Asymmetric one‐storey elastic systems with non‐linear viscous and viscoelastic dampers: Earthquake response

Investigated are earthquake responses of one‐way symmetric‐plan, one‐storey systems with non‐linear fluid viscous dampers (FVDs) attached in series to a linear brace (i.e. Chevron or inverted V‐shape braces).Thus, the non‐linear damper is viscous when the brace is considered rigid or viscoelastic (VE) when the brace is flexible. The energy dissipation capacity of a non‐linear FVD is characterized by an amplitude‐dependent damping ratio for an energy‐equivalent linear FVD, which is determined assuming the damper undergoes harmonic motion. Although this formulation is shown to be advantageous for single‐degree‐of‐freedom (SDF) systems, it is difficult to extend its application to multi‐degree‐of‐freedom (MDF) systems for two reasons: (1) the assumption that dampers undergo harmonic motion in parameterizing the non‐linear damper is not valid for its earthquake‐induced motion of an MDF system; and (2) ensuring simultaneous convergence of all unknown amplitudes of dampers is difficult in an iterative solution of the non‐linear system. To date, these limitations have precluded the parametric study of the dynamics of MDF systems with non‐linear viscous or VE dampers. However, they are overcome in this investigation using concepts of modal analysis because the system is weakly non‐linear due to supplemental damping. It is found that structural response is only weakly affected by damper non‐linearity and is increased by a small amount due to bracing flexibility. Thus, the effectiveness of supplemental damping in reducing structural responses and its dependence on the planwise distribution of non‐linear VE dampers were found to be similar to that of linear FVDs documented elsewhere. As expected, non‐linear viscous and VE dampers achieve essentially the same reduction in response but with much smaller damper force compared to linear dampers. Finally, the findings in this investigation indicate that the earthquake response of the asymmetric systems with non‐linear viscous or VE dampers can be estimated with sufficient accuracy for design applications by analysing the same asymmetric systems with all non‐linear dampers replaced by energy‐equivalent linear viscous dampers. Copyright © 2003 John Wiley & Sons, Ltd.     

Earthquake response of elastic SDF systems with non‐linear fluid viscous dampers

The steady‐forced and earthquake responses of SDF systems with a non‐linear fluid viscous damper (FVD) are investigated. The energy dissipation capacity of the FVD is characterized by the supplemental damping ratio ζsd and its non‐linearity by a parameter designated α. It is found that the structural response is most effectively investigated in terms of ζ_sd and α because (1) these two parameters are dimensionless and independent, and (2) the structural response varies linearly with the excitation intensity. Damper non‐linearity has essentially no influence on the peak response of systems in the velocity‐sensitive spectral region, but differences up to 14% were observed in the other spectral regions. The structural deformation is reduced by up to 25% when ζ_sd= 5%; and by up to 60% when ζ_sd= 30%. Non‐linear FVDs are advantageous because they achieve essentially the same reduction in system responses but with a significantly reduced damper force. For practical applications, a procedure is presented to estimate the design values of structural deformation and forces for a system with non‐linear FVD directly from the design spectrum. It is demonstrated that the earthquake‐induced force in a non‐linear FVD can be estimated from the damper force in a corresponding system with linear FVD, its peak deformation, and peak relative velocity; however, the relative velocity should not be approximated by the pseudo‐velocity as this approximation introduces a large error in the damper force. Finally, a procedure is presented to determine the non‐linear damper properties necessary to limit the structural deformation to some design value or the structural capacity for a given design spectrum. Copyright © 2002 John Wiley & Sons, Ltd.     

Simplified seismic analysis of one‐way asymmetric elastic systems with supplemental damping

This study investigates the effectiveness of the modal analysis using two‐degree‐of‐freedom (2DOF) modal stick to deal with the seismic analysis of one‐way asymmetric elastic systems with supplemental damping. The 2DOF modal stick possessing the non‐proportional damping property enables the modal translation and rotation to not be proportional even at elastic state. The analytical results of one‐storey and three‐storey buildings obtained by the proposed method are compared with those obtained by direct integration of the equation of motion and conventional approximate method, which neglects the off‐diagonal elements in the transformed damping matrix. It is found that the proposed simplified method, compared to conventional approximate methods, can significantly improve the accuracy of the analytical results and, at the same time, without obviously increasing computational efforts. Copyright © 2006 John Wiley & Sons, Ltd.     

