The flexibility coefficient is popularly used to implement the macroevaluation of shape, safety, and economy for arch dam. However, the description of flexibility coefficient has not drawn a widely consensus all the time. Based on a large number of relative instance data, the relationship between influencing factor and flexibility coefficient is analyzed by means of partial least-squares regression. The partial least-squares regression equation of flexibility coefficient in certain height range between 30 m and 70 m is established. Regressive precision and equation stability are further investigated. The analytical model of statistical flexibility coefficient is provided. The flexibility coefficient criterion is determined preliminarily to evaluate the shape of low- and medium-sized arch dam. A case study is finally presented to illustrate the potential engineering application. According to the analysis result of partial least-squares regression, it is shown that there is strong relationship between flexibility coefficient and average thickness of dam, thickness-height ratio of crown cantilever, arc height ratio, and dam height, but the effect of rise-span ratio is little relatively. The considered factors in the proposed model are more comprehensive, and the applied scope is clearer than that of the traditional calculation methods. It is more suitable for the analogy analysis in engineering design and the safety evaluation for arch dam.
As a superior type, arch dam has been extensively used in dam construction. But its design and calculation methods are more complex than that of earth dam and gravity dam. There are the following problems. First of all, to implement the comparative analysis for different design schemes of arch dam, some shape data are lack of reference. Secondly, it is difficult to estimate the earthwork volume index of dam body which is used to determine the dam shape and assess the economy. The problem has an impact on selection of dam site and determination of project scale during engineering preplanning. With the help of flexibility coefficients, macroevaluation of arch dam’s shape, security, and economy has recently become important research topic in the field of dam.
Lombardi [
On the whole, the existing definition and calculation method on the flexibility coefficient are accuracy and concision. They can embody the flexibility degree of arch dam at the horizontal direction. However, there are some questions to analyze and perfect. For example, the differences of flexibility coefficient between various canyon shapes are great which also have not some certain roles. In the condition of similar shape and height, a large difference in flexibility coefficient will affect engineering analogy analysis and arch shape design. Sometimes safety degree of the arch dam is unconscionable to reflect through Lombardi damage line building by flexibility coefficient.
Based on above problems in existing research, a large number of statistical data on low- and medium-sized arch dams are collected and implemented the regression analysis. The partial least-squares regression method is used to analyze the statistical data of the related factors on flexibility coefficient. The calculation model of flexibility coefficient is built. The statistical flexibility coefficient is proposed.
As a commonly multivariate statistical analysis method, PLSR (partial least-squares regression) combines the basic functions in multiple linear regression analysis, principal component analysis, and typical correlation analysis. It can be used to solve effectively the multicollinearity between the independent variables. After a partial least-squares regression analysis, the regression model between independent variable and dependent variable can be not only obtained but also the correlation between variables can be analyzed. It makes the analysis more richer and makes the interpretation of the regression model deeper.
A multiple linear regression model can be described as follows:
The least-square estimation of regression coefficient vector
Partial least-squares regression extracts the principal component
Assumed that dependent variable is
The data of
All the
Adopting all sample points, regression model is established fetching
For the principal component
A great amount of research indicates that when
In the partial least-squares regression, the principal component
The explanatory capacity of
The explanatory capacity of
The cumulate explanatory capacity of
The explanatory capacity of
The cumulate explanatory capacity of
In order to analyze the relationship between independent variable
The explanatory capacity can be measured by variable importance in the projection
It can be seen from the partial least-square principle that interpretation of
In addition, the square sum
According to the definition and existing research results of flexibility coefficient, dependent variable factor set of flexibility coefficient is selected as follows: Dam height is between 30 m–70 m. According to the design code of concrete arch dam (SL282-2003), arch dam thickness is divided. The ratio of above arch dams is thin arch dam : medium arch dam : thick arch dam = 38% : 61% : 1%. According to arch ring type, various types of arch dams above in the ratio are parabolic variable-thickness double-curved arch : double-curvature constant thickness arch dam with single-centered arc : others (such as mixture-type arch dam and circular variable thickness arch dam with five-centered arc) = 68% : 9% : 23%. Dams of discharging through crest orifice, which account for 66% of the total, are usually used.
The actual project data of dependent variables for flexibility coefficient.
