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Statistical analysis and structural identification in concrete dam monitoring
Engineering Structures 29 (2007) 110–120 www.elsevier.com/locate/engstruct
Statistical analysis and structural identification in concrete dam monitoring A. De Sortis a,∗ , P. Paoliani b a Department of Civil Protection, Via Vitorchiano 4, 00189 Roma, Italy b Large Dams National Authority, Via Curtatone 3, 00185 Roma, Italy
Received 18 October 2005; received in revised form 24 March 2006; accepted 18 April 2006 Available online 30 June 2006
Abstract Large dam monitoring systems are usually based on both boundary conditions (temperature, rainfall, water level, etc.) and structural responses (i.e. displacements, rotations, pore pressures, etc.). Statistical analysis tools are widely used to compare the current response of the dam to a whole string of recorded data, in order to highlight in a timely manner eventual unwanted behaviours. The main drawback of this approach is that the structural response quantities are related to the external loads using analytical functions, whose parameters do not have physical meaning. Another option is using the structural identification technique, based on finite element models of the structure, that can be usefully adopted to obtain an estimate of true physical parameters, such as the elastic Young modulus, an overall indicator of structural integrity. In this paper, some hollow buttress gravity dams, built in Italy some decades ago, have been considered. The horizontal upstream–downstream crest displacement with respect to the foundation of each buttress, generally recorded with an acceptable degree of accuracy by pendulum instruments, is mainly induced by variations of air temperature and reservoir water level. Eventual non-reversible effects, caused by the accumulation of permanent drifts due to concrete deterioration, can also be evidenced. Two different procedures have been compared: a statistical approach and a structural identification technique. Besides the additional advantages in terms of information about the structural integrity of the dams, the structural identification results provide also a higher degree of accuracy in predicting the future behaviour of the structure. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Structural monitoring; Statistical analysis; Structural identification; Structural integrity
1. Introduction Large dams, more than other engineering constructions, have strong interactions with environmental, hydraulic and geomechanical factors (i.e. air and water temperature, water level, pore pressure and uplift, rock deformability and so on), each of which influences the structural behaviour. Therefore the detection of significant indicators of structural behaviour poses a constant challenge for engineers involved in large dam safety control. Actually it is not easy to get reliable measures by means of instruments often placed in hostile environments, where humidity, temperature change, freeze and uncomfortable access put to the test any kind of mechanical and electronic tool. So, even if good records are available, data handling is essential to attain dam safety control [1]. ∗ Corresponding author. Tel.: +39 06 6820 4181; fax: +39 06 6820 2877.
E-mail addresses: [email protected] (A. De Sortis), [email protected] (P. Paoliani). c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.04.022
Statistical analysis is a well-known method, useful for setting the rough string of measures in mathematical expressions. The advantages of the method consist in simplicity of formulation, speed of execution, availability of any kind of correlation between governing and dependent parameters. The main drawback is that statistical coefficients do not have physical meaning, which prevents the engineer from going back to unexpected structural response causes. So it seems of interest to analyse the collected measures using also structural identification techniques, in order to correlate the dam behaviour with intrinsic parameters of the dam, such as structure size, boundary conditions, and elastic properties. In the following a comparison is given of statistical and identification procedures, using the recorded response of some hollow buttress gravity dams. In Italy there are 11 hollow buttress gravity dams, whose maximum height varies from 30 to over 100 m. Fig. 1 shows downstream and upstream views of two dams. The dams are
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Fig. 1. Downstream view of Sabbione dam (a) and upstream view of Malga Bissina dam (b).
The statistical and identification processing is outlined in the following for Ancipa dam, while the other structures will be referred to in the concluding remarks. In any case, the analyses performed deal with few and reliable measures, such as air mean daily temperature (T ), reservoir water level above the foundation (Q) and upstream–downstream crest displacements (D) with respect to the foundation of each buttress. The last choice was made in order to take into account just the contribution of the structure deformation to the crest displacement. An example of data records is shown in Fig. 3, where the available measures spanning over 40 years of monitoring are reported; some lack of data concerning the crest upstream–downstream displacement (D record) is due to instrument faults and rehabilitation works. 2. Statistical analysis For a long time now, statistical procedures have been applied to dam safety to find out the contribution of external loads to structure deformation and to identify irreversible components in the structural response. An analytical formulation gives the upstream–downstream crest displacement D j (where j is the time step, a day in the present analyses) as the sum of three terms: the first is due to air temperature change (D Tj ) and the Q
Fig. 2. Malga Bissina dam: vertical (a) and horizontal (b) sections (units: m).
each subdivided into buttresses having (Fig. 2) T shape in plan, with the larger side located upstream, triangular outline and the same slope for the upstream and downstream faces (inclination with respect to the vertical: almost 24◦ ). The thickness of the buttress walls increases from top to bottom with a constant slope. The total width of the buttress (18 or 22 m, depending on the dam), the shape and dimension of the crest and the slopes are the same for all the buttresses, thus implying that a shorter element can be obtained by “cutting” a taller one. Hence, the geometry of a buttress can be uniquely determined from its height. Permanent vertical joints are located between the elements. In the case of structural regular behaviour, the amount of joint opening assures that the response of each buttress is independent from those of the adjacent elements, a hypothesis that forms the basis of the subsequent analyses. When joints come into contact, the structural dam behaviour no longer lies in the upstream–downstream plane and a 3D analysis of the whole dam is required. The reliability of the statistical data process depends on long records of relevant and reliable measures. For this reason a first attempt to apply statistical and identification procedures has been performed on hollow buttress gravity dams, built in the 1950’s, for which records of the upstream–downstream crest displacements, measured with a high degree of accuracy by direct pendulum devices, span over decades. Thanks to its standardized shape, this dam typology is suitable for comparison of different structures in different environmental conditions. In the past, several studies have been devoted to this dam typology, as reported in [2–5].
second is related to the hydrostatic pressure (D j ); the third term takes into account unexpected behaviour, in the following called the trend line (D Aj ): Q
D j = D Tj + D j + D Aj . The simplest expressions for each term are: 2π j 2π j T D j = a cos + b sin 365 365 Q
(1)
(2)
D j = c1 Q j + c2 Q 2j
(3)
D Aj = d0 + d1 j.
(4)
The unknown coefficients (a, b, c1 , c2 , d0 and d1 ) are computed by minimizing the difference between the real measures and the analytical expression (1) by using the least squares minimum method. For the set of data shown in Fig. 3, the procedure gives the results reported in Fig. 4. Different time intervals (ranging from 2 to 15 years, as highlighted in Fig. 5) were tested for the assessment of the correlation coefficients. As expected, the longer the time interval, the more accurate the prediction of the crest displacement. Fig. 5 shows that after 10 years it is possible to obtain an accurate and stable regression. The statistical procedures allow one to single out the influence of each term of the regression, air temperature change, hydrostatic pressure and the trend line, as illustrated in Figs. 6 and 4. The same procedure has been applied also to other buttresses, as discussed in the concluding remarks. 3. Structural identification In statistical analysis the measured response of the structure is approximated using the analytical expression
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Fig. 3. Upstream–downstream crest displacements (D, in mm, positive upstream), mean daily air temperature (T , in ◦ C) and water level above the foundations (Q, in m) at Ancipa dam, main buttress.
Fig. 4. Statistical procedure: measured, computed crest upstream–downstream displacements and the trend line contribution.
