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Deep Symbolic Regression for Numerical Formulation of Fundamental Period in Concentrically Steel-Braced RC Frames
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This research explores the use of Deep Symbolic Regression (DSR) to develop a sophisticated predictive model for the fundamental period of vibration in concentrically steel-braced reinforced concrete (RC) frames. Traditional empirical models often overlook complex interactions within structural dynamics during seismic events, a gap this study addresses by deriving tailored equations for various bracing configurations such as Cross bracing, Diagonal bracing, and Chevron bracing. The model development incorporates an iterative refinement process utilizing DSR techniques to enhance accuracy and applicability in predicting seismic responses. Further refinement and optimization are achieved using the L-BFGS-B algorithm, ensuring robustness and adherence to safety standards. Validation against actual structural data reveals that our proposed equations achieve high predictive accuracy, with R-squared values up to 0.8247 and RMSE values as low as 0.2119, consistently presenting lower error metrics across various configurations compared to those found in established seismic design standards, such as ASCE, Eurocode, and Japan’s Building Standards. Comparative analyses and Bland-Altman plots confirm that the models not only match but often surpass the accuracy of traditional formulas, validating their potential as reliable tools in structural engineering for earthquake resilience planning. The findings demonstrate DSR’s potential to revolutionize traditional practices in formulating empirical equations, offering a scientifically rigorous, data-driven methodology for more accurately predicting the dynamic responses of structures under seismic loads.
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Symbolic regression is emerging as a promising machine learning method for learning succinct underlying interpretable mathematical expressions directly from data. Whereas it has been traditionally tackled with genetic programming, it has recently gained a growing interest in deep learning as a data-driven model discovery tool, achieving significant advances in various application domains ranging from fundamental to applied sciences. In this survey, we present a structured and comprehensive overview of symbolic regression methods, review the adoption of these methods for model discovery in various areas, and assess their effectiveness. We have also grouped state-of-the-art symbolic regression applications in a categorized manner in a living review.
Symbolic regression (SR) is the study of algorithms that automate the search for analytic expressions that fit data. While recent advances in deep learning have generated renewed interest in such approaches, the development of SR methods has not been focused on physics, where we have important additional constraints due to the units associated with our data. Here we present Φ-SO, a physical symbolic optimization framework for recovering analytical symbolic expressions from physics data using deep reinforcement learning techniques by learning units constraints. Our system is built, from the ground up, to propose solutions where the physical units are consistent by construction. This is useful not only in eliminating physically impossible solutions but also because the grammatical rules of dimensional analysis enormously restrict the freedom of the equation generator, thus vastly improving performance. The algorithm can be used to fit noiseless data, which can be useful, for instance, when attempting to derive an analytical property of a physical model, and it can also be used to obtain analytical approximations of noisy data. We test our machinery on a standard benchmark of equations from the Feynman Lectures on Physics and other physics textbooks, achieving state-of-the-art performance in the presence of noise (exceeding 0.1%) and show that it is robust even in the presence of substantial (10%) noise. We showcase its abilities on a panel of examples from astrophysics.
The study is directed toward the precise prediction of the fundamental period of steel-braced reinforced concrete (RC) moment-resisting frames through the utilization of stacked generalization, an advanced algorithmic ensemble machine learning technique. To facilitate this, a meticulously curated database comprising 17,280 building models has been automated using the ETABS Application Programming Interface (API). The database encompasses both concentrically braced frames and eccentrically braced frames and employs eigenvalue modal analysis to capture the fundamental periods, incorporating diverse bracing configurations and pivotal building parameters. Utilizing SHapley Additive exPlanations (SHAP), the study rigorously scrutinizes influential parameters that affect the fundamental period. The research introduces three stacking ensemble models, with the most effective model employing Random Forest as the meta-model and an ensemble of Extra Trees, Gradient Boosting, XGBoost, LightGBM, CatBoost, and kNN as base models. Hyperparameter tuning was accomplished through Bayesian Optimization, and a thorough sensitivity analysis was conducted. In rigorous evaluations conducted on the test dataset, the proposed model achieved an exceptionally high coefficient of determination (R²) of 0.9889, coupled with an impressively low root mean square error of 0.056. Further validation through multi-dimensional metrics confirmed the model’s robust generalization capabilities. Comparative validation against a few popular building code provisions and research models revealed that the proposed model markedly surpasses these benchmarks in predictive accuracy.
