Equivalent Linearization Research Articles

Wind-induced vibration control and parametric optimization of connected high-rise buildings with tuned liquid-column-damper–inerter

A novel passive control device, defined as tuned liquid-column-damper–inerter (TLCDI), with nonlinear characteristics is developed for control of the wind-induced responses of adjacent high-rise buildings. TLCDI exploits the combined advantages of liquid sloshing inside TLCD container as well as mass amplification effects of the inerter to control the wind induced responses. Firstly, a model for adjacent high-rise buildings linked by a TLCDI is established subjected to wind loading. Subsequently, an equivalent linearization method is used to study the model, and the results are verified by a numerical method. Finally, mitigation effects of the TLCDI with optimal designs on wind-induced responses are investigated by comparison studies with the results of two tuned liquid column dampers (TLCDs) installed on benchmark buildings. The results indicate that the TLCDI, which is much lighter than two TLCDs, dramatically decreases the wind-induced acceleration responses of the linked buildings by 37.7% (Building-1) and 42.0% (Building-2) at the across-wind direction (90° wind direction).

Nonlinear seismic responses of the soft rock site for nuclear power plants

With the development of the nuclear power industry, high-quality bedrock nuclear power stations have become scarce. Thus, it is inevitable that nuclear power stations will be built on soft rock sites. This involves the issue of the seismic safety of soft rocks as sites for nuclear power plants. Based on the soft rock site involved in the site selection for nuclear power plants, this paper uses ground motions with different spectral characteristics. Using the DEEPSOIL software for soil layer analysis and the Abaqus finite element software, one-dimensional equivalent linear and nonlinear methods are employed to compare and analyze the characteristics of the nonlinear seismic response of the soft rock site for nuclear power plants. The soft rock site has a significant amplification effect on the input ground motion. The equivalent linear and nonlinear methods have different levels of influence on the acceleration response of the soft rock site. Under the action of far-field earthquakes, surface acceleration and its response spectrum, which were calculated using the abovementioned methods, are relatively close. As the input ground motion intensity increases, the nonlinearity of the soft rock increases, and the amplification effect of the soft rock site weakens. However, the peak ground acceleration of the surface obtained using the two methods gradually differs. Moreover, the nonlinearity is greater than the equivalent linear. Therefore, the nonlinear characteristics of soft rocks should be considered in the site selection and seismic analysis of nuclear power plants.

Harmonic-Modal Hybrid Reduced Order Model for the Efficient Integration of Non-Linear Soil Dynamics

Nonlinear behavior of soils during a seismic event has a predominant role in current site response analysis. Soil response analysis, and more concretely laboratory data, indicate that the stress-strain relationship of soils is nonlinear and exhibits hysteresis. An equivalent linearization method, in which non-linear characteristics of shear modulus and damping factor of soils are modeled as equivalent linear relations of the shear strain is usually applied, but this assumption, however, may lead to a conservative approach of the seismic design. In this paper, we propose an alternative analysis formulation, able to address forced response simulation of soils exhibiting their characteristic nonlinear behavior. The proposed approach combines ingredients of modal and harmonic analyses enabling efficient time-integration of nonlinear soil behaviors based on the offline construction of a dynamic response parametric solution by using Proper Generalized Decomposition (PGD)-based model order reduction technique.

A numerical and scaled experimental study on ride comfort enhancement of a high-speed rail vehicle through optimizing traction rod stiffness