(1). The regression algorithm calculates the values of the unknown coefficients in order to minimize the difference between measured and analytical responses. The coefficients of the approximating functions (2) and (3) do not have a physical meaning and hence they cannot furnish any relevant information about the integrity conditions of the structure. On the other hand, structural identification techniques can be useful in the integrity assessment of concrete dams [6,7]. The aim of the structural identification approach proposed in this paper is to rewrite the expressions (2) and (3) using coefficients related to: (i) the geometry of the buttress (in particular its height); (ii) physical properties of the concrete (elastic modulus and thermal coefficient); and (iii) external loads (air
Fig. 5. Statistical procedure: percentage errors in assessing the regression coefficients as a function of the time interval extension used in the analyses.
temperature change and water pressure). General closed form solutions are not available for the problem at hand (particular cases are discussed in the following). Therefore parametric analyses using numerical models have been performed to obtain relationships between external loads and crest displacements. The unknown parameters are then identified applying wellknown optimization techniques. The details are reported in the following paragraphs. 3.1. Effects of hydrostatic pressure Two different sets of numerical models (Fig. 7) subjected to water load have been used: (i) a 2D finite difference model, approximating the size and the stiffness of the real buttress with
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Fig. 6. Statistical procedure: contribution of the air temperature change and hydrostatic pressure to the upstream–downstream crest displacement.
a plane structure; and (ii) a 3D finite element model, that gives an accurate description of the real structure. Both the 2D and 3D models were restrained at the base. A parametric analysis was conducted varying the values of the elastic modulus and buttress height over the whole ranges typical for this kind of structure. For each model, the variation of the upstream–downstream crest displacement with the water level above the foundations was calculated. It was found that the structural response is governed by the following non-dimensional parameters: Q Q D E ˆ Q d = qˆ = H ρw H 2 where D Q is the crest displacement due to water load, ρw is the water density, H is the height of the buttress and E is the Young modulus of the concrete. The relationships
113
Fig. 7. Structural identification: 3D (a) and 2D (b) models for non-dimensional analyses.
between the non-dimensional parameters are plotted in Fig. 8. The crest displacement is assumed positive upstream. The results obtained with 2D models for Q/H = 1 can be compared with the analytical expression proposed by Marcello and Spagnoletti [2]: D Q E 1 1 2 2 = + 2ν(1 + θ ) − θ (5) 8θ θ 2 ρw H 2 Q =1 H
where θ (i.e. the slope of the upstream or downstream faces) in the present case equals 0.45 and ν is the Poisson ratio. The values of (5) for ν = 0.15 and ν = 0.3 are reported in Fig. 8. In the non-dimensional plot the crest displacements computed with 2D models and related to different heights of the structure and mesh densities lie within a narrow band. The non-dimensional crest displacements computed with 3D models are almost coincident. In the following, a polynomial
Fig. 8. Structural identification: results from 2D and 3D models for hydrostatic pressure.
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Table 1 Q Values of the coefficients ci of the regression (6)
i
1
2
3
4
5
6
Q ci
0.04794
−0.78648
3.8031
−9.7953
9.9221
−5.1635
regression based on 3D models will be adopted, with expansion Q coefficients ci : 6 X DQ E Q = ci 2 ρw H i=1
Q H
i
.
(6)
The expansion coefficients are reported in Table 1. The expression (6) is valid when the buttress lies on a horizontal foundation plane. This situation is typical of the elements near the centre of the valley, while the lateral elements are characterized by significant differences between the elevations of right and left side foundations. The 3D models have been modified to account for this aspect and the analyses have been repeated, producing the results reported in Fig. 9. The parameter governing this aspect is η = 1z/H , where 1z is the difference between the elevations of left and right side foundations. It was found that the expression (6) can be replaced with a more general one: 3 Q ≤ η2 0, Q H D E i (7) = X 6 3 3 Q Q ρw H 2 Q 2 2. − η , > η c i H H i=1 Expression (7) reduces to (6) when η = 0. The regression curves are compared in Fig. 9 with results from 3D finite element models; a satisfactory agreement is seen.
In the methodology proposed herein the structural response is assumed linear elastic and the Young modulus is considered uniform over the whole buttress. For undamaged structures the first hypothesis holds, due to the very low allowable stresses assumed in the design. The second assumption results from the availability of one measurement point (i.e. the top of the buttress), while a larger set of measurements should be used to reliably estimate the spatial variation of mechanical properties. In damaged structures cracks act as sources of nonlinearity. In these cases the identification procedure presented herein leads to a reduced value of the global elastic modulus, thus highlighting the presence of a deterioration process, but the damaged zone cannot be localized. 3.2. Thermal effects While the hydrostatic load on a buttress can be accurately modelled on the basis of the reservoir water level, an accurate description of the thermal load requires a detailed knowledge of the temperature values at several points of the structure. This level of knowledge is usually not available and, as often as not, only the maximum and minimum daily air temperatures are known. Thus, a rough simplification of the real phenomenon was attempted, in order to relate the crest upstream–downstream displacement to the mean daily air temperature. Initially, the following assumptions were made: (i) same temperature for the points on the submerged portion of the upstream face; this temperature value is proportional to the temperature of the water; (ii) the water temperature assumed constant with depth and time; (iii) to the points lying on the dry portion of the upstream face and on the downstream face the same temperature is assigned; this temperature is proportional to the mean daily air temperature. Under these hypotheses, the
Fig. 9. Structural identification: results from 3D models for hydrostatic pressure on elements with different foundation elevations.
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Fig. 10. Structural identification: results from 2D and 3D models for thermal effects.
temperature time change of each point of the upstream and downstream faces is related only to the time variation of both the water level and the mean daily air temperature. The distribution of temperature at the internal points of the structure and the effects in terms of crest displacement have been studied with the two different sets of numerical models described previously (Fig. 7). In 2D models, a coupled thermal–structural steady state analysis was made: boundary conditions included only the temperature of the faces, while the internal distribution of the temperatures was calculated using the code. In 3D models a linear distribution of the temperatures along the buttress horizontal sections was assumed, so the temperatures of the internal points were included in the boundary conditions and plain structural analyses were performed. A parametric analysis was conducted varying the thermal expansion coefficient, buttress height and temperature difference between dry and wet contours. The variation of the upstream–downstream crest displacement with the water level above the foundation was calculated for each model. It was found that the structural response is governed by the following non-dimensional parameters: Q DT T ˆ qˆ = d = α H 1T H where D T is the crest displacement due to thermal effects, α is the thermal expansion coefficient, H is the height of the buttress and 1T is the temperature difference between dry and wet contours. The relationships between the non-dimensional parameters are plotted in Fig. 10. Also in this case the crest displacement is assumed positive in the upstream direction. The results obtained with 2D and 3D models are very close, also considering the differences in thermal boundary conditions. In the following, a
linear regression based on 3D models will be adopted: Q DT T =c α H 1T H
(8)
where c T = 1.1835. The expression (8) is valid when the buttress lies on a horizontal foundation plane. Again the 3D models have been modified for considering buttresses with different foundation elevations and the analyses have been repeated, yielding the results reported in Fig. 11. It was found that expression (8) can be replaced with a more general one: 5 Q 0, ≤ η T D H 6 = (9) Q 5 Q 5 α H 1T T c − η , > η. H 6 H 6 Expression (9) reduces to (8) when η = 0. The regression curves are compared in Fig. 11 with the results obtained using 3D finite element models; a satisfactory agreement is seen. On assuming Q/H = 0, the expressions (8) and (9) give D T = 0, i.e. no thermal effect is present for very low reservoir levels, whatever the air temperature history is. This can be explained from considering that the length of the upstream face is almost the same as that of the downstream one. Thus, if the temperature variations are the same on the two faces, their deformations are almost identical. This means that the crest moves up or down, with negligible displacement component in the upstream–downstream direction. This behaviour can actually be observed in most cases. When only one face is exposed to solar radiation, the hypothesis of uniform dry contour temperature is not consistent. To account also for this Q behaviour, expression (9) can still be used, with H = 1 d and 1T = 1T , i.e. the temperature difference between the
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Fig. 11. Structural identification: results from 3D models for thermal effects on elements with different foundation elevations.
downstream and upstream faces of the buttress. The latter phenomenon is evident only for low reservoir level and vanishes when the reservoir is full, so the following expression, obtained Q d, from (9), can be proposed, with H = 1 and 1T = 1T Q multiplied by the term 1 − H : DT = cT d α H 1T
5 Q 1− η 1− . 6 H
(10)
Considering all the above described effects, the following general expression can be proposed for the structural identification procedure: Q 5 Q T d 1 − 5η 1 − ≤ η , c α H 1T 6 H H 6 D T = c T α H 1T Q − 5 η (11) H 6 Q Q 5 + ∆T d 1 − 5η 1− , > η. 6 H H 6 3.3. Identification algorithm On the basis of the results of the parametric analyses reported in the previous paragraphs, the displacement Dt of the buttress crest relative to its foundation at time t can be expressed as the sum of three terms (superscripts T, Q and A refer respectively to hydrostatic, temperature and other effects): Q
Dt = Dt (E) + DtT (t0 , t˜, Tw , β, ) + DtA (d0 , d1 ) i 6 ρw H 2 X Q Qt Q c Dt (E) = E i=1 i H
(12) (13)
DtT (t0 , t˜, Tw , β, ) ( T bt (t0 , t˜) − Tw = c αH βT bt (t0 , t˜) 1 − + T
bt (t0 , t˜) Q H
DtA (d0 , d1 ) = d0 + d1 t.
bt (t0 , t˜) Q H !)
