Identification of Time-Variant Modal Parameters Using Time-Varying Autoregressive with Exogenous Input and Low-Order Polynomial Function

Abstract:   This work presents an approach that accurately identifies instantaneous modal parameters of a structure using time-varying autoregressive with exogenous input (TVARX) model. By developing the equivalent relations between the equation of motion of a time-varying structural system and the TVARX model, this work proves that instantaneous modal parameters of a time-varying system can be directly estimated from the TVARX model coefficients established from displacement responses. A moving least-squares technique incorporating polynomial basis functions is adopted to approximate the coefficient functions of the TVARX model. The coefficient functions of the TVARX model are represented by polynomials having time-dependent coefficients, instead of constant coefficients as in traditional basis function expansion approaches, so that only low orders of polynomial basis functions are needed. Numerical studies are carried out to investigate the effects of parameters in the proposed approach on accurately determining instantaneous modal parameters. Numerical analyses also demonstrate that the proposed approach is superior to some published techniques (i.e., recursive technique with a forgetting factor, traditional basis function expansion approach, and weighted basis function expansion approach) in accurately estimating instantaneous modal parameters of a structure. Finally, the proposed approach is applied to process measured data for a frame specimen subjected to a series of base excitations in shaking table tests. The specimen was damaged during testing. The identified instantaneous modal parameters are consistent with observed physical phenomena.     

Wavelet‐based approach of time series model for modal identification of a bridge with incomplete input

Identification of modal parameters of a bridge from its earthquake responses is crucial for performing damage assessment of the structure. However, all the input base excitations of the bridge may not be measured because of economic concerns and sensor malfunctions. Consequently, evaluating the modal parameters of a bridge under the consideration of incomplete input measurements is a challenging and important task. An approach that combines the continuous Cauchy wavelet transform with an autoregressive time‐varying moving average with exogenous input (AR‐TVMA‐X) model is proposed in this study to identify the modal parameters of a multispan bridge under multiple support earthquake excitations with incomplete measurements. The efficiency and efficacy of the proposed approach are first validated using numerically simulated responses of a three‐span continuous beam subjected to multiple support nonstationary excitations. A standard procedure of using the proposed approach to identify the modal parameters is established according to comprehensive studies on the effects of noise in the data, the number of supports whose excitations are used in the AR‐TVMA‐X model, and the orders of the AR‐TVMA‐X model on the accuracy of identifying the modal parameters. This procedure is further applied to process the earthquake responses of a two‐span cable‐stayed 510‐m‐long bridge to demonstrate the engineering applicability of the proposed approach.     

Identification of the mode shapes of spatial tall multi‐storey buildings due to earthquakes: the new ‘modal time‐histories’ method

In the present article, a new method is presented which attempts to identify the dynamic characteristics (eigen frequencies, eigen periods, mode shapes and modal damping ratios) of spatial asymmetric tall multi‐storey buildings through measured seismic responses (accelerations). This new method is entitled ‘method of modal time‐histories’, because its main target is to identify the modal time‐histories of accelerations that are obtained by accelerograms recorded on the points of buildings where suitable accelerometers have been installed. In the case of an earthquake, the multi‐channel local network of accelerometers records the time‐histories of the accelerations of the building. In addition, in order to have a successful outcome, the instrumentation form of the multi‐channel local network on the building can potentially play the most important role. This paper, first, presents a relevant mathematical analysis that is adapted to the instrumentation form applied to the multi‐storey buildings and, second, outlines the new method, which consists of nine steps. Finally, in order to illustrate the theory, a suitable numerical example of an instrumented asymmetric five‐storey r/c building that has been oscillated by a weak earthquake is also provided. On the one hand, the identification of the dynamic characteristics of spatial asymmetric buildings contributes to the removal of the uncertainties of building models in order to perform advanced non‐linear analyses about inherent building seismic capacity. On the other hand, this method supports the simple monitoring of a building's ‘structural integrity’. Copyright © 2010 John Wiley & Sons, Ltd.     

Earthquake responses of RC moment frames subjected to near‐fault ground motions

There are three objectives in this paper. The first objective is to compare the dynamic behaviour of a reinforced concrete building structure subjected to near‐fault and far‐field ground motions. A twelve‐storey and a five‐storey reinforced concrete building with moment resisting frames were selected in this study. The Chi‐Chi earthquake was selected as a first set in this study to test near‐fault earthquake characteristics. Further, another earthquake record of an event at the same site was selected to test the far‐field earthquake characteristics for comparison. Through nonlinear time history analyses, the results show that the near‐fault earthquake results in much more damage than the far‐field earthquake. The second objective of this paper is to compare the predictions for ductility demand by the nonlinear time history analyses with those obtained by the pushover analysis procedure. The third objective is to explore the parameters that will more significantly affect the the building structure's dynamic response characteristics of base shear reduction and displacement amplification. Copyright © 2001 John Wiley & Sons, Ltd.     

