Damage estimation on concrete gravity dams through artificial accelerograms

The aim of this paper is to analyse the damage on gravity dams through artificial earthquakes from two methods. The first procedure defines the performance and the response curve of concrete gravity dams using a harmonic function which establishes linear displacements. The other procedure to obtain the artificial earthquake defines the power spectral density function consistent with the response spectrum. This artificial accelerogram is necessary to quantify the response curve of concrete gravity dams in the time domain. The seismic activity in Spain is not frequent, therefore it is often difficult to select real accelerograms to perform a complete seismic analysis, which makes artificial accelerograms extremely useful. Finally, combining these two procedures, a damage index is determined for assessing the crack’s magnitude. These both efficient and practical procedures are useful to develop further complicated analysis.


Introduction
This paper describes two different methodologies, one defines the artificial accelerations under a novel analytical model of Power Spectral Density (PSD) function; the other one estimates the damage on concrete gravity dams.The main advantages of the first methodology may comprise its good applicability and compatibility with different spectra.Furthermore, the properties of the artificial accelerograms can be modified depending on the analysis to be carried out.Unfortunately, the artificial accelerograms do not account the seismological and geotechnical context.
On the other hand, the main advantage of the second method is that it is not necessary a nonlinear model to define the material behaviour.
Except for few regions in the world where a set of earthquake records are available, artificial accelerograms are used to carry out the time-history analysis.Some applications are shown in literature, for example for artificial accelerograms generated via wavelet transform changing the frequency content [1,2].
Earthquake damage assessments can be difficult to carry out given that, during the earthquake shaking, the frequency and the amplitude of the seism change with the stiffness reduction.The definition of the dynamic stiffness is difficult to be asserted because the material behaviour is related to complex aspects to be defined.In literature there are some MATEC Web of Conferences 211, 14001 (2018) https://doi.org/10.1051/matecconf/201821114001VETOMAC XIV studies to calculate the nonlinear parameters based on the plastic damage model and residual seismic bearing capacity for dams [3] or based on ductility and hysteretic energy for structure [4].
In this paper, in order to implement both methodologies, an ideal concrete gravity dam has been studied.A damage index through the artificial acceleration is estimated from an ad hoc generated stochastic process model.

Artificial accelerograms
The artificial accelerations coherent with the elastic spectrum has been calculated by using the PSD function defined by [5] and developed in [6].
The definition of the artificial accelerograms has been carried out by using a methodology that is well applicable to the Spanish elastic spectrum with three branches defined in Spanish code [7].
The accelerations a(t) in the time-domain t are calculated with the model defined in the literature [8]: Equation 1 is formed by three parts, i.e. the modulation function I(t), the amplitude of the accelerogram Ai obtained from literature [5] and the sinusoidal function with a circular frequency ωi and a phase angle ϕi.The modulation function simulates the stationary and transitory condition of a real earthquake and it has a trapezoidal shape [6].The sinusoidal function represents the sum of a series of simple harmonic components n, which define the vibrations of the accelerogram.
The methodology is explained as it follows.Firstly, an elastic spectrum Se(T) is chosen and is defined by following parameters: structural periods TA and TB, soil coefficient C, contribution coefficient K and Peak Ground Acceleration (PGA).The viscous damping ratio and the dynamic amplification factor are assumed 5% and 2.5, respectively.The elastic spectrum is defined by: K should account the influence of different types of expected earthquakes and in the Spanish code [7] ranges 1.0-1.3.However, given the complexity at defining the seismic acceleration, the authors believe that K has no meaning and it cannot be defined for a simple coefficient.
Then PSD function in frequency-domain ω is calculated as follows: where ωA and ωB are the circular frequencies of the structure related to the TA and TB, respectively and e1, e2 and e3 are coefficients that mainly depend on ω, viscous damping ratio and dynamic amplification factor.PSD0 represents the peak value of the PSD function.
Once I(t) and Ai are defined, finally, the artificial accelerogram a(t) by Eq. 1 is computed.The obtained maximum acceleration is 0.924 g, which is inconsistent with the PGA = 0.34 g because the calculation is stochastic.In order to solve this inconsistence, Acceleration Scale Factors (ASF) have been introduced.The accelerations have been scaled of 0.924/0.34= 2.72 and 2.72/2 = 1.36, therefore, the used accelerations are a1(t), a2(t) and a3(t) with the maximum acceleration 0.924 g, 0.68 g and 0.34 g, respectively.Thus, the dam is analysed under the action of the same earthquake with different amplitudes.
Figure 1 shows the results and Table 1 shows the parameters that have been used in the analysis.
Fig. 1.Elastic spectrum (left), PSD function (middle) and three artificial accelerograms (right) a1(t) (dashed light grey), a2(t) (solid grey) and a3(t) (dashed black) developed by software [9].The results for the artificial accelerograms are satisfactory.However, despite the accelerations referred to the elastic spectrum present in the Spanish code [7], the accelerograms do not account the characteristic of an earthquake, e.g.energy content and relation between the magnitude and distance of events.
Other inconvenient is that this methodology is stationary in time, in terms of the amplitude of motion but real accelerograms are often more irregular.However, in structural engineering the maximum values are of interest therefore the method is efficacious.

