Hybrid surrogate-based models for the evaluation of the fundamental period of regular URM buildings

Hybrid surrogate-based models for the evaluation of the fundamental period of regular URM buildings Vasco Bernardo a,* , Alfredo Campos Costa b , Paulo B. Lourenço a a ISISE, Department of Civil Engineering, University of Minho, 4800058, Guimar˜ aes, Portugal b National Laboratory for Civil Engineering, Av. do Brasil 101, 1700075, Lisbon, Portugal ARTICLE INFO Keywords: URM buildings Fundamental period Surrogate model Seismic safety ABSTRACT The fundamental period can be interpreted as the structures’ DNA since it carries essential information about their behavior and response to dynamic loads, such as seismic events. Most practitioners are familiar with using the fundamental period to calculate the seismic response coefficient for base shear estimation of regular buildings using the equivalent lateral procedure, the so-called Lateral Force Method in the current version of Eurocode 8 – part 1. The most straightforward method for determining the fundamental period usually relies on empirical formulas derived from the data observation of several instrumented buildings. Despite the usefulness and adequacy of these formulas in the regions where these buildings are located, their generalization for other regions with different seismicity levels and different building typologies may not be suitable. The present work aims to develop surrogate models for estimating the fundamental period of regular URM buildings in Portugal. This is achieved by combining representative numerical models with experimental data within a Bayesian framework. The results support the technical community in the validation of structural numerical models and provide a reliable estimation of the fundamental period, consistent with the region and structural typologies. 1. Introduction The fundamental period T1 is a critical parameter in structural engineering, playing an important role in the design and assessment of buildings subjected to dynamic loads, such as seismic forces. The fundamental period depends on several factors, including structural typology, material properties, connection between structural parts, building height, geometric layout, design assumptions, aggregate effect, and soil-structure interaction. In the literature, several studies highlight the influence of T1 on the seismic behavior and its impact on the accuracy of design seismic loads (e.g., Gilles et al. [1], Asteris et al. [2,3], Kose [4], Bernardo et al. [5], Penna et al. [6], Piro at al. [7]). Over the past few decades, the implementation of codes and standards has enhanced design and construction techniques, significantly improving structural safety. Although the seismic structural safety procedures in the current modern building codes recommend using nonlinear analysis methods, most practitioners continue to rely on the simplicity of linear-elastic analysis. The Lateral Force Method outlined in the current EN 1998–1 [8], Eurocode 8 – Part 1 (EC8), adopts linear-elastic analysis and relies on the estimation of T1 to calculate the seismic response coefficient for estimating base shear in regular buildings. An extensive review concerning the evaluation of building period formulas for seismic design is presented by Kwon & Kim [9]. In engineering practice, the determination of T1 is usually performed through straightforward methods or conventional eigenvalue analysis. The former usually relies on empirical formulas derived from several instrumented buildings subjected to ground motion during seismic events (e.g., Goel & Chopra [10,11], Mucciarelli el al. [12], Eleftheriadou & Karabinis [13], Wang et al. [14]) or ambient vibrations tests (e. g., Gallipoli et al. [15,16], Hatzigeorgiou & Kanapitsas [17], Kaplan et al. [18], Guler et al. [19]). Other studies have developed tridimensional numerical models to provide consistent empirical formulas for estimating T1 (e.g., Saadatkhah et al. [20], Ruggieri et al. [21], Marasco & Cimellaro [22], Ditommaso et al. [23]) or, more recently, machine learning algorithms (e.g., Yahiaoui et al. [24], Mirrashid & Naderpour [25], Asteris & Nikoo [26], Charalampakis at al. [27]). For the case of URM buildings, a recent investigation conducted by Gallipoli [15] derives empirical formulas to estimate T1 as function of the buildings height, based on ambient noise measurements on the Southern and Northeastern Italy. Regarding eigenvalue analysis, as implemented in the commercial software and commonly used by practitioners, disregarding stiffness and mass of non-structural elements can result in a * Corresponding author. E-mail address: [email protected] (V. Bernardo). Contents lists available at ScienceDirect Structures journal homepage: www.elsevier.com/locate/structures https://doi.org/10.1016/j.istruc.2024.107531 Received 28 June 2024; Received in revised form 22 September 2024; Accepted 9 October 2024 Structures 69 (2024) 107531 Available online 15 October 2024 2352-0124/© 2024 The Author(s). Published by Elsevier Ltd on behalf of Institution of Structural Engineers. