Equivalent Linearization Method Research Articles

Neural network prediction model for site response analysis based on the KiK-net database

Currently, the most widely used site response analysis methods include linear, equivalent linear and nonlinear approaches. However, the results of these analyses gradually deviate from the true values as the site softens or the earthquake intensity increases. To overcome this issue, this study employs a 1D Convolutional Neural Network model to simulate the relationships between the surface and downhole motions. To ensure comprehensive coverage of various sites and earthquake conditions, a careful selection of stations and seismic intensities resulted in 56 stations and 34,538 seismic records from KiK-net. Additionally, 30 seismic intensity measures and three site condition parameters were chosen. The convolutional operation was utilized to consider the correlations between the parameters, enabling the prediction of surface motion intensity measures. The performance of the neural network model was assessed based on residual analysis, and the model’s generalization ability was tested using samples that were not included in the training dataset. The results demonstrate that the neural network can effectively predict surface motion intensity measures. Furthermore, in comparison with the equivalent linear and nonlinear methods that do not rely on site-specific detailed soil data, the 1D Convolutional Neural Network exhibits superior performance and maintains high prediction accuracy even for soft soil and strong earthquake scenarios, confirming the enhanced predictive capability of the proposed model.

Study on Dynamic Post-Buckling Stability of Thin-Walled Cylinders Subjected to Horizontal Vibration

Abstract As a lesson learned from the Fukushima nuclear accident, the importance of accident mitigation for beyond design basis events (BDBEs) is recognized. Excessive earthquake is a typical BDBE. During such events, the fast reactor main vessel (FRMV) is vulnerable to buckling because of its thin-walled cylindrical structure. Buckling deformation without causing immediate collapse should not be considered as a limit state in BDBEs. Instead, the objective is to achieve a stable post-buckling state. In the present study, dynamic post-buckling behavior of thin-walled cylinders under intense horizontal vibration is experimentally studied, which simulates loop-type FRMV design. Finite element method (FEM) simulation is also conducted and validated by experimental data. Based on results analysis, a global response stability after buckling is confirmed. The buckled model shows resilience to withstand catastrophic collapse even when intense excitation continues to apply. Using equivalent linear response method, dynamic mechanisms behind the stability are revealed. Due to substantial degradation of structural stiffness, the post-buckling response shifts into the out-of-phase domain such that response divergence is prevented. Meanwhile, the response is notably delayed from the input, which impedes the conversion of input energy into kinetic energy, reducing the inertial impact to the structure. These mechanisms are found to be independent of input waveforms and can be applied to a general seismic scenario. Moreover, the limit for global response stability after buckling and the effect of input frequency on the post-buckling behavior are clarified.

Equivalent Linearization and Parameter Optimization of the Negative Stiffness Bistable Damper

The negative stiffness bistable damper (NSBD) was proposed to suppress structural dynamic responses in our previous study. The vibration mitigation performance of the NSBD is influenced by its design parameters, including negative stiffness, cubic stiffness, and damping coefficients. However, it is extremely challenging to directly acquire the ideal design parameters of the NSBD owing to its inherent nonlinearity. To address this disadvantage, the optimal design approach for the NSBD, based on the equivalent linearization method (ELM) and genetic algorithm (GA), is presented in this paper. The nonlinear NSBD system can be transformed to a linear system utilizing the ELM based on the pseudo-excitation method (PEM). The linearization model that corresponds to the nonlinear NSBD is fairly accurate in its approximation and can be indicated from the numerical results. Then, the main structure’s peak response is minimized through the optimization of the design parameters of the NSBD using the H∞ norm and GA. Moreover, the proposed approach’s effectiveness is assessed using the optimal parameters to calculate the displacement responses of a tall building equipped with the NSBD during various seismic excitations. As revealed by the numerical results, the displacement of the tall building can be effectively restrained by the optimized NSBD.

