The effect of column mass on shear building vibrations

In the usual assignment of mass for shear building models, the mass of the columns is lumped with the mass of the floor at each floor level. In order to account for frequencies of modes higher than N in an N-storey building, however, additional column masses must be identified. In this note, such additional masses are located at the mid-height of each column, and the characteristics of the shapes and frequencies of the resulting higher mode set are investigated.     

Elastic earthquake analysis of torsionally coupled multistorey buildings

With the aid of perturbation analysis of vibration frequencies and mode shapes it is shown that any lower vibration mode of a torsionally coupled building may be approximated as a linear combination of three vibration modes of the corresponding torsionally uncoupled system (a system with coincident centres of mass and resistance but all other properties are identical to the actual system): one translational mode along each of the two principal axes of resistance and one mode in torsional vibration. This result provides the motivation for a simpler—relative to the standard—procedure for analysing the response of torsionally coupled multistorey buildings to earthquake ground motion. To illustrate the application and accuracy of this procedure two numerical examples are presented.     

Influence of lateral-load-resisting system on the earthquake response of structures—a system identification study

Analysis and comparison of the dynamic responses of three well instrumented (with accelerographs) high-rise buildings shaken during the 1984 Morgan Hill earthquake are presented. The buildings examined in the present work are (i) the Town Park Towers Apartment building, a 10-storey, concrete shear wall building; (ii) the Great Western Savings and Loan building, a 10-storey building with concrete frames and shear walls; and (iii) the Santa Clara County Office building, a 13-storey, moment-resistant steel frame building. The structures are located within 2 km of each other and, as may be confirmed by visual inspection of the recorded seismograms, experienced similar ground motions. One-dimensional and three-dimensional linear structural models are fitted to the observations using the modal minimization method' for structural identification, in order to determine optimal estimates of the parameters of the dominant modes of the buildings. The time-varying character of these parameters over the duration of the response is also investigated. Comparison of the recorded earthquake response of the structures reveals that the type of lateral-load-resisting system has an important effect on the dynamic behaviour of the structures because it controls the spacing of the characteristic modes on the frequency axis. The Santa Clara County Office building has closely spaced natural frequencies and exhibits strong torsional response and modal coupling. Its dynamic behaviour is contrasted with that of the Great Western Savings and Loan building which has well separated natural frequencies and exhibits small torsional response and no modal coupling. Strong modal coupling causes a beating-type phenomenon and makes earthquake response of structures different from that envisioned by codes.     

On the coupled torsional and sway vibrations of a class of shear buildings

This paper is concerned with the free vibrations of a restricted class of multi-storey shear buildings in which inertial coupling exists between the torsional and the two sway vibrations. The restrictions imposed are that (a) the shear centres of all storeys lie on a vertical straight line, (b) the principal axes of shear are in the same directions in all storeys, (c) the centres of mass of all floors lie on another vertical straight line, (d) the radius of gyration about the shear centre of every floor mass is the same and (e) the ratios of the two shear stiffnesses to the torsional stiffness do not vary from storey to storey. In consequence of the last restriction it is proved that the 3n natural frequencies, normal modes and generalized masses, where n is the number of storeys, are expressible very simply in terms of products of the three natural frequencies, normal modes and generalized masses of the single-storey, three-dimensional building formed by removing everything above the first floor, with the n natural frequencies, normal modes and generalized masses of a certain n-storey, two-dimensional shear frame. In the special case of a uniform building, a simple closed form solution, valid for any number of storeys, is given.     

A case study of mimo system identification applied to building seismic records

A large number of high quality strong-motion records of building response are obtained from recent earthquake events in California. The accelerometers are typically deployed at several levels from the basement up. In order to learn as much as possible about the building behaviour from these records, a multiple input-multiple output (MIMO) system identification procedure is implemented to analyse these records in a systematic way. The procedure is an extension of the least-square-output-error method applied to a classically damped linear second order system. The time varying behaviour is modelled through a time window approach. The procedure includes (i) preliminary record analysis, (ii) input and model selection, (iii) parameter identification and (iv) drift analysis. The records of a 15-storey reinforced concrete building obtained during the Whittier earthquake are analysed. It is found that the fundamental period is much longer than that of a regular building. The torsional response is significant and is caused by both the translational and the torsional motions at the ground level. The maximum drift occurs at the ground storey. The second and the third translational modes in each direction are as important as the first modes in making up the ground storey drift. When the maximum drift occurs in one direction the corresponding drift in the orthogonal direction can be as high as 30 per cent of the maximum drift.     