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). bias of T1 and inaccuracies in the results, leading to an incorrect seismic verification. Therefore, the applicability of proper empirical formulas can serve as a starting point for the design and assessment procedure and contribute to the calibration of the numerical models. Despite efforts to provide more accurate T1 empirical formulas, current code-based period determination formulas for specific structural types, such shear walls system in URM buildings, have not been thoroughly evaluated or calibrated. Additionally, most instrumented buildings used for calibrating these formulas are located in regions with strong seismic activity [9], which may not accurately represent buildings in areas with lower to moderate seismicity, due to local seismic culture. Consequently, the site and structural dependencies influencing seismic performance require more comprehensive evaluation to ensure accurate period estimations across diverse seismic regions. In this study, hybrid-surrogate models (SM) are proposed to estimate T1 of regular URM buildings in Portugal. A synthetic database of representative buildings is used to predict the numerical T1, which is then combined with experimental data from ambient vibration tests within a Bayesian framework. The geometry and material properties influencing T1 are evaluated using conventional machine-learning algorithms. Two SM are developed as (i) a function of the building’s height (SM-I) and (ii) combining material properties and geometric layout (SMII). The results aim to support the technical community in validating structural numerical models for seismic safety assessment and strengthening, and in providing a more reliable estimation of the T1 consistent with the regional and structural characteristics. 2. Database of representative archetypes of the URM building sock The representative archetypes considered in the current study were derived from a previous research conducted by Bernardo et al. [28]. The data collection includes buildings up to five stories high obtained from the original blueprints. Table 1 summarizes the statistical properties of the geometric parameters collected: plan dimensions – L x and L y ; ground and upper floor stories height – H 0 and H n ; openings ratio – OR: front (ORF) and back (ORB) facade; interior walls density – IWD; walls thickness Th – facades (1), lateral side (2), interior (3), partition (4); average walls thickness reduction on the façade – AWR). Table 2 depicts the five-stories-high archetypes derived from the statistical information. Buildings with lower number of floors follow the same layout. Regarding the material properties, a wide range of mechanical properties were considered based on literature review (see Bernardo et al. [29]). The uncertainty in the materials was propagated through Monte Carlo simulations using predefined statistics [29]. By combining different geometries and material properties, a large synthetic database comprising 18.000 structures was built to represent the population of URM buildings. Additional information is given in [28] and [29]. 3. Structural numerical models 3.1. Modelling strategy and assumptions Tridimensional MDOF models were developed to simulate the synthetic database and estimate the natural frequencies. The numerical models are based on the equivalent frame method, which uses a nonlinear macro-element formulation available in the TreMuri research version [30]. The method only reproduces the in-plane behavior of the walls. Elements with openings are modeled by assembling the piers and spandrel beams (macroelements) with nonlinear behavior, connected by nondeformable rigid nodes. The failure modes at macroelement level consider the shear response and flexural-rocking of the panel, with constitutive models depicted in Fig. 1. Further details can be found in Penna et al. [30]. 3.2. Modal analysis – fundamental period results The estimation of the fundamental period T1 is based on the Jacobi inverse algorithm implemented in the research version of TreMuri [31]. The results are presented in the histograms of Fig. 2 for the entire database and disaggregated by number of stories. The numerical database is only composed by buildings up to five stories high, since it represents the vast majority of the masonry budling stock without RC frame [32]. The mean μ values range from 0.059 s to 0.289 s (16.9 Hz to 3.40 Hz) for buildings one to five stories high, respectively, with coefficient of variation (cov) between 0.26 and 0.29. The modelling approach used to estimate T1 is validated by the studies carried out by Bernardo et al. [5] and Salvalaggio et al. [33] for the same building typology, using the Applied Element Method formulation [34]. Furthermore, the range of frequencies obtained are also in line with the literature, namely the results compiled from operational modal analyses carried out by Oliveira [35] for the same buildings typologies and three to six stores high, varying from 2.3 Hz to 8.0 Hz. This data will also be used to update the numerical results of T1 through a Bayesian approach (see section 5), which also allows to overcome the issues related with the estimation of T1 from different modeling approaches [36,37]. 