Abstract
This study proposes a method for evaluating crack variations on structural performance of concrete dams using observational and computational output. The structural performance is represented by the system entropy in concrete dams, and the stress-strain field is established by incorporating observations and simulations for entropy calculations. Considering the characterization ability of the sample points, the weighted entropy sequence is constructed with a selected index system. Thereafter, the weighted fusion entropy is successively separated and analyzed with rescaled range (R/S) analysis, and the Hurst exponent difference and corresponding warning indicators are calculated to quantify the destructiveness of the crack variation. The application of the presented method is elaborated in a concrete arch dam, and the behavior of J01-crack exhibits no significant destructiveness on its structural condition. The analysis results verified the feasibility of the proposed method in destructiveness evaluation of crack behavior on the structural performance of concrete dams.
1. Introduction
Abundant hydropower resources provide clean and renewable energy for economic and social development; concrete dams play a significant role in the exploitation and utilization of hydroelectricity. Their failure poses a significant potential risk to the life and property of the lower reach [1–3]. Therefore, immediate and constant attention should be paid to the structural performance and safety conditions of concrete dams to avoid deteriorations [4–6]. The behavior variation of structural cracks can frequently result in the irreversible degradation of their macroscopic performance, which is one of the most common threats in concrete dams [7–9]. Consequently, it is necessary to investigate and propose reasonable evaluation methods for the destructive effect of crack variation on the structural performance of concrete dams, which is essential for forming a systematic mechanism of health monitoring and reliability guarantee for concrete dams involving fractures.
Numerical simulation [10–13] and observational investigation [14–17] are the two most representative methods for analyzing the macroscopic performance of concrete dams involving fractures, considering the dimensional inconsistency between laboratory experiments and practical engineering. With the incorporation of extended finite element method (XFEM) [18–20], virtual crack closure technique (VCCT) [21–23], and cohesive interface elements [24, 25], a series of computational methods have been proposed to reveal the service conditions of concrete dams in jeopardy cracking. Wang et al. [26] combined the XFEM and the finite volume method (FVM) to simulate the hydraulic fracturing (HF) process in high concrete dams, and engineering application verifies that the hybrid approach is an efficient tool for HF simulation in concrete dam. Lotfi and Espandar [27] proposed an original finite element (FE) program based on combined discrete and nonorthogonal smeared crack techniques, and the nonlinear dynamic behavior of a typical thin arch dam was investigated to validate the accuracy and feasibility of the developed model. Qin et al. [28] incorporated techniques including batch embedding of cohesive elements and parameter randomization of material properties and proposed a systematic approach with discrete-continuum mesh scheme for the numerical modelling of fracturing behavior and macroscopic performance in concrete dams. Though numerical simulation realizes the multiscale analysis of fracturing and degrading process in concrete dams, it is still a hypothetical characterization of its structural behavior, ignoring the environmental stochasticity and structural complexity.
As another powerful technique for structural analysis, field observation is the most direct indicator of behavior variation in concrete dams for its authenticity. Numerous studies have been conducted with motoring data for the structural control of concrete dams. Wei et al. [29] established a spatiotemporal hybrid model for the deformation monitoring of concrete arch dams, with analyzing and predicting the chaotic effect of residual series by a support vector machine optimized by a particle swarm optimization algorithm, and the engineering application testifies to the fitting and prediction accuracy of the proposed model. Hu and Wu [30] investigated the influence of a large-scale horizontal crack on the deformation behavior of a concrete dam using numerical analysis and established a statistical model of dam deformation considering the influence of crack, and the numerical results demonstrated that the crack variation and reinforcement measure exhibited a significant impact on the entire deformation condition. Ren et al. [31] presented an ensemble learning-based interval prediction model to quantify the uncertainty in the deformation behavior of concrete dams, and a gradient descent procedure was adopted to conduct parameter training. A case study confirmed that the proposed method provides high-quality prediction intervals of dam displacements, and it can be generalized for other types of structural behavior. However, the majority of previous literature focused on the structural analysis of concrete dams with monophyletic information, without thorough utilization and mutual certification of multisource information.