!
(14) (15)
For the sake of simplicity, in the above expressions the case η = 0 has been considered. Expression (13) has been obtained directly from (6), in which the function Q t is the water level above the foundation at time t. Expression (14) has been obtained from (11) assuming: bt (t0 , t˜) − Tw 1T = β T t˜ Z 1 t−t0 + 2 b Tτ dτ. Tt (t0 , t˜) = t˜ t−t0 − t˜
(16) (17)
2
Expressions (16) and (17) mean that the temperature of the dry contour of the buttress is proportional, through a parameter to be identified, β, to a suitable mean value of the air bt . This value is obtained by integrating the temperature T mean daily temperature Tτ over a time interval called t˜. This operation affects the computed crest displacement and results in a smoothing of the measured temperature time history. In fact, abrupt temperature changes do not produce significant effects on the crest displacement, due to the thermal inertia of the structure. Also the presence of the parameter t0 , that governs the time shift between the structure response and the external temperature, is due to the thermal inertia of the concrete. The parameter Tw is the constant temperature of the wet contour of the structure. The above parameters t˜, t0 and Tw are included in
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the identification procedure. Furthermore, to obtain expression bt : (14) the term Q in (11) has been replaced by Q bt (t0 , t˜) = 1 Q t˜
Z
t−t0 + 2t˜
t−t0 − 2t˜
Q τ dτ
(18)
for which the same considerations as were already reported for bt are valid. Finally, to obtain (14) from (11) the following T substitution has been made, introducing the parameter to be identified: d = T bt (t0 , t˜). 1T
(19)
For the trend line term (15), a simple linear expression has been selected, where the parameters d0 and d1 (respectively the initial mean value of the crest displacement and the daily gradient of the irreversible displacement) have to be identified. Obviously, different analytical expressions can be used, depending on the characteristics of the problem at hand. Let us collect the measured crest displacement time history in a vector called z, the parameters to be identified in the vector x: x = {E t0 t˜ Tw β d0 d1 }
T
(20)
and the differences between measured and analytical displacements in the vector e. It can be written as e = z − h(x)
(21)
where the vector h(x) collects the time history of the analytical crest displacements calculated with expression (12) for the x values of the parameters. To assemble the above vectors, the time variable contained in (12) has to be discretized. Usually the step selected corresponds to one day. The vector e can be considered a function of the unknown parameters and can be expanded using Taylor’s theorem with initial point e0 = z − h(x0 ), corresponding to an initial guess x0 for the parameter set: e(x) = e0 + J(x − x0 ).
(22)
For the elements of the Jacobian matrix J the following definitions hold: ∂h i (x) Ji j = − = −Hi j . (23) ∂ x j x=x0 The matrix H = −J is the sensitivity matrix. The influence of a parameter on the response can be expressed by the corresponding diagonal term of the Fischer matrix A: A = HT H.
(24)
The lower the diagonal term, the less important the parameter for the response. The analytical expressions for the derivatives (23) are reported in the Appendix. The algorithm adopted in this paper for the nonlinear least squares minimization is based on the objective function: f (x) =
1 T 1 e e = [e0 + J(x − x0 )]T [e0 + J(x − x0 )]. 2 2
(25)
Fig. 12. Structural identification: measured and identified crest displacements and the drift term.
From expression (25) we get: ∇ f (x) = JT [e0 + J(x − x0 )]
(26)
∇ 2 f (x) = JT J.
(27)
Solving for the minimum by setting ∇ f (x) = 0 we obtain the approximate solution: xmin = x0 + (HT H)−1 HT [z − h(x0 )].
(28)
The solution (28) can be considered as a new guess x0 and a new iteration can be started, calculating new estimates of h(x0 ) and H and putting them into (28). The procedure continues until convergence occurs. Several improvements of the above outlined procedure are available in the literature [8,9]. In the present paper, the base formulation has been adopted, as it proved to be reasonably reliable. The identification procedure has been carried out using a purposely developed computer code. 3.4. Results For the set of data shown in Fig. 3, previously analysed by means of a statistical approach, the identification algorithm gives the results reported in Fig. 12. The procedure allowed us to single out the influence of each term of the regression, air temperature change, hydrostatic pressure and trend line, as illustrated in Figs. 12 and 13. On comparing Fig. 6 with Fig. 13 it is evident that the thermal contribution estimated with the structural identification procedure follows the air temperature cycles more closely than that estimated with the statistical analysis, in which a sinusoidal behaviour has been simulated. To deal with this shortcoming, the use of alternative analytical expressions for (2) involving higher order terms has been attempted, but no significant improvement has been achieved. The identification procedure has been applied also to other buttresses and the results are discussed in the following. 4. Conclusions Both statistical analysis and structural identification have been applied to 13 buttresses belonging to the three dams listed
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Fig. 13. Structural identification: contribution of the thermal load and water load to the total crest displacement. Table 2 Dimensional parameters of the dams considered Dam
Height (m)
Top elevation (m ab. sea level)
Reservoir volume (×106 m3 )
Malga Bissina Ancipa Sabbione
84.0 111.9 63.6
1790.0 952.4 2461.6
62.37 31.05 44.83
in Table 2. A comparison of the accuracy of the two methods is reported in Table 3. It can be observed that the coefficients of correlation and the standard deviations of the error are similar for statistical and identification analyses, the latter method showing a better accuracy. The error of the estimate is close to the resolution error of the field measures, drawing attention to the excessive approximation in field operation and instrument care. Statistical and identification procedures lead to comparable errors in the assessment of upstream–downstream crest displacements and therefore both methods are useful in anticipating the structural response for the hollow gravity dam previously discussed. In Fig. 14 two applications of the structural identification technique are shown: the upstream–downstream crest displacements measured during a five-year interval are plotted using a narrow predictive band (±2 times the error standard deviation), computed on the basis of the previous 10 years’ monitoring. The predictive band for the shorter buttress, characterized by a smaller displacement amplitude, appears larger than the corresponding one for the taller element, because the measuring errors have a greater influence on the accuracy of the identification solution. Table 4 reports the values of the eight parameters introduced in the procedure and, in brackets, the corresponding normalized diagonal terms of the Fischer matrix (24). The normalization has been performed by dividing Aii by the number of measurement days M, in order to compare different dams characterized by different lengths of the measurements interval. Finally the square root was calculated. Firstly one can observe the almost uniform values of the Young moduli E for each dam; at the same time, due to the shorter time interval of available measures, for the Sabbione
Fig. 14. Structural identification: comparison between the measured displacement and predicted confidence interval for buttresses 5 (a) and 1 (b) of Ancipa dam.
dam the value of E identified appears to be too high and the Fischer terms reveal an inaccurate prediction. As previously stated, a longer time integration interval allows a better estimation of the parameters and reduces the ill-conditioning aspects always inherent in the inverse approach. Furthermore the accuracy of the displacement measures has a similar influence, as can be stated on observing the higher values of the Fischer terms for the highest buttresses. Worthy of interest are the values identified for the parameters t0 (time shift between the structure response and the external air temperature), t˜ (the mean air temperature is calculated over this time interval), Tw (mean temperature of the wet contour of the structure) and (parameter for correlating the temperature difference between the upstream and downstream faces of the buttress with the mean air temperature) which are well related to the different environmental conditions and thermal inertia of each dam. Structural identification allows a useful determination of the equivalent Young modulus (relative to the global behaviour of the buttress, rather than to the local properties of the material), and of its change in time and for each different buttress of the same dam. So it can be used in investigating dam structural defects and ageing. The method could also be applied in evaluating the effectiveness of remedial works, comparing the parameters identified before and after the intervention, and
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A. De Sortis, P. Paoliani / Engineering Structures 29 (2007) 110–120 Table 3 Comparison between statistical analysis and structural identification Dama
Butt.b
Hc
Statistical analysis
Structural identification
Re
d1 f
σe g
σe D (%)
Re
d1 f
σe g
σe D (%)
3.1 4.7 5.8 8.2 8.4
0.983 0.947 0.959 0.968 0.967
0.10 0.07 0.15 0.18 0.27
0.71 1.32 1.25 1.13 1.28
22.9 28.1 21.5 13.8 15.2
0.988 0.987 0.988 0.989 0.989
0.11 0.11 0.08 0.20 0.29
0.44 0.61 0.61 0.75 0.84
14.2 12.3 10.5 9.1 10.0
A
1 2 3 4 5
S
6 7 8
56.7 57.2 49.8
3.2 3.4 2.4
0.959 0.958 0.948
0.13 0.08 0.02
0.47 0.52 0.42
14.7 15.3 17.5
0.956 0.652 0.944
0.04 0.00 0.03
0.47 0.55 0.43
14.7 16.2 17.7
M
4 6 7 8 9
67 79.5 81.5 78.5 78.5
4.9 7.3 7.4 7.4 6.7
0.877 0.909 0.915 0.916 0.896
0.09 0.02 0.06 0.05 0.11
0.98 1.23 1.23 1.24 1.17
20.0 16.8 16.6 16.8 17.5
0.933 0.950 0.953 0.954 0.933
0.05 0.03 0.02 0.00 0.06
0.72 0.91 0.91 0.90 0.94
14.7 12.5 12.3 12.1 14.0
a b c d e f g
43 58 73 95 104
Dd
A = Ancipa (15 years’ monitoring), S = Sabbione (5 years), M = Malga Bissina (9 years). Buttress number. Buttress mean height (m). Upstream–downstream crest displacement semi-amplitude (mm). Correlation coefficient. Annual gradient of the irreversible displacement (mm/year). Standard error of estimate (mm).