Seismic response of linear and non‐linear asymmetric systems with non‐linear fluid viscous dampers

This investigation is concerned with the seismic response of one‐story, one‐way asymmetric linear and non‐linear systems with non‐linear fluid viscous dampers. The seismic responses are computed for a suite of 20 ground motions developed for the SAC studies and the median values examined. Reviewed first is the behaviour of single‐degree‐of‐freedom systems to harmonic and earthquake loading. The presented results for harmonic loading are used to explain a few peculiar trends—such as reduction in deformation and increase in damper force of short‐period systems with increasing damper non‐linearity—for earthquake loading. Subsequently, the seismic responses of linear and non‐linear asymmetric‐plan systems with non‐linear dampers are compared with those having equivalent linear dampers. The presented results are used to investigate the effects of damper non‐linearity and its influence on the effects of plan asymmetry. Finally, the design implications of the presented results are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

Understanding and predicting effects of supplemental viscous damping on seismic response of asymmetric one‐storey systems

Supplemental damping could mitigate the earthquake‐induced damage in buildings with asymmetric plan, known to be more vulnerable to damage than comparable symmetric‐plan buildings. This investigation aims to improve the understanding of how and why planwise distribution of fluid viscous dampers (FVDs) influences the response of linearly elastic, one‐storey, asymmetric‐plan systems. Starting with vibration mode shapes, we predict this influence on the modal damping ratios, and in turn on the individual modal responses and the total response. These predictions are confirmed by the computed responses, which demonstrated that the reduction in earthquake response of the system achieved by supplemental damping is strongly influenced by its planwise distribution, which is characterized by four parameters. Identified are asymmetric distributions of supplemental damping that are more effective in reducing the response compared to symmetric distribution. The percentage reduction achieved by a judiciously selected asymmetric distribution can be twice or even larger compared to symmetric distribution. Copyright © 2001 John Wiley & Sons, Ltd.     

A method of calculating the non‐linear seismic response of a building braced with viscoelastic dampers

Buildings are continually subject to dynamic loads, such as wind load, seismic ground motion, and even the load from internal utility machines. The recent trend of constructing more flexible high‐rise buildings underscores the importance of including viscoelastic dampers in building designs. Viscoelastic dampers are used to control the dynamic response of a building. If the seismic design is based only on the linear response spectrum, considerable error may occur when calculating the seismic response of a building; rubber viscoelastic dampers show non‐linear hysteretic damping that is quite different from viscous damping. This study generated a non‐linear response spectrum using a non‐linear oscillator model to simulate a building with viscoelastic dampers installed. The parameters used in the non‐linear damper model were obtained experimentally from dynamic loading tests. The results show that viscoelastic dampers effectively reduce the seismic displacement response of a structure, but transmit more seismic force to the structure, which essentially increases its seismic acceleration response. Copyright © 2003 John Wiley & Sons, Ltd.     

Probabilistic seismic response assessment of linear systems equipped with nonlinear viscous dampers

The paper aims at evaluating the influence of damper properties on the probabilistic seismic response of structural systems equipped with nonlinear viscous dampers. For this purpose, a linear single‐degree‐of‐freedom system with an added linear or nonlinear viscous damper is considered, and the response statistics are evaluated for a set of natural records describing the ground motion uncertainty. A dimensional analysis of the seismic problem is carried out first to identify the minimum set of characteristic parameters describing the system and controlling the seismic response. An extensive parametric study is then performed to estimate the influence of the damper properties on the statistics of the main response quantities of interest (i.e. maximum displacements, accelerations and damper forces), for a wide range of values of the characteristic parameters. Finally, a set of case studies is investigated to show some interesting issues concerning the influence of the damper nonlinear behaviour on the evaluation of the system reliability and to highlight some limitations of current deterministic approaches neglecting the probabilistic properties of the response. Copyright © 2014 John Wiley & Sons, Ltd.     