Based on the computational and observational output, continuous research on evaluation methods of fracturing behavior in concrete dams has been conducted. Gu et al. [32] proposed an evaluation method for abnormal fracture behavior with the fuzzy cross-correlation factor exponent in dynamics, and it can be applied directly with the monitored crack opening. Hu et al. [8] combined irrigation test, X-ray test, statistical model, and inversion analysis, to evaluate the influence of penetrating cracks on seepage behavior in concrete dams, which is a meaningful and practical attempt with both detection and observation. Dai et al. [33] proposed a genetic optimized online sequential extreme learning machine to fit and predict the observational crack behavior, and the BCa method is introduced to establish the confidence interval for abnormity diagnosis. Based on the aforementioned background, the existed research focused on the fracturing process and structural variation in concrete dams, while the relationship between them is still ambiguous. Considering that the fracture behavior produces little influence on the displacement behavior, the monitored stress-strain data of concrete dams are an intuitionistic description of their internal condition, but it is not a feasible approach to directly evaluate the influence of crack variation on concrete dams with field stress-strain observations. To integrate global observations, the service conditions of concrete dams are represented by structural energy. As important indicators of energy distribution and variation, dissipative structure theory and system entropy can be introduced to characterize and analyze the structural performance of concrete dams during fracture variation; they have already been extensively and successfully applied in the behavior analysis of civil engineering. Lőrincz et al. [34] summarized a large number of experimental results and engineering observations and established a three-level stability discriminant index system for bulk particle structures with the gradation entropy parameter, which can be applied to the failure analysis of earth-rock dams. It was found that bulk particle structure requires sufficient coarse particles to build a stable particle skeleton. Siqing [35] discussed that the evolution characteristics of dissipative structures, the nonlinear characteristics of the rock mass system, and entropy variation process during its collapse are studied from a multiscale perspective. The formation process and modeling method of dissipative structures are illustrated by considering landslides as an example.
In this study, the system entropy in concrete dams was selected to characterize the structural performance of concrete dams over fracture variation, which is an indicator of the energy distribution in concrete dams. To calculate the system entropy, the spatial stress-strain field of concrete dams is established with a combination of authenticity in field observations and sufficiency in computational output. To consider the difference in the representational ability of the sample points, the weighted entropy sequence is constructed with a specified index system and weighting method. Subsequently, a novel and cyclic utilization of R/S analysis [36–38] was performed to evaluate the influence of crack variation on the structural performance of concrete dams, and the Hurst exponent difference was calculated and analyzed to quantify the destructive effect. In summary, the proposed evaluation method for destructive effect caused by crack variation with weighted entropy for concrete dams is illustrated in Figure 1.
The remainder of this paper is organized as follows. Section 2 illustrates the process of energy transformation and entropy evolution in concrete dams with fracture propagation according to the uniaxial compression of concrete specimens, in preparation for the behavior analysis of concrete dams. The specific procedures of the characterization method for concrete dams with system entropy are elaborated in Section 3, based on the stress-strain field and weighted entropy established with observational and computational information. In Section 4, the impact of crack variation on the structural performance of concrete dams is evaluated with innovative application of R/S analysis, and the Hurst exponent difference and corresponding warning indicators are calculated for influence quantification. Finally, in Section 5, a concrete arch dam with fractures resulting from construction in China is selected for engineering application, and the feasibility and validity of the proposed method are verified through impact assessment of the selected fracture.
2. Energy Transformation during Fracture Propagation in Concrete Dams
The development of structural fractures in concrete dams is essentially a process that includes the initiation, propagation, and connection of internal microcracks, which demonstrates a significant deterioration in its structural performance. The deterioration of a concrete dam manifests as the variation of the stress-strain curve extrinsically, but a consequence from the transmission and dissipation of internal energy in concrete dam intrinsically. The energy transmission mechanism in concrete dams during fracture propagation was investigated using a typical uniaxial compressive experiment on a dam concrete specimen.
As illustrated in Figure 2, the fracture propagation and energy transmission in a dam concrete specimen under uniaxial compression can be divided into four stages:①When the applied load is less than 30% of the ultimate load, the internal microcracks of the concrete specimen are relatively stable, and the stress-strain curve is in a linear elastic relationship. Work done by the external force is mainly reserved as elastic strain energy in the concrete specimen, and no macroscopic irreversible processes occur within the concrete specimen, which is in global equilibrium.②When the applied load reaches more than 30% of the ultimate load, macro cracks in the interfacial transition zone expand gradually, new cracks are generated concurrently, a few cracks penetrate the cement mortar, and the stress-strain curve starts to exhibit nonlinear characteristics; a larger portion of work done by external force is still reserved as elastic strain energy, but the other part is dissipated by the slowly developed fracture process zone, and the concrete specimen progresses to a stationary state slightly deviating from the equilibrium state.③When the applied load increases to 70%–90% of the ultimate load, cracks in the cement mortar increase significantly and connect with the existing cracks in the interfacial transition zone; the stress-strain curve approaches a horizontal segment; the majority of the work done by external force is dissipated by the fracture process zone, the elastic strain energy stored in the concrete is released rapidly, and the concrete specimen develops away from the equilibrium state to a new stationary state.④When the applied load increases unceasingly, these continuous cracks rapidly develop into large penetrations in the concrete specimen; thus, it cannot absorb external energy and is confronted with complete failure with no bearing capacity.