In this research, the effect of rail vehicle carbody’s flexural modes on the ride comfort of an example high-speed railway vehicle is investigated. The vehicle is modeled as a rigid multi-body system, where the rigid body vertical, longitudinal, pitch, and roll degrees of freedom of the carbody and bogie frames and the rigid body vertical and roll degrees of freedom of the wheelsets are considered. An Euler–Bernoulli beam theory is used to account for the flexural motion of the carbody. The longitudinal interaction between carbody and bogie through the traction rod is modeled as a nonlinear spring element. The corresponding equations of motion of the system in the frequency domain are obtained by using the equivalent linearization method. The effect of the traction rod is explored by using this model. Also, the optimal stiffness of the traction rod element that minimizes the flexural vibrations of the carbody is obtained through a genetic algorithm. With the optimal stiffness for the traction rod, the ride quality index at the center of the carbody floor is improved by 41% at a speed of 300 km/h. For the validation of numerical results, a scaled model of the vehicle with a scale factor of 24.5 was constructed, and its associated results are presented. The model was excited by random input signals, which were generated based on the power spectral density of the track irregularity function. The agreement between the simulation results and the scaled experimental outcome when compared with the measured data from other sources is found to be satisfactory. In the framework of the physical scaled model, the filtering effect due to the vehicle bogie base is also examined.

An iterative equivalent linearization approach for stochastic sensitivity analysis of hysteretic systems under seismic excitations based on explicit time-domain method

The sensitivity analysis of hysteretic systems under nonstationary random excitations is of great concern to the stochastic optimal design and control of structures. The equivalent linear equation of motion is first constructed for the hysteretic system by the equivalent linearization method (ELM), and the sensitivity equation of the equivalent linear system is then derived by the direct differentiation technique. These two equations are further combined into an overall equivalent linear equation, which depends on the statistical moments and sensitivities of responses of the system and should be solved on an iterative basis. The analysis problem is thus transformed to a series of nonstationary random vibration analyses of the iterative linearized systems, which can be solved with the explicit time-domain method (ETDM) recently proposed for linear random vibration problems. The ETDM has high computational efficiency owing to the explicit formulations of statistical moments of responses. Therefore, a numerical algorithm is developed by combining ELM and ETDM for efficient stochastic sensitivity analysis of hysteretic systems. A 100-degree-of-freedom hysteretic system is investigated to illustrate the accuracy and efficiency of the proposed method, and a 10-degree-of-freedom base-isolated system is further analyzed to show the feasibility of the proposed method for stochastic structural optimization.

Nonlinear random vibrations of functionally graded porous nanobeams using equivalent linearization method

In the present study, the nonlinear random vibration of a porous functionally graded nanobeam supported by a viscoelastic foundation has been investigated by making use of the statistical linearization method for the first time. Nonlocal Euler-Bernoulli beam theory with von Kármán type nonlinearity is utilized to derive the governing equation of the motion of the FG porous nanobeam. Two types of porosity distribution have been considered for the FG beam. To discretize the governing equation Galerkin's method is applied. The mean square value of the response of the resulted Duffing's equation in the presence of a random input with nonzero mean value has been extracted using statistical linearization method. The results of the present study are compared with the analytical results of the FPK approach and the results of the perturbation method for the classical limit case of the homogeneous beam. It is revealed through detailed numerical analysis that the statistical linearization method (in contrast to the perturbation method) gives reliable results and there is an excellent agreement between its results and the analytical results. The effect of various parameters on the mean value of the response is also investigated numerically and illustrated graphically. It is found that the mean square value of vibrations increases when porosity, power-law index, nonlocal parameter, or the mean value of the input increases.

Optimal design of nonlinear TMD with Bingham-type damping for base-excited structures

ABSTRACT This paper presents an optimal design procedure for a nonlinear TMD with Bingham-type damping that can lead to a more economic and realistic design of the damper. The optimization problem is solved in the Genetic Algorithm (GA) framework. Considering small nonlinearities, the nonlinear TMD design using GA is validated through standard equivalent linearization method. The efficacy of the TMD designs is presented through a numerical simulation study on four different single-degree-of-freedom structure-TMD systems subjected to recorded earthquake ground motions. Results indicate that the optimally designed nonlinear TMD with Bingham-type damping can effectively reduce the size of the TMD damping element at the cost of a very slight degradation in control performance as compared to the linear TMD. The improved performance of the nonlinear TMD with Bingham-type damping over the linear TMD that develops post-design nonlinearities is also shown. Further, a robust design procedure of the nonlinear TMD using GA is presented that can cater to perturbations in the natural frequency of the primary structure. Thus, the optimal design procedure using GA for the nonlinear TMD offers several advantages as compared to the statistical linearization approach.