Table 4 Values of the parameters obtained with structural identification (in parentheses: the corresponding normalized diagonal terms of the Fischer matrix) Dama
Butt.b
1 2 A
3 4 5 6
S
7 8 4 6
M
7 8 9
E (MPa) √ A11 /M mm
t0 (days) √ A22 /M
t˜ (days) √ A33 /M mm day
Tw (◦ C) √ A44 /M
β √
A55 /M
√
A66 /M
GPa
mm day
mm ◦C
(mm)
(mm)
22 524 (0.013) 14 786 (0.086) 15 745 (0.138) 18 131 (0.222) 18 900 (0.263)
24 (0.023) 23 (0.038) 25 (0.046) 28 (0.058) 29 (0.061)
57 (0.006) 53 (0.011) 55 (0.013) 65 (0.015) 67 (0.016)
5.6 (0.31) 11.0 (0.48) 9.8 (0.65) 12.3 (0.91) 10.3 (1.01)
0.50 (4.4) 0.68 (6.9) 0.61 (9.3) 0.57 (13.0) 0.53 (14.5)
0.14 (3.2) −0.12 (3.2) −0.13 (3.2) −0.16 (3.2) −0.05 (3.2)
37 038 (0.010) 32 375 (0.013) 38 316 (0.006)
4 (0.052) 5 (0.051) 3 (0.040)
27 (0.022) 31 (0.021) 29 (0.016)
1.2 (0.41) 1.5 (0.41) 1.1 (0.33)
0.38 (2.5) 0.39 (2.5) 0.35 (2.0)
0.41 (1.7) 0.50 (1.7) 0.37 (1.7)
17 028 (0.110) 13 918 (0.261) 13 661 (0.288) 13 692 (0.262) 19 095 (0.134)
8 (0.055) 8 (0.083) 9 (0.076) 8 (0.082) 9 (0.062)
21 (0.021) 23 (0.032) 27 (0.030) 23 (0.032) 25 (0.024)
5.3 (0.57) 5.6 (0.71) 5.5 (0.73) 7.0 (0.70) 7.1 (0.70)
0.46 (4.9) 0.55 (6.1) 0.54 (6.2) 0.56 (6.0) 0.46 (6.0)
−0.08 (2.1) −0.08 (2.1) −0.11 (2.1) −0.12 (2.1) −0.10 (2.1)
a A = Ancipa (15 years’ monitoring), S = Sabbione (5 years), M = Malga Bissina (9 years). b Buttress number.
d0 (mm) √ A77 /M mm mm
mm d1 year √ A88 /M mm mm year
−63.2 (1) −70.1 (1) −64.6 (1) −48.3 (1) −51.0 (1)
0.11 (8.7) 0.11 (8.7) 0.08 (8.7) 0.20 (8.7) 0.29 (8.7)
2.2
0.03 (2.8) −0.04 (2.8) −0.05 (2.8)
(1) 2.6 (1) 1.5 (1) 2.1 (1) 3.4 (1) 3.0 (1) 4.0 (1) 3.8 (1)
0.05 (4.8) −0.03 (4.8) 0.02 (4.8) −0.01 (4.8) 0.06 (4.8)
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can also be extended to other phenomena and events, such as earthquake or rapid water level drawdown, to check, for instance, how the elastic constant of the concrete depends on the load applying velocity. Acknowledgements The authors wish to thank all the people that allowed us to get reliable measures in dam monitoring, especially those at ENEL and from the other companies that provided the data for the statistical and identification procedures outlined in this paper. Appendix. Elements of the sensitivity matrix
∂ Dt =1 ∂d0 ∂ Dt Ht8 = =t ∂d1 o bt 1n ∂T = Tt−t − t˜ − Tt−t + t˜ 0 2 0 2 ∂t0 t˜ o n bt 1 ∂Q = Q t−t − t˜ − Q t−t + t˜ 0 2 0 2 ∂t0 t˜ o 1 Z t−t0 + 2t˜ n bt ∂T 1 − 2 = T Tτ dτ t˜ + T t−t0 − 2t˜ ∂ t˜ 2t˜ t−t0 + 2 t˜ t−t0 − 2t˜ o 1 Z t−t0 + 2t˜ bt 1 n ∂Q = Q t−t + t˜ + Q t−t − t˜ − 2 Q τ dτ. 0 2 0 2 ∂ t˜ 2t˜ t˜ t−t0 − t˜ Ht7 =
2
In the following, the derivatives of the analytical expression (12) with respect to the parameters are reported. Their knowledge is needed to assemble the sensitivity matrix (23). i 6 ρw H 2 X ∂ Dt Q Qt =− Ht1 = c ∂E H E 2 i=1 i ( bt Q bt b ∂ Dt ∂T bt − Tw 1 ∂ Q t Ht2 = = cT α H β + βT ∂t0 ∂t0 H H ∂t0 ! ) b b b 1 ∂ Qt ∂ Tt Qt b + 1− − Tt ∂t0 H H ∂t0 ( b bt bt Q ∂ Dt ∂T bt − Tw 1 ∂ Q t + βT Ht3 = = cT α H β H ∂ t˜ ∂ t˜ ∂ t˜ H ! ) bt bt bt 1 ∂Q Q ∂T b 1− − Tt + H H ∂ t˜ ∂ t˜ ∂ Dt bt = −c T α Q ∂ Tw ∂ Dt = cT α Ht5 = ∂β Ht4 =
b ∂ Dt bt 1 − Q t Ht6 = = cT α H T ∂ H
!
References [1] ICOLD. Bulletin 118: Automated dam monitoring systems — guidelines and case histories. Paris: International Commission on Large Dams; 2000. [2] Marcello C, Spagnoletti S. On the structural behaviour of hollow buttresses gravity dams — Theoretical behaviour. L’Energia elettrica 1960;37(10) [in Italian]. [3] Appendino M, Di Monaco F, Garino A, Manzo F, Scarinci S. Specific and general trends of the ageing of buttress dams as revealed by investigations carried out on Ancipa dam. In: Proceedings of the 17th conference on large dams. 1991. [4] Brindisi P, De Sortis A, Di Lemma G, Orsini G. Seismic analysis of hollow buttress gravity dams. In: Proceedings of the 12th European conference on earthquake engineering. 2002. [5] De Sortis A, Paoliani P. Statistical and structural identification techniques in structural monitoring of concrete dams. In: Proceedings of the 73rd annual meeting of international commission on large dams. 2005. [6] Ardito R, Bartalotta P, Ceriani L, Maier G. Diagnostic inverse analysis of concrete dams with statical excitation. Journal of Mechanical Behavior of Materials 2004;15(6):381–9. [7] Fedele R, Maier G, Miller B. Identification of elastic stiffness and local stresses in concrete dams by in situ tests and neural networks. Structure and Infrastructure Engineering 2005;1(3):165–80. [8] Levenberg K. A method for the solution of certain problems in least squares. Quarterly of Applied Mathematics 1944;2:164–8. [9] Marquardt D. An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics 1963;11:431–41.
