1 Introduction

This paper describes the seismic evaluation of the Rules dam (located in Granada, Spain), a concrete arch-gravity dam, via time-history analysis (THA). Probabilistic Seismic Hazard Assessment (PSHA) and Deterministic Seismic Hazard Assessment (DSHA) have been used to define the seismic data input. The former is necessary to calculate the Operating Basis Earthquake (OBE), which defines any low-level damage to the dam, whether insignificant or absent. The latter is used to calculate the Safety Evaluation Earthquake (SEE), which defines significant levels of damage, such as events resulting from uncontrolled water flow that has not been released, which could lead to catastrophic consequences. The main difference between the probabilistic and deterministic analyses is that the former considers the earthquake frequency, whereas the latter provides a straightforward framework for evaluating the worst possible case of ground motion. The Cornell Method [1] is the procedure that defines the PSHA; it is based on three assumptions: the events are independent and stationary in time; the probability distribution of the magnitude is defined by an exponential distribution, and the seismicity is uniformly distributed in each seismogenic zone (in contrast to Kernel Method [2], which disregards the seismogenic zones). A DSHA considers the seismic-geological context and the historical earthquake data, which represent different ground motion accelerations. The current data analysis considers the dam location and any uncertainties of the procedure [3]. The probability density function (PDF) and the pseudo-spectra acceleration (PSA) for different return periods have been calculated; the results have been compared to similar results from a previous analysis. Some uncertainties of the PSHA are due to incorrect values from the catalogue. For example, different magnitudes are due to the existence of different types of seismic waves, equations and records. To perform an accurate analysis, the data must be corrected via homogenization, declustering and completeness [4]. These corrections have been explained below.

To evaluate the seismic response of dams, a mathematical model in two and three dimensions has been developed. This model consists of a 3D analysis that considers the block interactions and a 2D analysis that considers vertical and horizontal stresses. The vertical stresses have been calculated using the upstream and downstream faces of the dam wall. To account for the interaction between fluid–structure–foundation observed by the system, geometrical and material parameters have been considered.

The Rules dam was built in 2003, and the new seismogenic zones were created in 2015; as a result, the considerations exposed here may not have been considered in the original project. Many arch dams in the world may face similar problems. Therefore, seismic hazard reassessment is advisable, particularly for existing category A dams [5] in areas of high seismicity. Many important regions of the world are vulnerable to seismic activity; consequently, risk assessments must be made to ensure that vulnerable areas are protected.

2 Seismic risk analysis: explanation of criteria

2.1 Probabilistic seismic hazard analysis

The main characteristic of probabilistic seismic hazard analysis is that it considers earthquake frequency. The basic Cornell method is used in this study; it is based on the Poisson process and Gutenberg–Richter law [6, 7]. In this section, the probabilistic seismic hazard analysis and its chosen criteria will be explained. The new Seismogenic Zones (ZS) of the Iberian Peninsula, established in 2015, are also considered [8]; this map is formed by 55 superficial zones and four deep zones. Earthquakes have the same probability of occurrence at any point inside the zone regardless of its size. This is an important condition when applying the Cornell method, particularly since additional seismogenic zones exist in previous literature [9]. To start, hazard analysis from the historic catalogue must be examined. In this study, the Instituto Geográfico Nacional (IGN) catalogue has been used [10]. This catalogue contains information about the intensity, magnitude and depth of earthquakes in the Andalusia area. The relative range of the coordinates is − 8° to 0° (longitude) and 35° to 39° (latitude). There are other catalogues, e.g., Andalusian Institute of Geophysics (IAG) [11], that can be integrated if the data are incomplete. However, the authors determined that the IGN catalogue data and the number of the used zones are sufficient to carry out this analysis. To perform the probabilistic analysis, four attenuation relations (SP96, Am96, Am05, BT03) [12,13,14,15] have been used (see Table 1). The use of these attenuation equations is justified by the fact that they were calibrated using data from Italian records. Note that the Italian tectonic activity is the result of a somewhat similar geological context to that in our area of study [16]. In general, these equations have been largely used in the South-European context.