For practical concrete dams, the degradation process over the entire life-cycle service of arch and gravity dams is presented in Figures 3(a) and 3(b). Their energy transmission process can be divided into the corresponding stages presented in Figure 3(c), and the portion of transferred energy can be positively characterized by the width of the connecting line. As shown in Figure 3, in stage ① with low external load, few cracks exist in concrete dams, and almost all the work done by the external load is stored as elastic strain energy in dam structure. In stage ② with increasing external load, the cracks in concrete dams witness stable propagation and augmentation, partial work done by external force is consumed in the fracture process zone, and this part increases with the applied load. In stage ③, with an external load reaching the ultimate loading capacity of the concrete dam, fracture destabilization and structural collapse occur in concrete dams, and the majority of the work done by external forces is dissipated in the fracture process zone, and the reserved strain energy is released.
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Based on the aforementioned analysis, the fracture propagation is concomitant with the process composed of reservation-dissipation-liberation of work done by an external load, and it is also closely related to the deterioration of structural performance and destabilization of systematic equilibrium in concrete dams. Considering the dynamic irreversibility and complex nonlinearity of energy transmission, the magnitude and distribution of system energy in concrete dams should be comprehensively analyzed to evaluate the influence of fracture behavior on the service performance of concrete dams.
The dissipative structure theory can be used to analyze the state evolution process of a complex system when it deviates from the equilibrium state. A concrete dam is a nonlinear dynamic open system, and its behavior variation process meets the application conditions of the dissipative structure theory. From the perspective of system energy, the initiation and expansion of cracks and exertion of their influence on concrete dams lead to a process of entropy reduction of the concrete dam system and development to the equilibrium state of minimum entropy generation. As an important indicator of a dissipative structural system, the entropy value of a concrete dam system is a measure of the uniformity of the energy distribution, which is an upward convex function with a single peak value. When there is no structural fracture in the concrete dam, the higher the uniformity of the energy distribution in the concrete dam, the greater the entropy value of the structural system. Furthermore, the higher the concentration of the energy distribution in the concrete dam, the lower the entropy value of the structural system when there is a structural crack with continuous expansion in a concrete dam. When the strain energies of each monitoring area are equal, the system entropy of the dam structure reaches its maximum value. Therefore, the system entropy value of a concrete dam can be selected as a representation of its structural performance, and the influence of crack behavior on the structural performance of the concrete dam and its evolution trend can be analyzed on this basis.
3. Characterization Method of Concrete Dam with Observational Entropy
3.1. Entropy Calculation of Concrete Dam
Based on numerical simulations, the system entropy of concrete dams can be easily calculated, because the stress-strain value at an arbitrary point is known in the computational output. The element strain energy density in the FE model can be calculated using the following equation:where is the element strain energy density, and and are the element stress and strain, respectively. The element strain energy can then be calculated using the following equation:where is the element strain energy, and is the element volume.
The total strain energy of concrete dams can be obtained by the accumulation of all element strain energies using the following equation:where N is the number of elements in the FE model of concrete dams.
Hence, the system entropy of a concrete dam based on the numerical results can be calculated using the following equation:
However, for practical concrete dam engineering, it is difficult to calculate the system entropy with high accuracy based on observational information because there are limited stress-strain monitoring instruments embedded in concrete dams, particularly for highly degraded dams after long-term operation. In view of the above problem, a method for establishing the stress-strain field is investigated in the following section, and a characterization method for the structural performance of concrete dams with measured entropy is proposed.
3.2. Establishment of Measured Stress-Strain Field
The limited measured stress-strain data in a concrete dam provide practical calibration for the establishment of its stress-strain field, whereas the computational results can generate complete stress-strain data to reveal its spatial distribution characteristics. To incorporate the authenticity of observational data and sufficiency of numerical results, the universal Kriging interpolation method [39–41] was selected to construct the measured stress-strain field of a concrete dam with these two types of source data.