Dynamic reliability analysis of nonlinear structures using a Duffing-system-based equivalent nonlinear system method

To improve the analysis accuracy of dynamic reliability of a nonlinear system, an equivalent nonlinear system method is presented. In this method, general nonlinear systems are converted to equivalent Duffing nonlinear systems according to the minimum mean square error criterion, whose exact analytical solution of the random steady-state responses can be determined by a FokkerPlanckKolmogorov equation. Then the exact results of stochastic responses are used to analyze structural dynamic reliability. To use the equivalent nonlinear system method to analyze structural dynamic reliability is not only convenient for calculation but highly accurate. In addition, the presented equivalent nonlinear system has a parameter ε, which controls the degree of nonlinearity. Thus, it is easy to obtain the corresponding analysis results from converting the original system to equivalent nonlinear systems with different degrees of nonlinearity by changing the value of ε. Especially, when ε is zero, the equivalent nonlinear system will degenerate to the linear system, and leading to the analysis results of the equivalent linearization method. An example shows that the analysis results of the proposed equivalent nonlinear system method are reliable, and the calculation appears to be more accurate than that of the equivalent linearization method.

Probabilistic analysis on parametric random vibration of a marine riser excited by correlated Gaussian white noises

A path integration method is used to discuss the stochastic response of a marine riser excited parametrically and externally by correlated Gaussian white noises. On the basis of the nonlinear vibration equation of a marine riser, the first-order vibration mode is studied and its governing equation is formulated. The response probability density function (PDF) is further developed by a path integration method. During the solution procedure, the Gaussian closure method and the short-time Gaussian approximation method are used to develop the transition probability density function. The results of different time intervals are compared with the results of Monte-Carlo simulation (MCS) to determine the adequate time interval. After that, the path integration method with a Gauss–Legendre scheme is used to obtain the stationary PDF solution of displacement and velocity. The path integration solutions (PIS) are compared with those of the equivalent linearization method (EQL) and MCS. Five demonstrative cases are analyzed to evaluate the effectiveness of the path integration method. The influences of nonlinearity and excitations are considered. It shows that the results of PIS coincide better with the results of MCS than those of EQL. The analysis shows that parametric excitation causes the softening behavior of the PDF distribution compared with those of EQL. The increase of excitation correlation results in the nonsymmetrical PDF distribution of response. The increase of nonlinearity in displacement can lead the PDF of displacement to showing a hardening behavior.

Seismic response control of adjacent high-rise buildings linked by the Tuned Liquid Column Damper-Inerter (TLCDI)

A novel passive control device termed Tuned Liquid Column Damper-Inerter (TLCDI) is adopted to control the seismic response of adjacent high-rise buildings. The inerter of the proposed TLCDI scheme exploits the large relative acceleration between the two adjacent buildings, producing a significant resisting force. Firstly, the governing nonlinear equations of motion of the TLCDI-linked adjacent high-rise buildings under earthquake excitation are formulated. For design purposes, the inherent nonlinear behavior of the TLCDI is linearized by adopting an equivalent stochastic linearization method. Subsequently, a benchmark structure consisting of two adjacent high-rise buildings linked by a corridor with a roller support and equipped with a TLCDI is examined. Eight characteristic parameters of the TLCDI are optimized through two distinct constrained multi-objective optimization problems by an evolutionary algorithm aimed at minimizing either the peak acceleration or the interstory-drift ratio of the two adjacent benchmark structures, respectively. Mitigation effects of the optimally designed TLCDIs on the responses of the benchmark structure are assessed considering a set of ten natural ground-motion records. It is found that the optimum TLCDI achieves mean reduction ratios of 15.7 and 26.1% on the peak acceleration of both the buildings, respectively. Conversely, the TLCDI fails to significantly reduce the interstory drift ratios. Robustness analysis of the optimally designed TLCDI is investigated under perturbations to the frequency and damping ratios. Some practical considerations are finally discussed concerning the feasibility of employing the proposed TLCDI in the considered benchmark project.