According to (
Through calculating and analyzing, the PLSR equations of standardized data and raw data are (
Multiple correlation coefficients
According to ( As can be seen from Table As can be seen from Table The explaining capacity of
The explanatory capacity of main components to dependent variables and independent variables.
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0.631 | 0.223 | 0.308 | 0.226 | 0.400 | 0.001 | 0.181 |
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0.258 | 0.029 | 0.546 | 0.729 | 0.248 | 0.012 | 0.230 |
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0.038 | 0.004 | 0.117 | 0.022 | 0.303 | 0.869 | 0.447 |
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0.012 | 0.165 | 0.003 | 0.000 | 0.005 | 0.032 | 0.103 |
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0.156 | 0.061 | 0.190 | 0.191 | 0.276 | 0.488 | |
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0.230 | 0.000 | 0.058 | 0.767 | 0.677 | 0.204 | |
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0.441 | 0.003 | 0.011 | 0.011 | 0.000 | 0.119 | |
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0.152 | 0.293 | 0.138 | 0.000 | 0.004 | 0.066 |
The accumulative explanatory capacity of main components to dependent variables and independent variables.
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The explanatory capacity on |
0.237 | 0.315 | 0.189 | 0.076 |
The accumulative explanatory capacity on |
0.237 | 0.552 | 0.741 | 0.817 |
The accumulative explanatory capacity on |
0.488 | 0.692 | 0.811 | 0.877 |
Based on the above analysis, the data have relatively good linear trend and the PLSR equation has high precision. They can well reflect the average law between
According to ( The VIP values of The VIP value of
Variable importance in the projection.
From the dam structure, clearly, the average dam thickness and dam height have a great influence on the flexibility; the thickness-height ratio and the arc length-height ratio of crown cantilever, which reflect the thickness of arch dam and valley shape, have a major impact on the shape of arch dam, then affect the flexibility coefficient. The ratio of arc and chord of the dam crest is
According to the complexity of flexibility coefficient factors and the requirement of sample data, the method of stability in this paper is that after extracting a certain amount of the date, build the model by remaining data, make a coefficient compared, and judge stability of the equation. The specific implementation is to remove five sample points by three times and build the model with the remaining data.
After removing the extracted sample points and judging the main ingredients number of remaining data, the PLSR model can be made. In order to compare easily, the coefficient, result of regression model of the standardized data, can be used to be compared. The calculated specific factors are shown in Table
The coefficients of regression equation based on original data and extracted sample data.
Variable | Group | |||
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Original coefficient | Coefficient of sample 1 | Coefficient of sample 2 | Coefficient of sample 3 | |
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−0.0831 | −0.0887 | −0.0976 | −0.1084 |
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0.0287 | 0.0352 | 0.0212 | 0.0271 |
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−0.1681 | −0.1656 | −0.1690 | −0.1518 |
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0.2603 | 0.2560 | 0.2503 | 0.2426 |
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−0.7530 | −0.7343 | −0.7346 | −0.7157 |
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−0.5071 | −0.5159 | −0.5185 | −0.4716 |
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0.3780 | 0.3668 | 0.3704 | 0.3498 |
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0.4155 | 0.4030 | 0.4067 | 0.3851 |
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−0.1859 | −0.2082 | −0.1933 | −0.1900 |
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0.2196 | 0.2250 | 0.2043 | 0.2144 |
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0.2799 | 0.2771 | 0.2730 | 0.2672 |
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0.1776 | 0.1726 | 0.1684 | 0.1730 |
The above regression analysis is implemented to obtain the PLSR equation of flexibility coefficient in certain dam height range between 30 m and 70 m. From stability and regression accuracy, it can be seen that the PLSR equation is rational. Accordingly, calculation model of the flexibility coefficient
From the analysis of interpretation, it can be seen that explaining function of the average thickness of dam to the flexibility coefficient is strongest. According to [
Scatter diagrams of flexibility coefficient and average thickness.
Based on the above evaluation criteria and the calculation model (
The arch dam project began in 1974 and basically completed in 1979. The dam is a concrete double-curvature masonry arch dam, whose total storage capacity is 120.5 ten thousand m3, crest elevation is 121.0 m, bottom elevation is 86.0 m, the maximum dam height is 35 m, thickness of the dam crest is 2 m, thickness of the dam bottom is 7 m, thickness-height ratio is 0.2, chord length of dam crest is 128.2 m, width-height ratio is 3.66, central angle of dam crest is 120°, and central angle of dam bottom is 60°. At the corresponding dam height of 4 m, setting a horizontal fracture, cutting beam-based, and using bridge-type rubber to stop water are to be done. The spill way whose net width is 30 m is arranged on the dam crest. Curved form of free jump is used to overflow.