Table 1 Attenuation equations applied to define the PSA (g)
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The homogenization of the magnitude was made to provide the unique moment magnitude (Mw); this is necessary because the data from the IGN catalogue contain five different magnitudes and macro-seismic intensity values. To perform this operation, updated equations and methods from recent literature have been used [17,18,19]. To use the Poisson process (independence between events), fore-shocks, after-shocks and swarms (in time and space) must be eliminated. The analysis of completeness must be made to correctly estimate the mean annual rate of exceedance (λc). This analysis considers the periods that contain an adequate number of seismic events. For example, in the uncorrected catalogue of this analysis, there are only 96 seismic events between 1406 and 1795, whereas in the period between 1796 and 2013, there are more than 16,795 events. If only 96 events were considered in a large period, λc would be underestimated. In Fig. 1a, 12,058 uncorrected events within a radius of 150 km from the case study (black point) are shown. To make these operations (catalogue declustering, homogenization and completeness) practical, only the events in the 11 seismogenic zones (Zesis) have been considered. These 11 zones are within a radius of 150 km, which is the recommended distance to adopt the attenuation equations. To eliminate the duplicates, there are some approaches suggested in the literature [20]. Blocks of events with the same latitude and longitude (or with an allowable displacement difference of 0.5 decimal degrees) were considered; later, the largest magnitude events were identified, and any events below this threshold magnitude were eliminated. The maximum reasonable time interval was 90 days. To perform the completeness analysis, the cumulative earthquakes-per-year method has been used. The method uses linear regressions by dividing the magnitude into variance groups, ΔMw. For both cases (declustering and completeness), a visual method has been utilized. The Gutenberg and Richter exponential distribution was applied to define a (vertical intercept) and b (slope) parameters (see Table 2). The a-parameter represents the seismic activity; the greater the a-value is (in this analysis the range is 1.36–7.52), the higher the seismicity of the zone. In Table 2, the mean annual rate of exceedance, b-value, β-value, m,max and m,min are shown; λc depends on the earthquake’s time distribution, completeness time interval and number of reported events. Thus, if λc increases, then the number of events increases, or the time interval decreases. The b value describes the relative likelihood of small and large earthquakes. In this analysis, this value ranges between 0.40 and 2.09. When the values have a shallow slope (approximately 0.40–0.97), the small earthquakes have a lower frequency compared to the strong earthquakes in the zone; thus, the seismic hazard is high in this case. When b is approximately more than 1.00, the seismic hazard is low. The m,max and m,min values are needed to define the PDF and represent its upper and lower limits (see Fig. 1b). The PDF indicates the likelihood that a particular magnitude exceeds a certain m,min. For ZS 30 and ZS 43 (dashed line), the PDF also has an upper limit m,max. The upper and lower truncated curves likely have small magnitudes, and therefore, the curves are slightly higher. The PDF of ZS 39 is high; it is characterized by events that release energy from small earthquakes. Table 2 shows the values of this study; in the last four columns, the difference (Δ) between the data from this study and the Zesis study (the negative values indicate that the Zesis data are lower than our data) is provided, and some differences can be seen (values are underlined). When considering Δm,max, the macroseismic intensity changes for a magnitude range of ± 0.5, while the physical significance of Δλc and Δb changes to ± 0.15 and ± 0.35, respectively.

Fig. 1
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Earthquakes within a radius of 150 km from the Rules dam (a) and probability density function for five seismogenic zones—its mean value is shown in brackets (b)

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Table 2 Data of the probabilistic analysis
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The disaggregation analysis has been performed to separate the magnitude and distance contributions that generated acceleration. The values of the analysis are chosen according to the dam’s fundamental period (Td), return period (Tr) and Peak Ground Acceleration (PGA). The last parameter, obtained by software (Crisis) [21], differs greatly from the design values obtained by PSHA due to wide intervals of Tr. Therefore, the results depend on the first two periods. Three different return periods have been considered: 475-, 1000- and 5000-years; the following values are obtained as a result: Mw = 4.7 and Repi. = 7.5 km (epicentral-site distance); Mw = 5.9 and Repi. = 7.5 km; Mw = 6.1 and Repi. = 7.5 km, respectively. The 1950-year period is a fourth return period that has been considered; it will not be shown in the PSHA since the PSA values are approximately 1.65 more than the values of the 475-year period.