Supercritical and subcritical aeroelastic behaviors of a three-dimensional wing coupled with a nonlinear energy sink

This paper studies the flutter control of a three-dimensional wing using a nonlinear energy sink (NES), which has a purely cubic stiffness and a linear damper. The structural model is derived by Lagrange's equations and the aerodynamics is modeled using the auto regressive with exogenous input method. Both models are solved in a tightly coupling manner to obtain time domain results. Then, the structural describing function and the aerodynamic continuous-time state space model are used to derive the equivalent linearized aeroelastic system, so that eigenvalue analyses can be used to obtain stable and unstable limit cycle oscillations (LCOs). The LCOs obtained by time domain simulations agree well with the equivalent linearization method, while differences are observed for aperiodic motions which are only predicted in time domain for small damping coefficients. Results show the square of aeroelastic amplitude is inversely proportional to the NES stiffness coefficient for both periodic and aperiodic motions. The NES can induce LCOs below flutter speed for small damping coefficients, indicating a subcritical bifurcation which can change to a supercritical one by increasing the damping coefficient. It is discovered that the upper bound of partial suppression region is quantified by a turning point speed. The regions of complete suppression, partial suppression and no suppression are clarified.

Equivalent linearization of hyperelastic rubber rings in dynamic vibration absorber boring bar and simulation of its dynamic properties

A slight perturbation in high-speed precision cutting could cause violent vibration of boring bars, resulting in unexpected wear and failure of the structures. In this paper, the dynamical characteristic of a boring bar with a dynamic vibration absorber (DVA) is investigated. Considering that the nonlinear characteristics of the rubber ring play a significant role in improving the capacity of DVA, we proposed an equivalent linearization method for modeling hyperelastic rubber rings based on the experimental data of uniaxial tensile test. The relationship between the stiffness of rubber rings and the precompression parameter δ is established for two types of rubber. The equivalent linearization model shows good accuracy and calculation efficiency while the mass of the DVA is confined to a narrow space and the assumption of small strain is inherently satisfied. Numerical results show that the time cost of the equivalent linearization model is only 5% of the full hyperelastic model. Furthermore, the effect of different spring stiffnesses and damping on the dynamic characteristics of the boring bar is compared by using the equivalent linearization model, and the vibration peak of the DVA boring bar is found to be reduced about 45%. Our findings establish a simplified modeling scheme for choosing rubber rings in design of dynamic vibration absorbers with acceptable accuracy and little time cost, which could also give some reference for modeling of metal rubber and gel-like materials.

An Equivalent Linear Method to Predict Nonlinear Bending Mechanics of Dredging Floating Hose String

Dredging hoses are flexible and are particularly suitable for slurry transportations for mud or sand in dredging projects. To achieve sufficient bending stiffness and to prevent the pipe body from collapsing, this type of hose segment is a composite structure that is embedded with several cord reinforcement layers and steel wires in its rubber layer. To quickly evaluate the nonlinear bending mechanical properties of rubber hoses, this study proposes the equivalent stiffness method of linear superposition, which is verified by test data and numerical results. The results show that the equivalent bending stiffness method proposed in this study is in good agreement with numerical and experimental results. Then, by comparing the calculation results of the hose string, it was demonstrated that the linear stiffness superposition method proposed in this study can also accurately predict the bending mechanical behavior characteristics of string hose, and provide reliable guidance for hose design in practice.

Transformation of the Equivalent Linear Method for solving soil-structure interaction problems in the time domain

Transformation of the Equivalent Linear Method for solving soil-structure interaction problems in the time domain

Vibration Response Analysis of a Time-varying Stiffness Wheel with Contrate Gear under Unbalanced Excitation

The vibration and response of multi-contact interface micro-motion time-varying stiffness wheel are analyzed, and the nonlinear vibration response characteristics caused by the unbalance force of multi-contact interface contrate gear face of a gas turbine rotor are studied. Firstly, the local sliding model of dry friction-damped contact is extended to establish a holistic and local unified sliding model. The equivalent stiffness and damping of the damping device are calculated by means of the equivalent linearization method and the first harmonic balance method. Finally, the finite element model of multi-contact contrate gear rotor wheel is established to analyze the influence of different unbalanced masses on vibration response under nonlinear frictional contact. The results show that the response caused by the unbalance mass of multi-contact interface wheel with contrate gear is mainly radial response, and the peak value of resonance response increases with the increase of unbalance force.