The estimation of stress-strain data at a point in a concrete dam, , obtained from universal Kriging interpolation, can be represented by the following equation:where is the estimation of the stress-strain data at , is the drift term of the interpolation function in universal Kriging method, which represents the deterministic trend term of stress-strain field of this concrete dam, and its expected value is the semivariogram of the distance, that is, ; is the residual term of the interpolation function in universal Kriging method, and it denotes the stochastic residual term of the stress-strain field of the concrete dam, and its expected value is zero, that is, .
reflects the spatial distribution characteristics of the stress-strain field in this concrete dam, and the determination of its mathematical expression is usually complicated and can be treated using neighborhood model. In the neighborhood centered on in this concrete dam, can be expressed by the following equation:where is a known function related to the spatial position of , and are the undetermined coefficients. Consequently, is a linear combination of samples inside the neighborhood of a certain radius around .
Based on the above analysis, a numerical simulation of the structural condition of concrete dams should be performed firstly. According to the computational results of the stress-strain field of a concrete dam, the deterministic trend term can be estimated by linear fitting of equation (6) to describe the spatial surface reflecting the main characteristics of the stress-strain field in concrete dams. By eliminating of , the uncertain residual term, , can be interpolated by ordinary Kriging, which can be expressed as the following equation:where denotes the weight of the ith known monitoring point. When the estimated value of the residual term, , satisfies the optimal estimation, equation (8) can be obtained as follows:
By substituting into equation (8), it can be transformed into the following equation:where is the spatiotemporal variation function value of the ith and jth given stress-strain monitoring points, and is the spatiotemporal variation function value of the ith given and target stress-strain monitoring points.
Based on the given observational information of stress-strain data in concrete dams, all the existing monitoring points can be utilized to form arbitrary point pairs, and the estimation, , of the variation function, , can be calculated using the following equation:where is the spatial distance between the measured points, and are the stress-strain observations at point and point with a space interval , and is the total number of monitoring point pairs that meet spatial distance requirements.
Corresponding to different spatial distances, can be obtained by using equation (10), and the experimental variogram scattering plot can be drawn with as the horizontal axis and as the vertical axis. Based on the characteristics of the experimental variogram scattering plot, and can be determined by selecting the appropriate selection of variogram model.
The Lagrange multiplier method was introduced to calculate the weight, , by solving the following equations:
After the calculation of , the residual term of the stress-strain data can be estimated by substituting the weight into equation (7). The missing stress-strain data can be interpolated by substituting equations (6) and (7) into (5), and the measured stress-strain field of concrete dams can be constructed.
3.3. Entropy Calculation with Constructed Stress-Strain Field
Based on the constructed stress-strain field of a concrete dam, the stress-strain data of the target points can be interpolated. The strain energy in the monitoring area of each representative point can then be calculated using the following equation:where is the strain energy in the monitoring area of the ith point, and are the stress and strain in this area, respectively, and is the volume of the monitoring area. Then, the total strain energy of the concrete dams can be obtained by the addition of strain energy in all measuring area.
Considering the layout of monitoring instruments in practical engineering, it is difficult to perform integral calculations directly based on equation (12). For concrete dam engineering, the strain energy density can be calculated using the measured stress-strain value, and the entropy value can then be transformed. The strain energy density can be calculated using equation (13) based on the measured data:where , , and , , are the measured stress-strain on three spatial directions. The strain energy in each monitoring area is then calculated using the following equation:where is the assumed average strain energy density, and is the number of monitoring areas.
The key intermediate variable can then be obtained using the following equation:
The observational system entropy of a concrete dam can be calculated using the following equation:
3.4. Weighted Entropy of Concrete Dam
The conventional calculation method for the system entropy of concrete dams is investigated in Subsection 3.3, and however, only the monitoring area of the representative points is considered. For a practical concrete dam with fractures, the characterization ability of the structural performance and fracture influence varies for different representative points, which cannot be revealed in equation (16). Considering the variation in the characterization ability of different representative points, an index system is established to assign weights to each representative point, and the weighted entropy of the concrete dam can be defined using the following equation:where is the weight of each representative point in the concrete dams to reflect the characterization ability for the fracture influence and structural performance of concrete dams. The determination method for the point weight was studied as follows.