The shape calculation of above arch dam is implemented. The results are shown in Table
The calculated results of statistical factors for one arch dam.
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34 | 12.1 | 13768 | 3824.11 | 3.6 | 0.206 |
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5.129 | 3.965 | 1.336 | 0.263 | 3607.94 | 48453.13 |
The values of factors in Table
Dam stress is calculated and analyzed to check for the rationality of the above results.
The arch ring is a circular and single-centered ring with constant thickness. The dam is divided into 6 arches and 13 beams. Three of arches are located in the river bed. Analysis planar graph can be seen from Figure
Arch dam plan.
(2) Characteristic elevation and water level are shown in Table
Characteristic elevation and water level (m).
Maximum dam height | 35.0 | Check water level | 121.2 |
Crest elevation | 121.0 | Normal water level | 119.5 |
Bottom elevation | 86.0 | Lowest operating water level | 90.2 |
Sediment elevation | 95 | Design water level | 120.95 |
(3) Physical and mechanical parameters are given in Table
Physical and mechanical parameters of dam body and foundation.
Dam foundation | Dam body | |
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Deformation modulus | 7.02 × 106 MPa | 107 MPa |
Poisson’s ratio | 0.28 | 0.225 |
Density | 2.4 t/m3 | 2.3 t/m3 |
Concrete thermal diffusivity | / | 3.0 m2/month |
Coefficient of thermal expansion | / | 0.000008/ |
The temperature considering perennial mean temperature and sunshine effects is 16°C; the temperature considering annual temperature amplitude (temperature rise) and sunshine effects is 11.7°C; the temperature considering surface temperature of reservoir water and sunshine effects is 16°C; the temperature considering surface temperature of reservoir water (temperature drop) and sunshine effects is 11°C; the temperature considering surface temperature of reservoir water (temperature rise) and sunshine effects is 10.6°C; the water temperature of reservoir bottom is 11°C.
The following six operational conditions are selected.
Normal water level, sediment, and dead weight.
Check water level, sediment, dead weight, and temperature rise.
Lowest operating water level, sediment, dead weight, and temperature drop.
Lowest operating water level, sediment, dead weight, and temperature drop.
Normal water level, sediment, dead weight, and temperature rise.
Design water level, sediment, dead weight, and temperature rise.
Maximum dam surface stress of every condition can be seen in Table
Maximum dam surface stress of every condition (MPa).
Case | Upstream surface | Downstream surface | ||||||
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Maximum principal tensile stress | Position | Maximum principal pressure stress | Position | Maximum principal pressure stress | Position | Maximum principal pressure stress | Position | |
Case |
2.54 | 7R 0C | 1.88 | 3R 0C | 0.69 | 3R 0C | 3.22 | 7R 0C |
Case |
1.66 | 7R 0C | 2.00 | 1R-7C | 0.74 | 3R-5C | 2.49 | 7R 0C |
Case |
0.66 | 1R-7C | 2.24 | 6R-2C | 1.42 | 6R-2C | 1.09 | 7R 0C |
Case |
0.42 | 4R 0C | 3.70 | 6R-2C | 3.11 | 6R-2C | 1.28 | 1R-7C |
Case |
1.34 | 7R 0C | 1.77 | 1R-7C | 1.16 | 3R-5C | 2.19 | 7R 0C |
Case |
1.60 | 7R 0C | 1.96 | 1R-7C | 0.82 | 3R-5C | 2.44 | 7R 0C |
For low and medium arch dam dominated by tensile stress, contour map of principal tensile stress is only given (see Figure
Contour map of principal tensile stress on the dam surface.
The following can be seen from stress distributions. For the first, second, and sixth conditions, the upstream tensile stress exceeds the allowed value. The stress value is largest at the first condition, which is at the normal water level, and is distributed approximately in the whole river bed of the upstream dam bottom. The downstream tensile stresses of three conditions meet the standard value. For the third and fourth conditions, the upstream tensile stresses meet the standard value. But the downstream tensile stresses at the middle-lower part of the abutment exceed the standard value. The maximum tensile stress occurred at the fourth condition, up to 3.11 MPa, which is far more than the norms. For the fifth condition, the upstream tensile stress at the bottom of dam exceeds the standard value, and the downstream tensile stress at the abutment exceeds the standard value.
