Data analysis using ARX models applied to static structural health monitoring of the Monastery of Sant Cugat

12th International Conference on Structural Analysis of Historical Constructions SAHC 2020 P. Roca, L. Pelà and C. Molins (Eds.) DATA ANALYSIS USING ARX MODELS APPLIED TO STATIC STRUCTURAL HEALTH MONITORING OF THE MONASTERY OF SANT CUGAT N. MAKOOND1*, L. PELÀ1, C. MOLINS1 AND P. ROCA1 1Department of Civil and Environmental Engineering Universitat Politècnica de Catalunya (UPC-BarcelonaTech) Jordi Girona 1-3, 08034 Barcelona, Spain e-mail: [email protected], luc[email protected], clim[email protected]u, [email protected] Keywords: Masonry, environmental effects, dynamic regression models, diagnosis Abstract. The church of the monastery of Sant Cugat close to Barcelona is a medieval construction characterized by a complex structural behaviour stemming from the interaction among various structural elements built over different periods. Despite having survived for several centuries, such structures are often affected by slow irreversible deterioration mechanisms that can jeopardise their stability in the future. In order to identify such mechanisms at an early stage, and to better understand the cause of visible pathologies, a static structural health monitoring (SHM) system was installed in the church since 2017. Although this monitoring strategy, aimed at the continuous measurement of key slow-varying parameters, has been used successfully in the past to facilitate the diagnosis of this structural typology, the interpretation of data collected by such systems remains a challenging task. One of the main reasons for this is the fact that many monitored damage and deformation features are sensitive to changes caused by environmental conditions. To address this issue, this paper presents the application of a fully automated data analysis procedure to the records collected from the SHM system installed in the church of the monastery of Sant Cugat. The procedure consists of two parts. The first relies on the identification of models that comprehend an Auto-Regressive output and an eXogenous input (ARX) to represent the dynamics of each monitored response using suitable environmental parameters as predictors. The identified models are then used to estimate filtered evolution rates. The second part of the procedure involves classifying each monitored response into pre-defined evolution states based on outcomes from the first part. The main results from the application to the case of the church of the monastery of Sant Cugat are presented and the implications for the diagnosis of the structure are discussed. 1 INTRODUCTION The monastery of Sant Cugat is located near Barcelona in Catalonia, Spain. The masonry structure at the site today consists of different parts built over different time-periods mostly from the mid-12th century to the 15th century. The monastery is composed of a cloister and a church, with the latter being the main focus of this study. The church exhibits several pathologies notably in the form of cracks and inclinations. Hypotheses have been developed N. Makoond, L. Pelà, C. Molins and P. Roca 2 regarding the possible causes of these pathologies and an extensive restoration and strengthening campaign was even completed in 1996. However, the true cause of many pathologies, as well as the current evolutionary condition of possible deterioration mechanisms, remain unknown. Since identifying active mechanisms is crucial for a complete diagnosis of the church’s current structural condition, a long-term static structural health monitoring (SHM) system was installed in the structure in March 2017. The initial interpretation task in the case of static SHM involves the identification of either a stationary or an evolutionary condition from the recorded data of each monitored response. However, since monitored features are often sensitive to both damage and environmental changes, there usually exists no clear distinction between irreversible changes linked to active deterioration mechanisms and reversible ones caused by daily and seasonal environmental fluctuations. This, together with the complex interaction among structural elements, can make the interpretation of static SHM data very challenging in actual practice. Nevertheless, in spite of these difficulties, there exist many examples of successful applications of static SHM for the diagnosis of masonry heritage structures. Many of these have relied on directly fitting linear or periodic models to the recorded time series of the monitored responses. Others have explicitly taken measured environmental parameters such as temperature into consideration by using a simple linear model to predict and filter out the estimated change of the monitored structural response caused by environmental changes. Because the most common monitored structural responses in the case of masonry