Sloshing response of layered liquids in rigid tanks

The sloshing action of layered liquids in rigid cylindrical and long rectangular tanks is investigated, considering both their free vibrational characteristics and their response to a horizontal component of base shaking. Special attention is given to the maximum surface displacement induced by the base motion. The analysis is formulated for systems with N superimposed layers of different thicknesses and densities, and is illustrated by a numerical example. In addition, comprehensive numerical data are presented for two-layered and some three-layered systems which elucidate the underlying response mechanisms and the effects and relative importance of the numerous parameters involved. It is shown that for each horizontal natural mode of vibration, there are N distinct vertical modes, the frequencies of which are lower than the natural frequency of a homogeneous liquid of the same total depth. It is further shown that the maximum surface sloshing displacement of the base-excited layered system is typically larger than that of the corresponding homogeneous system, and that the results for the long rectangular and cylindrical tanks are quite similar.     

When the doorbell rings–a case of building response to a long distance earthquake

Responding to ground tremors caused by the magnitude 7–0 Liwa earthquake in Sumatra on 16 February 1994, a doorbell system on the 15th floor of a 17-storey building in Singapore rang repeatedly at an epicentral distance of more than 750 km. This paper first reviews briefly the regional seismicity, surface geology and effects of local soil amplification for Singapore. It then estimates the building response through the linear and non-linear analyses of impact conditions for the double-pendulum doorbell system. Based on the impact analysis results, it is shown that the acceleration response at the 15th floor reached at least 0–02 g and that the base shear coefficient was no less than 1–0 per cent. The response was comparable to the notional horizontal load which usually governs the design of most buildings in Singapore. In view of the fast growing economy and the rapid urbanization of Singapore, it is prudent to re-evaluate systematically the effects of a long distance, large Sumatra earthquake on Singapore.     

Coupling phases and amplitudes of minor constituents with temperature perturbations in the lower thermosphere

Based on the linearized theory of atmospheric gravity waves (AGW) and considering the effects of temperature perturbation on the chemical rate coefficients, the formulae of coupling relations between n_j/n_j and T′/T driven by AGW (n_j and T denote the background quantities) are described, the coupling phases and amplitudes of minor constituents O₃, NO, H, OH, and O are analyzed in detail for the mesopause (86 km) and just upside of the O layer (100 km) at daytime. A general principle is outlined: the coupling phases are strongly dependent on density scale heights and perturbation scales, while the amplitudes are little affected by these two factors. A criterion to distinguish the coupling behavior is given: when the minor constituent number density scale height H_j satisfies 1-H_m/H_j>0 (H_m denotes the scale height of the major constituent), the dynamical perturbation process always keeps the n_j/n_j in phase with T’/T, i.e. keeps the n_j/n_j in antiphase with that of the major constituent. The results obtained indicate that both the temperature dependence of reaction rate coefficients and the profile slopes of the O distribution may have a major influence on the behavior of the coupling relations.     

Minimal structural response under random excitation using the vibration absorber

The equations of motion are derived for the first mode response of a linear multistorey structure having a linear vibration absorber attached to the roof. Furthermore, the variance of the first mode response to a gaussian white noise lateral base acceleration (as a model of earthquake excitation) is determined. Smallest possible values of the variance of the response along with corresponding absorber parameters are established using an optimization program. It is demonstrated that the absorber is quite effective in reducing first mode response for 5- and 10-storey structures even with relatively small values of the absorber mass. Moreover, minimal responses for the randomly excited single-degree-of-freedom system have been determined, and a design example is presented. The absorber system has potential application not only in earthquake engineering but also in aerospace and terrestrial vehicle design.     

