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Interpretation of concrete dam behaviour with artificial neural network and multiple linear regression models
Engineering Structures 33 (2011) 903–910
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Interpretation of concrete dam behaviour with artificial neural network and multiple linear regression models J. Mata ∗ Monitoring Division, Concrete Dams Department, National Laboratory for Civil Engineering, Av. do Brasil 101, 1700-066, Lisbon, Portugal
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Article history: Received 15 July 2010 Received in revised form 9 November 2010 Accepted 1 December 2010 Available online 11 January 2011 Keywords: Concrete dam Dam behaviour Ceteris paribus Artificial neural network Multiple linear regression
abstract The safety control of large dams is based on the measurement of some important quantities that characterize their behaviour (like absolute and relative displacements, strains and stresses in the concrete, discharges through the foundation, etc.) and on visual inspections of the structures. In the more important dams, the analysis of the measured data and their comparison with results of mathematical or physical models is determinant in the structural safety assessment. In its lifetime, a dam can be exposed to significant water level variations and seasonal environmental temperature changes. The use of statistical models, such as multiple linear regression (MLR) models, in the analysis of a structural dam’s behaviour has been well known in dam engineering since the 1950s. Nowadays, artificial neural network (NN) models can also contribute in characterizing the normal structural behaviour for the actions to which the structure is subject using the past history of the structural behaviour. In this work, one important aspect of NN models is discussed: the parallel processing of the information. This study shows a comparison between MLR and NN models for the characterization of dam behaviour under environment loads. As an example, the horizontal displacement recorded by a pendulum is studied in a large Portuguese arch dam. The results of this study show that NN models can be a powerful tool to be included in assessments of existing concrete dam behaviour. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The main objective of the safety control of a concrete dam is to guarantee the functions for which it was built by maintaining its functionality and its structural integrity. The safety control is supported by monitoring activities and is based on models. The ultimate purpose of the models is to predict the behaviour of a concrete dam and to identify whether the behaviour of the structure is still similar to past behaviour under the same loads or if there is any difference. If indeed the evolution is divergent between the model prediction and actual behaviour, then the assumptions of the models have changed and the reason for the change should be identified to assess the consequences. Models based on mechanical principles are often difficult to construct and it is necessary to deal with the uncertainty in the parameters. In general, it is interesting to find out how changes in the input variables affect the values of the response variables. An empirical formulation for structural response is usually obtained as the sum of three terms: the temperature variation, the hydrostatic pressure variation and other unexpected unknown causes such
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as the result of time effects. The uncertainty of the model is represented by the residual term of the model. Some structural identification techniques have been successfully obtained by De Sortis and Paoliani [1] and Léger and Leclerc [2], although using a very complex procedure. On the other hand, with a large amount of observation data it is possible to define the characterization of a normal dam’s behaviour by using statistical models without the knowledge of mechanical principles [3]. Nowadays, there is great experience in using MLR model methods for the characterization of a concrete dam’s behaviour. The NN models have been applied in different areas, including dam engineering. Some works related to this subject can be mentioned such as Perner et al. [4], Gomes and Awruch [5], Fedele et al. [6], Feng and Zhou [7], Bakhary et al. [8], Wang and He [9], Wen et al. [10], Liu et al. [11], Joghataie and Dizaji [12] and Yi et al. [13]. Both MLR and NN approaches have potential value for assessing the behaviour of the control variables that support the safety assessment of the concrete dam as is shown with a ceteris paribus1 analysis in this study. In the period of normal operation of a
1 Ceteris paribus is a Latin phrase, that can be translated as ‘‘all other things being equal’’.
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concrete dam, the main actions and the structural response are well characterized and there is a strong correlation between these two. The study of a structural response of horizontal displacements with ceteris paribus analysis for the temperature effect is presented for different water levels. In the same way, a similar study for the hydrostatic pressure effect is carried out for different levels of temperature. 2. Statistical models The variations of hydrostatic pressure and temperature are the main actions to be taken into account when analysing the results of the concrete dam observations. The simultaneous effects of hydrostatic pressure and temperature variations create an observed effect which is the result of both loads. It is also important to take in account any effects of the loads separately in interpreting a concrete dam’s behaviour. Such isolation can be promptly obtained when the variation of only one load appears between seasons. However, the most general way to separate their effects is the use of statistical methods. These methods use simultaneous consideration of a large number of observations, allowing the establishment of the correlations between the observed behaviour and corresponding loads. The MLR model is one of the statistical techniques most widely used for analysing multifactor effects. A MLR model is a statistical technique for investigating and modeling the relationship between variables [14,15]. In dam engineering, MLR models have a long history and were initially known as quantitative analysis models [16]. A MLR model does not imply a cause effect relationship between the variables and in almost all applications of regression, the regression equation is only an approximation to the true relationship between the variables. In recent years, the field of Artificial Intelligence has introduced some tools that are able to perform cognitive tasks such as pattern recognition and function approximation [17]. This is the case of the Multilayer Perceptron (MLP) models that were used in this work. Generally, MLR and NN models are valid only within the region of the observed data. 2.1. Multiple linear regression A MLR model is a method used to model the linear relationship between a dependent variable and one or more independent variables. The dependent variable is sometimes also called the predictand or response, and the independent variables the predictors. These models consider that the effects associated with a limited time period at a specific point can be approximated by Eq. (1). E (h, θ, t ) = Eh + Eθ + Et
(1)
where E (h, θ , t ) is the observed effect; Eh is the elastic effect of hydrostatic pressure; Eθ is the elastic effect of temperature, depending on the thermal conditions; Et is the effect function of time, considered irreversible. Eq. (1) is based on several simplifying assumptions concerning the behaviour of materials, such as: (i) the analysed effects refer to a period in the life of a concrete dam, for which there is no relevant structural changes; (ii) the effects of the normal structural behaviour for normal operating conditions can be represented by two parts. A part of the elastic nature (reversible and instantaneous, resulting from the variations of the hydrostatic pressure and the temperature) and another part of inelastic nature (irreversible) such as a time function; (iii) the effects of the hydrostatic pressure and temperature changes can be studied separately. The effects of hydrostatic pressure variation, Eh (h), are usually represented by polynomials, depending on the height of water in
the reservoir h, Eq. (2). Eh (h) = β1 h + β1 h2 + β2 h3 + β4 h4 .