Controlling Dynamic Response of Structures Using Hybrid Passive Energy Dissipation Device

ABSTRACT This paper presents a new hybrid passive control device, consisting of a viscoelastic and a friction damper connected in parallel, to mitigate the effects of wind and seismic events. The two components are synergized with the aid of a novel locking mechanism, which enables the action of one or both the components depending upon the device deformation. This provides an innovative two-phase energy dissipation capacity, and exploiting the individual strengths of the two dampers together. The response of a single degree of freedom (SDOF) hybrid damped system is represented in terms of a linear system using equivalent linearization technique, which includes all associated parameters such as the added stiffness, the added damping and the locking mechanism. In order to ascertain the influence of various parameters of the hybrid device, normalized force and normalized displacement are defined in terms of linear SDOF hybrid damped system and further used to obtain force reduction factor for multi degree freedom (MDOF) system with hybrid device. The proposed design methodology of controlling the structural response using the hybrid device is then applied on a six-story reinforced concrete (RC) building. The response of the conventional moment resisting frame (MRF) system and moment resisting frame system with hybrid device (MRHDF) are compared using the nonlinear time history analysis. The performance of the MRHDF system is found to be superior than the MRF system. The hybrid device can effectively reduce the dynamic response of the building, especially in terms of inter story drift, residual drift, base shear and floor acceleration. In addition the proposed design methodology is very simple as compared to the response modification factor approach. This configuration of hybrid device shows improved performance for meeting multiple objectives under wind/seismic events of real-life structures.

Distributed Model-Free Bipartite Consensus Tracking for Unknown Heterogeneous Multi-Agent Systems with Switching Topology.

This paper proposes a distributed model-free adaptive bipartite consensus tracking (DMFABCT) scheme. The proposed scheme is independent of a precise mathematical model, but can achieve both bipartite time-invariant and time-varying trajectory tracking for unknown dynamic discrete-time heterogeneous multi-agent systems (MASs) with switching topology and coopetition networks. The main innovation of this algorithm is to estimate an equivalent dynamic linearization data model by the pseudo partial derivative (PPD) approach, where only the input–output (I/O) data of each agent is required, and the cooperative interactions among agents are investigated. The rigorous proof of the convergent property is given for DMFABCT, which reveals that the trajectories error can be reduced. Finally, three simulations results show that the novel DMFABCT scheme is effective and robust for unknown heterogeneous discrete-time MASs with switching topologies to complete bipartite consensus tracking tasks.

Optimization of SMA based damped outrigger structure under uncertainty

Outriggers have been proven to be an efficient system to reduce the dynamic responses of core-tube type high-rise buildings by utilizing the axial stiffness of the peripheral columns. However, outer columns and outriggers are subjected to excessive lateral force demands. To reduce the demand, the damped outrigger is used where dampers are installed between the perimeter columns and outriggers. To improve the performance and enhance the residual deformation, in this paper, the use of shape memory alloy (SMA) springs is introduced to dissipate energy. The SMA dissipates energy through a hysteretic phase transformation of its microstructure triggered by cyclic loading. In the mathematical model implementation, the nonlinear behavior of superelastic SMA is linearized by the stochastic equivalent linearization method. Stochastic uncertainty in the ground motion is considered, and the performance is assessed by minimizing the failure probability of the structure. To solve the optimization problem, Kriging is used as a surrogate model. For better accuracy of the surrogate model, an efficient global optimization scheme is used. The purpose of the optimization adopted in this study is to find optimal design parameters associated with the initial stiffness of the SMA and the location of the outrigger. The results clearly demonstrated the efficacy of the proposed SMA based outrigger.