Identifying the Modal Parameters of a Structure from Ambient Vibration Data via the Stationary Wavelet Packet

Ambient vibration tests are conducted widely to estimate the modal parameters of a structure. The work proposes an efficient wavelet‐based approach to determine the modal parameters of a structure from its ambient vibration responses. The proposed approach integrates the time series autoregressive (AR) model with the stationary wavelet packet transform. In addition to providing a richer decomposition and allowing for an improved time–frequency localization of signals over that of the discrete wavelet transform, the stationary wavelet packet transform also has significantly higher computational efficiency than the wavelet packet transform in terms of decomposing time‐shifted signals because the former has a time‐invariance property. The correlation matrices needed in determining the coefficient matrices in an AR model are established in subspaces expanded by stationary wavelet packets. The formulation for estimating the correlation matrices is shown for the first time. Because different subspaces contain signals with different frequency subbands, the fine filtering property enhances the ability of the proposed approach to identify not only the modes with strong modal interference, but also many modes from the responses of very few measured degrees of freedom. The proposed approach is validated by processing the numerically simulated responses of a seven‐floor shear building, which has closely spaced modes, with considering the effects of noise and incomplete measurements. Furthermore, the present approach is employed to process the velocity responses of an eight‐storey steel frame subjected to white noise input in a shaking table test and ambient vibration responses of a cable‐stayed bridge.     

Semi‐active control of seismic response of tall buildings with podium structure using ER/MR dampers

A tall building with a large podium structure under earthquake excitation may suffer from a whipping effect due to the sudden change of building lateral stiffness at the top of the podium structure. This paper thus explores the possibility of using electrorheological (ER) dampers or magnetorheological (MR) dampers to connect the podium structure to the tower structure to prevent this whipping effect and to reduce the seismic response of both structures. A set of governing equations of motion for the tower–damper–podium system is first derived, in which the stiffness of the member connecting the ER/MR damper to the structures is taken into consideration. Based on the principle of instantaneous sub‐optimal active control, a semi‐active sub‐optimal displacement control algorithm is then proposed. To demonstrate the effectiveness of semi‐active control of the system under consideration, a 20‐storey tower structure with a 5‐storey podium structure subjected to earthquake excitation is finally selected as a numerical example. The results from the numerical example imply that, as a kind of intelligent control device, ER/MR dampers can significantly mitigate the seismic whipping effect on the tower structure and reduce the seismic responses of both the tower structure and the podium structure. Copyright © 2001 John Wiley & Sons, Ltd.     

An empirical relationship to determine lateral seismic response of mid‐rise building frames under influence of soil–structure interaction

In this study, to determine the elastic and inelastic structural responses of mid‐rise building frames under the influence of soil–structure interaction, three types of mid‐rise moment‐resisting building frames, including 5‐storey, 10‐storey and 15‐storey buildings are selected. In addition, three soil types with the shear wave velocities less than 600 m/s, representing soil classes C_e, D_e and E_e according to AS 1170.4–2007 (Earthquake action in Australia, Australian Standards), having three bedrock depths of 10 m, 20 m and 30 m are adopted. The structural sections are designed after conducting nonlinear time history analysis, on the basis of both elastic method and inelastic procedure considering elastic‐perfectly plastic behaviour of structural elements. The frame sections are modelled and analysed, employing finite difference method adopting FLAC2D software under two different boundary conditions: (a) fixed base (no soil–structure interaction) and (b) considering soil–structure interaction. Fully nonlinear dynamic analyses under the influence of different earthquake records are conducted, and the results in terms of the maximum lateral displacements and base shears for the above mentioned boundary conditions for both elastic and inelastic behaviours of the structural models are obtained, compared and discussed. With the results, a comprehensive empirical relationship is proposed to determine the lateral displacements of the mid‐rise moment‐resisting building frames under earthquake and the influence of soil–structure interaction. Copyright © 2012 John Wiley & Sons, Ltd.     