(2)
The effect of the temperature changes can be considered as a proportional attenuation of the air temperature changes with a phase shift with depth along section. Very simple MLR models usually do not use temperature measurements because it is assumed that the thermal effect Eθ (d) can be represented by the sum of sinusoidal functions with one-year period and six-month period [18,2]. Thus, the effect of temperature variations is defined by a linear combination of sinusoidal functions, which only depends on the day of the year, Eq. (3). Eθ (d) = β5 sin(d) + β6 cos(d) + β7 sin2 (d) + β8 sin(d) cos(d) (3) 2π·j
where d = 365 and j represents the number of days between the beginning of the year (January 1) until the date of observation (0 ≤ j ≤ 365). To represent the time effects, Et (t ), it is usual to consider the functions presented in Eq. (4), where t is the number of days since the beginning of the analysis. Et (t ) = β9 t + β10 e−t .
(4)
Suppose that there are p independent variables and n observations, (X1 , . . . , Xp , Y ) where Y represents the observed effect and X are the functions of water level height, temperature and time, respectively. The model relating the independent variable to the dependent variable is Y = β0 + β1 X1 + β2 X2 + · · · + βj Xj + · · · + βp Xp + ϵ . The model is obtained by a system of n equations that can be expressed in a matrix notation as Y = Xβ + ϵ , where Y is a (n × 1) vector of the dependent variable or response, X is a (n ×(p + 1)) matrix of the levels of the p independent variables, β is a ((p + 1)× 1) vector of the regression coefficients, and ϵ is a (n × 1) vector of random errors. This method assumes that the expected value of the error term is zero, which is E (ϵ) = 0; the variance V (ϵ) = σ 2 and that the errors are uncorrelated [14,19,20]. MLR models are based on least squares: the model is fit such that the sum-of-squares of differences of observed and predicted ∑n 2 T values, L = ϵ = ϵ ϵ = (Y − Xβ)T (Y − Xβ), is minimized. i =1 i The regression coefficient estimator, βˆ , is the solution for β in the ∂L = 0. equation ∂β
In matrix notation, the least squares estimator of β is βˆ = (X X)−1 XT Y, the fitted model is Yˆ = Xβˆ and the vector of the ˆ residuals is denoted by ϵˆ = Y − Y. T
2.2. Multilayer Perceptron The NN model is a simplified mathematical model of a natural neural network. NN models are inspired on the efficiency of the brain process. Many of the important issues concerning the application of artificial neural networks can be introduced in the simpler context of polynomial curve fitting [17,21]. NN models have been employed successfully to solve complex problems in various fields of application including classification, pattern recognition, prediction, optimization, function approximation and control systems [22,23]. The increasing interest for this area derives from the learning ability of these models, which relate the variables without imposing relationships between them. A neuron is the main element of an artificial neural network. It is an operator with inputs and outputs, associated with a transfer function, f , interconnected by synaptic connections or weights, w . Fig. 1 illustrates how information is processed through a single neuron.
J. Mata / Engineering Structures 33 (2011) 903–910
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Fig. 3. Learning and prediction sets.
Fig. 1. Neuron.
Fig. 4. Cross validation criteria.
Fig. 2. Multilayer Perceptron.
Different NN models have been proposed since its conception in the 1940s, but the MLP is the most widely used. A MLP has neurons arranged in layers. The first layer receives the inputs and the last layer produces the outputs. The middle layers have no connection with the external world and are called hidden layers. Each neuron in one layer is connected to every neuron in the next layer. The information is constantly fed forward from one layer to the next layer. The incoming connection has two values associated with it, an input value and a weight. MLP models learn by an iterative process, by adjusting the weights so as to be able to correctly learn the training data and hence, after the testing phase, to predict unknown data. Knowledge is usually stored as a set of connecting weights. Fig. 2 illustrates a generic example of a MLP, with an input layer having N input parameters, one hidden layer, l, with Q neurons and an output layer, L, with one output. p The parameters have the following meaning: xi is the input network i, from pattern p; L is the output layer; l is the hidden layer number; N is the number of inputs in the input layer; Q is the number of neurons in the hidden layer l; wijl is the synoptic weight between input network i from layer l − 1 at processing element l ,p j; sj is the activation value at neuron j from layer l, from pattern l ,p
p; and, yj is the output of the neuron j from layer l, from pattern p. l ,p
The output yj , from a hidden layer, is the input from the following l ,p
= fjl (slj,p ) and the calculation of the ∑N l−1,p l l ,p activation value, was defined as sj = wij . i=1 yi p p p The set of patterns can be written as (x1 , . . . , xi , . . . , xN , dp ) p with p = 1, . . . , P, where d is the desired target from pattern p. layer and is calculated by yj l ,p sj ,
Training is the process of modifying the connection weights in some orderly fashion using a suitable learning method. The NN model uses a learning mode in which an input is presented to
the network along with the output, and the weights are adjusted so that the network attempts to produce the desired output. Weights after training contain meaningful information whereas before training they are random and have no meaning. The backpropagation algorithm, one of the most famous training algorithms for the MLP, is a gradient descent technique intended to minimize the error or cost function, in which it adjusts the weights by a small amount at a time. The generalized backpropagation delta learning rule algorithm was adopted. The cost function usually considered, C ,is defined by the mean square ∑P 1 L,p p error, as C = P1 ( y − d ) . p=1 2 The period analysed is usually separated in two data sets, (Fig. 3). The first set, designated as learning set {t0 , t1 }, is used for the model to learn the behaviour in this time. The second set, designated as predicting set {t2 , t3 }, is used to evaluate the prediction capabilities of the model obtained from the learning set. The cross validation is the stopping criteria used. A randomization of the learning set is previously done, which made it possible to define the training set, the cross validation set and the test set, with a number of examples equal to 65%, 15% and 20% of the learning set, respectively. The test set is used as an auxiliary element that makes it possible to carry out the evaluation of the quality of the NN model for the learning set. First, a NN model is trained with a small number of processing elements in the hidden layer and the performance over the cross validation set and the test set is determined. Then, the procedure is repeated with one more processing element in the hidden layer and the performance is again calculated. After a sufficient number of architectures are trained, the best architecture is chosen. In each iteration, the performance for the training set is usually better than before, but if at any time the error for the cross validation set increases, the NN model may lose its generalization capacity (Fig. 4). The training stops when the error for the cross validation set begins to increase, a better generalization thus being ensured. Each initialization of the weights can lead to a different local minimum. Therefore, several random initializations were established for each NN architecture. 3. Case study The Alto Rabagão dam (Fig. 5) is located in Portugal and has as its main water line the Rabagão River in the Cávado hydrographical basin. The Alto Rabagão dam consists of three main structures:
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Fig. 5. Alto Rabagão dam.