In the current research, the elastoplastic behaviour of symmetrical and asymmetrical reinforced concrete buildings is explored by dynamic analysis. The used ground excitations are of sequential type, which is found in the literature to possibly strongly affect the dynamic structural behaviour. The contemporary seismic codes neglect the impact of sequential earthquakes on the seismic response, highlighting a scientific gap necessary to be studied. Within the scope of this study, ordinary 3D reinforced concrete low-rise building frames are forced to sequential ground excitations, as well as to a respective single-occurrence corresponding ground excitation, for comparability reasons. In the present dynamic analyses, the two horizontal directions of the excitations, along with the vertical one, are included in the analysis input. The nonlinear behaviour of reinforced concrete sections under strong strain is considered in the present analyses. The geometrical in-plan asymmetry of the 3D models is expressed by a simply defined ratio. Selected unitless resulting plots of the current dynamic analyses are presented and appropriately discussed given the relative geometrical asymmetry. The role of sequential ground excitations on the dynamic response is recognized, along with the role of simple geometrical symmetry or asymmetry, in the resulting response plots. Thus, useful conclusions are acquired, pointing to remarks on the geometrical structural design helpful for the development of recommendations of seismic provisions.
Seismology and earthquake engineering are essential fields for structural engineers worldwide. The performance of reinforced concrete (RC) structures is greatly influenced by the selection of the proper lateral force resisting method. This research compares the seismic behavior of RC structures with several types of bracing systems. Inverted V-braced frames (IVBF), V-braced frames (VBF), X-braced frames (XBF), diagonally braced frames (DBF), three-member gate braced frames (TMGBF), modified inverted V-braced frames (MIVBF), K-braced frames (KBF), and Z-braced frames (ZBF) are the types of braces under investigation. The SAP2000 software is used to model and evaluate ten-storey RC-frame structures with first-storey heights of 3.5, 4, and 4.5 m. Pushover and nonlinear time-history analyses were carried out. The frames have three bays in both directions, with the steel bracing in the central bay of each frame. The structural responses of buildings are expressed in terms of capacity curve, drift ratio, global damage index, inter-storey drift ratio, base shear, storey displacements , the number of hinges, plastification, and the roof displacement time history. The results showed that adding bracing to frames considerably increased their seismic resilience. In addition, the bracing elements were found to be quite successful in reducing drifts, as seen by the reduction of inter-storey drifts compared to unbraced frames. Steel braces also contributed to reducing the global damage index (GDI) significantly.
https://authors.elsevier.com/a/1cSXkW4G4R6KK (free access till 13th March).
The fundamental period is an essential parameter in the force-based design of buildings as it defines the spectral acceleration and thus the base shear force to which the building should be designed. The fundamental period can be computed based on modeling or utilizing simplified empirical relationships defined in seismic design codes. Previous studies show that the simplified equations should be region-specific and should represent the general design and the construction characteristics of the region. In this study, ambient vibration measurements were carried out on 24 newly constructed, mid-rise reinforced concrete (RC) buildings in Eskisehir, Turkey. The relationships between the fundamental periods and the building heights were examined. Through regression analysis, a simple equation was derived to estimate the elastic fundamental vibration period of mid-rise RC buildings with respect to building height. The proposed relationship was compared to the equation defined in the new Turkey Building Earthquake Code (TBEC-2018) and with the other simplified equations in different design codes and the related literature. The results showed that the fundamental period estimates of the TBEC-2018 equation are much longer than the fundamental periods of the measured buildings in this study. This overestimation may lead to unconservative base shear forces. Therefore, the proposed equation can be used in the force-based design of mid-rise RC frame buildings for conservative estimates of the fundamental period. The contribution of infill walls to the lateral stiffness of the buildings was also investigated, and some preliminary suggestions are made for the related parts in TBEC-2018.
