Seismic Response of Base-Isolated High-Rise Buildings under Fully Nonstationary Excitation

Stochastic seismic responses of base-isolated high-rise buildings subjected to fully nonstationary earthquake ground motion are computed by combining the pseudoexcitation and the equivalent linearization methods, and the accuracy of results obtained by the pseudoexcitation method is verified by the Monte Carlo method. The superstructure of a base-isolated high-rise building is represented by a finite element model and a shear-type multi-degree of freedom model, respectively. The influence of the model type and the number of the modes of the superstructure participating in the computation of the dynamic responses of the isolated system has been investigated.The results of a 20-storey, 3D-framewith height to width ratio of 4 show that storey drifts and absolute accelerations of the superstructure for such a high-rise building will be substantially underestimated if the shear-type multi-degree of freedom model is employed or the higher modes of the superstructure are neglected; however, this has nearly no influence on the drift of the base slab.


Introduction
Seismic isolation is a technology that decouples a building structure from the damaging earthquake motion.It is a simple structural design approach to mitigate or reduce potential earthquake damage.In base-isolated structures, the seismic protection is obtained by shifting the natural period of the structure away from the range of the frequencies for which the maximum amplification effects of the ground motion are expected; thus, the seismic input energy is significantly reduced.At the same time, the reduction of the high deformations attained at the base of the structure is possible, thanks to the energy dissipation caused by the damping and the hysteretic properties of these devices, further improving the reduction of responses of the structures [1].Since the 1995 Hyogoken-nanbu earthquake, the number of seismic isolated buildings has been increasing remarkably, including residential buildings, nuclear power plants, office buildings, hospitals, and schools [2,3].Base isolation is also an attractive retrofitting strategy to improve the seismic performance of existing bridges and monumental historic buildings [1,3,4].
It is believed that isolation technology is very effective in improving the seismic performance of low-and mediumrise buildings, but it is not envisaged for high-rise buildings.However, a lot of base-isolated high-rise buildings have been built in recent decades.Sendai MT building is an 18-storey office building with a height of 84.9 m in Sendai city, which was the first base-isolated building with a height exceeding 60 m [5]; Thousand Tower, a 41-storey building with a height of 135 m, was constructed in 2002 [5]; and another super highrise building in Japan was built in 2006, with a height of 177.4 m and height to width ratio of 5.7, which is the highest base-isolated building in the world so far [6].
A substantial amount of work has been done on baseisolated high-rise buildings.Since isolators are easily damaged by uplift when such high-rise buildings are subjected to major earthquakes, Roussis and Constantinou [7] proposed some new devices to avoid damage caused by uplift of the isolators.Hino et al. [8] studied the limitation of the height to width ratio of the base-isolated buildings by the Monte Carlo method.Ariga et al. [9] investigated the resonant behavior of base-isolated high-rise buildings under long-period ground motions.Takewaki [10] investigated the robustness of base-isolated high-rise buildings under codespecified ground motions and concluded that base-isolated high-rise buildings have lower robustness than base-isolated low-rise buildings.Takewaki and Fujita [11] studied the earthquake input energy to base-isolated high-rise buildings by both time-domain and frequency-domain methods.Yamamoto et al. [12] studied the input energy and its rate to a base-isolated building during an earthquake in the frequency domain.Pourzeynali and Zarif [13] optimized the parameters of the base isolation system, using genetic algorithms, to simultaneously minimize the displacement of the top storey and that of the base isolation system.Recently, Islam et al. [14,15] studied the nonlinear performances of base-isolated multistorey buildings in both time domain and frequency domain.
However, in most of the above-mentioned research, the structure was simplified as a shear-type multi-degree of freedom (MDOF) model (i.e., each storey of the superstructure was simplified as one degree of freedom in the horizontal direction, and the vertical deformation of the columns was neglected).Such modeling is not suitable when the height to width ratio (defined by the ratio of the building height to the building width) of the superstructure exceeds 4, as by then the flexural effect of the column cannot be neglected.In the present work, the superstructure of a base-isolated high-rise building is represented by a finite element (FE) model and a shear-type MDOF model, respectively, to study the modeling effect.The stochastic response of this building is evaluated, by combining the pseudoexcitation method (PEM) and the equivalent linearization method (ELM), since this combined approach does not require a huge storage of data and vast computational effort [16].
The main objectives of this paper are (i) to evaluate the dynamic responses of base-isolated high-rise buildings under fully nonstationary earthquake excitations by combining the PEM and ELM and verify the accuracy of the results by the Monte Carlo method; (ii) to investigate the influence of the flexure of the beams and extensibility of the columns on the responses of such structures; and (iii) to study the influence of the higher modes of the superstructure on the responses of base-isolated high-rise buildings.