Vibration control study on a supertall building

Vibration control of a 288‐m supertall building with connective structure is studied in this work. Fluctuating wind time series of the structure in forward and reverse Y‐direction in a 10‐year frequency are simulated by modified auto‐regressive method (AR method) to perform wind vibration analysis, and six minor and major earthquake waves are provided by building designer to perform earthquake analysis. Three vibration control schemes with nonlinear viscous dampers are proposed to control structural dynamic responses under wind and earthquake excitations. The dynamic responses of the structure with the proposed control schemes in wind and earthquake excitations are investigated and their vibration control effects are analysed comparatively. The study results show that the modified AR method is reliable and effective for simulating the fluctuating wind exerted on the building. The excessive dynamic responses induced by wind and earthquake excitations can be controlled effectively by the proposed schemes. The peak acceleration of top storey can be reduced by almost 40% for the proposed control schemes in wind excitation. The elastic working state of the connective body between the high tower part and the low tower part in major earthquakes can also be ensured totally. So, the validity and feasibility of the proposed schemes in reducing structural vibration responses can be fully approved. Some suggestions about structural analysis and design under wind and earthquake excitations are proposed. Copyright © 2010 John Wiley & Sons, Ltd.     

Effects of infills on high‐rise buildings: a case study

In this study, the effects of infill walls to the response of a selected building under earthquake loading were investigated. Although various suggestions have been offered, designers usually neglect the effect of infill walls on building behaviour when designing a building. In this study, the effects of infill walls on a building, which consists of two storeys of basement, one storey of ground floor, one storey of mezzanine floor and 10 storeys of flats, were investigated. Three‐dimensional models of the building with and without infill walls were modelled in SAP2000. Then, nonlinear time history analysis was performed on the models with and without infill walls. Infill walls were modelled both as mass and structural elements. The results of two analyses were compared. Consequently, the effect of infill walls on the behaviour of buildings such as period, maximum roof displacement, base columns end‐forces and soft‐storey formation coefficient was determined. Addition of infill wall to the structures caused changes in maximum roof displacement, modal periods, maximum base column end‐forces, shear force and soft‐storey formation coefficient. Copyright © 2008 John Wiley & Sons, Ltd.     

An adaptive numerical scheme based on the Craig–Bampton method for the dynamic analysis of tall buildings

A new numerical scheme is proposed to perform a nonlinear dynamic analysis for tall buildings. The structural components (beams and columns) of tall buildings gradually enter the inelastic phase under strong seismic excitation. Because the distribution of nonlinear components is initially unknown due to the randomness of earthquake inputs, a group of linear and nonlinear substructures are automatically figured out during the time‐history analysis of a structure. Then a modified Craig–Bampton method is proposed to condense the DOFs of the linear substructures in modal coordinates at each time step while keeping the governing equation of the nonlinear substructure in physical coordinates. The dominant modes of the linear substructures are selected to capture the main dynamic characteristics of the structure. The time step integration analysis is used to solve the governing equation of the structures in hybrid coordinates. A 20‐story building is employed as the numerical simulation test to validate the feasibility and effectiveness of the proposed numerical scheme. This scheme provides a new method for the nonlinear dynamic analysis of tall buildings with acceptable simulation accuracy and high computational efficiency.     

Semi‐active predictive control of non‐linear structures with controlled stiffness devices and friction dampers

A predictive semi‐active control system used to improve the response of non‐linear multistorey structures to earthquakes is presented. A system of controlled stiffness devices (CSD) and friction dampers (FD) is studied. The system combines the forces produced by the semi‐active FD and additional stiffness supplied by the CSD in order to obtain an optimal structural response. A predictive algorithm is used to overcome the time delay problem in the control system. The control forces in the FD are calculated at every time step by applying the instantaneous optimum control law according to the structural behaviour predicted for the next time step at each storey level of the structure. The proposed system can be efficiently used for structural control because the forces developed in it are independent of the structure's displacements or velocities. Its efficiency is demonstrated by a numerical simulation of a seven‐storey building subjected to various seismic excitations. The simulation shows that the behaviour of a structure with the proposed control system is significantly improved compared to an uncontrolled one, or controlled by friction dampers or by semi‐active controlled stiffness devices only. The structural response with a predictive controlled system is similar to that with an instantaneous one. Copyright © 2004 John Wiley & Sons, Ltd.     