Dynamic responses of a bituminous concrete core rockfill dam on layered overburden under seismic wave input

The dynamic nonlinearity, overburden stratification characteristics and oblique incidence of seismic waves in soil greatly increase the difficulty of determining the free field of a dam foundation, which restricts seismic input accuracy. Based on the potential function theory, this paper constructs a transfer relationship between the wave amplitude matrix at the top layer of a dam foundation with overburden and the wave amplitude matrix at any layer and obtains the shear strain of the soil through Fourier inverse transformation. Then, the equivalent linearization method is adopted to depict the nonlinearity of the soil. An analytical calculation method for the free field of a dam foundation with layered overburden under inclined seismic waves is proposed. This method can not only forward the free field according to the incident wave in the deep part of the foundation but also directly invert the free field according to the seismic motion on the ground surface, and the inversion efficiency is high. Using a nonlinear viscous-spring boundary element developed by the authors, combined with the free field, a wave input method for a dam foundation with a layered overburden under inclined seismic waves is established. The maximum error between the numerical solution under wave input and the analytical solution is only 5%. The computational precision of the method is high. Finally, the dynamic responses of a bituminous concrete core rockfill dam on a layered overburden are studied. The results show that an upstream buoyancy and oblique incidence are the main reasons for a larger acceleration on the upstream side of the dam than on the downstream side. As the incident angle increases, the differential distribution becomes more significant, and the amplification coefficient of the anti-seepage system also increases. When the incident angle is 25°, the horizontal and vertical amplification coefficients at the top of the core wall increase by 12.8% and 25.2% compared to the vertical incidence, respectively. The accelerations of the dam are overestimated under uniform seismic input, and the amplification coefficients of the dam crest are overestimated by 52.3% and 39.8% compared with the wave input, respectively. The wave input reasonably solves the problem of seismic input for a dam foundation with layered overburden, considers the inclined seismic waves, and can reflect the seismic waves propagation characteristics. The wave method provides a solid foundation for the seismic safety of rockfill dams with layered overburden.

A framework for nonlinear peak response evaluation from linear response spectra of base-isolated structures using the bilinear hysteretic model

This paper proposes a simple computational framework that can account for both properties of the base-isolated structure using the bilinear hysteretic model and reliably evaluate the nonlinear peak response without nonlinear response history analysis. The proposed computational framework requires the construction of new non-proportional damping models that match the base-isolated structure to accurately decompose the nonlinear multi-degree-of-freedom structure into multiple single-degree oscillators with intrinsic frequencies and damping ratios by equivalent linearization and complex-mode superposition methods. Each oscillator is excited by a response spectrum matching its own damping ratio, and this procedure is iteratively computed until the displacement error of the isolation layer achieves a preset value. Simultaneously, the peak nonlinear response of interest is calculated using the complex-mode-superposition complete square combination rule. The effect of peak response on nonlinear parameters is analyzed using a typical numerical example, and the relative error values for the applicable range are presented. Parametric research demonstrates that the new non-proportional damping matrix accurately reflects the modal characteristics of the base-isolated structure and prevents Rayleigh damping from suppressing the first-order modal characteristics. The particular example evaluation shows that the computational framework can effectively evaluate the nonlinear peak response of the BIS, with the isolation layer displacement error controlled within +5 %, the bottom shear force error within +20 %, and the top acceleration error between −10 % and +25 %.

Effect of site specific soil characteristics on the nonlinear ground response analysis and comparison of the results with equivalent linear analysis

Damages occurring during earthquakes may vary depending on the ground conditions in which the earthquake waves pass, the magnitude of the earthquake, the focal depth of the shaking and as well as the structural characteristics. Regional seismicity, ground movement and behavior of local soil conditions are important in earthquake-resistant building designs. Local site conditions consist of the layering, bedrock depth, and dynamic and topographic characteristics of soil that alter the bedrock waves through the soil profile during an earthquake. The change occurs in terms of amplitude, frequency and the time when the peak happens during the wave propagation. This initiates a big difference between the surface and bedrock motion. The behavior of the soil under cyclic loads resulting from seismic action is non-linear. This study aims to demonstrate the effects of strong ground motion by taking into account the nonlinear behavior of the soil layers. In addition, the results obtained from the equivalent linear and non-linear methods were compared. The results of the study showed that the characteristics of soil layers and strong ground motion (frequency content and duration) significantly affect the field response analysis and generally larger spectral parameters (about %20) have been obtained with the equivalent linear method compared to the nonlinear behavior. Finally, empirical models to estimate the soil amplification reflect different compared to the site specific analysis.