Due to the high computational cost of finite element analyses, particularly in optimization tasks, establishing a high-fidelity surrogate model is of immense importance to engineers. Analyses of fiber steering composite cylinders are costly and often involve high nonlinearity. Among different metamodeling approaches, machine learning techniques provide effective models that can capture high nonlinearity. Therefore, in this study, different machine learning techniques are employed to establish a relationship between the fiber angle and buckling capacity of the cylinders under bending-induced loads. The utilized data set contains a total of 11,000 cases, including seven attributes (i.e., fiber angles) for 11 different aspect ratios (L/RL/R). The numerical results demonstrate that compared with the Random Forest Regressor, Decision Tree algorithm, and Multiple Linear Regression, the Deep Learning model obtains stable results with smaller errors and higher generalization.
More than 30 buildings around Taiwan have been selected to monitor the floor responses under seismic excitation. The structural array monitoring system in each building controls at most 27 channels of accelerometers distributed in several floors. Those buildings were triggered by many events during the past five years of operation. In each building, the records at the basement can be considered as the ground excitation, and the others at the upper floors are the structural responses. The frequency transfer functions of those buildings can be identified by ARX models, and then the fundamental vibration periods are estimated. The identified fundamental vibration periods using different events are compared in order to ensure the reliability of system identification. An empirical formula in predicting the fundamental vibration period is presented through the regression analysis to the identified fundamental vibration periods of 21 reinforced concrete (RC) moment-resisting frame (MRF) buildings. It is found that the height of a building plays an important role in predicting the fundamental vibration period, compared with the length, width, and time after completion of the building. It is also found that the RC MRF buildings in Taiwan tend to be stiffer than those in the U.S. Copyright (C) 2000 John Wiley & Sons, Ltd.
Simple empirical relationships are available in many design codes to relate the height of a building to its fundamental period of vibration. These relationships have been realised for force-based design and so produce conservative estimates of period such that the lateral shear force will be conservatively predicted from an acceleration spectrum. Where assessment of a structure is concerned, however, it is the displacement demand that gives an indication of the damage that can be expected; this displacement would be underestimated with the use of the aforementioned period-height formulae. Furthermore, the period of vibration of interest in assessment is the yield period, which is calculated using the yield stiffness, also often referred to as the cracked or elastic stiffness. The derivation of a yield period-height formula for use in displacement-based assessment of European buildings is thus the focus of this work. Analytical fibre element models of RC frames of varying height have been developed and the yield period has been sought using eigenvalue, pushover and dynamic analyses.
2 SUMMARY Currently, seismic design of new European buildings follows a force-based approach, whilst the assessment of existing buildings is moving towards a displacement-based philosophy. In force- based design, conservative estimates of the period of vibration should be produced such that the base shear force will be conservatively predicted from an acceleration spectrum, and thus the use of gross section (uncracked) stiffness in analytical calculations is perhaps acceptable. For the assessment of buildings, the use of the uncracked stiffness in the determination of the period is certainly inappropriate considering cracking of critical elements such as beams generally occurs under gravity loading alone. Even if cracking is not found to have occurred before the design seismic level of excitation (considered unlikely as this level of excitation would with all probability have been preceded by a number of lower intensity events), it will occur early on in the response to excitation and thereafter the stiffness will reduce rapidly leading to the loss of the tension stiffening effect of the concrete. Thus, the reliable stiffness of the members of an existing RC frame can only be confidently taken as the yield/cracked stiffness. The uncracked and yield period of existing European reinforced concrete buildings of varying height is analytically calculated herein using eigenvalue analysis. A simplified equation is proposed to relate the yield period of vibration of existing buildings to their height for use in large-scale vulnerability assessment applications.
Earthquake design codes generally provide empirical formulae or various approximations for determining fundamental periods of buildings for the evaluation of statically equivalent seismic forces. It is thus important to use realistic periods for the design and performance assessment of existing buildings. This study deals with the evaluation of fundamental vibration periods of reinforced concrete (RC) buildings having a frame-type structural system, and considers the effects of infill walls. The problem is evaluated by utilizing experimental and analytical approaches. A height-dependent relationship to estimate the fundamental period of vibration of Turkish RC moment-resisting frames is derived for a fully elastic condition. Furthermore, a constant parameter is defined to magnify the empirical relationship in order to attain code-specified periods obtained by considering moderate-intensity earthquakes.