Application of non‐linear multiple tuned mass dampers to suppress man‐induced vibrations of a pedestrian bridge

This paper presents an application of multiple tuned mass dampers (MTMDs) with non‐linear damping devices to suppress man‐induced vibrations of a 34m long pedestrian bridge. The damping force generated by each of these damping devices is simply a drag force from liquid acting on an immersed section. The quadratic non‐linear property of these devices was directly determined from free vibration tests of a simple laboratory set‐up. Dynamic models of the bridge and pedestrian loads were constructed for numerical investigation based on field measurement data. The control effectiveness of non‐linear MTMDs was examined along with its sensitivity against estimation errors in the bridge's natural frequency and magnitude of pedestrian load. The numerical results indicated that the optimum non‐linear MTMD system was as effective and robust as its linear counterpart. Then, a six‐unit non‐linear MTMD system was designed, constructed, and installed on the bridge. Field measurements after the installation confirmed the effectiveness of non‐linear MTMDs, and the measurement results were in good agreement with numerical predictions. After the installation, the average damping ratio of the bridge was raised from 0.005 to 0.036 and the maximum bridge accelerations measured during walking tests were reduced from about 0.80–1.30 ms^?2 to 0.27–0.40 ms^?2, which were within an acceptable range. Copyright © 2003 John Wiley & Sons, Ltd.     

Simplified analysis of asymmetric structures with supplemental damping

This study investigated the effects of neglecting off‐diagonal terms of the transformed damping matrix on the seismic response of non‐proportionally damped asymmetric‐plan systems with the specific aim of identifying the range of system parameters for which this simplification can be used without introducing significant errors in the response. For this purpose, a procedure is presented in which modal damping ratios computed by neglecting off‐diagonal terms of the transformed damping matrix are used in the traditional modal analysis. The effects of the simplification are evaluated first by comparing the aforementioned modal damping ratios with the apparent damping ratios obtained from the complex‐valued eigenanalysis. The variation of a parameter that was defined by Warburton and Soni as an indicator of the errors introduced by the simplification is examined next. Finally, edge deformations obtained from the simplified procedure are compared with those obtained from the direct integration of the equations of motion. It is found that the simplified procedure may be used without introducing significant errors in response for most practical values of the system parameters. Furthermore, estimates of the edge deformations, in general, tend to be on the conservative side. Copyright © 2001 John Wiley & Sons, Ltd.     

Analytical expression between the impact damping ratio and the coefficient of restitution in the non‐linear viscoelastic model of structural pounding

Earthquake‐induced structural pounding has been recently intensively studied with the help of different models of impact force. It has been verified through comparisons, that the non‐linear viscoelastic model may be considered somewhat the most accurate one among them. The aim of this short paper is to derive an approximating formula relating the impact damping ratio, as a parameter of the model mentioned, with a more widely used coefficient of restitution. The accuracy of the derived analytical formulation has been confirmed through the comparison with the results of numerical simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Simplified non‐linear seismic analysis of infilled reinforced concrete frames

The N2 method for simplified non‐linear seismic analysis has been extended in order to make it applicable to infilled reinforced concrete frames. Compared to the simple basic variant of the N2 method, two important differences apply. A multi‐linear idealization of the pushover curve, which takes into account the strength degradation which occurs after the infill fails, has to be made, and specific reduction factors, developed in a companion paper, have to be used for the determination of inelastic spectra. It is shown that the N2 method can also be used for the determination of approximate summarized IDA curves. The proposed method was applied to two test buildings. The results were compared with the results obtained by non‐linear dynamic analyses for three sets of ground motions, and a reasonable accuracy was demonstrated. A similar extension of the N2 method can be made to any structural system, provided that an appropriate specific R–µ–T relation is available. Copyright © 2004 John Wiley & Sons, Ltd.     