Herein, the spatial distance, concomitant variation, and sequential correlation indexes were selected to determine the point weight in weighted entropy, representing the structural performance of concrete dams.(1)Spatial distance index The closer the representative point is to the fracture tip region, the more significant the influence of fracture variation. Therefore, the reciprocal of the Euclidean distance between the representative point and crack tip (or a certain point in the region near the crack tip) is selected as the spatial distance indicator, which can be calculated using the following equation:where and are the coordinates of the representative point and the fracture tip, respectively.(2)Concomitant variation index When the fracture suffered unstable propagation, it led to variations in the observations at representative points. Therefore, the absolute changing value of the representative points induced by fracture propagation was calculated using equation (19) as the concomitant variation index:where and are the strain values of the ith representative point at the fracture unstable propagation and the previous moments, respectively.(3)Sequential correlation index The correlation coefficient between the measured value sequence of each representative point and the crack opening sequence indicates a direct representation of their interaction relationship, and hence, their Pearson correlation coefficient can be generated using the following equation as the correlation index:where and are the measured value sequence of each representative point and the crack opening sequence, respectively; is the covariance of and ; and are the variance of and , respectively.
The improved projection pursuit method, whose feasibility has been validated in the process of observational information in structural engineering, is adopted to decide the point weight with these three indicators, and the main procedures are as follows:①Index selection and normalization. evaluation samples of stress-strain representative points for concrete dams were selected to construct the evaluation index set, , as shown in the following equation: As mentioned above, the spatial distance, concomitant variation, and sequential correlation indices are selected for weight determination, and they should be normalized initially using the following equation:where and are the maximum and minimum value of the jth indicator; is the normalized jth indicator of the ith point.②Establishment of the projection index function. The normalized evaluation index set is integrated into the projection value, , with the unit-length vector, , as the projection direction: According to the overall distribution characteristics and local aggregation degree of the evaluation data, the projection index function can be expressed as the following equation:where and are the standard deviation and local density of , respectively, and can be calculated using the following equation:where is the mean of ; is the window radius of the local density, in this study; is the distance between the projection values and ; and is the unit step function, that is, when and when .③Optimization of projection direction. When the evaluation index set is given, the projection index function, is affected only by the projection direction. The optimal projection direction, , is estimated by maximizing the projection index function in this study, which is transformed into the following optimization problem:④Calculation of point weights. The projection value is calculated by substituting the optimal projection direction into equation (23), and the weight of each representative point in the concrete dams can be determined using the following equation:
4. Analysis on Structural Influence of Crack Behavior on Concrete Dam
For the structural system of an actual concrete dam, the weighted entropy sequence constructed in the previous section can be regarded as its characterization with a certain degree of freedom. For a concrete dam with solid integrity, the weighted entropy of its structural performance has a strong autocorrelation, and its dynamic characteristics have smaller fractal dimension. When this concrete dam operates with fractures that jeopardize structural integrity, the nonlinearity of its structural performance increases, the autocorrelation of the weighted entropy sequence decreases, and the fractal dimension of the concrete dam system increases. Based on these principles, the destructiveness of fracture variation on the structural performance of concrete dams can be evaluated by analyzing the fractal features of the weighted entropy sequence.
For a sequence formed by a group of independent random variables with a mean value of zero and a variance of one, its variation range within a certain time span, , is linearly related to , and the sequence increments are independent of each other. For the system entropy sequence of a concrete dam, its variation range in time span is not directly proportional to , which reveals that the sequential value of the concrete dam system entropy is not independent, but interactional, that is, its autocorrelation coefficient is not zero, and this kind of sequence has the feature of long-term autocorrelation. Therefore, it is assumed that the time series formed by the system entropy of a concrete dam is a generalized stationary random process, and the Hurst exponent is highly significant for a further understanding of the autocorrelation and persistence of the weighted entropy sequence in concrete dams, and the variation in its structural performance can be dynamically evaluated. In this study, the R/S method is adopted to solve the Hurst exponent of the weighted entropy sequence, and the calculation process was elaborated as follows.