Fig. 6. Maximum and minimum daily air temperature between 1980 and 2005.
the central structure that consists of a double curvature arch which is supported on two artificial abutments and two gravity sections that are located on the prolongation of each abutment. The construction began in 1961 and the completion date was 1964. The dam started operating in 1964 and the maximum reservoir water level was achieved in March 1966. The maximum dam height is 94 m in the double curvature arch and the total crest length is 1897 m. The minimum reservoir water level for hydropower operation is 829 m and the maximum reservoir water level is 880 m, with a total capacity of 569 × 106 m3 . The monitoring system of the Alto Rabagão dam consists of several devices which make it possible to observe and to measure quantities such as: concrete and air temperatures, reservoir water level, horizontal and vertical displacements, rotations, movements of joints, strain, stress, uplift pressure, foundation displacements and seepage. Among the different loads acting on concrete dams, it is usual to distinguish, as the most important ones for structures in normal operation, the hydrostatic pressure and the temperature variation (Fig. 6). The structural response, for instance, the displacement in any point of the dam, is strongly related to the corresponding variation in the water level in the reservoir. Fig. 7 shows this strong correlation for the displacements measured at block KJ of the crest
arch. The observations presented in Fig. 7 will be used for the computation of the MLR and NN models presented in this work. Signs (+) indicate displacements towards downstream and signs (−) towards upstream. 3.1. Multiple linear regression model Through semi-empirical knowledge, the wide knowledge in analysing the behaviour of concrete dams provides a functional relationship between the independent variables and the dependent variable. In this case study, the MLR model with the best performance for the upstream–downstream crest displacement of the FP1 pendulum yMLR FP1 was obtained as the sum of the hydrostatic pressure term β4 × h4 (where h is the reservoir water level height and can vary between 0 and 94 m) and the temperature terms β5 × sin(d) + β6 × cos(d). The time effect did not seem to have a significant importance in the period examined by this study. Mention must be made of the fact that if any effect of time exists, it can be easily detected through a graph of the evolution of the residuals over time. The data used to determine the model parameters comprised the period from January 1980 to December 2002 (914 observations) and the data considered for prediction comprised the period from January 2003 to September 2005 (69 observations), Fig. 7.
J. Mata / Engineering Structures 33 (2011) 903–910
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Fig. 7. Horizontal upstream–downstream crest displacement recorded by the FP1 pendulum at KJ block.
Fig. 8. Displacement by MLR model for the horizontal upstream–downstream crest displacement at KJ block.
Fig. 10. Displacement by NN model for the horizontal upstream–downstream crest displacement at KJ block. Table 1 Performance parameters for MLR and NN models. Learning set
MLR NN
Fig. 9. Architecture of proposed NN model.
The least squares method was applied, the solution presented in Eq. (5) being obtained, so the horizontal upstream–downstream crest displacement at KJ block can be computed by Eq. (5). −7 4 yMLR h + 2.40 cos(d) FP1 (h, d) = 3.33 × 10
+ 4.55 sin(d) − 1.47.
(5)
Fig. 8 shows the good agreement between the model and the data set.
Prediction set
|ϵ|Max
E (|ϵ|)
r
|ϵ|Max
E (|ϵ|)
r
5.7 5.7
1.3 1.2
0.97 0.98
4.3 4.3
1.3 1.4
0.98 0.98
A hyperbolic tangent transfer function having been chosen to be the activation function for the hidden layer and the linear function for the output layer. The generalized backpropagation delta learning rule algorithm was used in the training process. The chosen network architecture was the best for all networks, from 3 until 30 neurons, at the hidden layer. To find out the optimum result, 5 initializations of random weights and a maximum of 5000 iterations were performed for each NN architecture. In this case study the NN model with best performance was a 3-12-1 MLP (less error for the cross validation set) and the good results are illustrated in Fig. 10. 4. Model comparison
3.2. Multilayer Perceptron model The NN model considered consists of an input layer with 3 input parameters (to represent the hydrostatic pressure and the temperature effects), an output layer (to represent the upstream–downstream crest displacement of the FP1 pendulum) and one hidden layer, as can be seen in Fig. 9. Every neuron in the network is fully connected with each neuron of the next layer.
Table 1 presents some performance parameters such as the mean absolute error, E (|ϵ|), the maximum absolute error, |ϵ|Max , and the correlation coefficient, r. MLR and NN models have good performances for both learning and prediction sets. Both methods proved to be adequate for predicting the observed values, each model, however, having its own peculiarities, as can be seen in Fig. 11.
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Fig. 11. Observed data, MRL and NN model comparison for the horizontal crest displacements on FP1.
Fig. 12. Residuals produced by MLR and NN models.
The residuals obtained make it possible to estimate the dispersion in results. These residuals are due to factors contributing to the measured effect, which are not included in the magnitudes concerned. The residual analysis shows a more uniform error distribution for the NN model than the MLR model. The MLR model has greater errors from August to October and in the period from March to May, Fig. 12. If the hypotheses that support the MLR models are true, the separation of effects is valid, which is advantageous to quantify the contribution that a particular action has on the structural response. Furthermore, in MLR models, functions are selected based on a semi-empirical knowledge of the behaviour of the structure, such as the allocation of a polynomial function h4 to provide the effect of the variation in the water level height on the horizontal displacements of arch dams. The NN models themselves identify the influence that each variable has on the structural response,
by taking advantage of parallel processing in the interaction of inputs. The effect that each variable has on the output of the NN model can also be obtained from ceteris paribus analysis. This study was carried out on the ceteris paribus analysis for the effect of the hydrostatic pressure changes and the effect of the temperature changes on the horizontal crest displacement at the KJ block. The ceteris paribus analysis for horizontal displacements due to the effect of the hydrostatic pressure was obtained by establishing the day and by presenting the domain of the reservoir water level height. In this study, the analyses were performed on the 20th,100th,180th and 260th day of the year. As can be seen in Fig. 13, the two models show a similar evolution. The ceteris paribus analysis for horizontal displacements due to the effect of the temperature was achieved for several reservoir water level heights previously established, covering the entire field for the temperature variation. This process was repeated several times for various reservoir water level heights (74 m, 79 m, 84 m
J. Mata / Engineering Structures 33 (2011) 903–910
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Fig. 13. Ceteris paribus of the water level effect for days equal to 20, 100, 180 and 260 of the year.
Fig. 14. Ceteris paribus of the temperature effect for water levels equal to 860, 865, 870 and 875 m.
and 89 m that correspond to a water level equal to 860 m, 865 m, 870 m and 875 m, respectively). The results illustrated in Fig. 14 show that the effect of the temperature changes on the structural response is practically independent from the reservoir water level variation. The effects for both models are similar. In the MLR model, the effect of the temperature changes is given by a sinusoidal curve with an amplitude and a well defined phase. However, it seems that in the NN model, the response due to the thermal effect has a wave with smaller opening in the warm months than in the cold months and consequently is better adapted to the observed values than the MLR model. This flexibility is an advantage that the NN models have over MLR models. One of the advantages of MLR and NN models, within the safety control of concrete dams, is that they make it possible to establish a functional relationship between the loading actions and the structural response. MLR models use empirical functions to quantify the effect of the independent variables on the structural response. In NN models, the relationship between variables is established by the architecture and the synaptic weights of the network by taking advantage of the parallel processing of input variables, hence it only being necessary to know the inputs and the outputs.