2.2 Deterministic seismic hazard analysis

The deterministic seismic hazard analysis is needed to define the worst earthquake. Since the analysis does not consider the probability of occurrence (return period), it is convenient to consider the standard deviations (σ) of the attenuation equations. The standard deviation values used range from ± 0.19 to ± 0.26. For a strategic structure, such as a dam, the return period is generally very large. For this reason, the probabilistic analysis alone can be unreliable. A deterministic analysis should be carried out using the seismic-geological context [22] and the historical earthquake data. In Table 3, five controlling earthquakes were chosen according to the disaggregation analysis and the PGA obtained from the PSHA. The earthquake details have been taken from the Engineering Strong-Motion (ESM) database [23], and their analyses have been performed by software (Seismosignal) [24]. The recorded data have considered the three orientations (east–west, north–south and up–down), but only the heaviest results are shown (Table 3). The seismic events are superficial, e.g., the hypo-central distance (depth) is up to 30 km. The other four events are historical earthquakes in Spain.

Table 3 Series of records used in the analysis
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3 Materials and methods

The mathematical model must be made in two and three dimensions because there is an interaction between arch and cantilever units. The actions create six components—three translational and three rotational (vertical movements and rotations in vertical tangential planes are neglected). The modelling is divided into three parts: 3D analysis considers the arch effects, 2D analysis studies the vertical and horizontal stresses, and 2D analysis via the gravity method studies the stability (only some results will be shown). To consider the interactions among liquid, structure and rock observed by the system, geometry and material parameters have been specified. Certain earthquakes have been chosen to perform the THA. The vertical stresses (normal and principal) have been calculated in the upstream and downstream faces of the dam wall; therefore, the exceedance that produced the plastic hinges was estimated.

To design the structure, the SEE and OBE levels were defined and calculated via deterministic and probabilistic approaches, respectively [25]. In Fig. 2, the annual probability of exceedance from Crisis is shown for ZS 35. The differences between the curves mainly depend on the attenuation equations used and on their standard deviations, e.g., if the standard deviation decreases, then the return period increases. The standard deviation values used a range from ± 0.19 to ± 0.29. For the analysis, zero has been used, and the source has been approximated via a circular region with a 47-km radius. The solid-triangle curve points, shown in Fig. 2, are higher because the equation was not well-constrained for low magnitudes; therefore, the curve overestimates the acceleration with a higher Tr. In Fig. 3, the PSAs calculated using values obtained by disaggregation analysis are shown (the pair values, Mw and Repi., used are the same for the 5000-year period). Because of the limited interaction of the software, the curves refer exactly to the following return periods: 455-, 1023-, 1949- and 4835-years. This difference is irrelevant for the analysis of the output data. The attenuation equations do not use the same magnitude and distances; therefore, the values are estimated. The distances, shown in Fig. 3, are as follows: 5.5 km for Am96 and 10 km for BT03. The Uniform Hazard Spectra (UHS), which considered all the zones and four controlling earthquakes, are shown in Figs. 4 and 5. The UHS provides the accelerations, which depend on the structure periods (T) for a fixed return period. The PGA for the 475-year return period in the analysis is 0.198 g (19.8% of gravity acceleration). This value concurs with literature [26]. The PGA for the 1000-year return period in the analysis is 0.252 g, which is the value used for the dam analysis. For the 5000-years return period, the PSA for T = 0.25 s is 0.928 g, which is similar to the deterministic PSA (1.00 g); this is because a probabilistic analysis for a large return period is unrealistic and often overestimated. In Fig. 6, the deterministic spectra obtained for four historical earthquakes, via two different attenuation relations, are shown. In the Spanish code, the PGA for this area is 0.17 g; however, in the full analysis, the soil with the greatest registered acceleration is 0.511 g, which is three times higher than the Spanish code.

Fig. 2
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PGA in function of the return period for the ZS 35 (site of the dam)

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Fig. 3
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Four synthetic spectra for 1950 years

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Fig. 4
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Probabilistic spectra using Tr for 475 years and 1000 years, and two controlling earthquakes

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Fig. 5
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Probabilistic spectra using Tr for 5000 years and two controlling earthquakes

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Fig. 6
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Deterministic spectra of four records using SP96 and using Am05 (in dashed line). In the brackets, the estimated Rjb distance used in the Am05 equation is shown

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4 Case study

4.1 The Rules dam on the Guadalfeo River

The structure of interest is a large, concrete, arch-gravity dam in the Granada province of southern Spain. This dam has a 620 m crown length and a 500-m radius. The maximum height of the vertical cantilever is 130.33 m (central black block in Fig. 7b), and the downstream and upstream slope faces are 1:0.60 and 1:0.18, respectively. The dam is made from 32 blocks. The seven dark grey blocks in the centre represent the spillway. The reservoir is shown in Fig. 7a [27]. The capacity and area for the maximum operating level of the reservoir are 117.07 Hm3 and 308 Ha, respectively, whereas the area of the water basin is 1070 km2.