4. Dynamic identification test data collection and benchmarking This section presents the fundamental periods obtained from the data collection of 29 ambient modal identification tests performed on URM buildings carried out by various authors [5,35,38,39]. The results are plotted in Fig. 3 as a function of the building height H. A power law function was fitted to these data, achieving a coefficient of determination (R 2 ) of 0.48. The adoption of the power law allowed the comparison between the estimation of T1 provided in the current version of EC8 (part-1), given by: T1=CtH0.75 (1) where the factor Ct depends on the type of structure and assumes a value of 0.05 for various structural systems, excluding steel or concrete moment-resisting frames. Alternatively, the value of Ct can be estimated considering the total effective area, Ac [m 2 ], for shear walls structural systems: Ct=0.075/ Ac √(2) with Ac=Σ[Ai(0.22 + (li/H)0.75 )] ,li/H≤0.9 (3) where Aiis the effective cross-sectional area of shear wall i and li is the length of the shear wall i, both related to the first story. In ASCE 7–22 and other previous codes (e.g., FEMA 302, BOCA-96 and NEHRP 00), the factor Ct in Eq. (1) is assigned a value of 0.02 for R/C or masonry wall shear structures. For the specific case of URM shear walls, the investigation by Know and Kim [9] specified a value of Ct= Table 1 Summary of the statistical properties for the geometric parameters collected [28]. Moments L x [m] L y [m] IWD [-] H 0 [m] H n [m] OR F [-] OR B [-] Th 1 [m] Th 2 [m] Th 3 [m] Th 4 [m] AWTR [m] Mean µ12.6 12.1 0.054 3.23 3.01 0.23 0.21 0.47 0.34 0.21 0.14 0.11 Std. deviation σ 5.00 4.1 0.01 0.42 0.24 0.08 0.08 0.14 0.11 0.05 0.02 0.06 V. Bernardo et al. Structures 69 (2024) 107531 2 0.015. Other approximate formulas can be found in the literature based on the lateral elastic displacement of the top of the building due to the gravity loads applied in the horizontal direction [8,9]. Since this last approximation requires a structural model, it is not included in Fig. 3; the remaining formulas are plotted in the same figure. EC8_1 and EC8_2 curves correspond to T1 obtained from Eq. (1) and Eq. (3), respectively. EC8_2 curve was estimated using in Eq. (3) the average Ac value from the buildings synthetic database. Analyzing Fig. 3, the exponent of the experimental fitting is similar to the one proposed in the literature. However, the low scaling factor in the formulas for shear wall structural systems (EC8_2, Kwon & Kim and ASCE 7–22) provides a lower bound for the building’s period, which may lead to an overestimation of the T1-response and a conservative seismic assessment considering the code response spectrum. The estimation of T1 based on the building’s geometry (EC8_2) and the crude approximation of ASCE 7–22 provide similar results and less error. The opposite is verified when employing the EC8_1 approximation, where the estimation of T1 may lead to unconservative safety verification. 5. Development of surrogate models to estimate T 1 This section proposes surrogate models (SMs) for estimating T1. An initial surrogate model (SM-0) is computed to predict the numerical results, which is updated with the experimental data (SM-I and SM-II) in section 5.3. 5.1. Selection of best predictors variables The development of the predictive SM for T1 should rely on a set of simple parameters at the building level that can be easily surveyed in practice without compromising the model’s response. The selection procedure to evaluate and identify the most effective predictors variables of T1 employed supervised machine learning algorithms, namely a wide neural network regression model available in deep-learning MATLAB toolbox. This neural network allows the model to capture relationships between the input and output variables by using less number of hidden layers but more number of neurons per layer [40]. The models were validated using cross-validation technique, where the data are partitioned into five subsets (folds), approximately with equal size. Considering as the true response (observations) the numerical results of f1=T1−1, fundamental frequency, presented in section 