For a weighted entropy sequence with length of , , the mean value sequence is defined with equation (28) for any positive integer, :
The cumulative deviation of this weighted entropy sequence, , can be calculated using the following equation:
The range, , and standard deviation, , of this weighted entropy sequence can be acquired using the following equations:
The statistical analysis of confirms that is subject to the relationship presented in the following equation:where is a constant, and is the Hurst exponent. The Hurst exponent in equation (32) is an indicator for measuring the long-term correlation and trend strength of weighted entropy sequences in concrete dams, and its value range is [0, 1]. In detail, when , the entropy sequence is independent of each other at all scales and is a standard random process; when , the entropy sequence is completely positively correlated and belongs to the deterministic system; when , the entropy sequence is a biased random process with positive persistence and trend enhancement; when , the entropy sequence is antipersistent and inversely correlated at all time scales. Furthermore, when approaches 0.5, the randomness of the entropy sequence becomes stronger, and when approximates 1 (or 0), the positive (or negative) trend of the entropy sequence becomes more significant.
Thereafter, equation (32) can be transformed into equation (33) with the logarithm operation:
On this basis, a log-log scattering plot is drawn with and as the independent and dependent variables, respectively, and the least square estimation is utilized to conduct linear fitting; the slope and intercept of the straight line are the value of and , respectively.
For the weighted entropy sequence of a concrete dam during operation with cracks, a series of demarcation points, including the time point of crack unstable variation, can be taken to separate the entropy sequence into two subsequences, and R/S analysis is performed on the two subsequences to obtain their Hurst exponent difference, and it is a significant indicator for the devastating effect of fracture variation on the structural performance of concrete dam. The detailed procedures for the calculation and establishment of the corresponding evaluation index are illustrated in Figure 4 and are as follows.①For the weighted entropy sequence of a concrete dam with cracks, , a positive integer, , is considered, and is regarded as a series of demarcation points of the weighted entropy sequence. The time points of crack instable variation, which can be determined using catastrophe theory or phase-space reconstruction, are included in this series.②For the demarcation point , R/S analysis is performed on the subseries of the weighted entropy sequence divided by it, and the Hurst exponent is denoted as and ; hence, their difference can be calculated using the following equation:③Based on the probability distribution function (PDF) of , the warning indicator can be determined given certain significance level . When , the Hurst exponent difference of the system entropy in concrete dams, exceeds , the structural performance of the dam is in a warning situation. Therefore, forewarning measures must be adopted against probable danger. The corresponding exceedance probability and warning indicator can be derived with the following formula:where is the PDF of the Hurst exponent difference .
5. Engineering Application
5.1. Project Profile
A key cascade hydropower station project is located in southwest China, which has comprehensive utilization benefits of power generation, flood control, agricultural irrigation, and sand containment. This hydropower station comprises a concrete double-curvature arch dam, plunge pool, spillway tunnel at the left bank, and underground water diversion power generation system at the right bank. The concrete arch dam is constituted of 43 dam sections, and the width of flood-discharging dam sections is 22∼26 m; its crest elevation and maximum dam height are 1245.5 and 294.5 m, respectively, and the arc length of the center line in the dam top is 892.8 m. The geographic location and downstream view of the arch dam are shown in Figure 5.
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According to the field investigation, there are dozens of cracks in several dam sections during the construction process. The main reason for the initiation and propagation of these cracks in this arch dam has been confirmed to be the high tensile stress induced by the temperature drop during the second stage of the postcooling process before joint grouting. After three stages of crack inspection, the distribution range of these cracks is identified: all the cracks appear in a total of 18 dam sections numbered 13#∼30#, and only sporadic cracks were found in dam section of 31# and 32#; the lowest and highest elevations of these cracks are 972 and 1127 m, respectively, and their maximum length is 145 m; the bottom edge of these cracks does not extend to the foundation or the induced crack in dam heel, and all cracks end below the anticracking reinforcement. The distribution of these cracks in the global dam body, typical elevation, and dam section [42] is shown in Figure 6, and the J01-crack is located in Block 15# and the elevation of 1013 m∼1112 m. Based on the observations of the crack opening and stress-strain data of this concrete arch dam, the detrimental effect induced by the behavior variation of J01-crack on the structural performance of this arch dam was analyzed using the method proposed in this study.
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5.2. Establishment of Measured Dam Stress-Stain Field
To establish the dam stress-stain field with field observation and numerical simulation, the monitored stress-strain data sequences of gauge points in this arch dam were collected and transformed initially, and a numerical model of this arch dam was constructed, as shown in Figure 7. The entire model had a total of 821914 nodes, 779914 elements, and 7305 folium crack elements. The arch dam, consisting of 19559 elements and 24889 nodes, is composed of 43 dam sections. The coordinate system of the model is selected as follows: the X-axis is perpendicular to the center line of the arch dam, and the positive direction points to the left bank (NE88°); the Y-axis is parallel to the center line of the arch dam, and the positive direction points upstream (SE178°); the Z-axis is positive in the vertical direction.