It is considered that the two models have identified the same type of relationship between the structural response to the temperature and the reservoir water level variation. However, the NN model has more capacity of learning this kind of scenario because of the parallel processing of the information. One of the disadvantages is that it is sometimes suggested that it is not based on the principles of mechanics but only the knowledge of statistical nature. Although this may be true, the physical behaviour is intrinsic to the value observed, i.e. the observations are statistical data but are still governed by the actual behaviour of the structure. Since data is available in sufficient quantity, the models presented can quickly obtain estimates of the relationship between variables, and also estimate the response of the structure for a new load. 5. Conclusions In this paper MLR and NN models are generated and calibrated on the basis of experimental data of time histories (over about 25 years) of reservoir level and external temperature and of structural responses (specifically crest displacements). Deviations of monitored dam behaviour from the above results give rise to alert of possible damage. The main limitation of the methodology
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presented here is that it does not take mechanical properties and possible damage directly into account. Once an alert is given, additional analysis is required for detailed diagnoses, which should rely on the results of non-destructive tests (statical and dynamical), computational mechanical modeling and inverse analysis. In the MLR model, the variables are assumed to be independent, the overlapping effects being valid, and it is possible to identify the contribution of each loading action to the structural response. The NN models simultaneously process the inputs and, therefore, the contribution of each load to the output model depends on the value of the other inputs. Thus, the contribution of each input variable to the structural response can be achieved through the implementation of a ceteris paribus analysis, as shown in this work. NN models showed flexibility and proved to be more adequate for months with extreme temperatures than the MLR models with the same variables. The two methods have the advantage of being easily implemented and of being simultaneously used, which increases the confidence in the use of these models. Finally, the results of this study reinforce the notion that statistical models are useful for establishing relations between loads and structural responses for the behaviour analysis in the safety control of concrete dams. Acknowledgements The author acknowledges the company EDP-Energias de Portugal that provided the data for the statistical procedures addressed in this paper. The author is also grateful to the colleagues of the Concrete Dams Department of the National Laboratory for Civil Engineering of Portugal and of the Instituto Superior Técnico— Technical University of Lisbon for their support of the work. References [1] De Sortis A, Paoliani P. Statistical analysis and structural identification in concrete dam monitoring. Eng Struct 2007;29(1):110–20. [2] Leger P, Leclerc M. Hydrostatic, temperature, time-displacement model for concrete dams. J Eng Mech 2007;133(3):267–77. [3] ICOLD. Methods of analysis for the prediction and the verification of dam behaviour. Tech. rep. Swiss Committee on Dams; 2003. [4] Perner F, Koehler W, Obernhuber P. Interpretation of Schlegeis dam crest displacement. In: Sixth benchmark workshop on numerical analysis of dams. International commission of large dams. Salzburg. 2001. p. 10.
[5] Gomes HM, Awruch AM. Comparison of response surface and neural network with other methods for structural reliability analysis. Struct Saf 2004;26(1): 49–67. [6] Fedele R, Maier G, Miller B. Health assessment of concrete dams by overall inverse analyses and neural networks. Int J Fract 2006;137(1):151–72. [7] Feng X, Zhou J. Structural health monitoring using artificial neural network and statistical process control in a changing environment. In: Ou JP, Li H, Duan ZD, editors. Structural health monitoring and intelligent infrastructure. Proceedings and monographs in engineering, water and earth sciences, vols. 1 and 2. London: Taylor & Francis Ltd.; 2006. p. 737–43. [8] Bakhary N, Hao H, Deeks AJ. Damage detection using artificial neural network with consideration of uncertainties. Eng Struct 2007;29(11):2806–15. [9] Wang BS, He ZC. Crack detection of arch dam using statistical neural network based on the reductions of natural frequencies. J Sound Vibration 2007; 302(4–5):1037–47. [10] Wen CM, Hung SL, Huang CS, Jan JC. Unsupervised fuzzy neural networks for damage detection of structures. Struct Control Health Monit 2007;14(1): 144–61. [11] Liu J, Wang G, Chen Y. Research and application of GA neural network model on dam displacement forecasting. ASCE conf. proc., vol. 323. Long Beach (CA): ASCE; 2008. p. 69. [12] Joghataie A, Dizaji MS. Nonlinear analysis of concrete gravity dams by neural networks. In: Ao SI, Gelman L, Hukins DWL, Hunter A, Korsunsky AM, editors. World congress on engineering 2009. Lecture notes in engineering and computer science, vols. I and II. Hong Kong: Int. Assoc. Engineers; 2009. p. 1022–7. [13] Yi XD, Xu F, Jiang CK. Research on dam deformation forecast model based on genetic algorithm neural network. In: Wen TD, editor. ISTM/2009: 8th international symposium on test and measurement, vols. 1–6. Hong Kong: International Academic Publishers Ltd.; 2009. p. 1536–9. [14] Montgomery DC, Runger GC. Applied statistics and probability for engineers. John Wiley & Sons, Inc.; 1994. [15] Draper NR, Smith H. Applied regression analysis. John Wiley & Sons, Inc.; 1981. [16] Rocha M, Serafim J, Silveira A. A method of quantitative interpretation of the results obtained in the observation of the dams. R.84, Q.21. In: VI ICOLD congress. vol. II. New York. 1958. p. 927–60. [17] Bishop CM. Neural networks for pattern recognition. Oxford: Clarendon Press; 1995. [18] Willm G, Beaujoint N. Les méthodes de surveillance des barrages au service de la production hidraulique déléctricité de france. Problémes anciens et solutions nouvelles. R.30, Q.34. In: IX ICOLD congress. vol. III. Istanbul. 1967. p. 529–50. [19] Johnson RA, Wichern DW. Applied multivariate statistical analysis. 4th ed. Prentice Hall; 1998. [20] Kutner MH, Nachtsheim CJ, Neter J, Li W. Applied linear statistical models. 5th ed. McGraw-Hill Higher Education; 2004. [21] Rocha M, Cortez P, Neves J. Evolution of neural networks for classification and regression. Neurocomputing 2007;70(16–18):2809–16. [22] Patterson DW. Artificial neural networks: theory and applications. Singapore: Prentice Hall; 1996. [23] Almeida LB. Section C.1.2—multilayer perceptron. Handbook of neural computation. Oxford. 1997. p. 34.