Probabilistic solution of nonlinear ship rolling in random beam seas

In this paper, the probability density function (PDF) and the mean up-crossing rate of nonlinear ship rolling in random beam seas are investigated. The excitation of stationary random sea waves is approximated as a second-order linear filtered white noise. The Fokker–Planck–Kolmogorov (FPK) equation governing the probability density function of ship rolling is a four-dimensional linear partial differential equation with varying coefficients, and obtaining its exact solution is much more sophisticated. The exponential-polynomial closure (EPC) method is applied to solve the corresponding FPK equation of the system. In numerical examples, linear-plus-cubic damping model and linear-plus-quadratic damping model with three different sea states are further examined. Comparison with the equivalent linearisation (EQL) method and Monte Carlo simulated results show that the proposed procedure is effective to obtain a satisfactory probability density function solution, especially in the tail region.

Stochastic sensitivity analysis of energy-dissipating structures with nonlinear viscous dampers by efficient equivalent linearization technique based on explicit time-domain method

Nonlinear fluid viscous dampers have been widely used in energy-dissipating structures due to their stable and high dissipation capacity and low maintenance cost. However, the literature on stochastic optimization of nonlinear viscous dampers under nonstationary excitations is limited. This paper is devoted to the stochastic response and sensitivity analysis of large-scale energy-dissipating structures equipped with nonlinear viscous dampers subjected to nonstationary seismic excitations. The analysis procedure is developed in the frame of the equivalent linearization method (ELM) in conjunction with the explicit time-domain method (ETDM). The equivalent linear system and the corresponding statistical moments of responses at a specific time instant are first obtained through a series of stochastic response analyses of the linearized systems. Then the sensitivities of the statistical moments of responses are determined via a series of stochastic sensitivity analyses of the equivalent linear system at the corresponding time instant. The above two iterative procedures are facilitated at high efficiency using ETDM with explicit formulations of the statistical moments of responses and the sensitivities of the statistical moments. This process is repeated for different time instants, and the time histories of the statistical moments and their sensitivities can be obtained. The stochastic response and sensitivity results are further utilized to conduct the stochastic optimal parametric design of the nonlinear viscous dampers. A one-storey building model equipped with a nonlinear viscous damper is analyzed to demonstrate the accuracy of the proposed method, and a suspension bridge with a main span of 1200 m equipped with 4 nonlinear viscous dampers is further investigated to illustrate the feasibility of the proposed method for stochastic optimal design of large-scale structures.

Composite behavior in RC buildings retrofitted using buckling-restrained braces with elastic steel frames

Seismically retrofitting reinforced concrete (RC) building with a combination of buckling-restrained braces (BRBs) and elastic steel frames offers a practical solution that provides additional lateral stiffness and energy dissipation capacity. However, the available methods to vertically distribute the BRB sizes based on equivalent linearization do not consider the additional stiffness due to the composite behavior between the RC frame and the elastic steel frame, which may lead to an overly conservative estimate of the BRB stiffness demands. This study proposes a retrofit design method incorporating the composite behavior. Numerical models considering the detailed composite behavior are developed and calibrated against quasi-static cyclic loading tests, and a simplified evaluation method is proposed. A four-story RC school building is used as a benchmark model, and the proposed retrofit design method is validated using nonlinear response history analysis. The analysis results suggest that taking the composite behavior into account by using the proposed retrofit design method more accurately estimates the lateral stiffness of the retrofitted structure and leads to a more economic retrofit.

A Site-Specific Response Analysis: A Case Study in Hanoi, Vietnam

A series of one-dimensional (1-D) site response analyses were performed using the nonlinear (NL) and equivalent linear (EQL) approaches to assess the applicability of the Vietnamese earthquake-resistance design code TCVN 9386: 2012. Six soil profiles were selected from three districts in Hanoi (Vietnam). A number of ground motions compatible with the rock design spectrum were used as input for carrying out analyses. The results highlight that the calculated response is higher than the design spectrum for site class C and lower for site class D. The normalized response spectra of the EQL approach results are higher than those of the NL approach. Moreover, the peak ground accelerations at the surface from EQL analyses are greater than those of the NL method because the latter generates a higher amount of nonlinearity. The results from the NL approach also illustrate that the deamplification phenomenon occurs in the soft soils of the Hanoi region (e.g., soil profile P3 and P5 of site class D). Additionally, the shear strains calculated from the NL method are closely matched with those from the EQL method, the difference between them increasing with a decrease in soil stiffness.

