• 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.

• Structural Engineering and Mechanics
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• v.62 no.4
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• pp.455-464
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• 2017

Rayleigh damping model is recommended in the recently developed Performance-Based Earthquake Engineering (PBEE) methodology, but this methodology does not provide sufficient information due to the complexity of the damping mechanism. Furthermore, each Rayleigh-type damping model may have its individual limitations. In this study, Rayleigh-type damping models that are used widely in engineering practice are discussed. The seismic performance of a large-span single-layer latticed dome subjected to earthquake ground motions is investigated using different Rayleigh damping models. Herein a simulation technique is developed considering low cycle fatigue (LCF) in steel material. In the simulation technique, Ramberg-Osgood steel material model with the low cycle fatigue effect is used to simulate the non-uniformly distributed material damping and low cycle fatigue damage in the structure. Subsequently, the damping forces of the structure generated by different damping models are compared and discussed; the effects of the damping ratio and roof load on the damping forces are evaluated. Finally, the low cycle fatigue damage values in sections of members are given using these damping models. Through a comparative analysis, an appropriate Rayleigh-type damping model used for a large span single-layer latticed dome subjected to earthquake ground motions is determined in terms of the existing damping models.

• Geomechanics and Engineering
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• v.18 no.6
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• pp.639-650
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• 2019

Time-domain commercial codes are widely used to evaluate the seismic behavior of tunnels. Those tools offer a good insight into the performance and the failure mechanism of tunnels under earthquake loading. Viscous damping is generally employed in the dynamic analysis to consider damping at very small strains in some cases, and the Rayleigh damping is commonly used one. Many procedures to obtain the damping parameters have been proposed but they are seldom discussed. This paper illustrates the influence of the Rayleigh damping formulation on the tunnel visco-elastic behavior under earthquake. Four Rayleigh damping determination procedures and three soil shear velocity profiles are accounted for. The results show significant differences in the free-field and in the tunnel response caused by different procedures. The difference is somewhat decreased when the soil site fundamental frequency is increased. The conventional method which consists of using solely the first soil natural mode to determine the viscous damping parameters may lead to an unsafe seismic design of the tunnel. In general, using five times site fundamental frequency to obtain the damping formulation can provide relatively conservative results.

• Geomechanics and Engineering
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• v.29 no.2
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• pp.155-170
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• 2022

An accurate analysis of structures supported on soft soils and subjected to seismic loading requires the consideration of the soil-foundation-structure interaction. An important aspect of this interaction lies with the energy dissipation due to soil material damping. Unlike advanced constitutive models that can induce energy loss, the use of simple elastoplastic constitutive models requires additional damping. The frequency dependent Rayleigh damping is a formulation that is frequently used in dynamic analysis. The main concern of this formulation is the correct selection of the target damping ratio and the frequency range where the response is frequency independent. The objective of this study is to investigate the effects of the Rayleigh damping parameters in soil-pile-structure and soil-inclusion-platform-structure systems in the presence of soft soil under seismic loading. Three-dimensional analyses of both systems are carried out using the finite difference software Flac3D. Different values of target damping ratios and minimum frequencies are utilized. Several earthquakes are used to study the influence of different excitation frequencies in the systems. The soil response in terms of accelerations, displacements and strains is obtained. For the rigid elements, the results are presented in terms of bending moments and normal forces. The results show that when the frequency of the input motion is close to the minimum (central) frequency in the Rayleigh damping formulation, the overdamping amount is reduced, and the surface spectral acceleration of the analyzed pile and inclusion systems increases. Thus, the bending moments and normal forces throughout the piles and inclusions also increase.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.26 no.6_spc
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• pp.635-643
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• 2016

A stability theory of a damped cantilever column under sub-tangential follower forces is first summarized based on the stability map. It is then demonstrated that internal and external damping can be exactly transformed to Rayleigh damping so that the damping coefficients can be effectively determined using proportional damping. Particularly a parametric study with variation of damping coefficients is performed in association with flutter loads of Beck's column and it is shown that two damping coefficients can be correctly estimated for real systems under the assumption of Rayleigh damping. Finally a frequency equation of a cantilever beam subjected to both a sub-tangentially follower force and two kinds of damping forces is presented in the closed-form and its stability maps are constructed and compared with FE solutions in the practical range of damping coefficients.

• Transactions of the Korean Society of Pressure Vessels and Piping
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• v.13 no.2
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• pp.38-43
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• 2017

This paper investigates how to determine the Rayleigh damping coefficients for dynamic time history seismic analysis of piping systems. Three methods are applied. The first one is a conventional method to use the natural frequencies of the mode 1 and 2, derived from dynamic analysis. The second method is to determine the Rayleigh damping coefficients based on frequency range of the acceleration histories. The last one is a iterative transient response analysis method using the transient analysis results without and with damping. It is found that the conventional method and the iterative transient response method yield the same results whereas the acceleration frequency-basis method provides more conservative result than the other methods. In addition, it is concluded that the iterative transient response method is recommended.