structures (cracks and inclinations) frequently show a predominantly linear dependence to the most influential monitored environmental parameter (temperature), the above-mentioned filtering method often leads to accurate estimates of underlying evolution rates. However, it is not very robust because the linear model is certainly an over-simplification of the true nature of the relationship. Notably, in the case of the relationship between temperature and structural parameters, a linear model is not able to consider effects caused by the thermal inertia of the material or by thermal gradients between interior and exterior temperature. Dynamic linear regression models that comprehend an Auto-Regressive output and an eXogenous input (ARX) appear to be an appealing alternative since they can account for more of the dynamics of relationships between environmental and structural parameters. As their name suggests, these black box models can exploit a large number of observations to reconstruct linear dependencies of monitored responses on their own rate of change, on the rate of change of selected predictors and on the present value of predictors. The first step of such a filtering procedure involves selecting which measured environmental variables will be used as predictors in the ARX model. Before the parameters of ARX models can be estimated, it is also very important to define the number of past response and predictor samples that will be used to describe the system. These are referred to as the model orders. Although there exist some examples of successful applications of ARX models to filter out environmental effects from dynamic SHM data [1,2], the application to static SHM data has been very limited [3]. As a result, there is very little guidance available on how to select appropriate model orders and there is a clear lack of tools for the interpretation of results from the identified ARX models. In order to address this issue, a fully automated data analysis procedure incorporating ARX models has recently been proposed [4]. The entire process includes a method to select optimal model orders from a pre-defined range as well as a procedure to classify each monitored N. Makoond, L. Pelà, C. Molins and P. Roca 3 response according to their estimated evolutionary state. This paper presents the application of this fully automated procedure for the diagnosis of the church of the monastery of Sant Cugat. Firstly, the SHM system currently installed in the monastery is briefly described. Some methods that attempt to filter out environmental effects from monitored responses are then described. The classification procedure that utilises results from all the filtering methods is then presented. Finally, the estimated evolutionary conditions and their corresponding rates are presented before discussing their implications for the diagnosis of the structure. 2 STRUCTURAL HEALTH MONITORING SYSTEM The SHM system installed in the church of the monastery of Sant Cugat consists of 14 crackmeters, 2 inclinometers, 3 thermistors and 3 humidity sensors, as shown in Figure 1. The cracks and inclinations to be monitored were chosen following a comprehensive damage survey and analysis. Of the 22 sensors, only 3 were placed on the exterior of the structure. Crackmeter FS 3-15 was fixed on the exterior wall of the central nave adjacent to the cimborio while the temperature and humidity sensors, TEMP-2.4 and HUMI-2.4, were installed on the exterior wall of the bell tower facing the cimborio (see Figure 1). Most of the sensors have been installed since March 2017. The two crackmeters placed on cracks beneath the rose window (FS-3.19 and FS-3.20) were installed in December 2017 and the one placed in the lintel of the main entrance (FS-3.21) was installed in April 2018. All the methods presented in this paper make use of data collected up to 02/02/2020. This constitutes 2.9 years of data for most sensors, 2.1 years for FS-3.19 and FS-3.20 and 1.8 years for FS-3.21. Figure 1: Plan view showing layout of the SHM system installed in Sant Cugat monastery. N. Makoond, L. Pelà, C. Molins and P. Roca 4 3 METHODS FILTERING OUT EFFECT OF TEMPERATURE The simplest analysis methods that have been applied to static SHM data consist of estimating underlying evolution rates by directly fitting the time series of each monitored structural response to a linear or to a nonlinear periodic model. Although there exist several successful examples of such applications, particularly when long monitoring periods are available, estimates from these types of analyses can easily be biased by underlying trends or irregular changes in environmental parameters. Many of the shortcomings of the above-mentioned methods can be addressed by taking advantage of environmental parameters monitored on site to better characterise their effect on monitored responses. As such, the SHM system installed in the monastery of Sant Cugat included sensors to measure two of the most relevant environmental parameters, i.e. temperature and relative humidity (see Figure 1). Although the methods presented in this section can be applied to any monitored environmental parameter, only the effect of temperature will be discussed in the case of Sant Cugat because a preliminary evaluation clearly revealed that the monitored responses are generally significantly more strongly correlated to temperature than to relative humidity. 