Wave propagation in a seven-story reinforced concrete building: III. Damage detection via changes in wavenumbers

This paper presents low frequency wavenumbers in a seven-storey reinforced concrete building estimated from its recorded response to eleven earthquakes, one of which (1994 Northridge) caused visible structural damage, and two of which are its aftershocks. The wavenumbers, K_i,j(f), are estimated from pairs (i,j) of records at neighboring recording sites in the building, distributed vertically or horizontally. Changes in K_i,j(f) from one event to another are compared in the undamaged (lower) and in the damaged (upper) part of the building, with the aim to find whether trends in K_i,j(f) can indicate damage. The results suggest significant and permanent increase of the wavenumbers in the damaged parts for the 1994 Northridge earthquake and its aftershocks, which is not the case for the other events in the damaged parts, and for all eleven events in the undamaged parts of the building. This increase in wavenumbers in the damaged parts can be explained by reduced wave velocities through the damaged structural members, and by scattering of waves from the discontinuities created by the damage. It is concluded from this qualitative analysis that wavenumbers estimated from strong motion recordings in a building can indicate location of damage, and that it would be useful to refine further this method (extend it to higher frequencies, and add the capability to quantify the damage). However, this would require more dense strong motion instrumentation in buildings than currently available. Deployment of dense arrays in selected buildings would provide data for further work on this subject.     

Modal identification of structures from ambient vibration,free vibration,and seismic response data via a subspace approach

This work presents a unified procedure for determining the natural frequencies, modal damping ratios and modal shapes of a structure from its ambient vibration, free vibration and earthquake response data. To evaluate the coefficient matrices of a state‐space model, the proposed procedure applies a subspace approach cooperating with an instrumental variable concept. The dynamic characteristics of a structure are determined from the coefficient matrices. The feasibility of the procedure is demonstrated through processing an in situ ambient vibration measurement of a five‐storey steel frame, an impulse response measurement of a three‐span continuous bridge, and simulated earthquake responses of five‐storey steel frames from shaking table tests. The excellent agreement of the results obtained herein with those published previously confirms the feasibility of the present procedure. Copyright © 2001 John Wiley & Sons, Ltd.     

Soil-structure interaction effects on an instrumented building

The paper highlights the use of fem and bi-directional lumped-mass-storey-stiffness numerical models for the study of the soil–structure interaction (ssi) effects on an instrumented building. Data on the structural response have been obtained through the project for seismic instrumentation of a 16-storey r/c cast-in-place dwelling building (Chisinau, Republic of Moldova) during a series of earthquakes (Gutenberg–Richter M _GR = 5.0−6.7). The effect of soil–structure interaction is clearly observed comparing the responses recorded on foundation and free-field. ssi becomes more pronounced for higher level of ground shaking amplifying the natural period of the structure and slightly suppressing high frequences on the foundation in comparison with the free-field motion.     

On non-linear dynamic analysis in the frequency domain: Algorithms and applications

It is well known that the solution of the forced vibration of a N-DOF dynamical problem is very cumbersome when conditions which allow the equations to uncouple do not exist. In the literature several techniques were proposed to overcome the problem, but they were mainly focused on a particular problem in turn. So, we deemed it useful to search for a unifying procedure able to deal with different sources of non-linear behaviour introducing only minor changes in the operation flow. In this respect, attention is paid to the Alternating Frequency/Time domain method (AFT) which draws its robustness from the speed and switching capabilities of the Fast Fourier Transform; moreover, taking advantage of the pseudo-force concept, we can arrive at a solution method featuring greater generality and able to solve different non-linear dynamical problems by means of specialization of the same conceptual framework (G-AFT or Generalized AFT). In the first section of the paper the theoretical background is discussed in detail and the proposed algorithms are presented. In the second one, several examples of technical relevance are documented and solved, highlighting the efficiency, convergence and accuracy of the presented algorithm. For cases such as an 11-storey building or a block simulated power plant we introduce the soil-structure interaction effect by means of non-proportional damping; the responses computed either by direct frequency analysis or by iteration are compared with existing solutions or with time domain solutions determined through the Newmark β method. An original example prepared by the authors and fully referenced is finally worked out in order to show the capability of the method when Coulomb damping is taken into account; this effect covers a significant practical relevance in the base isolation field.     