Fig. 7
figure 7

Position of the Rules dam (36°51′35″ N, 3°29′43″ W) (a) and FEA 3D model (b)

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4.2 Mathematical model

The numerical simulation was made using Finite Element Analysis (FEA) and the gravity method. The two- and three-dimensional FEA (Sap2000 software) [28] methods were used. The discretization of the 2D and 3D models are explained below.

In FEA (3D), the solid elements (32,980 elements) have a dimension of 3.00 m per 2.50 m, and there are 36,925 monolith joints. Those joints are necessary to analyse the nonlinear effects, such as concrete cracking and plastic hinges [29]. In the 3D model, the length of the block is 19.375 m. In FEA (2D), the solid elements (683 elements) have a dimension of 3.01 m per 2.79 m, and there are 765 joints. In the 2D model, the length of the block is 1 m. The solid element has eight nodes; each one has three degrees-of-freedom. In the gravity method (2D), the principal triangle of the central dam block has been considered. The height is 120 m, and the base is 93.6 m. The dam is divided into 48 lift joints with 2.5-m heights. The gravity method has been used only for the 2D analysis; it is based on rigid body equilibrium and on beam theory. The dam, via the 2D analysis, has been idealized as a triangular shape [30, 31] because the transverse behaviour is similar to a thick gravity dam with a large base thickness. The lift joints have homogeneous properties, and the loads are transferred to the foundation only by cantilever; therefore, it does not consider the arch-effect. The foundation and abutments have rocky stratigraphic profiles; the values of the site geology refer to an average shear wave velocity over 750 m/s. Since the foundation is rocky, the base has been considered fixed, i.e., the direction displacements of all points of the base are constrained. The gravity method does not consider the dam-foundation interaction, and therefore, the constraints are not well-defined (it is enough to insert the foundation parameters, which is explained below). The loads present in the FEA include the dead load, the hydrostatic pressure and the hydrodynamic pressures. In the gravity method, in addition to the three loads, the seismic dead load and uplift load have been applied. In FEA, the hydrodynamic and hydrostatic pressures act normal to the face and are uniform for each solid.

The pseudo-hydrodynamic pressure occurs because the water is linearly compressible and irrotational. Furthermore, the internal viscosity of water is neglected, and the effects of waves at the free surface are omitted [32]. The hydrodynamic pressure of incompressible water (pressure independent of frequency) and the absorption effects of the reservoir bottom [33] have also been studied. In this example, the input data used are Ec = 44.40 GPa (modulus of elasticity of the concrete), γc = 24 kN/m3 (mass density of the concrete), νc = 0.20 (Poisson’s ratio of the concrete) and ξd = 5% (dam damping). In the FEA models, to consider the voids (e.g., galleries, drains and spillways) in the dam body, γc is set to 20.64 kN/m3, i.e., less than 14% of 24 kN/m3. This difference has been obtained from the ratio of the real weight to the model weight: (3.6 × 103t)/(4.2 × 103t) = 0.86. To consider the interaction between the substructures, e.g., dam–reservoir–foundation–sediments [34] and dam–abutments, the following data have been used: Ef = 41.55 GPa (modulus of elasticity of the foundation rock), γf = 27.47 kN/m3 (mass density of the foundation rock), νf = 0.33 (Poisson’s ratio of foundation rock), γw = 9.81 kN/m3 (mass density of water), Cw = 1438 m/s (velocity of pressure waves) and αw = 0.41 (wave reflection coefficient for reservoir bottom materials).