Statistical analysis and structural identification in concrete dam monitoring A. De Sortis a,∗ , P. Paoliani b a Department of Civil Protection, Via Vitorchiano 4, 00189 Roma, Italy b Large Dams National Authority, Via Curtatone 3, 00185 Roma, Italy
Received 18 October 2005; received in revised form 24 March 2006; accepted 18 April 2006 Available online 30 June 2006
Abstract Large dam monitoring systems are usually based on both boundary conditions (temperature, rainfall, water level, etc.) and structural responses (i.e. displacements, rotations, pore pressures, etc.). Statistical analysis tools are widely used to compare the current response of the dam to a whole string of recorded data, in order to highlight in a timely manner eventual unwanted behaviours. The main drawback of this approach is that the structural response quantities are related to the external loads using analytical functions, whose parameters do not have physical meaning. Another option is using the structural identification technique, based on finite element models of the structure, that can be usefully adopted to obtain an estimate of true physical parameters, such as the elastic Young modulus, an overall indicator of structural integrity. In this paper, some hollow buttress gravity dams, built in Italy some decades ago, have been considered. The horizontal upstream–downstream crest displacement with respect to the foundation of each buttress, generally recorded with an acceptable degree of accuracy by pendulum instruments, is mainly induced by variations of air temperature and reservoir water level. Eventual non-reversible effects, caused by the accumulation of permanent drifts due to concrete deterioration, can also be evidenced. Two different procedures have been compared: a statistical approach and a structural identification technique. Besides the additional advantages in terms of information about the structural integrity of the dams, the structural identification results provide also a higher degree of accuracy in predicting the future behaviour of the structure. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Structural monitoring; Statistical analysis; Structural identification; Structural integrity
1. Introduction Large dams, more than other engineering constructions, have strong interactions with environmental, hydraulic and geomechanical factors (i.e. air and water temperature, water level, pore pressure and uplift, rock deformability and so on), each of which influences the structural behaviour. Therefore the detection of significant indicators of structural behaviour poses a constant challenge for engineers involved in large dam safety control. Actually it is not easy to get reliable measures by means of instruments often placed in hostile environments, where humidity, temperature change, freeze and uncomfortable access put to the test any kind of mechanical and electronic tool. So, even if good records are available, data handling is essential to attain dam safety control [1]. ∗ Corresponding author. Tel.: +39 06 6820 4181; fax: +39 06 6820 2877.
E-mail addresses: [email protected] (A. De Sortis), [email protected] (P. Paoliani). c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.04.022
Statistical analysis is a well-known method, useful for setting the rough string of measures in mathematical expressions. The advantages of the method consist in simplicity of formulation, speed of execution, availability of any kind of correlation between governing and dependent parameters. The main drawback is that statistical coefficients do not have physical meaning, which prevents the engineer from going back to unexpected structural response causes. So it seems of interest to analyse the collected measures using also structural identification techniques, in order to correlate the dam behaviour with intrinsic parameters of the dam, such as structure size, boundary conditions, and elastic properties. In the following a comparison is given of statistical and identification procedures, using the recorded response of some hollow buttress gravity dams. In Italy there are 11 hollow buttress gravity dams, whose maximum height varies from 30 to over 100 m. Fig. 1 shows downstream and upstream views of two dams. The dams are
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Fig. 1. Downstream view of Sabbione dam (a) and upstream view of Malga Bissina dam (b).
The statistical and identification processing is outlined in the following for Ancipa dam, while the other structures will be referred to in the concluding remarks. In any case, the analyses performed deal with few and reliable measures, such as air mean daily temperature (T ), reservoir water level above the foundation (Q) and upstream–downstream crest displacements (D) with respect to the foundation of each buttress. The last choice was made in order to take into account just the contribution of the structure deformation to the crest displacement. An example of data records is shown in Fig. 3, where the available measures spanning over 40 years of monitoring are reported; some lack of data concerning the crest upstream–downstream displacement (D record) is due to instrument faults and rehabilitation works. 2. Statistical analysis For a long time now, statistical procedures have been applied to dam safety to find out the contribution of external loads to structure deformation and to identify irreversible components in the structural response. An analytical formulation gives the upstream–downstream crest displacement D j (where j is the time step, a day in the present analyses) as the sum of three terms: the first is due to air temperature change (D Tj ) and the Q
Fig. 2. Malga Bissina dam: vertical (a) and horizontal (b) sections (units: m).
each subdivided into buttresses having (Fig. 2) T shape in plan, with the larger side located upstream, triangular outline and the same slope for the upstream and downstream faces (inclination with respect to the vertical: almost 24◦ ). The thickness of the buttress walls increases from top to bottom with a constant slope. The total width of the buttress (18 or 22 m, depending on the dam), the shape and dimension of the crest and the slopes are the same for all the buttresses, thus implying that a shorter element can be obtained by “cutting” a taller one. Hence, the geometry of a buttress can be uniquely determined from its height. Permanent vertical joints are located between the elements. In the case of structural regular behaviour, the amount of joint opening assures that the response of each buttress is independent from those of the adjacent elements, a hypothesis that forms the basis of the subsequent analyses. When joints come into contact, the structural dam behaviour no longer lies in the upstream–downstream plane and a 3D analysis of the whole dam is required. The reliability of the statistical data process depends on long records of relevant and reliable measures. For this reason a first attempt to apply statistical and identification procedures has been performed on hollow buttress gravity dams, built in the 1950’s, for which records of the upstream–downstream crest displacements, measured with a high degree of accuracy by direct pendulum devices, span over decades. Thanks to its standardized shape, this dam typology is suitable for comparison of different structures in different environmental conditions. In the past, several studies have been devoted to this dam typology, as reported in [2–5].
second is related to the hydrostatic pressure (D j ); the third term takes into account unexpected behaviour, in the following called the trend line (D Aj ): Q
D j = D Tj + D j + D Aj . The simplest expressions for each term are: 2π j 2π j T D j = a cos + b sin 365 365 Q
(1)
(2)
D j = c1 Q j + c2 Q 2j
(3)
D Aj = d0 + d1 j.
(4)
The unknown coefficients (a, b, c1 , c2 , d0 and d1 ) are computed by minimizing the difference between the real measures and the analytical expression (1) by using the least squares minimum method. For the set of data shown in Fig. 3, the procedure gives the results reported in Fig. 4. Different time intervals (ranging from 2 to 15 years, as highlighted in Fig. 5) were tested for the assessment of the correlation coefficients. As expected, the longer the time interval, the more accurate the prediction of the crest displacement. Fig. 5 shows that after 10 years it is possible to obtain an accurate and stable regression. The statistical procedures allow one to single out the influence of each term of the regression, air temperature change, hydrostatic pressure and the trend line, as illustrated in Figs. 6 and 4. The same procedure has been applied also to other buttresses, as discussed in the concluding remarks. 3. Structural identification In statistical analysis the measured response of the structure is approximated using the analytical expression
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Fig. 3. Upstream–downstream crest displacements (D, in mm, positive upstream), mean daily air temperature (T , in ◦ C) and water level above the foundations (Q, in m) at Ancipa dam, main buttress.
Fig. 4. Statistical procedure: measured, computed crest upstream–downstream displacements and the trend line contribution.