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Interpretation of concrete dam behaviour with artificial neural network and multiple linear regression models J. Mata ∗ Monitoring Division, Concrete Dams Department, National Laboratory for Civil Engineering, Av. do Brasil 101, 1700-066, Lisbon, Portugal
article
info
Article history: Received 15 July 2010 Received in revised form 9 November 2010 Accepted 1 December 2010 Available online 11 January 2011 Keywords: Concrete dam Dam behaviour Ceteris paribus Artificial neural network Multiple linear regression
abstract The safety control of large dams is based on the measurement of some important quantities that characterize their behaviour (like absolute and relative displacements, strains and stresses in the concrete, discharges through the foundation, etc.) and on visual inspections of the structures. In the more important dams, the analysis of the measured data and their comparison with results of mathematical or physical models is determinant in the structural safety assessment. In its lifetime, a dam can be exposed to significant water level variations and seasonal environmental temperature changes. The use of statistical models, such as multiple linear regression (MLR) models, in the analysis of a structural dam’s behaviour has been well known in dam engineering since the 1950s. Nowadays, artificial neural network (NN) models can also contribute in characterizing the normal structural behaviour for the actions to which the structure is subject using the past history of the structural behaviour. In this work, one important aspect of NN models is discussed: the parallel processing of the information. This study shows a comparison between MLR and NN models for the characterization of dam behaviour under environment loads. As an example, the horizontal displacement recorded by a pendulum is studied in a large Portuguese arch dam. The results of this study show that NN models can be a powerful tool to be included in assessments of existing concrete dam behaviour. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The main objective of the safety control of a concrete dam is to guarantee the functions for which it was built by maintaining its functionality and its structural integrity. The safety control is supported by monitoring activities and is based on models. The ultimate purpose of the models is to predict the behaviour of a concrete dam and to identify whether the behaviour of the structure is still similar to past behaviour under the same loads or if there is any difference. If indeed the evolution is divergent between the model prediction and actual behaviour, then the assumptions of the models have changed and the reason for the change should be identified to assess the consequences. Models based on mechanical principles are often difficult to construct and it is necessary to deal with the uncertainty in the parameters. In general, it is interesting to find out how changes in the input variables affect the values of the response variables. An empirical formulation for structural response is usually obtained as the sum of three terms: the temperature variation, the hydrostatic pressure variation and other unexpected unknown causes such
∗
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0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2010.12.011
as the result of time effects. The uncertainty of the model is represented by the residual term of the model. Some structural identification techniques have been successfully obtained by De Sortis and Paoliani [1] and Léger and Leclerc [2], although using a very complex procedure. On the other hand, with a large amount of observation data it is possible to define the characterization of a normal dam’s behaviour by using statistical models without the knowledge of mechanical principles [3]. Nowadays, there is great experience in using MLR model methods for the characterization of a concrete dam’s behaviour. The NN models have been applied in different areas, including dam engineering. Some works related to this subject can be mentioned such as Perner et al. [4], Gomes and Awruch [5], Fedele et al. [6], Feng and Zhou [7], Bakhary et al. [8], Wang and He [9], Wen et al. [10], Liu et al. [11], Joghataie and Dizaji [12] and Yi et al. [13]. Both MLR and NN approaches have potential value for assessing the behaviour of the control variables that support the safety assessment of the concrete dam as is shown with a ceteris paribus1 analysis in this study. In the period of normal operation of a
1 Ceteris paribus is a Latin phrase, that can be translated as ‘‘all other things being equal’’.
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concrete dam, the main actions and the structural response are well characterized and there is a strong correlation between these two. The study of a structural response of horizontal displacements with ceteris paribus analysis for the temperature effect is presented for different water levels. In the same way, a similar study for the hydrostatic pressure effect is carried out for different levels of temperature. 2. Statistical models The variations of hydrostatic pressure and temperature are the main actions to be taken into account when analysing the results of the concrete dam observations. The simultaneous effects of hydrostatic pressure and temperature variations create an observed effect which is the result of both loads. It is also important to take in account any effects of the loads separately in interpreting a concrete dam’s behaviour. Such isolation can be promptly obtained when the variation of only one load appears between seasons. However, the most general way to separate their effects is the use of statistical methods. These methods use simultaneous consideration of a large number of observations, allowing the establishment of the correlations between the observed behaviour and corresponding loads. The MLR model is one of the statistical techniques most widely used for analysing multifactor effects. A MLR model is a statistical technique for investigating and modeling the relationship between variables [14,15]. In dam engineering, MLR models have a long history and were initially known as quantitative analysis models [16]. A MLR model does not imply a cause effect relationship between the variables and in almost all applications of regression, the regression equation is only an approximation to the true relationship between the variables. In recent years, the field of Artificial Intelligence has introduced some tools that are able to perform cognitive tasks such as pattern recognition and function approximation [17]. This is the case of the Multilayer Perceptron (MLP) models that were used in this work. Generally, MLR and NN models are valid only within the region of the observed data. 2.1. Multiple linear regression A MLR model is a method used to model the linear relationship between a dependent variable and one or more independent variables. The dependent variable is sometimes also called the predictand or response, and the independent variables the predictors. These models consider that the effects associated with a limited time period at a specific point can be approximated by Eq. (1). E (h, θ, t ) = Eh + Eθ + Et
(1)
where E (h, θ , t ) is the observed effect; Eh is the elastic effect of hydrostatic pressure; Eθ is the elastic effect of temperature, depending on the thermal conditions; Et is the effect function of time, considered irreversible. Eq. (1) is based on several simplifying assumptions concerning the behaviour of materials, such as: (i) the analysed effects refer to a period in the life of a concrete dam, for which there is no relevant structural changes; (ii) the effects of the normal structural behaviour for normal operating conditions can be represented by two parts. A part of the elastic nature (reversible and instantaneous, resulting from the variations of the hydrostatic pressure and the temperature) and another part of inelastic nature (irreversible) such as a time function; (iii) the effects of the hydrostatic pressure and temperature changes can be studied separately. The effects of hydrostatic pressure variation, Eh (h), are usually represented by polynomials, depending on the height of water in
the reservoir h, Eq. (2). Eh (h) = β1 h + β1 h2 + β2 h3 + β4 h4 .