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Note that the influence of aging deformation should be considered in the numerical analysis of concrete dams, during the construction of measured stress-strain field. Therefore, a viscous-elastic-plastic constitutive model should be adopted to describe the mechanical properties of this arch dam, which is complicated and time consuming. For concrete dams, the aging component is relatively small compared with the hydraulic and seasonal components. Consequently, an elastic constitutive model can be used for the numerical computation of stress-strain data in concrete dams, and the calculated spatial characteristics satisfy the accuracy requirement. On this basis, the dam stress-strain field can be constructed using computational output with acceptable accuracy and promoted efficiency. The material parameters required for the numerical analysis are presented in Table 1.
Based on the impounding schedule, the upstream water level conditions were selected as 1050, 1090, 1130, 1160, 1190, 1210, 1230, 1240, and 1243 m. The annual average temperature of this arch dam is 19.1°C, the annual temperature variation range is 5.6°C, the annual average temperature of the reservoir water surface is 20.6°C, and the water temperature below the constant temperature layer is 10.0°C. The high and low temperatures of this arch dam under the design condition can be calculated using calculation method of the arch dam temperature load, as presented in Table 2, and the measured temperature distribution can then be determined.
Subsequently, the stress distribution in this arch dam was calculated under the aforementioned working conditions to analyze the spatial regularities of stress-strain field and determine the drift term in the interpolation function of universal Kriging method. The numerical results of the stress distribution in this arch dam at a water level of 1090 m are presented in Figure 8.
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According to the numerical results of the stress distribution in this arch dam, the first principal stress on the upstream surface of the dam body includes primarily compressive or small tensile stress, generally less than 0.5 MPa; the lower the elevation, the greater the compressive stress of the dam body; the local maximum principal stress at high elevation is tensile stress under a lower water level, and the distribution range of tensile stress decreases gradually at high elevations of the upstream surface with the rising water level. The third principal stress on the upstream surface is primarily compressive stress, the closer it is to the dam section of the riverbed, the greater the compressive stress, and the third principal stress increases with lower elevation and water level. The first principal stress on the downstream surface of the dam body constitutes a small value, and the stress distribution is relatively uniform. The first principal stress is in a state of slight tension near the foundation surface and compression in the middle part on the downstream side under a lower water level, and it changes to a state of compression near the foundation surface and slight tension in the middle part on the downstream side. The third principal stress on the downstream surface is compressive; its maximum value appears in the middle of the downstream surface, and it gradually decreases from the center to the surrounding and has a certain tendency to increase near the foundation surface. The distribution law of the third principal stress on the downstream surface gradually changes to increasing compression from the dam top to the toe.
The residual term in the interpolation function can then be calculated using the difference between the measured and calculated stresses. Consequently, the measured dam stress-stain field can be generated after the determination of interpolation function in universal Kriging method.
5.3. Calculation of Weighted Entropy in Dam System
To calculate the weighted entropy in this arch dam system, stress-strain data at seven monitoring points and seven interpolating points were collected with the established measured stress-strain field. Thereafter, the sequence of the strain energy density and intermediate variable, λlnλ, can be generated, as plotted in Figure 9.
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According to the observational crack opening of the J01-crack and stress-strain data of monitoring points, YB-S9-01-YB-S9-07, and interpolating points, N-S9-01-N-S9-07, the index system including the spatial distance index, concomitant variation index, and sequential correlation index can be obtained, respectively. On this basis, the weights of each measuring and interpolating points in this concrete arch dam were determined using the improved projection pursuit method, as shown in Table 3, and the weighted entropy sequence characterizing the structural performance of this arch dam was calculated, as plotted in Figure 10. As it can be observed, the weighted entropy sequence reveals the variation in the service condition of this concrete arch dam, and it is relatively stable in the target period. The greatest range is 0.0196, which can be regarded as the extreme fluctuation in the structural performance of this arch dam.