(1). The regression algorithm calculates the values of the unknown coefficients in order to minimize the difference between measured and analytical responses. The coefficients of the approximating functions (2) and (3) do not have a physical meaning and hence they cannot furnish any relevant information about the integrity conditions of the structure. On the other hand, structural identification techniques can be useful in the integrity assessment of concrete dams [6,7]. The aim of the structural identification approach proposed in this paper is to rewrite the expressions (2) and (3) using coefficients related to: (i) the geometry of the buttress (in particular its height); (ii) physical properties of the concrete (elastic modulus and thermal coefficient); and (iii) external loads (air
Fig. 5. Statistical procedure: percentage errors in assessing the regression coefficients as a function of the time interval extension used in the analyses.
temperature change and water pressure). General closed form solutions are not available for the problem at hand (particular cases are discussed in the following). Therefore parametric analyses using numerical models have been performed to obtain relationships between external loads and crest displacements. The unknown parameters are then identified applying wellknown optimization techniques. The details are reported in the following paragraphs. 3.1. Effects of hydrostatic pressure Two different sets of numerical models (Fig. 7) subjected to water load have been used: (i) a 2D finite difference model, approximating the size and the stiffness of the real buttress with
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Fig. 6. Statistical procedure: contribution of the air temperature change and hydrostatic pressure to the upstream–downstream crest displacement.
a plane structure; and (ii) a 3D finite element model, that gives an accurate description of the real structure. Both the 2D and 3D models were restrained at the base. A parametric analysis was conducted varying the values of the elastic modulus and buttress height over the whole ranges typical for this kind of structure. For each model, the variation of the upstream–downstream crest displacement with the water level above the foundations was calculated. It was found that the structural response is governed by the following non-dimensional parameters: Q Q D E ˆ Q d = qˆ = H ρw H 2 where D Q is the crest displacement due to water load, ρw is the water density, H is the height of the buttress and E is the Young modulus of the concrete. The relationships
113
Fig. 7. Structural identification: 3D (a) and 2D (b) models for non-dimensional analyses.
between the non-dimensional parameters are plotted in Fig. 8. The crest displacement is assumed positive upstream. The results obtained with 2D models for Q/H = 1 can be compared with the analytical expression proposed by Marcello and Spagnoletti [2]: D Q E 1 1 2 2 = + 2ν(1 + θ ) − θ (5) 8θ θ 2 ρw H 2 Q =1 H
where θ (i.e. the slope of the upstream or downstream faces) in the present case equals 0.45 and ν is the Poisson ratio. The values of (5) for ν = 0.15 and ν = 0.3 are reported in Fig. 8. In the non-dimensional plot the crest displacements computed with 2D models and related to different heights of the structure and mesh densities lie within a narrow band. The non-dimensional crest displacements computed with 3D models are almost coincident. In the following, a polynomial
Fig. 8. Structural identification: results from 2D and 3D models for hydrostatic pressure.
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Table 1 Q Values of the coefficients ci of the regression (6)
i
1
2
3
4
5
6
Q ci
0.04794
−0.78648
3.8031
−9.7953
9.9221
−5.1635
regression based on 3D models will be adopted, with expansion Q coefficients ci : 6 X DQ E Q = ci 2 ρw H i=1
Q H
i
.
(6)
The expansion coefficients are reported in Table 1. The expression (6) is valid when the buttress lies on a horizontal foundation plane. This situation is typical of the elements near the centre of the valley, while the lateral elements are characterized by significant differences between the elevations of right and left side foundations. The 3D models have been modified to account for this aspect and the analyses have been repeated, producing the results reported in Fig. 9. The parameter governing this aspect is η = 1z/H , where 1z is the difference between the elevations of left and right side foundations. It was found that the expression (6) can be replaced with a more general one: 3 Q ≤ η2 0, Q H D E i (7) = X 6 3 3 Q Q ρw H 2 Q 2 2. − η , > η c i H H i=1 Expression (7) reduces to (6) when η = 0. The regression curves are compared in Fig. 9 with results from 3D finite element models; a satisfactory agreement is seen.
In the methodology proposed herein the structural response is assumed linear elastic and the Young modulus is considered uniform over the whole buttress. For undamaged structures the first hypothesis holds, due to the very low allowable stresses assumed in the design. The second assumption results from the availability of one measurement point (i.e. the top of the buttress), while a larger set of measurements should be used to reliably estimate the spatial variation of mechanical properties. In damaged structures cracks act as sources of nonlinearity. In these cases the identification procedure presented herein leads to a reduced value of the global elastic modulus, thus highlighting the presence of a deterioration process, but the damaged zone cannot be localized. 3.2. Thermal effects While the hydrostatic load on a buttress can be accurately modelled on the basis of the reservoir water level, an accurate description of the thermal load requires a detailed knowledge of the temperature values at several points of the structure. This level of knowledge is usually not available and, as often as not, only the maximum and minimum daily air temperatures are known. Thus, a rough simplification of the real phenomenon was attempted, in order to relate the crest upstream–downstream displacement to the mean daily air temperature. Initially, the following assumptions were made: (i) same temperature for the points on the submerged portion of the upstream face; this temperature value is proportional to the temperature of the water; (ii) the water temperature assumed constant with depth and time; (iii) to the points lying on the dry portion of the upstream face and on the downstream face the same temperature is assigned; this temperature is proportional to the mean daily air temperature. Under these hypotheses, the
Fig. 9. Structural identification: results from 3D models for hydrostatic pressure on elements with different foundation elevations.
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Fig. 10. Structural identification: results from 2D and 3D models for thermal effects.
temperature time change of each point of the upstream and downstream faces is related only to the time variation of both the water level and the mean daily air temperature. The distribution of temperature at the internal points of the structure and the effects in terms of crest displacement have been studied with the two different sets of numerical models described previously (Fig. 7). In 2D models, a coupled thermal–structural steady state analysis was made: boundary conditions included only the temperature of the faces, while the internal distribution of the temperatures was calculated using the code. In 3D models a linear distribution of the temperatures along the buttress horizontal sections was assumed, so the temperatures of the internal points were included in the boundary conditions and plain structural analyses were performed. A parametric analysis was conducted varying the thermal expansion coefficient, buttress height and temperature difference between dry and wet contours. The variation of the upstream–downstream crest displacement with the water level above the foundation was calculated for each model. It was found that the structural response is governed by the following non-dimensional parameters: Q DT T ˆ qˆ = d = α H 1T H where D T is the crest displacement due to thermal effects, α is the thermal expansion coefficient, H is the height of the buttress and 1T is the temperature difference between dry and wet contours. The relationships between the non-dimensional parameters are plotted in Fig. 10. Also in this case the crest displacement is assumed positive in the upstream direction. The results obtained with 2D and 3D models are very close, also considering the differences in thermal boundary conditions. In the following, a
linear regression based on 3D models will be adopted: Q DT T =c α H 1T H
(8)
where c T = 1.1835. The expression (8) is valid when the buttress lies on a horizontal foundation plane. Again the 3D models have been modified for considering buttresses with different foundation elevations and the analyses have been repeated, yielding the results reported in Fig. 11. It was found that expression (8) can be replaced with a more general one: 5 Q 0, ≤ η T D H 6 = (9) Q 5 Q 5 α H 1T T c − η , > η. H 6 H 6 Expression (9) reduces to (8) when η = 0. The regression curves are compared in Fig. 11 with the results obtained using 3D finite element models; a satisfactory agreement is seen. On assuming Q/H = 0, the expressions (8) and (9) give D T = 0, i.e. no thermal effect is present for very low reservoir levels, whatever the air temperature history is. This can be explained from considering that the length of the upstream face is almost the same as that of the downstream one. Thus, if the temperature variations are the same on the two faces, their deformations are almost identical. This means that the crest moves up or down, with negligible displacement component in the upstream–downstream direction. This behaviour can actually be observed in most cases. When only one face is exposed to solar radiation, the hypothesis of uniform dry contour temperature is not consistent. To account also for this Q behaviour, expression (9) can still be used, with H = 1 d and 1T = 1T , i.e. the temperature difference between the
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Fig. 11. Structural identification: results from 3D models for thermal effects on elements with different foundation elevations.
downstream and upstream faces of the buttress. The latter phenomenon is evident only for low reservoir level and vanishes when the reservoir is full, so the following expression, obtained Q d, from (9), can be proposed, with H = 1 and 1T = 1T Q multiplied by the term 1 − H : DT = cT d α H 1T
5 Q 1− η 1− . 6 H
(10)
Considering all the above described effects, the following general expression can be proposed for the structural identification procedure: Q 5 Q T d 1 − 5η 1 − ≤ η , c α H 1T 6 H H 6 D T = c T α H 1T Q − 5 η (11) H 6 Q Q 5 + ∆T d 1 − 5η 1− , > η. 6 H H 6 3.3. Identification algorithm On the basis of the results of the parametric analyses reported in the previous paragraphs, the displacement Dt of the buttress crest relative to its foundation at time t can be expressed as the sum of three terms (superscripts T, Q and A refer respectively to hydrostatic, temperature and other effects): Q
Dt = Dt (E) + DtT (t0 , t˜, Tw , β, ) + DtA (d0 , d1 ) i 6 ρw H 2 X Q Qt Q c Dt (E) = E i=1 i H
(12) (13)
DtT (t0 , t˜, Tw , β, ) ( T bt (t0 , t˜) − Tw = c αH βT bt (t0 , t˜) 1 − + T
bt (t0 , t˜) Q H
DtA (d0 , d1 ) = d0 + d1 t.
bt (t0 , t˜) Q H !)