(2)
The effect of the temperature changes can be considered as a proportional attenuation of the air temperature changes with a phase shift with depth along section. Very simple MLR models usually do not use temperature measurements because it is assumed that the thermal effect Eθ (d) can be represented by the sum of sinusoidal functions with one-year period and six-month period [18,2]. Thus, the effect of temperature variations is defined by a linear combination of sinusoidal functions, which only depends on the day of the year, Eq. (3). Eθ (d) = β5 sin(d) + β6 cos(d) + β7 sin2 (d) + β8 sin(d) cos(d) (3) 2π·j
where d = 365 and j represents the number of days between the beginning of the year (January 1) until the date of observation (0 ≤ j ≤ 365). To represent the time effects, Et (t ), it is usual to consider the functions presented in Eq. (4), where t is the number of days since the beginning of the analysis. Et (t ) = β9 t + β10 e−t .
(4)
Suppose that there are p independent variables and n observations, (X1 , . . . , Xp , Y ) where Y represents the observed effect and X are the functions of water level height, temperature and time, respectively. The model relating the independent variable to the dependent variable is Y = β0 + β1 X1 + β2 X2 + · · · + βj Xj + · · · + βp Xp + ϵ . The model is obtained by a system of n equations that can be expressed in a matrix notation as Y = Xβ + ϵ , where Y is a (n × 1) vector of the dependent variable or response, X is a (n ×(p + 1)) matrix of the levels of the p independent variables, β is a ((p + 1)× 1) vector of the regression coefficients, and ϵ is a (n × 1) vector of random errors. This method assumes that the expected value of the error term is zero, which is E (ϵ) = 0; the variance V (ϵ) = σ 2 and that the errors are uncorrelated [14,19,20]. MLR models are based on least squares: the model is fit such that the sum-of-squares of differences of observed and predicted ∑n 2 T values, L = ϵ = ϵ ϵ = (Y − Xβ)T (Y − Xβ), is minimized. i =1 i The regression coefficient estimator, βˆ , is the solution for β in the ∂L = 0. equation ∂β
In matrix notation, the least squares estimator of β is βˆ = (X X)−1 XT Y, the fitted model is Yˆ = Xβˆ and the vector of the ˆ residuals is denoted by ϵˆ = Y − Y. T
2.2. Multilayer Perceptron The NN model is a simplified mathematical model of a natural neural network. NN models are inspired on the efficiency of the brain process. Many of the important issues concerning the application of artificial neural networks can be introduced in the simpler context of polynomial curve fitting [17,21]. NN models have been employed successfully to solve complex problems in various fields of application including classification, pattern recognition, prediction, optimization, function approximation and control systems [22,23]. The increasing interest for this area derives from the learning ability of these models, which relate the variables without imposing relationships between them. A neuron is the main element of an artificial neural network. It is an operator with inputs and outputs, associated with a transfer function, f , interconnected by synaptic connections or weights, w . Fig. 1 illustrates how information is processed through a single neuron.
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Fig. 3. Learning and prediction sets.
Fig. 1. Neuron.
Fig. 4. Cross validation criteria.
Fig. 2. Multilayer Perceptron.
Different NN models have been proposed since its conception in the 1940s, but the MLP is the most widely used. A MLP has neurons arranged in layers. The first layer receives the inputs and the last layer produces the outputs. The middle layers have no connection with the external world and are called hidden layers. Each neuron in one layer is connected to every neuron in the next layer. The information is constantly fed forward from one layer to the next layer. The incoming connection has two values associated with it, an input value and a weight. MLP models learn by an iterative process, by adjusting the weights so as to be able to correctly learn the training data and hence, after the testing phase, to predict unknown data. Knowledge is usually stored as a set of connecting weights. Fig. 2 illustrates a generic example of a MLP, with an input layer having N input parameters, one hidden layer, l, with Q neurons and an output layer, L, with one output. p The parameters have the following meaning: xi is the input network i, from pattern p; L is the output layer; l is the hidden layer number; N is the number of inputs in the input layer; Q is the number of neurons in the hidden layer l; wijl is the synoptic weight between input network i from layer l − 1 at processing element l ,p j; sj is the activation value at neuron j from layer l, from pattern l ,p
p; and, yj is the output of the neuron j from layer l, from pattern p. l ,p
The output yj , from a hidden layer, is the input from the following l ,p
= fjl (slj,p ) and the calculation of the ∑N l−1,p l l ,p activation value, was defined as sj = wij . i=1 yi p p p The set of patterns can be written as (x1 , . . . , xi , . . . , xN , dp ) p with p = 1, . . . , P, where d is the desired target from pattern p. layer and is calculated by yj l ,p sj ,
Training is the process of modifying the connection weights in some orderly fashion using a suitable learning method. The NN model uses a learning mode in which an input is presented to
the network along with the output, and the weights are adjusted so that the network attempts to produce the desired output. Weights after training contain meaningful information whereas before training they are random and have no meaning. The backpropagation algorithm, one of the most famous training algorithms for the MLP, is a gradient descent technique intended to minimize the error or cost function, in which it adjusts the weights by a small amount at a time. The generalized backpropagation delta learning rule algorithm was adopted. The cost function usually considered, C ,is defined by the mean square ∑P 1 L,p p error, as C = P1 ( y − d ) . p=1 2 The period analysed is usually separated in two data sets, (Fig. 3). The first set, designated as learning set {t0 , t1 }, is used for the model to learn the behaviour in this time. The second set, designated as predicting set {t2 , t3 }, is used to evaluate the prediction capabilities of the model obtained from the learning set. The cross validation is the stopping criteria used. A randomization of the learning set is previously done, which made it possible to define the training set, the cross validation set and the test set, with a number of examples equal to 65%, 15% and 20% of the learning set, respectively. The test set is used as an auxiliary element that makes it possible to carry out the evaluation of the quality of the NN model for the learning set. First, a NN model is trained with a small number of processing elements in the hidden layer and the performance over the cross validation set and the test set is determined. Then, the procedure is repeated with one more processing element in the hidden layer and the performance is again calculated. After a sufficient number of architectures are trained, the best architecture is chosen. In each iteration, the performance for the training set is usually better than before, but if at any time the error for the cross validation set increases, the NN model may lose its generalization capacity (Fig. 4). The training stops when the error for the cross validation set begins to increase, a better generalization thus being ensured. Each initialization of the weights can lead to a different local minimum. Therefore, several random initializations were established for each NN architecture. 3. Case study The Alto Rabagão dam (Fig. 5) is located in Portugal and has as its main water line the Rabagão River in the Cávado hydrographical basin. The Alto Rabagão dam consists of three main structures:
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Fig. 5. Alto Rabagão dam.