THE ESTIMATION OF SIGNAL SPECTRA AND RELATED QUANTITIES BY MEANS OF THE MULTIPLE COHERENCE FUNCTION *

A seismic trace recorded with suitable gain control can be treated as a stationary time series. Each trace, χ_j(t), from a set of traces, can be broken down into two stationary components: a signal sequence, α_j(t) *s(t—τ_j), which correlates from trace to trace, and an incoherent noise sequence, n_j(t), which does not correlate from trace to trace. The model for a seismic trace used in this paper is thus χ_j(t) =α_j(t) * s(t—τ_j) +n_j(t) where the signal wavelet α_j(t), the lag (moveout) of the signal τ_j, and the noise sequence n_j(t) can vary in any manner from trace to trace. Given this model, a method for estimating the power spectra of the signal and incoherent noise components on each trace is presented. The method requires the calculation of the multiple coherence function γ_j(f) of each trace. γ_j(f) is the fraction of the power on traced at frequency f that can be predicted in a least-square error sense from all other traces. It is related to the signal-to-noise power ratio ρ_j(f) by where K_j(f) can be computed and is in general close to 1.0. The theory leading to this relation is given in an Appendix. Particular attention is paid to the statistical distributions of all estimated quantities. The statistical behaviour of cross-spectral and coherence estimates is complicated by the presence of bias as well as random deviations. Straightforward methods for removing this bias and setting up confidence limits, based on the principle of maximum likelihood and the Goodman distribution for the sample multiple coherence, are described. Actual field records differ from the assumed model mainly in having more than one correctable component, components other than the required sequence of reflections being lumped together as correlated noise. When more than one correlatable component is present, the estimate for the signal power spectrum obtained by the multiple coherence method is approximately the sum of the power spectra of the correlatable components. A further practical drawback to estimating spectra from seismic data is the limited number of degrees of freedom available. Usually at least one second of stationary data on each trace is needed to estimate the signal spectrum with an accuracy of about 10%. Examples using synthetic data are presented to illustrate the method.     

地铁列车曲线运行引起学校建筑物振动响应分析

地铁以其快捷、准时、运量大等优点,已成为重要的轨道交通形式,但由此引起的环境振动问题日益突出。针对杭州市地铁3号线曲线地段的某中学建设工程,利用有限元软件ABAQUS,对车辆-普通整体道床轨道系统的竖向耦合模型进行振动响应分析,得到考虑轨道高低不平顺影响的轨道振动源强。应用有限元软件MIDAS GTS/NX,建立双孔平行曲线盾构隧道-土-桩-建筑物系统的三维有限元模型。以轨道支点力作为激励对地铁列车运行时的隧道-土-桩-建筑物系统的振动响应进行计算,研究地铁振动波在地层中的传播规律和建筑物的动力响应特性。根据相关环境振动控制标准对建筑物的振动舒适性进行评价。结果表明：轨道加速度和扣件动支点力的最大值分别约为40 m/s²和30 kN;地层和建筑物的振动以竖向为主,水平Y向振动略大于水平X向振动;地面加速度随着距隧道中心线距离的增加而逐渐衰减;各建筑物楼层的振动主频位于16~40 Hz;部分建筑物楼层的振动响应水平已超出了规范的限值要求,建议对地铁轨道或建筑物采取适当的减振措施。     

Response of secondary systems in structures subjected to transient excitation

An analytical method, based on matrix perturbation theory, is developed whereby a simple estimate can be obtained of the maximum dynamic response of lightly damped, light equipment (modelled as a n(²)-degree-of-freedom system) attached to a structure (modelled as a n(¹)-degree-of-freedom system) subjected to ground motion or impact. A natural frequency of the equipment is considered close or equal to a natural frequency of the structure. It is assumed that the information available to the designer is a time history of the ground motion or impact, or an associated design spectrum; the fixed base modal properties of the structure; and the fixed base modal properties of the equipment. The method employed avoids the direct conventional analysis of a n(²) + n(¹)-degree-of-freedom system either by modal or by matrix time-marching methods; as well as errors in estimates of peak response due to the possible unreliability of numerical schemes because of the lightness of the equipment, or due to uncertainty as to the appropriate procedure for summing the contributions of the two closely spaced modes which occur in the system. The proposed procedure is demonstrated for an example equipment-structure system. Computed results based on the method are in close agreement with results obtained through a Newmark time-integration scheme.     

