• Structural Engineering and Mechanics
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• v.8 no.3
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• pp.299-310
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• 1999

One problem with base isolators of the sliding type is that their dynamic responses are nonlinear, which cannot be solved in an easy manner, as distinction must be made between the sliding and non-sliding phases. The lack of a simple method for analyzing structures installed with base isolators is one of the obstacles encountered in application of these devices. As an initial effort toward simplification of the analysis procedure for base-isolated structures, an approach will be proposed in this paper for computing the equivalent damping for the resilient-friction base isolators (R-FBI), based on the condition that the sum of the least squares of errors of the linearized response with reference to the original nonlinear one is a minimum. With the aid of equivalent damping, the original nonlinear system can be replaced by a linear one, which can then be solved by methods readily available. In this paper, equivalent damping curves are established for all ranges of the parameters that characterize the R-FBI for some design spectra.

• Structural Engineering and Mechanics
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• v.85 no.5
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• pp.655-664
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• 2023

Reinforced concrete (RC) shear wall structures are one of the most widely used structural systems to resist seismic loading all around the world. Although there have been several efforts to provide conceptually simple procedures to reasonably assess the seismic demands of structures over recent decades, it seems that lesser effort has been put on a number of structural forms such as RC shear wall structures. Therefore, this study aims to represent a simple linear response spectrum-based method which can acceptably predict the nonlinear displacements of a non-ductile RC shear wall structure subjected to an individual ground motion record. An effective period and an equivalent damping ratio are introduced as the dynamic characteristics of an equivalent linear SDOF system relevant to the main structure. By applying the fundamental mode participation factor of the original MDOF structure to the linear spectral response of the equivalent SDOF system, an acceptable estimation of the nonlinear displacement response is obtained. Subsequently, the accuracy of the proposed method is evaluated by comparison with another approximate method which is based on linear response spectrum. Results show that the proposed method has better estimations for maximum nonlinear responses and is more utilizable and applicable than the other one.

• Journal of the Korean Geotechnical Society
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• v.24 no.3
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• pp.13-23
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• 2008

In a nonlinear site response analysis which is performed in time domain, small strain damping is modeled as viscous damping through use of various forms of Rayleigh damping formulations. Small strain damping of soil is known to be independent of the loading frequency, but the viscous damping is greatly influenced by the loading frequency. The type of Rayleigh damping formulation has a pronounced influence on the dependence. This paper performs a series of nonlinear analyses to evaluate the degree of influence of the viscous damping formulation on Korean soil profiles. Analyses highlight the strong influence of the viscous damping formulation for soil profiles exceeding 30 m in thickness, commonly used in simplified Rayleigh damping formulation overestimating energy dissipation at high frequencies due to artificially introduced damping. When using the full Rayleigh damping formulation and carefully selecting the optimum modes, the artificial damping is greatly reduced. Results are further compared to equivalent linear analyses. The equivalent linear analyses can overestimate the peak ground acceleration even for shallow profiles less than 20 m in thickness.

• Journal of Ocean Engineering and Technology
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• v.19 no.3
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• pp.25-30
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• 2005

To predict rolling performance for a barge-type FPSO, the evaluation of correct nonlinear roll damping is critical. The square section of FPSO causes a considerable viscous damping effect. Free roll decay tests were carried out to estimate nonlinear roll damping for a barge-type FPSO, under three different conditions. The roll motion RAO was deduced from model tests in the wave condition of the wideband spectrum. In numerical calculation, the quadratic damping was considered as equivalent linear damping, using the results of free roll decay test. Tested roll performance in the JONSWAP wave spectrum was compared with numerical results. These two results shaw good agreement, in spite of the proximity of peak wave period and roll natural period.

• Wind and Structures
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• v.6 no.4
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• pp.249-262
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• 2003

The purpose of this study is to propose a procedure for evaluating quantitatively the increase of the equivalent damping ratio of a structure with passive/active vibration control systems subjected to a stationary wind load. A Lyapunov function governing the response of a structure and its differential equation are formulated first. Then the state-space equation of the structure coupled with the secondary damping system is solved. The results are substituted into the differential equation of the Lyapunov function and its derivative. The equivalent damping ratios are obtained from the Lyapunov function of the combined system and its derivative, and are used to assess the control effect of various damping devices quantitatively. The accuracy of the proposed procedure is confirmed by applying it to a structure with nonlinear as well as linear passive/active control systems.