3.1 Preliminary evaluation of correlation Before identifying the parameters of any model to represent the dependence of a structural response on environmental parameters, it is important to identify which environmental parameters are more suitable to be used as predictors in the model. This can be achieved by computing the Pearson correlation coefficient (R) between measured environmental and structural parameters [4]. This coefficient can vary between -1 and +1 with absolute values closer to unity indicating a stronger linear correlation. The sign of the coefficient reveals the type of correlation. A negative sign implies that an increase of a parameter leads to a decrease of the other and vice-versa. The R-values between temperature and each structural response monitored in the monastery of Sant Cugat are shown in Figure 2. For each response, the preliminary evaluation of correlation was carried out with exterior temperatures recorded by the thermistor placed outside the structure, as well as with interior temperatures recorded by the nearest thermistor placed inside the structure. Figure 2: Correlation coefficients of monitored structural parameters with temperature computed over the entire monitoring period. Both inclinometers have a strong negative correlation with temperature. This indicates that -1.00 -0.60 -0.20 0.20 0.60 1.00 Exterior temperature Interior temperature N. Makoond, L. Pelà, C. Molins and P. Roca 5 both the bell tower and the pillar at the southwest corner of the cimborio tend to incline towards the south when temperatures increase. As expected, most monitored crack widths exhibit a negative correlation with temperature. This is the expected behaviour since increasing temperatures cause materials to expand thus reducing crack widths and vice versa. However, four of the monitored cracks show a positive correlation with temperature. It is possible that the unexpected thermal response of some of these cracks is linked to the structural intervention that was completed in 1996 [5]. This activity involved inserting several tie rods in the southern part of the church, as shown in Figure 3. If these elements were actively working, an increase in temperature would cause an expansion of the tie rod and a subsequent loss in tension, which could induce the opening of cracks. This type of response has been observed previously in a masonry tower as reported in [6]. Figure 3: Structural interventions completed in 1996 in the southern part of the church, with the position of some sensors monitoring cracks which exhibit a positive correlation with temperature. In addition to the insertion of tie rods, the structural intervention of 1995-1996 also involved N. Makoond, L. Pelà, C. Molins and P. Roca 6 consolidation with a heavily reinforced concrete overlay of the gothic vaults of the fourth and lateral aisle on the side of the bell tower (see Figure 3). The expansion of the reinforcement during increasing temperatures can exert a force on the concrete and subsequently on the masonry. This effect can also contribute to the positive correlation observed between temperatures and the crack widths monitored by FS-1.2, FS-2.6 and FS-2.7. 3.2 Filtering environmental effects through linear models The simplest method to characterise explicitly the direct effect of temperature uses a linear model between measurements of temperature and each structural parameter, as shown in Figure 4. Since only a single predictor can be used for such a model, only the interior or exterior temperature is used for each response depending on which one has the greatest correlation coefficient. Due to the simplicity of this model, it can be said that effects caused by structural mechanisms of interest are less likely to influence the identified models if data from only a single seasonal cycle is used for the estimation of model parameters. When this practice is carried out, the first year of monitoring is usually used for the estimation of model parameters, and is referred to as the estimation phase. However, it can be argued that using data from the entire monitoring period would allow the model to better capture the changing nature of the relationship in some cases. Therefore, as recommended in [4], both operations listed below were carried out for each response variable and the results from both methods were compared: 1. Linear filter (i): Linear regression between selected predictor and response variable, using data from the entire monitoring period. 