4, the key parameters influencing f1 were selected based on an ad-hoc process that identified variables capable of describing the elastic period according to fundamental principles of structural dynamics, particularly the stiffness and mass of the structural system [41]. Four groups of predictor models – model (i) to (iv) – were tested, each incorporating an increasing number of relevant variables to estimate f1: model (i) – height (H) and modulus of elasticity (E); model (ii) – H, E and mass ( ρ ); model (iii) – H, E, ρ and walls density (D); model (iv) H, E, M, D, and in-plan slenderness (λ), defined as the ratio of the building’s plan dimensions. It is important to note that other geometrical factors, such as the number and size of openings, are implicitly accounted for in the surrogate model outcome. The model considers the true response of f1 derived from the synthetic database, which reflects a 10 % margin of error for the entire URM Table 2 Representative archetypes and size [m] of the URM building stock. Fig. 1. Macroelement modeling: a) tridimensional model of the building; b) frame-type representation of façade; c) and d) constitutive models for shear and flexuralrocking, respectively. V. Bernardo et al. Structures 69 (2024) 107531 3 building stock, with a 95 % confidence level [32]. Fig. 4 shows the scatter plot of predicted response against true response to evaluate the model’s performance and the residuals between responses in the range of frequencies analyzed. The model’s evaluation is computed through the root mean squared error (RMSE) depicted in the graphs. A perfect prediction is defined by the diagonal line, which indicate the best selected variables to describe the problem. As can be seen, models (i) and (ii) present similar performance, with residuals tending to increase with frequency. To overcome this issue, geometric parameters are added to the model: the inclusion of variable walls density in model (iii) improves the response estimation (RMSE=0.49), while combining it with in-plan slenderness in model (iv) achieves almost perfect prediction (RMSE=0.06). 5.2. Predictive surrogate model The proposed SM for estimating T1, presented in this section, aims to be simple and suitable for practical application. Therefore, when examining the results obtained, one encounters a multidimensional problem that requires simplification, considering the best response predictors to express T1. In this sense, a first estimate to reduce the problem into one independent variable, x0, was defined by: x0=E∗[MPa] H[m] • ρ [kg/m3]•λ[ − ] (4) where E∗is the modulus of elasticity weighted by the walls density in the direction considered. By considering x0 as the input variable, Fig. 5a) shows its relationship with the fundamental frequency f1 for different number of stories (P1 to P5) For graphical convenience, the results are expressed in terms of f1. To reduce scatter and derive a unique model that couple geometric and material properties, exponents c and d were introduced to the geometric parameters H and λ, as follows: x=E∗ Hc• ρ •λd(5) By fitting a unique power law function y=axb to the point cloud, the optimization of constants c and d can be achieved by minimizing the Fig. 2. Fundamental period obtained from the numerical models and stratified by number of stories. Fig. 3. Fundamental period T 1 (s) obtained from operational modal analysis and comparison between formulas in literature. V. Bernardo et al. Structures 69 (2024) 107531 4 Fig. 4. Evaluation of model performance considering different predictor variables of f 1. Fig. 5. Surrogate model to estimate the fundamental frequency f 1: a) point cloud of f1 as a function of x0 and number of stories (P1 to P5); b) optimized model (SM-0). V. Bernardo et al. Structures 69 (2024) 107531 5 standard deviation of the logarithmic error between the numerical and predicted frequencies. The final surrogate model (SM-0) is depicted in Fig. 5b) as a function of x, considering the optimized values of approximately a=48.6,b=0.49,c=2.10 and d=0.30. This solution leads to coefficient of determination (R2)equal to 0.99, with zero mean error and standard deviation of the error around 0.07. 