• Structural Engineering and Mechanics
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• v.70 no.3
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• pp.289-301
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• 2019

Hysteretic energy is defined as energy dissipated through inelastic deformations during a ground motion by the system. It includes frequency content and duration of ground motion as two remarkable parameters, while these characteristics are not seen in displacement spectrum. Since maximum displacement individually cannot be the appropriate criterion for damage assessment, hysteretic energy has been evaluated in this research as a more comprehensive seismic demand parameter. An innovative methodology has been proposed to establish a new equivalent linear model to estimate hysteretic energy spectrum for bilinear SDOF models under two different sets of earthquake excitations. Error minimization has been defined in the space of equivalent linearization concept, which resulted in equivalent damping and equivalent period as representative parameters of the linear model. Nonlinear regression analysis was carried out for predicting these equivalent parameter as a function of ductility. The results also indicate differences between seismic demand characteristics of far-field and near-field ground motions, which are not identified by most of previous equations presented for predicting seismic energy. The main advantage of the proposed model is its independency on parameters related to earthquake and response characteristics, which has led to more efficiency as well as simplicity. The capability of providing a practical energy based seismic performance evaluation is another outstanding feature of the proposed model.

• Structural Engineering and Mechanics
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• v.55 no.6
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• pp.1241-1260
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• 2015

Any rational approach to define the configuration and size of viscous fluid dampers in a structure should be based on the dynamic properties of the system with the dampers. In this paper we propose an alternative representation of the complex eigenvalues of multi degree of freedom systems with dampers to calculate new equivalent natural frequencies. Analytical expressions for the dynamic properties of a two-story building model with a linear viscous damper in the first floor (i.e. with a non-proportional damping matrix) are derived. The formulas permit to obtain the equivalent damping ratios and equivalent natural frequencies for all the modes as a function of the mass, stiffness and damping coefficient for underdamped and overdamped systems. It is shown that the commonly used formula to define the equivalent natural frequency is not applicable for this type of system and for others where the damping matrix is not proportional to the mass matrix, stiffness matrix or both. Moreover, the new expressions for the equivalent natural frequencies expose a novel phenomenon; the use of viscous fluid dampers can modify the vibration frequencies of the structure. The significance of the new equivalent natural frequencies is expounded by means of a simulated free vibration test. The proposed approach may offer a new perspective to study the effect of viscous dampers on the dynamic properties of a structure.

• Coupled systems mechanics
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• v.7 no.4
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• pp.395-406
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• 2018

This paper proposes a linear damping model of tire-soil system using semi-empirical method. A test rig was designed and developed to measure the vertical equivalent linear damping ratio of tire only and tire-soil system using Free-Vibration Logarithmic Decay Method. The test was performed with two kinds of tractor tires using a combination of five inflation pressure levels, two soil depths and four soil moisture contents in the paddy soil. The results revealed that the linear damping ratio of tires increased with decreasing tire inflation pressure; the linear damping ratio of tire-soil system also increased with decreasing tire inflation pressure and increased with the increasing soil depth (observed at 80 and 120 mm). It also increased with a relative increase of soil moisture contents (observed at 37.9%, 48.8%, 66.7% and 77.4%). The results also indicated that the damping ratio of tire-soil system was higher than that of tire only. A linear damping model of tire-soil system is proposed as a damping model in parallel which is established based on experimental results and vibration theory. This model will have a great significance in study of tractor vibration.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1995.04a
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• pp.99-106
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• 1995

The seismic responses of a base Isolated Pressurized Water Reactor(PWR) are investigated using a mathematical model which expresses the superstructure as a linear lumped mass-spring and the seismic Isolator as an equivalent spring-damper. Time history analyses are performed for the 1940 El Centre earthquake with linear amplification. In the analysis 5% of structural damping is used for the superstructure. The effects of high damping rubber bearing on seismic response of the superstructure in base isolated system are evaluated for four stiffness model types. The acceleration responses in base isolated PWR superstructure with high damping rubber bearings are much smaller than those in fixed base structure. In the higher strain region where stiffness behaves non-linearly, the acceleration responses modelled by one equivalent stiffness are smaller than those in nonlinear spring model, and the higher stiffness spring model of isolator exhibits larger peak acceleration response at superstructure in the frequency range above 2.0 Hz. when subjected to linearly amplified 1940 El Centre earthquake.

• Structural Engineering and Mechanics
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• v.87 no.2
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• pp.173-189
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• 2023

For a given structural geometry, the stiffness and damping parameters of the soil and the dynamic response of the structure may change in the face of an equivalent linear soil behavior caused by a strong earthquake. Therefore, the influence of equivalent linear soil behavior on the impedance functions form and the seismic response of the soil-structure system has been investigated. Through the substructure method, the seismic response of the selected structure was obtained by an analytical formulation based on the dynamic equilibrium of the soil-structure system modeled by an analog model with three degrees of freedom. Also, the dynamic response of the soil-structure system for a nonlinear soil behavior and for the two types of impedance function forms was also analyzed by 2D finite element modeling using ABAQUS software. The numerical results were compared with those of the analytical solution. After the investigation, the effect of soil nonlinearity clearly showed the critical role of soil stiffness loss under strong shaking, which is more complex than the linear elastic soil behavior, where the energy dissipation depends on the seismic motion amplitude and its frequency, the impedance function types, the shear modulus reduction and the damping increase. Excellent agreement between finite element analysis and analytical results has been obtained due to the reasonable representation of the model.