Seismic Response of Long-Span Domes Supported by Multi-Storey Substructures

This paper investigates the seismic response characteristics of long-span domes. The natural periods of the prominent modes are longer than medium-span domes, which leads to a greater contribution from the higher modes to the response of the long-span dome. The acceleration distributions, particularly the vertical acceleration distributions are sensitive to the dominant mode shapes of these higher modes. This leads to inaccuracies when applying the previously proposed response evaluation methods. The vibration modes of multi-storey supporting substructures also affect the excited vibration modes of the roof. In this paper, the dynamic characteristics and seismic response of 150m-span domes supported by multi-storey substructures are studied. The effects of the post- yield stiffness of multi-storey substructures are also analysed by considering two structural systems, buckling- restrained braced frames (BRBF) and damped spine frames. A simple design procedure to evaluate the equivalent static loads using amplification factors and incorporating the effects of higher modes is proposed based on response spectrum analysis and equivalent linearisation procedures. The accuracy of the proposed method is evaluated by comparing the responses with those obtained from non-linear response history analysis.

Inerter-based tuned liquid column damper for seismic vibration control of a single-degree-of-freedom structure

The study investigates the effectiveness of a novel tuned liquid column damper-inerter (TLCDI) for the seismic vibration control of a single-degree-of-freedom (SDOF) structure. First, a mathematical model of the TLCDI-equipped SDOF structure under the condition of earthquake excitation is developed. Based on the equivalent linearization method, closed-form solutions of the model under a white-noise excitation are derived and verified by using a numerical method. Subsequently, a parametric optimization of the TLCDI is performed under white-noise excitation with peak relative displacement and absolute acceleration responses of the primary structure as the two objectives. The optimal parameters are used to assess the mitigation effects of the TLCDI in comparison with those of a conventional tuned liquid column damper (TLCD). The results obtained by using the equivalent linearization method are consistent with those obtained from the numerical method under both white-noise and recorded earthquake excitations. The TLCDI achieves average reductions in peak displacement and acceleration responses of 54.2% and 64.9 %, respectively, which are 18.2% and 9.5 % larger than the reduction in the corresponding responses achieved by TLCD under various recorded earthquake excitations. Furthermore, the performance of the TLCDI is proven to be more robust than that of the TLCD under perturbation in the frequency ratio.

Modified flag-shaped model for self-centering system and its equivalent linearization and structural optimization for stochastic excitation

The analysis and design of self-centering structural systems have attracted substantial attention from researchers for the seismic design of structures. The flag-shaped hysteretic model has been widely used in the design and structural response analysis of self-centering devices. In this research, a modified flag-shaped (MFS) model is proposed to describe the hysteretic characteristics of self-centering energy dissipation (SCED) braces, which are commonly used in self-centering structures. The MFS model comprises three parts: a linear elastic part, a bilinear elastic part, and an elasto-plastic part with a slip zone. The effectiveness of the MFS model is validated based on the results of the quasi-static and hybrid simulation tests available in the literature. Equivalent linearization is carried out on the MFS model for stochastic earthquakes, and closed-form solutions are obtained for the linearized parameters. A one-degree-of-freedom braced system is used to verify the equivalent linearization of the MFS model in comparison with Monte Carlo simulation results. Finally, an SCED-braced five-story frame structure is optimized by minimizing the maximum ductility demand for the SCED braces using the equivalent linearization of the MFS model. The optimized design by the MFS model is observed to exhibit uniform ductility demands along the height. This can be considered as the ideal optimal solution. The responses of the multi-degree-of-freedom structure also demonstrate the effectiveness of the proposed MFS model and its equivalent linearization.