Fig. 6. Maximum and minimum daily air temperature between 1980 and 2005.
the central structure that consists of a double curvature arch which is supported on two artificial abutments and two gravity sections that are located on the prolongation of each abutment. The construction began in 1961 and the completion date was 1964. The dam started operating in 1964 and the maximum reservoir water level was achieved in March 1966. The maximum dam height is 94 m in the double curvature arch and the total crest length is 1897 m. The minimum reservoir water level for hydropower operation is 829 m and the maximum reservoir water level is 880 m, with a total capacity of 569 × 106 m3 . The monitoring system of the Alto Rabagão dam consists of several devices which make it possible to observe and to measure quantities such as: concrete and air temperatures, reservoir water level, horizontal and vertical displacements, rotations, movements of joints, strain, stress, uplift pressure, foundation displacements and seepage. Among the different loads acting on concrete dams, it is usual to distinguish, as the most important ones for structures in normal operation, the hydrostatic pressure and the temperature variation (Fig. 6). The structural response, for instance, the displacement in any point of the dam, is strongly related to the corresponding variation in the water level in the reservoir. Fig. 7 shows this strong correlation for the displacements measured at block KJ of the crest
arch. The observations presented in Fig. 7 will be used for the computation of the MLR and NN models presented in this work. Signs (+) indicate displacements towards downstream and signs (−) towards upstream. 3.1. Multiple linear regression model Through semi-empirical knowledge, the wide knowledge in analysing the behaviour of concrete dams provides a functional relationship between the independent variables and the dependent variable. In this case study, the MLR model with the best performance for the upstream–downstream crest displacement of the FP1 pendulum yMLR FP1 was obtained as the sum of the hydrostatic pressure term β4 × h4 (where h is the reservoir water level height and can vary between 0 and 94 m) and the temperature terms β5 × sin(d) + β6 × cos(d). The time effect did not seem to have a significant importance in the period examined by this study. Mention must be made of the fact that if any effect of time exists, it can be easily detected through a graph of the evolution of the residuals over time. The data used to determine the model parameters comprised the period from January 1980 to December 2002 (914 observations) and the data considered for prediction comprised the period from January 2003 to September 2005 (69 observations), Fig. 7.
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Fig. 7. Horizontal upstream–downstream crest displacement recorded by the FP1 pendulum at KJ block.
Fig. 8. Displacement by MLR model for the horizontal upstream–downstream crest displacement at KJ block.
Fig. 10. Displacement by NN model for the horizontal upstream–downstream crest displacement at KJ block. Table 1 Performance parameters for MLR and NN models. Learning set
MLR NN
Fig. 9. Architecture of proposed NN model.
The least squares method was applied, the solution presented in Eq. (5) being obtained, so the horizontal upstream–downstream crest displacement at KJ block can be computed by Eq. (5). −7 4 yMLR h + 2.40 cos(d) FP1 (h, d) = 3.33 × 10
+ 4.55 sin(d) − 1.47.
(5)
Fig. 8 shows the good agreement between the model and the data set.
Prediction set
|ϵ|Max
E (|ϵ|)
r
|ϵ|Max
E (|ϵ|)
r
5.7 5.7
1.3 1.2
0.97 0.98
4.3 4.3
1.3 1.4
0.98 0.98
A hyperbolic tangent transfer function having been chosen to be the activation function for the hidden layer and the linear function for the output layer. The generalized backpropagation delta learning rule algorithm was used in the training process. The chosen network architecture was the best for all networks, from 3 until 30 neurons, at the hidden layer. To find out the optimum result, 5 initializations of random weights and a maximum of 5000 iterations were performed for each NN architecture. In this case study the NN model with best performance was a 3-12-1 MLP (less error for the cross validation set) and the good results are illustrated in Fig. 10. 4. Model comparison
3.2. Multilayer Perceptron model The NN model considered consists of an input layer with 3 input parameters (to represent the hydrostatic pressure and the temperature effects), an output layer (to represent the upstream–downstream crest displacement of the FP1 pendulum) and one hidden layer, as can be seen in Fig. 9. Every neuron in the network is fully connected with each neuron of the next layer.
Table 1 presents some performance parameters such as the mean absolute error, E (|ϵ|), the maximum absolute error, |ϵ|Max , and the correlation coefficient, r. MLR and NN models have good performances for both learning and prediction sets. Both methods proved to be adequate for predicting the observed values, each model, however, having its own peculiarities, as can be seen in Fig. 11.
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Fig. 11. Observed data, MRL and NN model comparison for the horizontal crest displacements on FP1.
Fig. 12. Residuals produced by MLR and NN models.
The residuals obtained make it possible to estimate the dispersion in results. These residuals are due to factors contributing to the measured effect, which are not included in the magnitudes concerned. The residual analysis shows a more uniform error distribution for the NN model than the MLR model. The MLR model has greater errors from August to October and in the period from March to May, Fig. 12. If the hypotheses that support the MLR models are true, the separation of effects is valid, which is advantageous to quantify the contribution that a particular action has on the structural response. Furthermore, in MLR models, functions are selected based on a semi-empirical knowledge of the behaviour of the structure, such as the allocation of a polynomial function h4 to provide the effect of the variation in the water level height on the horizontal displacements of arch dams. The NN models themselves identify the influence that each variable has on the structural response,
by taking advantage of parallel processing in the interaction of inputs. The effect that each variable has on the output of the NN model can also be obtained from ceteris paribus analysis. This study was carried out on the ceteris paribus analysis for the effect of the hydrostatic pressure changes and the effect of the temperature changes on the horizontal crest displacement at the KJ block. The ceteris paribus analysis for horizontal displacements due to the effect of the hydrostatic pressure was obtained by establishing the day and by presenting the domain of the reservoir water level height. In this study, the analyses were performed on the 20th,100th,180th and 260th day of the year. As can be seen in Fig. 13, the two models show a similar evolution. The ceteris paribus analysis for horizontal displacements due to the effect of the temperature was achieved for several reservoir water level heights previously established, covering the entire field for the temperature variation. This process was repeated several times for various reservoir water level heights (74 m, 79 m, 84 m
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Fig. 13. Ceteris paribus of the water level effect for days equal to 20, 100, 180 and 260 of the year.
Fig. 14. Ceteris paribus of the temperature effect for water levels equal to 860, 865, 870 and 875 m.
and 89 m that correspond to a water level equal to 860 m, 865 m, 870 m and 875 m, respectively). The results illustrated in Fig. 14 show that the effect of the temperature changes on the structural response is practically independent from the reservoir water level variation. The effects for both models are similar. In the MLR model, the effect of the temperature changes is given by a sinusoidal curve with an amplitude and a well defined phase. However, it seems that in the NN model, the response due to the thermal effect has a wave with smaller opening in the warm months than in the cold months and consequently is better adapted to the observed values than the MLR model. This flexibility is an advantage that the NN models have over MLR models. One of the advantages of MLR and NN models, within the safety control of concrete dams, is that they make it possible to establish a functional relationship between the loading actions and the structural response. MLR models use empirical functions to quantify the effect of the independent variables on the structural response. In NN models, the relationship between variables is established by the architecture and the synaptic weights of the network by taking advantage of the parallel processing of input variables, hence it only being necessary to know the inputs and the outputs.