5.3. Surrogate model updating with experimental data In this section, two hybrid-surrogate models are proposed, combining the numerical results with the experimental data of Section 4: i) Surrogate Model I (SM-I), derived from simple regressions and considering only H as the input variable; and ii) Surrogate Model II (SMII), formulated within a Bayesian update framework and accounting for various geometric parameters and material properties. Both models are supported by the optimized predictive surrogate model (SM-0) from the previous section. For the development of SM-I, the experimental results of T1 (T1_exp) and the numerical values (per story – P1 to P5) of T1 (T1_num) are initially considered, plotted in Fig. 6. Both T1 values were only dependent on the building’s height H, as information regarding the geometry layout and material properties of the surveyed buildings is absent. Linear regressions were fitted to the values of T1_exp (evidence) and T1_num (prior), with R2 indicated in the same figure. The mean relative error between the T1_num obtained from linear regression and the SM-0 of Fig. 5b) is around 5.6 %. To estimate the T1(posterior), the standard deviation of the error between the numerical and experimental fittings was computed equal to 0.025. By adding this value (deviation) to T1_num yields T1(posterior), in Fig. 6. Regarding SM-2, Bayesian inference was applied by considering the prior distribution as the values of T1_num and the likelihood function derived from the experimental dataset T1_exp (evidence) to obtain the posterior distribution of T1. In this case, since the evidence is discrete (count data), the likelihood function needs to be modeled accordingly. Poisson distribution is often suitable for this kind of data set; however, it can be approximated by a Normal distribution due to the Central Limit Theorem. This assumption was also confirmed using the Chi-square goodness-of-fit test (5 % of significance level) to support the null hypothesis. A Normal distribution was also fitted to T1_num, which is suitable for T1 values between 0.05 s and 0.4 s. The fitting of both distributions to the dataset is depicted in the probability plot of Fig. 7a) with the corresponding moments (mean μ and standard deviation σ ). Given that both the prior and likelihood functions are considered normal distributions, the posterior distribution will also be a normal distribution, with mean μ post and standard deviation σ post defined by [42]: μ post = σ 2 prior σ 2 prior + σ 2 exp n • μ prior + σ 2 exp n σ 2 prior + σ 2 exp n • μ exp (6) σ post = σ 2 prior • σ 2 exp σ 2 prior + σ 2 exp n √ √ √ √(7) where μ prior and σ prior are, respectively, the mean and standard deviation of the prior and normal distribution; μ exp and σ exp the mean and standard deviation of the evidence; n the evidence size. Fig. 7b) depicts the posterior Normal distribution of T1 combining the prior distribution and the likelihood function. Finally, SM-II can be computed by adding to SM-0 (Fig. 5b) the values of σ post and the standard deviation of the error between the optimized power regression model and the numerical observations equal to 0.07, resulting in a total dispersion of around 0.14. The final model is depicted in Fig. 8. As a simplification, the power regression coefficients of SM-II can be rounded to a constant a=51.0 and exponent b=0.50, resulting in a standard error lower than 1.5e-4. Note that, SM-II can be interpreted as an upper bound of SM-0 leading to more conservative results in the seismic assessment by adopting code spectrum. 6. Example of application As an example of application and comparison between the proposed surrogate models (SM-I and SM-II) and the code-based formulas in current version of EC8 (EC8_1 and EC8_2 – see Section 4), the building’s synthetic database and the EC8_1 seismic response spectrum (offshore and onshore epicenter) for the region of Lisbon at bedrock, with a 475 - years return period, were considered. The estimation of T1 is straightforward for the EC8_1 and SM-I models since it only depends on H. For the EC8_2, see Eq. (3), an average value of Ac(m2) = 14.6 was assumed, obtained from the buildings database [32]. In the case of SM-II application, mean values of the material properties were assumed (E= 2233,9MPa; ρ =1654,7 kg/m3), combined with various in plan slenderness λ (−) =[0.5, 1.0, 2.0]. T1 was estimated for 2.0<H(m) ≤ 40.0. Fig. 9 shows the values of spectral acceleration Sa(T1) for the different prediction models. As can be seen, for low-rise buildings and both offshore and onshore scenarios, EC8_1 and EC8_2 models seem to provide an upper and lower bound of the models proposed resulting in conservative and underestimated seismic safety verification, respectively. The opposite is verified for medium to high-rise buildings. Nevertheless, this last observation is not relevant for offshore scenario, since the vast majority of the URM buildings are up to 20 m high. However, for onshore seismic action, the differences between EC8 models can reach around 90 %, where the application of EC8_1 compromises the structural safety for approximately H(m)>9.0. Naturally, this conclusion for the EC8_2 model depends on the Ac adopted, i.e. for lower values of Ac the range of constant Sa(T1) is shorter and tends to converge to EC8_1. Finally, the results obtained from SM-I and SM-II are similar for H(m)<6.0 and seem to diverge for H(m)>9.0 (onshore scenario) as the values of λ decreases. The differences between the models result in a maximum deviation of Sa(T1) values of about 20 % with respect to SM-II (λ=0.5). 