2. Linear filter (ii): Linear regression between selected predictor and response variable, using data only from the first complete year of monitoring (estimation phase). Figure 4: Examples of linear regression between monitored structural parameters and temperature over the entire monitoring period. Once the regression procedure has been completed, measured values of each predictor can be substituted into the identified linear models to simulate changes of the structural parameter that have been caused by changes in temperature. Actual measurements of the structural response are then filtered by simply subtracting the simulated temperature effect, as shown in Figure 5. Estimates of the underlying evolution rates (ERlin(i) and ERlin(ii)) can then be obtained by carrying out a regression of the filtered residuals. A significant advantage of this method is that it enables the assessment of how well each linear model can predict the relationship between temperature and every structural parameter. If we assume that residuals are normally distributed when no significant structural mechanism is present, a prediction interval representing a specific level of confidence can be obtained based N. Makoond, L. Pelà, C. Molins and P. Roca 7 on the standard error of the estimate (σe), computed using the differences between predicted and measured values. Figure 5 shows the plot of the 95% prediction interval. Figure 5: Filtering of temperature effect based on identified general linear trend and estimation of evolution of INC-1.3 from filtered residuals. 3.3 Filtering environmental effects using ARX models It is clear that the relationship between temperature and structural parameters can often be sufficiently well represented by a linear model. However, it can fail to do so in many cases since it cannot account for certain effects influencing the system it aims to describe, such as those due to thermal inertia or caused by thermal gradients. Auto-Regressive models incorporating an eXogenous input (ARX) are better equipped to deal with such effects because they utilise measured values of past responses together with those of past and current or delayed predictors to describe the dynamics of a system. In this case, only measurements collected during an estimation phase are used to estimate the parameters of the models. As was the case for Linear filter (ii), this estimation phase should span a full year to capture most of the reversible components caused by environmental effects during a complete seasonal cycle. The identified models are then used together with data collected over the entire monitoring period (simulation phase) to simulate responses based on measured predictors. For each structural response monitored in the monastery of Sant Cugat, a single-input single output (SISO) ARX model was first employed using the same predictor as was used for the linear filters. Multiple-input single output (MISO) ARX models incorporating both interior and exterior temperatures as predictors were then implemented to allow the model to consider effects caused by thermal gradients. The complete procedure incorporating ARX models that was used for the analysis of data from the static SHM system is described in [4] and will not be reiterated in this article. Besides the measurements of the response and of selected predictors, the only other required input to the procedure is the range of model orders to be tested. The final quality of ARX models depend strongly on the model orders which define the number of past response and predictor samples used to describe the system. Although the procedure described in [4] suffers from the disadvantage of being more computationally expensive than using fixed model orders, if an adequate range is defined, it ensures an optimal N. Makoond, L. Pelà, C. Molins and P. Roca 8 choice of model orders for each response based on the characteristics of the data itself. This in turn ensures that a sufficiently accurate ARX model will be obtained. Several ranges were tested as part of this research and for medieval masonry structures such as the church of the monastery of Sant Cugat, it is recommended to set the lower limit of the range to one corresponding to at least 4 days while it is recommended to set an upper limit corresponding to at least 10 days [4]. Naturally, the final range selected is largely dependent on the computational expense that can be spared. For the case study forming part of this research, a range corresponding to a duration from 5 to 25 days was specified for the SISO ARX models, and one corresponding to a duration from 4 to 10 days was specified for the MISO ARX models. Once the effect of measured environmental parameters has been simulated with ARX models, the filtered evolution rate (ERSISO-ARX or ERMISO-ARX) of