Linear and non‐linear scaling of the Yangtze River flow

Power law correlation properties of sign and magnitude series have been studied based on the series of observation records of flow of the River Yangtze. The results obtained give improved insight into and understanding of the linear and non‐linear processes of the water cycle. With the newly developed Delayed Vector Variance method and the surrogate test, the documented linkage between the sign series and the linear process, and that between the magnitude series and non‐linear process can be verified. The spectra estimated by detrended fluctuation analysis method show different properties of intra‐annual and inter‐annual correlations in both sign and magnitude series. The linear process behaves as an 1/f noise at a time scale less than about 60 days, but shows features of anti‐persistence in terms of long‐term fluctuation. The magnitudes are clustered in three ways mainly caused by non‐linear processes, i.e. periodic clustering, strong short‐term clustering of 1/f noise at time scales less than 20 days, and long‐term clustering with weak persistence. Copyright © 2007 John Wiley & Sons, Ltd.     

A Hertz contact model with non‐linear damping for pounding simulation

This paper investigates the cogency of various impact models in capturing the seismic pounding response of adjacent structures. The analytical models considered include the contact force‐based linear spring, Kelvin and Hertz models, and the restitution‐based stereomechanical approach. In addition, a contact model based on the Hertz law and using a non‐linear hysteresis damper (Hertzdamp model) is also introduced for pounding simulation. Simple analytical approaches are presented to determine the impact stiffness parameters of the various contact models. Parameter studies are performed using two degree‐of‐freedom linear oscillators to determine the effects of impact modelling strategy, system period ratio, peak ground acceleration (PGA) and energy loss during impact on the system responses. A suite of 27 ground motion records from 13 different earthquakes is used in the analysis. The results indicate that the system displacements from the stereomechanical, Kelvin and Hertzdamp models are similar for a given coefficient of restitution, despite using different impact methodologies. Pounding increases the responses of the stiffer system, especially for highly out‐of‐phase systems. Energy loss during impact is more significant at higher levels of PGA. Based on the findings, the Hertz model provides adequate results at low PGA levels, and the Hertzdamp model is recommended at moderate and high PGA levels. Copyright © 2006 John Wiley & Sons, Ltd.     

Linear‐elastic lateral load analysis and seismic design of pin‐supported wall‐frame structures with yielding dampers

This paper describes an analytical investigation on a reinforced concrete lateral load resisting structural system comprising a pin‐supported (base‐rocking) shear wall coupled with a moment frame on 1 or both sides of the wall. Yielding dampers are used to provide supplemental energy dissipation through the relative displacements at the vertical connections between the wall and the frames. The study extends a previous linear‐elastic model for pin‐supported wall‐frame structures by including the effects of the dampers. A closed‐form solution of the lateral load behavior of the structure is derived by approximating the discrete wall‐frame‐damper interactions with distributed (ie, continuous) properties. The validity of the model is verified by comparing the closed‐form results with computational models using OpenSees program. Then, a parametric analysis is conducted to investigate the effects of the wall, frame, and damper stiffness on the behavior of the structure. It is found that the damper stiffness significantly affects the distribution of shear forces and bending moments over the wall height. Finally, the performance‐based plastic design approach extended to the wall‐frame‐damper system is proposed. Case studies are carried out to design 2 damped pin‐supported wall‐frame structures using the proposed approach. Nonlinear dynamic time‐history analyses are conducted to verify the effectiveness of this method. Results indicate that the designed structures can achieve the performance level with the story drift ratios less than target values, and weak‐story failure mechanism is not observed. The approach can be used in engineering applications.     