Design optimization of hybrid isolation bearing for base-isolated buildings catering to multi-level performance fortification criteria

The adaptive seismic isolation system has attracted significant attention from researchers due to its multi-stage behavior and which can be used for multi-staged performance-based design. This paper introduces the configuration and mechanical behavior of a novel hybrid isolation bearing (HIB) combined with sliding bearings and rubber springs. The proposed isolator has three levels of stiffness and can achieve decoupling of horizontal and vertical performance. From the static test conducted on HIB500_R200, it is evident that the HIB displays a three-stage stiffness characteristic. The experimental stiffness values match well with the theoretical values, with a maximal error of 7.18%. Subsequently, an optimal design function for HIB-based isolation structures aiming to satisfy multiple performance objectives is established based on isolation effectiveness and displacement, utilizing the mode-superposition response spectrum and equivalent linear method. The optimization function can effectively and swiftly provide the optimal friction coefficient and the diameter of the rubber spring. A Performing elastoplastic analysis of HIB-based structures adopted for optimization function and traditional design method is performed using Seismostruct software. The numerical analysis reveals that the maximum error between the seismic isolation coefficient and the ratio of shear displacement amplitude values, as well as their theoretical predictions, is only 3.4%. And it also has higher safety redundancy than the traditionally designed isolation structures under mega earthquakes.

A time and ensemble equivalent linearization method for nonlinear systems under combined harmonic and random excitation

An Equivalent Linearization technique, termed an Equivalent Linearization Time and Ensemble Expectation (EL-TEE) approach, is used to develop an alternative method for estimating the response of a nonlinear oscillator to a combination of deterministic harmonic and random white noise excitation. The approach is based on applying equivalent linearization and averaging over the time period of one harmonic excitation cycle. This gives a set of coupled nonlinear equations that can be solved for the response averaged over time and across the ensemble. The primary advantages of the proposed method are its computational speed, ability to return physically meaningful linearization matrices and that it can be applied to a wide variety of nonlinearities. The method is applied to three example test systems: the well-known single degree of freedom Duffing oscillator; a single degree of freedom system with a displacement constraint imposing a discontinuous nonlinearity; and a multi degree of freedom oscillator with a localized polynomial nonlinearity that has also been examined experimentally. It is shown that the response predicted matches well with Monte Carlo results from direct time integration at a fraction of the computational cost, and the method is capable of reproducing key results observed experimentally.

Optimized equivalent linearization for random vibration

A fundamental limitation of various Equivalent Linearization Methods (ELMs) in nonlinear random vibration analysis is that they are approximate by their nature. A quantity of interest estimated from an ELM has no guarantee to be the same as the solution of the original nonlinear system. In this study, we tackle this fundamental limitation. We sequentially address the following two questions: (i) given an equivalent linear system obtained from any ELM, how to construct an estimator such that, as the linear system simulations are guided by a limited number of nonlinear system simulations, the estimator converges on the nonlinear system solution? (ii) how to construct an optimized equivalent linear system such that the estimator approaches the nonlinear system solution as quickly as possible? The first question is theoretically straightforward since classic Monte Carlo techniques, such as the control variates and importance sampling, can improve upon the solution of any surrogate model. We adapt the well-known Monte Carlo theories into the specific context of equivalent linearization. The second question is challenging, especially when rare event probabilities are of interest. We develop specialized methods to construct and optimize linear systems. In the context of uncertainty quantification (UQ), the proposed optimized ELM can be viewed as a physical surrogate model-based UQ method. The embedded physical equations endow the surrogate model with the capability to handle high-dimensional uncertainties in stochastic dynamics analysis.