In addition, the maximum radial displacement to the downstream is 17.2 mm, to the upstream is 8.47 mm, which is a little high for arch dams whose heights are 30 m–35 m. That shows that the overall stiffness of the arch dam is not high and the ability of deformation resistance is finite.
From the above analysis, they can be seen that stress distributions of this arch dam are not good, and the radial displacement is relatively large. It is reason that there are unreasonable shape design, relatively small average thickness of arch dam, and relatively small overall stiffness. They also have been proved from comparisons of arch dams that are the same height range to the case.
From the stress analysis, it can be known that the dam is relatively thin and lower structure safety keeping in step with the introduced model. In addition, during dam safety evaluation, experts also think that the dam is a little thin and has a limited overload capacity, which showed that it is feasible to evaluate arch dam safety with the introduced model once more.
In recent years, flexibility coefficient, which is an objective index, is put forward to deal with problems, such as much subjective evaluation to the body safety of arch dam and lack of criteria of determining shape parameters of shape design. Flexibility coefficient has a unique advantage on the macroevaluation of arch dam shape, safety, and economy. According to large numbers of projects data, statistic rules of flexibility coefficient of arch dam are studied from the perspective of regression. The regressive equation of flexibility coefficient in certain height range, which is based on partial least-squares method, is established. Further, regressive precision and equation stability is analyzed deeply. And the calculation model of statistical flexibility coefficient is presented. A case application shows that the model has certain application value. After analyzing explanatory capacity of factors to dependent variable, the result shows that average thickness of dam, thickness-height ratio of crown cantilever, arc-height ratio, and dam height have the higher explanation ability than others. The relation between them should be focused mainly when calculating flexibility coefficient. Compared to traditional methods calculate of flexibility coefficient, the model in this paper has a comprehensive consideration, such as the valley shape coefficient that reflects the valley shape, thickness-height ratio that reflects arch dam thickness and thinness, and temperature-lowering load that has an important influence to arch dam stress, and the force of water and areas of upstream face in normal water level in the dam working. There is a wider application in calculating dam volume inversely. Traditional models do not distinguish dam height. But the rationality is worth to be discussed. It is because that the low- and medium-sized and the high-sized arch dam have different stress conditions and methods. While statistical flexibility coefficient presented by this paper has a clear operating range, it is more suitable to analogy analysis and study on variation of flexibility coefficient. Because the designing method of arch dam is more complex than gravity dam and earth dam. For specific valley conditions, project quantity can not be estimated quickly, which bring many difficulties to choose dam site and determine engineer scales during the preliminary planning. Now, we can apply calculation model introduced by this paper to select the specific flexibility coefficient. In coupled with the fitting values of other factors, the volume of dam body is inversely calculated. Choosing dam site and determine engineer scales preliminarily also provides certain references to shape data estimation. For the arch dam shape which is designing and optimizing, after calculating shape data and values of corresponding factors, its flexibility coefficient can be gotten by the introduced calculation model of flexibility coefficient. Then, considering dam volume and dam safety, its reasonable shape can be chosen based on the flexibility coefficient. The calculation model of statistical flexibility coefficient, which is based on PLSR, not only provides the more reasonable method to ascertain flexibility coefficient but also accomplishes some study related to principle component regression (PCR) and canonical correlation analysis (CCR). It can supply the better regressive equation that contains rich and deep data information. Moreover, it is studied that quadratic term and cubic term of related factors of flexibility coefficient affect the regressive equation on the basis of linear analysis. The result shows nonlinear parts of added factors have anunapparent influence to improve the precision of regressive analysis.
This research has been partially supported by National Natural Science Foundation of China (SN: 51179066, 51139001), Jiangsu Natural Science Foundation (SN: BK2012036), the Program for New Century Excellent Talents in University (SN: NCET-10-0359), the Special Fund of State Key Laboratory of China (SN: 2009586912), the Fundamental Research Funds for the Central Universities (Grant No. 2012B06614), Jiangsu Province “333 High-Level Personnel Training Project” (Grant No. BRA2011179) and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) (SN: YS11001).