!
(14) (15)
For the sake of simplicity, in the above expressions the case η = 0 has been considered. Expression (13) has been obtained directly from (6), in which the function Q t is the water level above the foundation at time t. Expression (14) has been obtained from (11) assuming: bt (t0 , t˜) − Tw 1T = β T t˜ Z 1 t−t0 + 2 b Tτ dτ. Tt (t0 , t˜) = t˜ t−t0 − t˜
(16) (17)
2
Expressions (16) and (17) mean that the temperature of the dry contour of the buttress is proportional, through a parameter to be identified, β, to a suitable mean value of the air bt . This value is obtained by integrating the temperature T mean daily temperature Tτ over a time interval called t˜. This operation affects the computed crest displacement and results in a smoothing of the measured temperature time history. In fact, abrupt temperature changes do not produce significant effects on the crest displacement, due to the thermal inertia of the structure. Also the presence of the parameter t0 , that governs the time shift between the structure response and the external temperature, is due to the thermal inertia of the concrete. The parameter Tw is the constant temperature of the wet contour of the structure. The above parameters t˜, t0 and Tw are included in
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the identification procedure. Furthermore, to obtain expression bt : (14) the term Q in (11) has been replaced by Q bt (t0 , t˜) = 1 Q t˜
Z
t−t0 + 2t˜
t−t0 − 2t˜
Q τ dτ
(18)
for which the same considerations as were already reported for bt are valid. Finally, to obtain (14) from (11) the following T substitution has been made, introducing the parameter to be identified: d = T bt (t0 , t˜). 1T
(19)
For the trend line term (15), a simple linear expression has been selected, where the parameters d0 and d1 (respectively the initial mean value of the crest displacement and the daily gradient of the irreversible displacement) have to be identified. Obviously, different analytical expressions can be used, depending on the characteristics of the problem at hand. Let us collect the measured crest displacement time history in a vector called z, the parameters to be identified in the vector x: x = {E t0 t˜ Tw β d0 d1 }
T
(20)
and the differences between measured and analytical displacements in the vector e. It can be written as e = z − h(x)
(21)
where the vector h(x) collects the time history of the analytical crest displacements calculated with expression (12) for the x values of the parameters. To assemble the above vectors, the time variable contained in (12) has to be discretized. Usually the step selected corresponds to one day. The vector e can be considered a function of the unknown parameters and can be expanded using Taylor’s theorem with initial point e0 = z − h(x0 ), corresponding to an initial guess x0 for the parameter set: e(x) = e0 + J(x − x0 ).
(22)
For the elements of the Jacobian matrix J the following definitions hold: ∂h i (x) Ji j = − = −Hi j . (23) ∂ x j x=x0 The matrix H = −J is the sensitivity matrix. The influence of a parameter on the response can be expressed by the corresponding diagonal term of the Fischer matrix A: A = HT H.
(24)
The lower the diagonal term, the less important the parameter for the response. The analytical expressions for the derivatives (23) are reported in the Appendix. The algorithm adopted in this paper for the nonlinear least squares minimization is based on the objective function: f (x) =
1 T 1 e e = [e0 + J(x − x0 )]T [e0 + J(x − x0 )]. 2 2
(25)
Fig. 12. Structural identification: measured and identified crest displacements and the drift term.
From expression (25) we get: ∇ f (x) = JT [e0 + J(x − x0 )]
(26)
∇ 2 f (x) = JT J.
(27)
Solving for the minimum by setting ∇ f (x) = 0 we obtain the approximate solution: xmin = x0 + (HT H)−1 HT [z − h(x0 )].
(28)
The solution (28) can be considered as a new guess x0 and a new iteration can be started, calculating new estimates of h(x0 ) and H and putting them into (28). The procedure continues until convergence occurs. Several improvements of the above outlined procedure are available in the literature [8,9]. In the present paper, the base formulation has been adopted, as it proved to be reasonably reliable. The identification procedure has been carried out using a purposely developed computer code. 3.4. Results For the set of data shown in Fig. 3, previously analysed by means of a statistical approach, the identification algorithm gives the results reported in Fig. 12. The procedure allowed us to single out the influence of each term of the regression, air temperature change, hydrostatic pressure and trend line, as illustrated in Figs. 12 and 13. On comparing Fig. 6 with Fig. 13 it is evident that the thermal contribution estimated with the structural identification procedure follows the air temperature cycles more closely than that estimated with the statistical analysis, in which a sinusoidal behaviour has been simulated. To deal with this shortcoming, the use of alternative analytical expressions for (2) involving higher order terms has been attempted, but no significant improvement has been achieved. The identification procedure has been applied also to other buttresses and the results are discussed in the following. 4. Conclusions Both statistical analysis and structural identification have been applied to 13 buttresses belonging to the three dams listed
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Fig. 13. Structural identification: contribution of the thermal load and water load to the total crest displacement. Table 2 Dimensional parameters of the dams considered Dam
Height (m)
Top elevation (m ab. sea level)
Reservoir volume (×106 m3 )
Malga Bissina Ancipa Sabbione
84.0 111.9 63.6
1790.0 952.4 2461.6
62.37 31.05 44.83
in Table 2. A comparison of the accuracy of the two methods is reported in Table 3. It can be observed that the coefficients of correlation and the standard deviations of the error are similar for statistical and identification analyses, the latter method showing a better accuracy. The error of the estimate is close to the resolution error of the field measures, drawing attention to the excessive approximation in field operation and instrument care. Statistical and identification procedures lead to comparable errors in the assessment of upstream–downstream crest displacements and therefore both methods are useful in anticipating the structural response for the hollow gravity dam previously discussed. In Fig. 14 two applications of the structural identification technique are shown: the upstream–downstream crest displacements measured during a five-year interval are plotted using a narrow predictive band (±2 times the error standard deviation), computed on the basis of the previous 10 years’ monitoring. The predictive band for the shorter buttress, characterized by a smaller displacement amplitude, appears larger than the corresponding one for the taller element, because the measuring errors have a greater influence on the accuracy of the identification solution. Table 4 reports the values of the eight parameters introduced in the procedure and, in brackets, the corresponding normalized diagonal terms of the Fischer matrix (24). The normalization has been performed by dividing Aii by the number of measurement days M, in order to compare different dams characterized by different lengths of the measurements interval. Finally the square root was calculated. Firstly one can observe the almost uniform values of the Young moduli E for each dam; at the same time, due to the shorter time interval of available measures, for the Sabbione
Fig. 14. Structural identification: comparison between the measured displacement and predicted confidence interval for buttresses 5 (a) and 1 (b) of Ancipa dam.
dam the value of E identified appears to be too high and the Fischer terms reveal an inaccurate prediction. As previously stated, a longer time integration interval allows a better estimation of the parameters and reduces the ill-conditioning aspects always inherent in the inverse approach. Furthermore the accuracy of the displacement measures has a similar influence, as can be stated on observing the higher values of the Fischer terms for the highest buttresses. Worthy of interest are the values identified for the parameters t0 (time shift between the structure response and the external air temperature), t˜ (the mean air temperature is calculated over this time interval), Tw (mean temperature of the wet contour of the structure) and (parameter for correlating the temperature difference between the upstream and downstream faces of the buttress with the mean air temperature) which are well related to the different environmental conditions and thermal inertia of each dam. Structural identification allows a useful determination of the equivalent Young modulus (relative to the global behaviour of the buttress, rather than to the local properties of the material), and of its change in time and for each different buttress of the same dam. So it can be used in investigating dam structural defects and ageing. The method could also be applied in evaluating the effectiveness of remedial works, comparing the parameters identified before and after the intervention, and
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A. De Sortis, P. Paoliani / Engineering Structures 29 (2007) 110–120 Table 3 Comparison between statistical analysis and structural identification Dama
Butt.b
Hc
Statistical analysis
Structural identification
Re
d1 f
σe g
σe D (%)
Re
d1 f
σe g
σe D (%)
3.1 4.7 5.8 8.2 8.4
0.983 0.947 0.959 0.968 0.967
0.10 0.07 0.15 0.18 0.27
0.71 1.32 1.25 1.13 1.28
22.9 28.1 21.5 13.8 15.2
0.988 0.987 0.988 0.989 0.989
0.11 0.11 0.08 0.20 0.29
0.44 0.61 0.61 0.75 0.84
14.2 12.3 10.5 9.1 10.0
A
1 2 3 4 5
S
6 7 8
56.7 57.2 49.8
3.2 3.4 2.4
0.959 0.958 0.948
0.13 0.08 0.02
0.47 0.52 0.42
14.7 15.3 17.5
0.956 0.652 0.944
0.04 0.00 0.03
0.47 0.55 0.43
14.7 16.2 17.7
M
4 6 7 8 9
67 79.5 81.5 78.5 78.5
4.9 7.3 7.4 7.4 6.7
0.877 0.909 0.915 0.916 0.896
0.09 0.02 0.06 0.05 0.11
0.98 1.23 1.23 1.24 1.17
20.0 16.8 16.6 16.8 17.5
0.933 0.950 0.953 0.954 0.933
0.05 0.03 0.02 0.00 0.06
0.72 0.91 0.91 0.90 0.94
14.7 12.5 12.3 12.1 14.0
a b c d e f g
43 58 73 95 104
Dd
A = Ancipa (15 years’ monitoring), S = Sabbione (5 years), M = Malga Bissina (9 years). Buttress number. Buttress mean height (m). Upstream–downstream crest displacement semi-amplitude (mm). Correlation coefficient. Annual gradient of the irreversible displacement (mm/year). Standard error of estimate (mm).