7. Final comments and conclusions The present study proposes hybrid surrogate models (SMs) for the estimation of the fundamental period T1 of regular URM buildings in Portugal. The SM should be based on a straightforward set of buildinglevel parameters that are easy to survey in practice, ensuring the model’s effectiveness is not compromised and that can be simply applied by the Fig. 6. Surrogate model I (SM-I) to estimate T 1 based on the building’s height H. V. Bernardo et al. Structures 69 (2024) 107531 6 practitioners. The models were developed by combining representative structural numerical models, which account for geometry and material properties uncertainties, with experimental data obtained from ambient vibration tests within a Bayesian update framework. Machine-learning algorithms was used to identify structural parameters, such as buildings height H, modulus of elasticity E, mass ρ , walls density D and in plan slenderness λ, selected as the best predictors for estimating numerical T1. Based on these results, multiple regression analysis was employed to determine T1 as a function of the selected variables. Two surrogate models were proposed: (i) SM-I, based only on the building height, and (ii) SM-II, combining geometric layout with material properties. The empirical formulas for the fundamental frequency f1=1/T1 are summarized below: •Surrogate model I (SM-I): f1[Hz] = 1 0.0176 •H[m](8) •Surrogate model II (SM-II): f1[Hz] = 51.0•[E∗[MPa] H[m]2.10 • ρ [kg/m3]•λ[− ]0.3]0.5 (9) Comparison between SM-I and SM-II and the code-based formulas in the current version of EC8 was also conducted in terms of spectral acceleration Sa(T1) for the Lisbon region, considering the code response spectrum (onshore and onshore seismic action, bedrock, 475-years return period) and the synthetic database of buildings. For low-rise buildings, the results indicated similar results between SM, with a maximum deviation of around 0.05 g between those and EC8, providing a) b) Fig. 7. a) Probability plot of the fitted functions to the dataset; b) Prior, likelihood and posterior distributions. Fig. 8. Surrogate model II (SM-II) to estimate T 1 based on the building’s geometry and materials. Fig. 9. Spectral acceleration S a(T1) as a function of H for different predicted models: a) offshore seismic action; b) onshore seismic action. V. Bernardo et al. Structures 69 (2024) 107531 7 an overestimation and underestimation of the seismic assessment using the general code formula (EC8_1) or accounting for the shear walls area (EC8_2), respectively. The opposite is observed for medium to high-rise buildings, where SM-1 seems to be an upper bound of SM-2, with a maximum Sa(T1) deviation of 0.06 g, namely for onshore seismic action. Finally, the proposed formulas support practitioners in the seismic safety verification and contribute to the development of the National Annex documents for Eurocode 8. Although the models were developed for the Portuguese building stock, the rational and methodology can be employed in other regions/countries, which essentially requires sitespecific information about the building stock. Further studies on soilstructure interaction are also suggested, requiring additional sitespecific experimental data to stratify the sample and provided soildependent formulas. Abbreviations None. Authors’ contribution Conceptualization and methodology – VB and ACC; formal analysis and investigation – VB; supervision, project administration and funding acquisition – PBL. All authors have read and agreed to the published version of the manuscript. CRediT authorship contribution statement Vasco Bernardo: Methodology, Investigation, Formal analysis, Conceptualization. Paulo B. Lourenço: Supervision, Project administration, Funding acquisition. Alfredo Campos Costa: Methodology, Conceptualization. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This study was funded by the STAND4HERITAGE project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant agreement No. 833123), as an Advanced Grant. This work was also partly financed by FCT / MCTES through national funds (PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under reference UIDB / 04029/2020 (doi.org/10.54499/UIDB/04029/2020), and under the Associate Laboratory Advanced Production and Intelligent Systems ARISE under reference LA/P/0112/2020. Data availability Data will be made available on request. References [1] Gilles D, McClure G, Chouinard LE. Uncertainty in fundamental period estimates leads to inaccurate design seismic loads. Can J Civ Eng 2011. [2] Asteris PG, Repapis CC, Repapi EV, Cavaleri L. Fundamental period of infilled reinforced concrete frame structures. Struct Infrastruct Eng 2017. 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