Shaking table model test and numerical analysis of a supertall building with high‐level transfer storey

The seismic behaviour of a 53‐storey tower with the height of 250 m is investigated in this paper. This supertall building is composed of a reinforced core, two outrigger trusses (a strengthened storey at 20th ～ 21st floor and a high‐level transfer storey at 37th ～ 38th floor) and eight composite (steel‐encased concrete) mega‐columns below the high‐level transfer storey in the exterior perimeter of the building. Due to the discontinuity of the eight composite mega‐columns, they are replaced by 33 upper‐close columns at the 37th ～ 38th level. This paper describes tests performed on a 1/30 scaled model of the building to study its dynamic characteristics and to evaluate its earthquake resistant capacity. The model test results indicate that the prototype structure is able to withstand an earthquake of intensity 7 without severe damage. In addition, a 3D finite element analysis of model structure was carried out and the analytical results were compared with the experimental ones to gain a better understanding of the structural behaviour. Copyright © 2010 John Wiley & Sons, Ltd.     

Mode shape linearization and correction in coupled dynamic analysis of wind‐excited tall buildings

The three‐dimensional mode shapes found in modern tall buildings complicate the use of the high‐frequency base balance (HFBB) technique in wind tunnel testing for predicting their wind‐induced loads and effects. The linearized‐mode‐shape (LMS) method was recently proposed to address some of the complications in the calculation of the generalized wind forces, which serve as the input to modal analysis for predicting wind‐induced dynamic responses of tall buildings. An improved LMS method, called the advanced linearized‐mode‐shape (ALMS) method, is developed in this paper by introducing torsional mode shape corrections to account for the partial correlation of torques over building height. The ALMS method has been incorporated into the accurate complete quadratic combination method in the coupled dynamic analysis to form a comprehensive procedure for the determination of equivalent static wind loads (ESWLs) for structural design of complex tall buildings. The improved accuracy in the prediction of generalized forces by the ALMS method has been validated by a 60‐storey benchmark building with multiple‐point simultaneous pressure measurements. A practical 40‐storey residential building with significant swaying and torsional effects is presented to demonstrate the effectiveness of the proposed wind load and response analysis procedure based on the HFBB data. Copyright © 2010 John Wiley & Sons, Ltd.     

Instantaneous earthquake input energy and sensitivity in base‐isolated building

The input energy and energy input rate to a base‐isolated (BI) building during an earthquake are considered and formulated in the frequency domain. The frequency‐domain approach for computation of input energy and energy input rate has different remarkable advantages compared with the conventional time‐domain approach. It is demonstrated that the input energy can be of a compact form via the frequency integration of the product between the input component (squared Fourier amplitude spectrum of acceleration) and the structural model component (so‐called energy transfer function). Furthermore, the energy input rate can also be of a similar form via the frequency integration of the product between the instantaneous power spectrum and the energy transfer function. With the help of this compact form, it is shown that the formulation in the frequency domain is essential for deriving arbitrary‐order closed‐form sensitivities of the input energy and energy input rate with respect to uncertain stiffness and damping coefficients in the BI storey. The closed‐form sensitivity expressions provide us with information on the most unfavourable variation of the uncertain parameters that leads to the maximum input energy and input rate. Copyright © 2009 John Wiley & Sons, Ltd.     

消能减震建筑结构的附加有效阻尼比估计

为估计一栋实际消能减震建筑结构在地震作用下的附加有效阻尼比,提出了一种基于有限元模型修正技术的阻尼比估计和验证方法。考虑到结构初始有限元模型存在较大模型误差,采用基于结构振动模态参数的直接模型更新方法修正初始有限元模型,其中,对于实测振型不完整问题,利用振型扩阶方法补充完整振型。基于模态应变能概念,利用整体结构模态参数识别值,推导了油阻尼器支撑系统附加给结构的有效阻尼比和有效频率的计算公式,并估计了主体结构的频率和阻尼比。以一组实际地震动监测数据为例,采用建议方法估计有效阻尼比和有效频率,通过对比修正模型的预测响应与实际监测数据,验证了建议方法的有效性。     

Seismic capacity of typical high‐rise buildings in Singapore

Pushover analysis is a simplified method to determine the lateral load capacity of buildings. However, recent studies have suggested that pushover analysis could underestimate the capacity by as much as 25%. Thus, this study uses dynamic collapse analysis to determine the overstrength of a 16‐storey and a 25‐storey building, which are typical in Singapore. The results are compared with previously performed pushover analyses to justify the adequacy of pushover analysis for determining the ultimate capacity of such buildings. It is found that the buildings in Singapore, which are not designed for earthquake loads, possess overstrength varying from 4 to 12 times the design strength depending on the type of building. Furthermore, the pushover analysis could underestimate the capacity of such buildings up to 14%. It is suggested that one may choose to adopt pushover analysis to evaluate the lateral load capacity of such high‐rise buildings to err on the conservative side. Copyright © 2012 John Wiley & Sons, Ltd.     