Evaluation of equivalent linearization methods of MDOF isolated structures in response spectrum analysis

In the seismic design of isolated structures, equivalent linearization of the hysteretic isolators is normally considered. There exist many equivalent linearization methods and evaluation of these methods has been widely studied. However, most of the previous studies concern a single degree of freedom system only, and the used analysis method is time history analysis. In such a case, the evaluation heavily depends on the ground motions selected. Note that the response spectrum analysis is a commonly used method for seismic design of structures. Therefore, it is more valuable to evaluate these equivalent linearization methods based on the response spectrum analysis. In this study, a multiple degree of freedom model is adopted due to the contribution of higher modes of the superstructure to the seismic responses, and the procedure of the response spectrum analysis in conjunction with the equivalent linearization method is presented in detail. Then, considering a series of parameters of isolators, a statistical study is performed to evaluate the accuracy of these equivalent linearization methods by comparing with the nonlinear time history analysis. Moreover, the effect of the damping reduction factor for the response spectrum on the accuracy of the equivalent linearization methods is discussed in terms of probability in a desirable error range. The analytical results show that the accuracy of the equivalent linearization methods is associated with the earthquake intensity. Meanwhile, for different response quantities the accuracy of each equivalent linearization method varies significantly. In addition, to estimate the seismic responses with a better accuracy in the response spectrum analysis, a suitable damping reduction factor should be considered.

Short-time frequency-domain method for truly nonlinear dynamic ground response analysis: The equivalent-nonlinear approach

Evaluating soil nonlinearity during cyclic loading is one of the most significant challenges in ground response analysis, especially when dealing with the inverse problem of deconvolution. Different schemes have already been developed for dynamic ground response analysis, both in the time and the frequency domain. The most accurate method to account for soil nonlinearity is the nonlinear dynamic analysis in the time domain. This approach is based on nonlinear constitutive models capable of accurately simulating highly nonlinear problems like soil liquefaction. However, the time-domain analysis is suitable only for the convolution analysis to define the ground motion at the free surface of a soil deposit from the bedrock motion. The frequency-domain analysis is the most common solution for the inverse problem called deconvolution, which is used to define the bedrock motion from the free surface ground motion. A well-known approach developed in the frequency domain for ground response analysis is the equivalent-linear method (EQL). This approach adopts an iterative procedure to define elastic shear modulus and damping ratio compatible with the induced strain level. Still, it presents some limitations, especially for highly nonlinear soil response, due to the use of strain-compatible but constant soil properties. This article presents a new scheme to conduct truly nonlinear dynamic analysis in the frequency domain based on the new concept of the short-time transfer function. Unlike the EQL method, which uses a constant transfer function, the proposed approach, called the “Equivalent-Nonlinear” method (EQNL), defines a soil transfer function evolving in time, depending on the shear stress and strain demands. The EQNL method approximates the response of a nonlinear system as an incrementally changing viscoelastic system and could represent a valuable tool for nonlinear deconvolution. This article shows the analytical formulation and the first set of validations of the EQNL approach, with detailed comparisons with the EQL and NL methods and vertical array data. These comparisons show the potentialities of the EQNL approach to reproduce the results of the nonlinear dynamic analysis. The EQNL approach has been implemented in MATLAB, and the source code is provided as supplementary material for this article. A more comprehensive validation is underway, aiming to better characterize the limitations and the capabilities of the method.

Design and analysis of LRB base-isolated building structure for multilevel performance targets

According to China's current code for the seismic design of buildings (GB50011-2010), in the course of designing base-isolated building structures, the isolation system and superstructure are designed separately. Hence, the horizontal seismic isolation coefficient in terms of the design-based earthquake (DBE) and horizontal deformation of the isolation system under the maximum considerable earthquake (MCE) cannot be considered simultaneously. Consequently, the optimum mechanical parameters for isolation bearing and the corresponding horizontal seismic isolation coefficient used in the seismic design of the superstructure cannot be obtained. To address these shortcomings, we propose an optimum seismic design procedure for LRB base-isolated building structures to meet multilevel performance fortification criteria, where explicit mathematical formulations for horizontal seismic isolation coefficients and the horizontal deformation of isolation systems in terms of the mechanical parameters of the LRB isolation system are established by combining equivalent linearization and modal decomposition response spectrum methods. Through a parametric study, we find the critical yield strength ratio, i.e., the ratio of the yielding shear of lead core to the total weight of the LRB base-isolated structure, which leads to the minimal value of the horizontal seismic isolation coefficient. Beyond this critical yield strength ratio, an optimum solution is unavailable, as the horizontal seismic isolation coefficient and horizontal displacement of the isolation system are of a conflicting nature. By converting the multi-criterion formulation into an auxiliary problem with a scalar objective, the optimum design method of an LRB base-isolated building structure oriented to multilevel performance fortification criteria is proposed, so as to obtain the optimum yield strength ratio and the optimum second shape coefficient for the seismic design of the superstructure. A numerical example demonstrates that the seismic horizontal seismic isolation coefficient and maximum horizontal displacement of the isolation system calculated from the optimum yield strength ratio and the optimum second shape coefficient agree well with the results from the time-history method. In a comparative study, LRB base-isolated building structures designed by the critical yield strength ratio, the optimum yield strength ratio and the optimum second shape coefficient, the optimum yield strength ratio and the corresponding fixed-base superstructure are subjected to MCE and mega earthquakes, where the LRB base-isolated structure shows good seismic performance in terms of the optimum yield strength ratio, even when subjected to mega earthquakes.