Table 4 Values of the parameters obtained with structural identification (in parentheses: the corresponding normalized diagonal terms of the Fischer matrix) Dama
Butt.b
1 2 A
3 4 5 6
S
7 8 4 6
M
7 8 9
E (MPa) √ A11 /M mm
t0 (days) √ A22 /M
t˜ (days) √ A33 /M mm day
Tw (◦ C) √ A44 /M
β √
A55 /M
√
A66 /M
GPa
mm day
mm ◦C
(mm)
(mm)
22 524 (0.013) 14 786 (0.086) 15 745 (0.138) 18 131 (0.222) 18 900 (0.263)
24 (0.023) 23 (0.038) 25 (0.046) 28 (0.058) 29 (0.061)
57 (0.006) 53 (0.011) 55 (0.013) 65 (0.015) 67 (0.016)
5.6 (0.31) 11.0 (0.48) 9.8 (0.65) 12.3 (0.91) 10.3 (1.01)
0.50 (4.4) 0.68 (6.9) 0.61 (9.3) 0.57 (13.0) 0.53 (14.5)
0.14 (3.2) −0.12 (3.2) −0.13 (3.2) −0.16 (3.2) −0.05 (3.2)
37 038 (0.010) 32 375 (0.013) 38 316 (0.006)
4 (0.052) 5 (0.051) 3 (0.040)
27 (0.022) 31 (0.021) 29 (0.016)
1.2 (0.41) 1.5 (0.41) 1.1 (0.33)
0.38 (2.5) 0.39 (2.5) 0.35 (2.0)
0.41 (1.7) 0.50 (1.7) 0.37 (1.7)
17 028 (0.110) 13 918 (0.261) 13 661 (0.288) 13 692 (0.262) 19 095 (0.134)
8 (0.055) 8 (0.083) 9 (0.076) 8 (0.082) 9 (0.062)
21 (0.021) 23 (0.032) 27 (0.030) 23 (0.032) 25 (0.024)
5.3 (0.57) 5.6 (0.71) 5.5 (0.73) 7.0 (0.70) 7.1 (0.70)
0.46 (4.9) 0.55 (6.1) 0.54 (6.2) 0.56 (6.0) 0.46 (6.0)
−0.08 (2.1) −0.08 (2.1) −0.11 (2.1) −0.12 (2.1) −0.10 (2.1)
a A = Ancipa (15 years’ monitoring), S = Sabbione (5 years), M = Malga Bissina (9 years). b Buttress number.
d0 (mm) √ A77 /M mm mm
mm d1 year √ A88 /M mm mm year
−63.2 (1) −70.1 (1) −64.6 (1) −48.3 (1) −51.0 (1)
0.11 (8.7) 0.11 (8.7) 0.08 (8.7) 0.20 (8.7) 0.29 (8.7)
2.2
0.03 (2.8) −0.04 (2.8) −0.05 (2.8)
(1) 2.6 (1) 1.5 (1) 2.1 (1) 3.4 (1) 3.0 (1) 4.0 (1) 3.8 (1)
0.05 (4.8) −0.03 (4.8) 0.02 (4.8) −0.01 (4.8) 0.06 (4.8)
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can also be extended to other phenomena and events, such as earthquake or rapid water level drawdown, to check, for instance, how the elastic constant of the concrete depends on the load applying velocity. Acknowledgements The authors wish to thank all the people that allowed us to get reliable measures in dam monitoring, especially those at ENEL and from the other companies that provided the data for the statistical and identification procedures outlined in this paper. Appendix. Elements of the sensitivity matrix
∂ Dt =1 ∂d0 ∂ Dt Ht8 = =t ∂d1 o bt 1n ∂T = Tt−t − t˜ − Tt−t + t˜ 0 2 0 2 ∂t0 t˜ o n bt 1 ∂Q = Q t−t − t˜ − Q t−t + t˜ 0 2 0 2 ∂t0 t˜ o 1 Z t−t0 + 2t˜ n bt ∂T 1 − 2 = T Tτ dτ t˜ + T t−t0 − 2t˜ ∂ t˜ 2t˜ t−t0 + 2 t˜ t−t0 − 2t˜ o 1 Z t−t0 + 2t˜ bt 1 n ∂Q = Q t−t + t˜ + Q t−t − t˜ − 2 Q τ dτ. 0 2 0 2 ∂ t˜ 2t˜ t˜ t−t0 − t˜ Ht7 =
2
In the following, the derivatives of the analytical expression (12) with respect to the parameters are reported. Their knowledge is needed to assemble the sensitivity matrix (23). i 6 ρw H 2 X ∂ Dt Q Qt =− Ht1 = c ∂E H E 2 i=1 i ( bt Q bt b ∂ Dt ∂T bt − Tw 1 ∂ Q t Ht2 = = cT α H β + βT ∂t0 ∂t0 H H ∂t0 ! ) b b b 1 ∂ Qt ∂ Tt Qt b + 1− − Tt ∂t0 H H ∂t0 ( b bt bt Q ∂ Dt ∂T bt − Tw 1 ∂ Q t + βT Ht3 = = cT α H β H ∂ t˜ ∂ t˜ ∂ t˜ H ! ) bt bt bt 1 ∂Q Q ∂T b 1− − Tt + H H ∂ t˜ ∂ t˜ ∂ Dt bt = −c T α Q ∂ Tw ∂ Dt = cT α Ht5 = ∂β Ht4 =
b ∂ Dt bt 1 − Q t Ht6 = = cT α H T ∂ H
!
References [1] ICOLD. Bulletin 118: Automated dam monitoring systems — guidelines and case histories. Paris: International Commission on Large Dams; 2000. [2] Marcello C, Spagnoletti S. On the structural behaviour of hollow buttresses gravity dams — Theoretical behaviour. L’Energia elettrica 1960;37(10) [in Italian]. [3] Appendino M, Di Monaco F, Garino A, Manzo F, Scarinci S. Specific and general trends of the ageing of buttress dams as revealed by investigations carried out on Ancipa dam. In: Proceedings of the 17th conference on large dams. 1991. [4] Brindisi P, De Sortis A, Di Lemma G, Orsini G. Seismic analysis of hollow buttress gravity dams. In: Proceedings of the 12th European conference on earthquake engineering. 2002. [5] De Sortis A, Paoliani P. Statistical and structural identification techniques in structural monitoring of concrete dams. In: Proceedings of the 73rd annual meeting of international commission on large dams. 2005. [6] Ardito R, Bartalotta P, Ceriani L, Maier G. Diagnostic inverse analysis of concrete dams with statical excitation. Journal of Mechanical Behavior of Materials 2004;15(6):381–9. [7] Fedele R, Maier G, Miller B. Identification of elastic stiffness and local stresses in concrete dams by in situ tests and neural networks. Structure and Infrastructure Engineering 2005;1(3):165–80. [8] Levenberg K. A method for the solution of certain problems in least squares. Quarterly of Applied Mathematics 1944;2:164–8. [9] Marquardt D. An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics 1963;11:431–41.