each response can be computed in the same way as it was with the linear filters described in Section 3.2. Several error metrics were used to assess the accuracy of the different models used to characterise the relationship between temperature and monitored responses. However, the standard error of the estimate (σe) is probably the one that is most easily interpreted since it is a measure of the dispersion of simulated values from measured ones expressed in the same units as the measurements. Figure 6 shows the values of σe for each monitored response using all the filtering procedures applied. Figure 6: Standard error of the estimate (σe) computed from residuals between measured and simulated responses using different models over estimation phase. It is clear to see that the ARX models are better suited to model the environmental variation since their residuals have a smaller dispersion than those of the linear models for 13 out of 16 monitored responses. Moreover, the added benefit of using both interior and exterior temperature as predictors is also apparent since the MISO models outperform the SISO ones for 10 of these 13 responses even if they have lower model orders. 4 AUTOMATED CLASSIFICATION PROCEDURE Although using SISO and MISO ARX models to filter out the effect of environmental parameters can greatly improve the accuracy of the estimated filtered evolution rates, it is still important to consider the uncertainties and errors associated to modelling the effect of temperature. As a result, the interpretation of results can still be challenging. Utilising the automated classification procedure elaborated in [4] can greatly facilitate this task. This multistep classification procedure involves 5 tests. All of these tests rely on comparisons between the estimated filtered evolution rate computed 0.006 0.015 0.104 0.006 0.002 0.004 0.048 0.002 0.007 0.057 0.050 0.016 0.041 0.011 0.074 0.058 0.000 0.050 0.100 0.150 [mm] or [°] Linear filter (i) Linear filter (ii) ARX (SISO) ARX (MISO) N. Makoond, L. Pelà, C. Molins and P. Roca 9 using the most sophisticated ARX model employed (ERARX) and the standard error of the estimate computed over the estimation phase from residuals associated to the same model (σeARX). Note that in the case of the analysis carried out for the SHM data collected in the monastery of Sant Cugat, the most sophisticated ARX models employed refers to the MISO models. Some tests also consider the normality of the residuals, the magnitude of daily fluctuations and results from all the procedures described in Section 3. A detailed description of each test is not given here but can be found in [4]. Based on the outcomes of the tests, each response is classified in one of the following 5 categories. 1. Stationary: Responses showing a clear stationary trend outside reversible variations caused by environmental parameters. 2. Evolutionary: Responses showing a clear evolutionary trend outside reversible variations caused by environmental parameters. 3. Apparently stationary: Responses showing a stationary trend but for which there is still a relatively large uncertainty associated to the estimation of the trend. 4. Apparently evolutionary: Responses showing an evolutionary trend but for which there is still a relatively large uncertainty associated to the estimation of the trend. 5. Inconclusive: Monitored parameters for which no clear conclusion can be made on its evolutionary state from the available monitoring data alone. 5 RESULTS AND DISCUSSION The estimated filtered evolution rates from all the methods described in Sections 3.2 and 3.3 are summarised in Table 1, together with the evolution state estimated from the classification procedure presented in Section 4. Table 1: Comparison of estimated evolution rates for monitored structural parameters of Sant Cugat monastery from methods filtering out the simulated effect of measured environmental parameters. Sensor Units Estimate of annual evolution rate [unit/year] Estimated condition Linear filter (i) Linear filter (ii) SISO ARX filter MISO ARX filter FS-1.1 mm 0.102 0.102 0.100 0.100 App. Evolutionary FS-1.2 mm 0.003 0.003 0.003 0.003 App. Evolutionary INC-1.3 ° -0.003 -0.003 -0.003 -0.003 Evolutionary FS-2.5 mm 0.029 0.029 0.025 0.026 Evolutionary FS-2.6 mm 0.001 0.001 0.000 0.001 App. Stationary FS-2.7 mm 0.001 0.001 0.001 0.001 Stationary INC-2.8 ° -0.006 -0.006 -0.007 -0.007 Evolutionary FS-2.11 mm 0.068 0.067 0.072 0.070 App. Evolutionary FS-3.15 mm 0.002 0.001 0.001 0.000 Inconclusive FS-3.13 mm -0.003 -0.002 -0.004 -0.009 App. Evolutionary FS-3.14 mm -0.013 -0.014 -0.013 -0.014 Inconclusive FS-3.17 mm -0.041 -0.042 -0.042 -0.047 App. Evolutionary FS-3.18 mm 0.001 0.001 -0.002 0.002 App. Stationary FS-3.19 mm -0.060 -0.063 -0.081 -0.060 App. Evolutionary FS-3.20 mm -0.023 -0.017 -0.040 -0.014 Inconclusive FS-3.21 mm 0.138 0.135 0.170 0.126 App. Evolutionary