5.4. Impact Assessment of Fracture Variation on Dam Structural Performance
Then, the existence and time point for the unstable variation of J01-crack are determined with the method incorporating wavelet multiresolution and Catastrophe theory, which is illustrated in reference [43]. The sequence of crack opening between 2010-10-01 and 2011-07-27 was selected for wavelet multiresolution analysis to obtain the aging component, and the Daubechies-4 function in wavelet multiresolution was utilized to perform four-scaledecomposition on the monitoring data. The components in the crack-opening change with environmental variables such as water level and air temperature have corresponding periodicity, and the components affected by random factors and observation error also exhibit the fluctuation of the short cycle; hence, they all belong to signal of a high frequency. However, the aging components in fracture behavior vary infrequently and are represented as low-frequency signal. Furthermore, the Catastrophe theory is introduced to determine the time points of unstable variation in crack behavior, and the calculated discriminative index in the Catastrophe theory and decomposed components with wavelet multiresolution are presented in Figure 11.
It can be observed that the components of high frequency include small values, demonstrating that the behavior of J01-crack is slightly influenced by environmental factors, and the crack opening is dominated by aging components. Based on the calculated decisive index, three moments including 2010/11/26, 2010/12/24, and 2011/5/26 are recognized with unstable propagation to evaluate the influence on the structural performance of the arch dam.
Based on the weighted entropy sequence of this concrete arch dam established in the previous section, 300 monitoring time points in the target phase were considered as the cut-off points, and R/S analysis was performed on the two subsequences divided by the cut-off points separately to solve the Hurst index difference sequence in this period, as shown in Figure 12.
The sequence of Hurst indexes difference was testified to obey Gaussian distribution, and the warning indicators were determined with significance level of 0.05 and 0.95. The Hurst exponent variation at these three points and the warning indicators are presented in Table 4. As can be observed from Table 4, the absolute value of Hurst exponent differences at 2010-11-26 and 2011-05-26 of the unstable variation in J01-crack is relatively small, while it reaches 0.0324 at 2010-12-21, indicating that the unstable expansion of J01-crack at this time point may lead to changes in the long-term correlation and dynamic characteristics of the weighted entropy sequences, but it is still within the scope of the normal condition, which is decided by typical low probability method. Therefore, the behavior of J01-crack exerts no significant destructiveness on the structural performance of this concrete arch dam.
6. Conclusion
This study establishes a systematic approach for the destructiveness evaluation of crack variation in structural performance of concrete dams using numerical simulation and observational analysis. The characteristics of energy transformation and evolution in concrete dams over the propagation of cracks are discussed initially, with an analogy between experimental specimens and practical engineering. Thereafter, a characterization method for the structural performance of concrete dams with system entropy is elaborated, and the stress-strain field in concrete dams is established using the universal Kriging interpolation method, based on the combination of authenticity in field observations and sufficiency in computational output. Furthermore, the weighted entropy sequence of concrete dams is calculated, to consider the characterization ability of various monitoring points, and their weights are determined by an improved projection pursuit method with an index system that includes the spatial distance, concomitant variation, and sequential correlation indexes.
To calculate the variation in the structural characteristic of concrete dams with cracks, the weighted entropy sequence was divided by time points in the study phase, and the Hurst exponent of the subsequences was calculated to determine its tendency. In addition, the Hurst exponent difference and corresponding warning indicators were calculated to quantify the destructiveness caused by crack variation. The feasibility of the proposed method was validated with application in a concrete arch dam, which is in operation with fractures induced by thermal tension during construction, and it was confirmed that the behavior of J01-crack has not jeopardized the structural performance of this concrete arch dam during the target period, which confirms the feasibility of the proposed method in evaluating the destructiveness of fracturing behavior in concrete dams with a limited number of strain gauges.
Furthermore, the main limitation of the proposed method is that the precision of the stress-strain field in concrete dams is drastically dependent on the sufficiency of observational information, which is especially difficult for old and damaged concrete dams. In addition, other advanced numerical techniques can also be utilized in the establishment of stress-strain field; however, the balance between efficiency and accuracy should be considered.
Data Availability
The data on cracks used to support the findings of this study are available from the corresponding author upon request. The other data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study is financially supported by National Key R&D Program of China (2022YFC3004405), National Natural Science Foundation of China (52209170, U2040224, 52079120), Key R&D Program of Henan Province (221111321100), Project funded by China Postdoctoral Science Foundation (2021M702949), the Belt and Road Special Foundation of the State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering (2021492111), Key Scientific Research Project of Colleges and Universities in Henan Province (23A570001).