Time-domain deconvolution procedure for elastoplastic materials: Application to the Treasure Island site during the 1989 Loma Prieta earthquake

When dynamic soil-structure interaction (DSSI) analyses are performed, e.g. using the finite element (FE) method, the input signal is required at the base of the model. Nevertheless, acceleration records are usually available at the surface and, therefore, the desired motion must be deconvolved to the base. The latter is usually performed through the solution of one-dimensional propagation of shear waves in an elastic medium, in the frequency domain. Herein, nonlinear behavior is generally incorporated through the equivalent-linear method, by iteratively reducing the stiffness and increasing the critical damping ratio as a function of the maximum strains attained in each iteration. However, if complex material models are adopted to characterize the soil, the input motion derived with the equivalent linear method will not be compatible due to the simplified approach used to represent the nonlinear behavior. In this article, the use of a procedure to perform a time-domain deconvolution in non-linear elastoplastic materials is demonstrated. The goal is to generate input accelerograms at the base of a FE model to perform DSSI analyses. The procedure is based on the iterative modification of the motion at the base according to the relative differences between the propagated and target surface spectra. To illustrate the use of the methodology, it was applied to a FE model of the Treasure Island site (San Francisco, US), to derive the required motion at the base from a record of the Loma Prieta earthquake.•This article provides a useful guideline to optimize the use of the deconvolution procedure to derive input motions for dynamic FE analyses considering nonlinear elastoplastic materials.

Seismic isolated bridge performance evaluation and design proposal using artificial neural network

AbstractSeismic isolators such as rubber bearings and other dampers are critical components of seismic design. The hysteresis behavior of such bearings is complicated, and engineers are using a bilinear model or equivalent linearization method in design practice. The response evaluation and initial design parameter assumptions are highly dependent on engineers' expertise and need repeated analysis. Therefore, this study proposed a machine learning‐based approach to conduct bridge seismic isolation performance evaluation and design parameters proposal without repeating the nonlinear simulations. The training data were generated by conducting a nonlinear time history analysis of the bridge structure model with input design earthquakes. Four trained three‐layer neural network models could perform bridge seismic response evaluation, predict the structure's seismic response, and suggest design proposal using some input parameters.

Dynamic Modelling and Mutual Coordination of Electricity and Watershed Networks for Spatio-Temporal Operational Flexibility Enhancement Under Rainy Climates

This paper proposes a watershed-electricity nexus model to unlock the flexibility of watershed networks (WSNs) for supporting the operation of power distribution networks (PDNs) under rainy climates. The proposed model exploits the spatio-tem-poral flexibility of geographically dispersed pump clusters to pro-vide reserve services to PDNs, and a hyperbolic partial differential function derived from Saint-Venant hydrodynamic equations is formed to describe the dynamic processes of river stream flows. Be-sides, a flexibility evaluation method based on a composite sensitiv-ity matrix of water levels with respect to power injections is pre-sented to quantify the time-varying adjustable power domain of pump loads. Then, a multi-stage interactive coordinated scheduling strategy is developed for the mutual operation of WSNs and PDNs, where drainage pumps are jointly optimized to provide flexible power reserves, while an optimal PDN economic dispatch is per-formed to improve the power supply voltage of pump loads. Fur-thermore, an equivalent mixed-integer linear programming refor-mulation method is derived to cope with the original nonlinear par-tial differential optimization problem for computational tractabil-ity improvements. Comparative results have validated the effective-ness of the proposed strategy in eliminating voltage violations and shaving peak loads.