Non‐iterative time integration scheme for non‐linear dynamic FEM analysis

This paper proposes a non‐iterative time integration (NITI) scheme for non‐linear dynamic FEM analysis. The NITI scheme is constructed by combining explicit and implicit schemes, taking advantage of their merits, and enables stable computation without an iteration process for convergence even when used for non‐linear dynamic problems. Formulation of the NITI scheme is presented and its stability is studied. Although the NITI scheme is not unconditionally stable when applied to non‐linear problems, it is stable in most cases unless stiffness hardening occurs or the problem has a large velocity‐dependent term. The NITI scheme is applied to dynamic analysis of the non‐linear soil–structure system and computation results are compared with those by the central difference method (CDM). Comparison shows that the stability of the NITI scheme is superior to that of the CDM. Accuracy of the NITI scheme is verified because its results are identical with those by the CDM in which the time step is set as 1/10 of that for the NITI scheme. The application of the NITI scheme to the mesh‐partitioned FEM is also proposed. It is applied to dynamic analysis of the linear soil–structure system. It yields the same results as a conventional single‐domain FEM analysis using the Newmark β method. This result verifies the usability of mesh‐partitioned FEM analysis using the NITI scheme. Copyright © 2003 John Wiley& Sons, Ltd.     

Estimation of seismic demands on isolators in asymmetric buildings using non‐linear analysis

A procedure based on rigorous non‐linear analysis is presented that estimates the peak deformation among all isolators in an asymmetric building due to strong ground motion. The governing equations are reduced to a form such that the median normalized deformation due to an ensemble of ground motions with given corner period T_d depends primarily on four global parameters of the isolation system: the isolation period T_b, the normalized strength η, the torsional‐to‐lateral frequency ratio Ω_θ, and the normalized stiffness eccentricity e_b/r. The median ratio of the deformations of the asymmetric and corresponding symmetric systems is shown to depend only weakly on T_b, η, and Ω_θ, but increases with e_b/r. The equation developed to estimate the largest ratio among all isolators depends only on the stiffness eccentricity and the distance from the center of mass to the outlying isolator. This equation, multiplied by an earlier equation for the deformation of the corresponding symmetric system, provides a design equation to estimate the deformations of asymmetric systems. This design equation conservatively estimates the peak deformation among all isolators, but is generally within 10% of the ‘exact’ value. Relative to the non‐linear procedure presented, the peak isolator deformation is shown to be significantly underestimated by the U.S. building code procedures. Copyright © 2003 John Wiley & Sons, Ltd.     

Probabilistic economic seismic loss estimation in steel buildings using post‐tensioned moment‐resisting frames and viscous dampers

The potential of post‐tensioned self‐centering moment‐resisting frames (SC‐MRFs) and viscous dampers to reduce the economic seismic losses in steel buildings is evaluated. The evaluation is based on a prototype steel building designed using four different seismic‐resistant frames: (i) conventional moment resisting frames (MRFs); (ii) MRFs with viscous dampers; (iii) SC‐MRFs; or (iv) SC‐MRFs with viscous dampers. All frames are designed according to Eurocode 8 and have the same column/beam cross sections and similar periods of vibration. Viscous dampers are designed to reduce the peak story drift under the design basis earthquake (DBE) from 1.8% to 1.2%. Losses are estimated by developing vulnerability functions according to the FEMA P‐58 methodology, which considers uncertainties in earthquake ground motion, structural response, and repair costs. Both the probability of collapse and the probability of demolition because of excessive residual story drifts are taken into account. Incremental dynamic analyses are conducted using models capable to simulate all limit states up to collapse. A parametric study on the effect of the residual story drift threshold beyond which is less expensive to rebuild a structure than to repair is also conducted. It is shown that viscous dampers are more effective than post‐tensioning for seismic intensities equal or lower than the maximum considered earthquake (MCE). Post‐tensioning is effective in reducing repair costs only for seismic intensities higher than the DBE. The paper also highlights the effectiveness of combining post‐tensioning and supplemental viscous damping by showing that the SC‐MRF with viscous dampers achieves significant repair cost reductions compared to the conventional MRF. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal design of tuned mass dampers for a multi‐storey cross laminated timber building against seismic loads