Results: damage estimation on dams
The earthquake damage estimation for concrete gravity dams is based on the methodology proposed in the literature [10].The methodology defines the performance and response curve of the structure.The procedures have been explained below but for more details the reader may refer to relative literature [10].
In this analysis the central block of an ideal concrete gravity dam has been studied.The volume value is coherent with the mean volume value of the concrete gravity dam placed in Spain.These data are available in database [11,12].
Table 2 shows the data of the structure with the stiffness that is calculated by: 4  /   .From database [11,12], for concrete gravity dams with high > 61.0 m, the mean height and the mean base are 86.35m and 67.55 m, respectively.The ratio is 86.35/67.55= 1.28.The trigonometric function in the time-domain t to define the Cumulative Inelastic Area (CIA) of the performance curve is: where T1 is the structure's fundamental period, Cu and Cy are the ultimate and yielding crest displacement capacities, respectively.
The procedure to define the response curve is divided into: (i) estimation of Cu and Cy; (ii) definition of T1; (iii) quantification of the number of Demand Capacity Ratio (DCR) to carry out the analysis step-by-step; (iv) calculation of the CIA of the performance curve; (v) drawing of the performance curves.The area is between (Cu/Cy)i of step i and (Cu/Cy)max.In this analysis seven points (i = 7) have been used.
To define the response curve the process is similar to the employed process to calculate performance curve.However, it is not the same because, due to the great irregularity of the earthquake accelerations, CIA is complicated to calculate.The following approximated equation has been used by authors to solve this inconvenient: CIA = np(2 x 0.05)/(2 x 3) = 0.1(np/6).
The procedure to define the performance curve is: (i) defining the Cy; (ii) individualizing the sum of the elastic displacement peaks np that exceed Cyi; (iii) multiplying np by two times the step time of 0.05 s.
To eliminate the descending branches of the time-history and to consider only the positive peak times, a division by two and three, respectively, has been made.The displacement is between Cyi of step i and Cy,max.Seven points (i = 7) have been used.DCR is between 1.0 and the maximum elastic displacement divided by Cy. Figure 2 shows the elastic displacements used to obtain the response curve and the sinusoidal function CDH defined by Eq. 4 to obtain the performance curve.Cy is fixed, whereas Cu/Cy ratio change.Considering Cy = 1.5 cm, the Cu ranges 2.25 to 6.0 cm. Figure 3 shows the results of the analysis through the response curves (solid and dashed lines) and response curves (dotted lines) for CIA vs. DCR.
The Damage Index (DI) is defined by: min{1, Ar/Ap}, where Ap is the area under the performance curve (the central dotted lines in the graphics in Figure 3) and Ar is the area under the response curve.For DI < 0.5 the dam suffers minor damage, for 0.5 < DI < 1.0 the  3 shows the DI obtained in the analysis.For a3(t) the dam central block does not suffer damage because the accelerations are too low to excite the rigid structure.Damages usually occur in the base and in the crown of the dam, and in the points where the slope changes.For severe damage, the cracks propagate along the whole upstream and downstream face.

Conclusions
This paper describes two methodologies related to the definition of artificial acceleration by using the model of PSD function and to the estimation of the damage of concrete gravity dams.

Table 2 .
://doi.org/10.1051/matecconf/201821114001VETOMAC XIV Consistently with a considered ideal volume of 2400.0 m 3 , the dam could have a height of 78.0 m and a base of 61.0 m.Data of the central block of the concrete gravity dam.

Fig. 3 .
Fig. 3. Results of the analysis of the dam's central block.

Table 1 .
Data of the analysis.

Table 3 .
DI obtained in the analysis for Cu/Cy and for both a1(t) and a2(t).