5 Results of the dynamic analysis

The recorded data must be carefully analyzed by performing the appropriate corrections and focusing on seismic demands [35, 36]. The controlling earthquakes have been used for this THA. By defining the significant durations [37] for each record, it is possible to select the most influential interval. Figure 8 shows the THA of the vertical stresses (σy) at the wall’s heel of the dam’s central block using three records; it shows the cases of empty and full reservoirs and both dynamic actions (seismic deadweight and hydrodynamics). In this case, the seismic deadweight includes the self-weight. In the analysis, there are slight differences between the analytical (normal stress) and the computational analysis (principal stress). The analytic pressure values are in accordance with those reported in the literature [38]. The principal stresses, due to the wall inclination, are obtained by software [39]. The full reservoir analysis demonstrates that the stresses are lower due to the hydrostatic and uplift negative stresses. However, during the seism, uplift pressures within the crack can be assumed to be zero, and therefore, the σy total increases. The constitutive law for compression and tension used for concrete is in accordance with the European Committee for Standardization [40]. The maximum tensile value adopted, which generates significant nonlinear plastic deformations, is fctd = 1547 kN/m2 (horizontal dashed line in Figs. 8, 9). Other tensile strengths, which can be used for gravity dams with lower characteristics (fctd = 960 kN/m2), are shown in Figs. 8 and 9. For the unfavourable (unf.) full reservoir, when all forces have the same direction, the stresses exceed the ± 960 kN/m2 value by 174 times in the heel and 558 in the toe. The stress analysis determines potential crack (the crack opens and closes during the earthquake) and plastic hinge formations. For each vibration, the stress analysis accumulates inelastic deformations [41], producing a hysteretic behaviour that depends on dissipated energy. Figure 9 shows the THA of the stresses in the wall’s toe of the dam’s central block using two records. The seismic dead weight also includes the self-weight.

Fig. 8
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Dynamic stress at the heel of the wall (singular point in the upstream)

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Fig. 9
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Dynamic stress at the toe of the wall (singular point in the downstream)

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The structural analysis provided the following results: Td is 0.284 s, the second dam period is 0.245 s, and the third dam period is 0.208 s. The first dam-reservoir period is 0.344 s, and the system’s (dam-foundation-reservoir) fundamental period, Ts, is 0.393 s (the system damping, ξs, is 8.5%). The operating level considered is 113.00 m. The seismic dead weight and the hydrodynamic normal stresses, in relation to the maximum values calculated in this analysis, are less than 31.6% (at one-third of y) and 68.4% (at two-thirds of y) and 42.1% (at one-third of y) and 89.5% (at two-thirds of y), respectively. The y axis is zero in the bottom of the reservoir. The self-weight and hydrostatic horizontal stresses are − 71.78 kN/m2 (at the toe), − 239.26 kN/m2 (at the heel), − 1028.24 kN/m2 (at the toe), and − 585.09 kN/m2 (at the heel), respectively. Along the base of the dam, the seismic dead stresses increase, whereas the hydrostatic stresses decrease. The hydrodynamic and hydrostatic pressure effects and the sediment effects have been studied by other authors [42] and, therefore, are not shown here.

6 Conclusions

In this paper, the seismic risk of the Rules dam is evaluated. The possible maximum accelerations that can act against the dam’s body are evaluated, and the dam’s behaviour has been made explicit. The main conclusions are summarized as follows:

  • The analysis has been performed in the site in which the dam is placed. The analysis carried out in this study shows that there are some differences in relation with the Zesis data. However, a precise study “ad hoc” is needed. Different values were only obtained in four zones (Δλc, Δb and Δm,max range between − 0.31 and 0.23, − 0.75 to 0.67, − 0.2–1.7, respectively). The seismic hazard is high and has a maximum moment magnitude of 6.8. The main differences are due to the recovered data and process uncertainties, e.g., the choice of the operations: homogenization, declustering and completeness.

  • The PSHA and the DSHA have been used to define the design parameters. In the Spanish code, the PGA for the area is 0.17 g. In the PSHA, the PGA range is 6–135% higher than 0.17 g. In the DSHA, the range goes up to 200%. By considering the return period of the seismic action, for the no-collapse requirement of 1000 years, 0.25 g of ground acceleration has been calculated for the analysis. This study does not consider the site effects, e.g., amplification, topographic effects, and effect of foundation inhomogeneity [43]. This paper encourages the development of these issues in future research.

  • The obtained system values are as follows: a fundamental period of 0.39 s and 8.5% damping. The system represents the interaction between the dam, foundation and water. In the heel, the increase due to dynamic action is 41.67% of the self-weight normal stress and 41.19% of the water principal stress. In the toe, the horizontal stress due to the hydrostatic action in the dam is 36% of the vertical stress. In the dynamic analysis, the maximum vertical principal stress is 3.8 MPa, while in the pseudo-static analysis—a rigid system with a period of vibration equal to zero—it is 2.6 MPa. An important number of inelastic deformations have been observed. The seismic input is obtained from a seismic hazard reanalysis. Since the structure was built (in 2003) before the modern seismogenic zone was created, these calculations and considerations may not be accurate since they were made in the original dam project.