A frequency‐domain noniterative algorithm for structural parameter identification of shear buildings subjected to frequent earthquakes

System identification is the key technique for damage detection in application of structural health monitoring. In contrast to modal parameters, changes in structural parameters (stiffness and damping) are more sensitive and straightforward for damage detection of a building under severe environments such as earthquakes. In this study, we first present the fundamental theory for direct identification of structural parameters by using the frequency‐domain responses of a shear building in frequent earthquakes. Shear buildings are widely adopted for structural analysis of low‐ and middle‐rise buildings in practice. Modal information, in terms of spectrum ratios, is implicitly used in the proposed noniterative algorithm to greatly improve the estimation accuracy as well as to avoid any human intervention. The fundamental theory is validated by the numerical and physical examples. The numerical examples are further used to verify the high efficiency, accuracy, and robustness of the proposed algorithm against noised responses. The proposed algorithm is highly efficient because no iterative computation is necessary, while the necessary Fourier transform of the dynamic responses is not very time consuming. Furthermore, the proposed algorithm is highly accurate and robust because (a) the fundamental theory behind the algorithm is straightforward: the identification values should have the same value irrespective of circular frequencies, according to the theory; (b) error in modal parameter identification is completely avoided because it is unnecessary to identify the exact values of the frequencies as in many existing methods.     

Seismic tests on a 16‐storey cast‐in‐place r/c building,and stochastic simulation results

Records of the dynamic response of a 16‐storey cast‐in‐place reinforced concrete apartment building subjected to moderate‐magnitude (M_w < 7·0) earthquakes were performed before and after three recent significant Vrancea earthquakes. They are compared with random vibration simulation results based on a newly developed computer program and with an FEM linear model of the building, employing time‐history analysis. Theoretical investigations fitted to the results of the tests confirmed the validity of the approaches used and demonstrate satisfactorily good compatibility of the simulated and observed structural responses. The seismic records from the test building (str. Roz 15, Kishinev, Moldova), consisting of displacements and accelerations, were taken in ‘free‐field’, and at the foundation, the 8th storey and the top of the building. Copyright © 2001 John Wiley & Sons, Ltd.     

A new pushover procedure for two‐way asymmetric‐plan tall buildings under bidirectional earthquakes

Conventional pushover analyses despite of extensive applications are unable to estimate the general responses of asymmetric‐plan tall buildings because of ignoring the effects of higher modes and torsion. A consecutive modal pushover procedure is one of the recent nonlinear static pushover procedures that used to analyse the seismic response of one‐way asymmetric‐plan tall buildings under one‐directional seismic ground motions. In this paper, a modified consecutive modal pushover procedure (MCMP) has been proposed to estimate the seismic demands of two‐way asymmetric‐plan tall buildings under two horizontal components of earthquakes simultaneously. The accuracy of the MCMP procedure is evaluated using different buildings and comparing with the results of FEMA (Federal Emergency Management Agency) procedures, the practical modal pushover procedure and nonlinear time history analyses as an exact solution. The results show the proposed MCMP procedure is able to estimate the displacements and storey drifts accurately and introduces a great improvement in predicting the plastic hinge rotations. Copyright © 2013 John Wiley & Sons, Ltd.     

Practical modelling of high‐rise dual systems with reinforced concrete slab‐column frames

This paper discusses practical modelling issues pertinent to the design of an irregularly shaped reinforced concrete (RC) high‐rise building currently under development in New York City. The structure analysed consists of a 60‐storey residential tower and a 25‐storey hotel building structurally connected to each other. For the seismic force resistance, a dual system combining ordinary RC shear walls and intermediate slab–column moment frames was used at the upper portion, while a building frame system of ordinary RC shear walls was used at the lower portion of the structure. A variety of models were used to simulate the behaviour of various elements of the structure, with special attention given to overall systemic effects of different member stiffnesses considered to account for distinct stress levels under service and ultimate loads. The models used for slab–column frames and shear walls were verified by comparing with other available models or laboratory tests. The in‐plane flexibility of floor diaphragms at the interface between the two substructures with different geometries was simulated to identify the most critical wind conditions for each structural member. Finally, building dynamic analyses were performed to demonstrate the modelling issues to be considered for the lateral force design of irregular high‐rise buildings. Copyright © 2009 John Wiley & Sons, Ltd.