorption coefficient was set to 0.65 for the dam and the spillway. No solar radiation was considered for the rock foundation in order to take into account the vegetation cover.

4.5.6. Shading of Beam Radiation

It can be observed from Error! Reference source not found. that self- and external obstructions can prevent the direct component of light from reaching the dam faces. Therefore, it is important to determine which part of the dam faces will be shaded at any particular time to correctly predict the solar radiation flux.

One of the most popular methods is the ray-tracing technique. The basic idea of this method is to start a ray at the object and send it to the sun. If this backward ray reaches the sun without hitting any object along its way, then the beam radiation strikes the object, otherwise, it will be in shadow. The problem of the intersection of a ray with an object is one of the classical problems in the field of Computer Graphics, where many algorithms for ray-tracing have been developed. For this work, the ray-triangle intersection algorithm presented in [22] was adopted.

4.6. Preliminary Results

The transient thermal analysis was performed with the same version of the in-house code PAT used in [23]. This code applies a fully-implicit Euler backward finite difference scheme for the time discretization and a finite element scheme for the spatial discretization. The analysis was conducted considering an incremental time of 1 h.

Before addressing the whole thermo-mechanical results in section 6, some previous interesting analysis will be presented.

On the one hand, in order to analyze the uneven distribution of the received solar irradiation along the downstream face of the dam, the concept of daily and annual solar irradiance is used. In this sense, the total amount of solar irradiance for one day or for the year is calculated by integrating (i.e. summing) the individual hourly solar irradiance values obtained within the corresponding period. Error! Reference source not found. illustrates the values obtained for the daily solar irradiation on the downstream face on the winter and summer solstices and on the equinoxes.

Figure 8. Daily solar irradiance calculated at the solstices and equinoxes over the downstream face of the dam.

Figure 8. Daily solar irradiance calculated at the solstices and equinoxes over the downstream face of the dam.

As can be seen, the right flank of the dam received much more sunshine than the left flank from autumn to spring. As a result of that, the total annual irradiance received by the right flank of the downstream face of the dam is more than twice as large than the amount received by the left flank, Error! Reference source not found..

Figure 9. Annual solar irradiance over the downstream face of the dam.

Figure 9. Annual solar irradiance over the downstream face of the dam.

On the other hand, due to the lack of thermal monitoring and in an attempt to validate the numerical results, a comparison between the image obtained by a thermal survey and the corresponding FEM values are shown in Error! Reference source not found..

The thermal survey was carried out by LNEC two weeks before the demolition of the dam, on 4^th of June of 2013 at approximately 3 pm. The object of the thermal survey was to evaluate the suitability of a thermal camera to detect resurgences of water on the downstream surface of dams [24]. For the survey, the camera was mounted on a tripod placed on the left bank and 26 images were recorded. Later, these images were stitched creating the panoramic image represented in the figure. Although there is not an associated color scale, the blue color corresponds to cold areas, the red color corresponds to hot areas and the white color is reserved for the hotter areas.

The FEM results correspond to the temperature distribution obtained for 4^th of June at 3 pm, day and time of the thermal survey. As can be inferred from Error! Reference source not found., near the summer solstice, the sun at 3 pm is further west than the one represented in the picture. In that position the sun strikes the downstream face of the dam in an almost symmetric way.

Figure 10. Thermal panoramic image and FEM simulation on 4^th June at 3 pm.

Figure 10. Thermal panoramic image and FEM simulation on 4^th June at 3 pm.

It is worth noting that the numerical simulation uses average values of the thermal actions. However, in spite of the variability of the air temperature, nebulosity and wind speed, it is still possible to recognize a close resemblance between the two images. Both of them show a hotter zone above the walkway located at level 655.0 m, with the maximum near the crest, and a cooler zone below the walkway with an increment of the temperature from the center of the dam to the abutments.

5. Structural Analysis

6. Results

Figure 11. Temperature and volumetric AAR expansion increment calculated on 29^th of January and 13^th of August of 1951.

Figure 11. Temperature and volumetric AAR expansion increment calculated on 29^th of January and 13^th of August of 1951.

Figure 12. Displacement contour plots calculated on 12^th of March from 1950 to 1978.

Figure 12. Displacement contour plots calculated on 12^th of March from 1950 to 1978.

Figure 13. Radial displacements evolution from 1950 to 1978 for the object points CD 663, DE 663 and EF 663.

Figure 13. Radial displacements evolution from 1950 to 1978 for the object points CD 663, DE 663 and EF 663.

Figure 14. Vertical displacements evolution from 1950 to 1978 for the object points III, IV and V.

Figure 14. Vertical displacements evolution from 1950 to 1978 for the object points III, IV and V.

7. Discussion

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