Wave dispersion in high‐rise buildings due to soil–structure interaction

Nonparametric techniques for estimation of wave dispersion in buildings by seismic interferometry are applied to a simple model of a soil–structure interaction (SSI) system with coupled horizontal and rocking response. The system consists of a viscously damped shear beam, representing a building, on a rigid foundation embedded in a half‐space. The analysis shows that (i) wave propagation through the system is dispersive. The dispersion is characterized by lower phase velocity (softening) in the band containing the fundamental system mode of vibration, and little change in the higher frequency bands, relative to the building shear wave velocity. This mirrors its well‐known effect on the frequencies of vibration, i.e. reduction for the fundamental mode and no significant change for the higher modes of vibration, in agreement with the duality of the wave and vibrational nature of structural response. Nevertheless, the phase velocity identified from broader band impulse response functions is very close to the superstructure shear wave velocity, as found by an earlier study of the same model. The analysis reveals that (ii) the reason for this apparent paradox is that the latter estimates are biased towards the higher values, representative of the higher frequencies in the band, where the response is less affected by SSI. It is also discussed that (iii) bending flexibility and soil flexibility produce similar effects on the phase velocities and frequencies of vibration of a building. Copyright © 2014 John Wiley & Sons, Ltd.     

Wave propagation in a seven-story reinforced concrete building: I. Theoretical models

For transient, high frequency, and pulse like excitation of structures in the near field of strong earthquakes, the classical design approach based on relative response spectrum and mode superposition may not be conservative. For such excitations, it is more natural to use wave propagation methods. In this paper (Part I), we review several two-dimensional wave propagation models of buildings and show results for theoretical dispersion curves computed for these models. We also estimate the parameters of these models that would correspond to a seven-story reinforced concrete building in Van Nuys, California. Ambient vibration tests data for this building imply vertical shear wave velocity β_z=112 m/s and anisotropy factor β_x/β_z=0.55 for NS vibrations, and β_z=88 m/s and β_x/β_z=1 for EW vibrations. The velocity of shear waves propagating through the slabs is estimated to be about 2000 m/s. In the companion paper (Part II), we estimate phase velocities of vertically and horizontally propagating waves between seven pairs of recording points in the building using recorded response to four earthquakes.     

Ambient vibration tests of a seven-story reinforced concrete building in Van Nuys, California, damaged by the 1994 Northridge earthquake

Results of two detailed ambient vibration surveys of a 7-story reinforced concrete building in Van Nuys, California, are presented. Both surveys were conducted after the building was severely damaged by the 17 January 1994, Northridge earthquake (M_L=5.3, epicenter 1.5 km west from the building site) and its early aftershocks. The first survey was conducted on 4 and 5 February 1994, and the second one on 19 and 20 April 1994, about one month after the 20 March aftershock (M_L=5.3, epicenter 1.2 km north–west from the building site). The apparent frequencies and two- and three-dimensional mode shapes for longitudinal, transverse and vertical vibrations were calculated. The attempts to detect the highly localized damage by simple spectral analyses of the ambient noise data were not successful. It is suggested that very high spatial resolution of recording points is required to identify localized column and beam damage, due to the complex building behavior, with many interacting structural components. The loss of the axial capacity of the damaged columns could be seen in the vertical response of the columns, but similar moderate or weak damage typically would not be noticed in ambient vibration surveys. Previous analysis of the recorded response of this building to 12 earthquakes suggests that, during large response of the foundation and piles, the soil is pushed sideways and gaps form between the foundation and the soil. These gaps appear to be closing during “dynamic compaction” when the building site is shaken by many small aftershocks. The apparent frequencies of the soil–foundation–structure system appear to be influenced significantly by variations in the effective soil–foundation stiffness. These variations can be monitored by a sequence of specialized ambient vibration tests.     

A ground motion selection procedure for enforcing hazard consistency and estimating seismic demand hazard curves

This paper develops a procedure to select unscaled ground motions for estimating seismic demand hazard curves (SDHCs) in performance‐based earthquake engineering. Currently, SDHCs are estimated from a probabilistic seismic demand analysis, where several ensembles of ground motions are selected and scaled to a user‐specified scalar conditioning intensity measure (IM). In contrast, the procedure developed herein provides a way to select a single ensemble of unscaled ground motions for estimating the SDHC. In the context of unscaled motions, the proposed procedure requires three inputs: (i) database of unscaled ground motions, (ii) I M , the vector of IMs for selecting ground motions, and (iii) sample size, n; in the context of scaled motions, two additional inputs are needed: (i) a maximum acceptable scale factor, SF_max, and (ii) a target fraction of scaled ground motions, γ. Using a recently developed approach for evaluating ground motion selection and modification procedures, the proposed procedure is evaluated for a variety of inputs and is demonstrated to provide accurate estimates of the SDHC when the vector of IMs chosen to select ground motions is sufficient for the response quantity of interest. Copyright © 2015 John Wiley & Sons, Ltd.