• Structural Engineering and Mechanics
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• v.16 no.2
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• pp.177-193
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• 2003

This paper presents an illustrative example of the advantages offered by inserting added viscous dampers into shear-type structures in accordance with a special scheme based upon the mass proportional damping (MPD) component of the Rayleigh viscous damping matrix. In previous works developed by the authors, it has been widely shown that, within the class of Rayleigh damped systems and under the "equal total cost" constraint, the MPD system provides best overall performance both in terms of minimising top-storey mean square response to a white noise stochastic input and maximising the weighted average of modal damping ratios. A numerical verification of the advantages offered by the application of MPD systems to a realistic structure is presented herein with reference to a 4-storey reinforced-concrete frame. The dynamic response of the frame subjected to both stochastic inputs and several recorded earthquake ground motions is here analysed in detail. The results confirm the good dissipative properties of MPD systems and indicate that this is achieved at the expense of relatively small damping forces.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.29 no.6
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• pp.583-590
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• 2016

This paper presents a dynamic crack propagation algorithm with Rayleigh damping effect based on the MLS(Moving Least Squares) Difference Method. Dynamic equilibrium equation and constitutive equation are derived by considering Rayliegh damping and governing equations are discretized by the MLS derivative approximation; the proportional damping, which has not been properly treated in the conventional strong formulations, was implemented in both the equilibrium equation and constitutive equation. Dynamic equilibrium equation including time relevant terms is integrated by the Central Difference Method and the discrete equations are simplified by lagging the velocity one step behind. A geometrical feature of crack is modeled by imposing the traction-free condition onto the nodes placed at crack surfaces and the effect of movement and addition of the nodes at every time step due to crack growth is appropriately reflected on the construction of total system. The robustness of the proposed numerical algorithm was proved by simulating single and multiple crack growth problems and the effect of proportional damping on the dynamic crack propagation analysis was effectively demonstrated.

• Wind and Structures
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• v.11 no.6
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• pp.497-512
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• 2008

In this paper, optimum methods of wind-induced vibration control of high-rise buildings are mainly studied. Two optimum methods, genetic algorithms (GA) method and Rayleigh damping method, are firstly employed and proposed to perform optimum study on wind-induced vibration control, six target functions are presented in GA method based on spectrum analysis. Structural optimum analysis programs are developed based on Matlab software to calculate wind-induced structural responses. A high-rise steel building with 20-storey is adopted and 22 kinds of control plans are employed to perform comparison analysis to validate the feasibility and validity of the optimum methods considered. The results show that the distributions of damping coefficients along structural height for mass proportional damping (MPD) systems and stiffness proportional damping (SPD) systems are entirely opposite. Damping systems of MPD and GAMPD (genetic algorithms and mass proportional damping) have the best performance of reducing structural wind-induced vibration response and are superior to other damping systems. Standard deviations of structural responses are influenced greatly by different target functions and the influence is increasing slightly when higher modes are considered, as shown fully in section 5. Therefore, the influence of higher modes should be considered when strict requirement of wind-induced vibration comfort is needed for some special structures.

• Structural Engineering and Mechanics
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• v.63 no.6
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• pp.735-742
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• 2017

Steel cables as the most important components are widely used in the certain types of structures such as cable-supported bridges, but the long-span structures may result in an increase in fatigue under high stress and corrosion of steel cables. The traditional steel cable is becoming a more evident hindrance. Fiber Reinforced Polymer (FRP) cables with lightweight, high-strength are widely used in civil engineering, but there is little research in vibrational characteristics of FRP cables, especially on the damping characteristic. This article studied the two methods to evaluate dynamical damping characteristic of basalt FRP(BFRP) and glass FRP(GFRP) cables. First, the vibration tests of the B/G FRP cables with different diameter and different cable force were executed. Second, the cables forces were calculated using dynamic strain, static strain and dynamic acceleration respectively, which were further compared with the measured force. Third, experimental modal damping of each cables was calculated by the half power point method, and was compared with the calculation by Rayleigh damping theory and energy dissipation damping theory. The results indicate that (1) The experimental damping of FRP cables decreases with the increase of cable force, and the trend of experimental damping changes is roughly similar with the theoretical damping. (2) The distribution of modal damping calculated by Rayleigh damping theory is closer to the experimental results, and the damping performance of GFRP cables is better than BFRP cables.