The precise prediction of the fundamental vibrational period for reinforced concrete (RC) buildings with infilled walls is essential for structural design, especially earthquake-resistant design. Machine learning models from previous studies, while boasting commendable accuracy in predicting the fundamental period, exhibit vulnerabilities due to lengthy training times and inherent dependence on pre-trained models, especially when engaging with continually evolving data sets. This predicament emphasizes the necessity for a model that adeptly balances predictive accuracy with robust adaptability and swift data training. The latter should include consistent re-training ability as demanded by real-time, continuously updated data sets. This research implements an optimized Light Gradient Boosting Machine (LightGBM) model, highlighting its augmented predictive capabilities, realized through the astute use of Bayesian Optimization for hyperparameter tuning on the FP4026 research data set, and illuminating its adaptability and efficiency in predictive modeling. The results show that the R2 score of LightGBM model is 0.9995 and RMSE is 0.0178, while training speed is 23.2 times faster than that offered by XGBoost and 45.5 times faster than for Gradient Boosting. Furthermore, this study introduces a practical application through a streamlit-powered, web-based dashboard, enabling engineers to effortlessly utilize and augment the model, contributing data and ensuring precise fundamental period predictions, effectively bridging scholarly research and practical applications.
In the past few decades, multi-story X-braced frames have frequently been used to retrofit RC buildings and have become the subject of more attention and implementation to improve the stiffness and strength of structures. In this study, a comprehensive series of inelastic analyses were undertaken to better understand the nonlinear behavior of reinforced concrete frames retrofitted with different bracing configurations subjected to cyclic lateral load. At first, finite element model of a one-story reinforced concrete frame was developed, the analyses were carried out and the system behavior was verified by comparing the results of numerical model with experimental test. Then, a two-story frame is considered and retrofitted by implementing different bracing configurations including V-bracing, Ʌ-bracing, one-story X-bracing and two-story X-bracing type and the responses of the models are investigated subjected to cyclic loads. The results indicate that the two-story X-brace has a better performance compared to other specimens and the stiffness, strength, and behavior of this system is more desirable.
A rational approach is presented for minimizing the dynamic response of reinforced concrete framed structures forced by a seismic base acceleration. Reference is made to EC8 regulations, but the presented approach may in principle be applied to structures ruled by any regulation code. Governing equations are set in the frequency domain (and not in the periods domain as usual) so as to enable the adoption of sound approaches for analysis and design of dynamic structures that are typical of automatics. Among these, a novel usage of the H∞-norm concept is proposed that determines a rational design approach capable to minimize the structural response with reference to any quantity of engineering interest, eg, the overall compliance and the displacement of a specific point or the interstorey drift. A numerical investigation on a 6-storey 3-bay frame is performed, and relevant analysis and design results are presented in much detail to validate the theoretical framework.
Most seismic codes specify empirical formulas to estimate the fundamental vibration period of buildings. Evaluated first in this paper are the formulas specified in present U.S. codes using the available data on the fundamental period of buildings "measured" from their motions recorded during eight California earthquakes, starting with the 1971 San Fernando earthquake and ending with the 1994 Northridge earthquake. It is shown that current code formulas for estimating the fundamental period of concrete shear wall buildings are grossly inadequate. Subsequently, an improved formula is developed by calibrating a theoretical formula, derived using Dunkerley's method, against the measured period data through regression analysis. Also recommended is a factor to limit the period calculated by a "rational" analysis, such as Rayleigh's method.
Large-scale shake table tests were conducted to examine the dynamic response of steel concentrically braced frames. The specimen was a single-bay, single-story frame with a pair of square-tube braces placed in a chevron arrangement. The gusset plates connecting the brace to the framing elements were provided with an elliptical clearance to permit free out-of-plane rotation. The specimen was subjected repeatedly to unidirectional ground motion with increasing magnitude until the braces fractured. The test results demonstrated excellent performance of the gusset plate connections. The specimen response was reproduced by a numerical model using fiber elements. This model was able to predict the occurrence of brace fracture and post-fracture behavior of the frame. Observations from the shake table tests and aspects of the numerical analysis are reported in this paper.
L-BFGS-B is a limited-memory algorithm for solving large nonlinear optimization problems subject to simple bounds on the variables. It is intended for problems in which information on the Hessian matrix is difficult to obtain, or for large dense problems. L-BFGS-B can also be used for unconstrained problems and in this case performs similarly to its predessor, algorithm L-BFGS (Harwell routine VA15). The algorithm is implemented in Fortran 77.
An algorithm for solving large nonlinear optimization problems with simple bounds is described.
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