It is considered that the two models have identified the same type of relationship between the structural response to the temperature and the reservoir water level variation. However, the NN model has more capacity of learning this kind of scenario because of the parallel processing of the information. One of the disadvantages is that it is sometimes suggested that it is not based on the principles of mechanics but only the knowledge of statistical nature. Although this may be true, the physical behaviour is intrinsic to the value observed, i.e. the observations are statistical data but are still governed by the actual behaviour of the structure. Since data is available in sufficient quantity, the models presented can quickly obtain estimates of the relationship between variables, and also estimate the response of the structure for a new load. 5. Conclusions In this paper MLR and NN models are generated and calibrated on the basis of experimental data of time histories (over about 25 years) of reservoir level and external temperature and of structural responses (specifically crest displacements). Deviations of monitored dam behaviour from the above results give rise to alert of possible damage. The main limitation of the methodology
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presented here is that it does not take mechanical properties and possible damage directly into account. Once an alert is given, additional analysis is required for detailed diagnoses, which should rely on the results of non-destructive tests (statical and dynamical), computational mechanical modeling and inverse analysis. In the MLR model, the variables are assumed to be independent, the overlapping effects being valid, and it is possible to identify the contribution of each loading action to the structural response. The NN models simultaneously process the inputs and, therefore, the contribution of each load to the output model depends on the value of the other inputs. Thus, the contribution of each input variable to the structural response can be achieved through the implementation of a ceteris paribus analysis, as shown in this work. NN models showed flexibility and proved to be more adequate for months with extreme temperatures than the MLR models with the same variables. The two methods have the advantage of being easily implemented and of being simultaneously used, which increases the confidence in the use of these models. Finally, the results of this study reinforce the notion that statistical models are useful for establishing relations between loads and structural responses for the behaviour analysis in the safety control of concrete dams. Acknowledgements The author acknowledges the company EDP-Energias de Portugal that provided the data for the statistical procedures addressed in this paper. The author is also grateful to the colleagues of the Concrete Dams Department of the National Laboratory for Civil Engineering of Portugal and of the Instituto Superior Técnico— Technical University of Lisbon for their support of the work. References [1] De Sortis A, Paoliani P. Statistical analysis and structural identification in concrete dam monitoring. Eng Struct 2007;29(1):110–20. [2] Leger P, Leclerc M. Hydrostatic, temperature, time-displacement model for concrete dams. J Eng Mech 2007;133(3):267–77. [3] ICOLD. Methods of analysis for the prediction and the verification of dam behaviour. Tech. rep. Swiss Committee on Dams; 2003. [4] Perner F, Koehler W, Obernhuber P. Interpretation of Schlegeis dam crest displacement. In: Sixth benchmark workshop on numerical analysis of dams. International commission of large dams. Salzburg. 2001. p. 10.
[5] Gomes HM, Awruch AM. Comparison of response surface and neural network with other methods for structural reliability analysis. Struct Saf 2004;26(1): 49–67. [6] Fedele R, Maier G, Miller B. Health assessment of concrete dams by overall inverse analyses and neural networks. Int J Fract 2006;137(1):151–72. [7] Feng X, Zhou J. Structural health monitoring using artificial neural network and statistical process control in a changing environment. In: Ou JP, Li H, Duan ZD, editors. Structural health monitoring and intelligent infrastructure. Proceedings and monographs in engineering, water and earth sciences, vols. 1 and 2. London: Taylor & Francis Ltd.; 2006. p. 737–43. [8] Bakhary N, Hao H, Deeks AJ. Damage detection using artificial neural network with consideration of uncertainties. Eng Struct 2007;29(11):2806–15. [9] Wang BS, He ZC. Crack detection of arch dam using statistical neural network based on the reductions of natural frequencies. J Sound Vibration 2007; 302(4–5):1037–47. [10] Wen CM, Hung SL, Huang CS, Jan JC. Unsupervised fuzzy neural networks for damage detection of structures. Struct Control Health Monit 2007;14(1): 144–61. [11] Liu J, Wang G, Chen Y. Research and application of GA neural network model on dam displacement forecasting. ASCE conf. proc., vol. 323. Long Beach (CA): ASCE; 2008. p. 69. [12] Joghataie A, Dizaji MS. Nonlinear analysis of concrete gravity dams by neural networks. In: Ao SI, Gelman L, Hukins DWL, Hunter A, Korsunsky AM, editors. World congress on engineering 2009. Lecture notes in engineering and computer science, vols. I and II. Hong Kong: Int. Assoc. Engineers; 2009. p. 1022–7. [13] Yi XD, Xu F, Jiang CK. Research on dam deformation forecast model based on genetic algorithm neural network. In: Wen TD, editor. ISTM/2009: 8th international symposium on test and measurement, vols. 1–6. Hong Kong: International Academic Publishers Ltd.; 2009. p. 1536–9. [14] Montgomery DC, Runger GC. Applied statistics and probability for engineers. John Wiley & Sons, Inc.; 1994. [15] Draper NR, Smith H. Applied regression analysis. John Wiley & Sons, Inc.; 1981. [16] Rocha M, Serafim J, Silveira A. A method of quantitative interpretation of the results obtained in the observation of the dams. R.84, Q.21. In: VI ICOLD congress. vol. II. New York. 1958. p. 927–60. [17] Bishop CM. Neural networks for pattern recognition. Oxford: Clarendon Press; 1995. [18] Willm G, Beaujoint N. Les méthodes de surveillance des barrages au service de la production hidraulique déléctricité de france. Problémes anciens et solutions nouvelles. R.30, Q.34. In: IX ICOLD congress. vol. III. Istanbul. 1967. p. 529–50. [19] Johnson RA, Wichern DW. Applied multivariate statistical analysis. 4th ed. Prentice Hall; 1998. [20] Kutner MH, Nachtsheim CJ, Neter J, Li W. Applied linear statistical models. 5th ed. McGraw-Hill Higher Education; 2004. [21] Rocha M, Cortez P, Neves J. Evolution of neural networks for classification and regression. Neurocomputing 2007;70(16–18):2809–16. [22] Patterson DW. Artificial neural networks: theory and applications. Singapore: Prentice Hall; 1996. [23] Almeida LB. Section C.1.2—multilayer perceptron. Handbook of neural computation. Oxford. 1997. p. 34.