In this paper, the effectiveness of different design solutions for tuned mass dampers (TMD) applied to high‐rise cross‐laminated (X‐Lam) timber buildings as a means to reduce the seismic accelerations was investigated. A seven‐storey full‐scale structure previously tested on shaking table was used as a reference. The optimal design parameters of the TMDs, i.e. damping and frequency ratios, were determined by using a genetic algorithm on a simplified model of the reference structure, composed by seven masses each representing one storey. The optimal solutions for the TMDs were then applied to a detailed finite element model of the seven‐storey building, where the timber panels were modelled with shell elements and the steel connectors with linear spring. By comparing the numerical results of the building with and without multiple TMDs, the improvement in seismic response was assessed. Dynamic time‐history analyses were carried out for a set of seven natural records, selected in accordance with Eurocode 8, on the simplified model, and for Kobe earthquake ground motion on the detailed model. Results in terms of acceleration reduction for different TMD configurations show that the behaviour of the seven‐storey timber building can be significantly improved, especially at the upper storeys. Copyright © 2016 John Wiley & Sons, Ltd.     

Gopher bioturbation: field evidence for non‐linear hillslope diffusion

It has generally been assumed that diffusive sediment transport on soil‐mantled hillslopes is linearly dependent on hillslope gradient. Fieldwork was done near Santa Barbara, California, to develop a sediment transport equation for bioturbation by the pocket gopher (Thomomys bottae) and to determine whether it supports linear diffusion. The route taken by the sediment is divided into two parts, a subsurface path followed by a surface path. The first is the transport of soil through the burrow to the burrow opening. The second is the discharge of sediment from the burrow opening onto the hillslope surface. The total volumetric sediment flux, as a function of hillslope gradient, is found to be: q_s (cm³ cm⁻¹ a⁻¹) = 176(dz/dx)³ − 189(dz/dx)² + 68(dz/dx) + 34(dz/dx)^0·4. This result does not support the use of linear diffusion for hillslopes where gopher bioturbation is the dominant mode of sediment transport. A one‐dimensional hillslope evolution program was used to evolve hillslope profiles according to non‐linear and linear diffusion and to compare them to a typical hillslope. The non‐linear case more closely resembles the actual profile with a convex cap at the divide leading into a straight midslope section. Copyright © 2000 John Wiley & Sons, Ltd.     

Boundary crossing and non‐linear theory in earth‐system sciences – a proof of concept based on tsunami and post‐eruption scenarios on Java Island,Indonesia

Although volcanic eruptions are well‐known to be the trigger of some weather and climatic changes, land‐cover changes by pyroclastic‐flows and lahars do not get this recognition, neither do major hazards such as tsunami. These two earth processes are even lesser considered as being able to modify other earth processes they are not directly connected to, such as landslides or river discharge in non‐connected basins more than a hundred kilometres away. In this contribution the authors argue that these ideas are mainly driven by the process of being ‘educated’ in a single academic discipline and once put to the test interactions and retroactions between earth processes and atmospheric processes are far more reaching than commonly thought. For this study, the site of Java Island (Indonesia) was chosen to conduct (1) an analysis of a major tsunami impact – in the same area as the 2006 Java tsunami and (2) an analysis of the post‐eruption impacts of Merapi Volcano after a major eruption – excluding any ejecta in the atmosphere for the sake of the demonstration. The atmospheric feedback simulations were conducted using the regional climate model (RegCM‐4) with calibration from weather stations in Java Island. As a result, both simulations have proven that large scale deposits of pyroclasts (not introducing the ejectas sent in the atmosphere) and tsunamis can have outstanding impacts on the atmospheric situation and the bio‐geomorphologic evolution of the landscape in the following weeks to months. Interestingly enough these impacts are not limited to the area impacted by the earth process and the effect are not linear in time as they work following thresholds. These rainfalls ‘tele‐impacts’ are important enough to, in turn, modify earth‐surface processes in areas remote from the original phenomenon. This system acts in the same manner as a famous butterfly in Africa that could trigger a hurricane on the other side of the Atlantic Ocean. Copyright © 2012 John Wiley & Sons, Ltd.