Governing Equations of Motion for
the Base-Isolated System where M  , C  , and K  are the  ×  mass, damping, and stiffness matrices of the superstructure, respectively; ü  , u  , and u  are the acceleration, velocity, and displacement vector of order  relative to the base slab;   is the displacement of the base slab relative to the ground displacement   ; and r  is the -dimensional influence coefficient vector.The governing equation of motion of the base slab, with rotation and vertical deflection neglected, can be expressed as [1]   ( ẍ where  0 denotes the post-to preyielding stiffness ratio;  is a hysteretic component, which is a function of the time history of   and ẋ  ; and   ,   , and   are the mass, supplemental damping, and preyielding stiffness of the base slab, respectively.One has where   is the supplemental damping ratio of the base slab;   is the postyielding stiffness of the base slab; and   is the total mass of the superstructure and base slab.In (2)  is related to   and ẋ  through the following nonlinear differential equation [17]: Note that in (4),  and  control the shape of the hysteretic loop;  controls the restoring force amplitude; and  controls the smoothness of the transition from elastic to plastic response.These parameters are related by   =  √/( + ), where   is the yield displacement of the isolators.

Static Correction Procedure.
In properly designed baseisolated systems, the superstructure remains elastic even when subjected to a major earthquake ground motion.Therefore the modal superposition method is used to reduce the number of degrees of freedom of the system, with the first  ( ≪ ) modes participating in the dynamic computation.This improves the computational efficiency but introduces truncation errors because of neglecting the influence of the higher modes.The static correction procedure is employed to take into consideration the contribution of the higher modes.The displacement of the superstructure corresponding to the higher modes is obtained based on the fact that the high frequency modes react essentially in a static manner when excited by low frequencies [18].Assume that the displacement of the superstructure u  consists of two parts, that is, the dynamic part u   and the static part u   [19,20]: The dynamic part of the displacement can be expressed as where Φ  is the mass normalised eigenvector corresponding to the th eigenvalue  2  ; frequencies and modes satisfy is the  ×  eigenvector matrix; and  is the  × 1 generalised modal displacement vector.
Premultiplying (1) by Φ  gives where in which L is the mode participation factor of order .
The flexibility matrix of the superstructure can be expressed as The remaining static displacement corresponding to the higher modes is where

Equivalent Linearization Method (ELM).
Using the ELM, (4) can be rewritten as follows when  = 1 [17]: with where [ ] denotes the expectation operator and   and  ẋ  are the standard deviations of  and ẋ  , respectively.

State Space
Method.Equations ( 2) and ( 7) can be compacted together into the single equation below where in which I  is the  ×  identity matrix.
The second-order ( 14) can be replaced by the first-order differential equations by the state space method where I +1 is the ( + 1) × ( + 1) identity matrix and ( + 1)dimensional vector a is given by The state variable vector v, of order Thus ( 12) and ( 16) can be replaced by the first-order differential equation where in which ( + 1)-dimensional vector b is given by b = {−  , 0, 0, . . ., 0} .

Nonstationary Stochastic Analysis by PEM
Because of the uncertainty of earthquakes, stochastic seismic analysis is a powerful tool in earthquake engineering and has experienced extensive development in recent decades.Jangid [21] studied the performance of the isolated buildings and bridges, and the stochastic responses of the isolated structures subjected to uniformly modulated nonstationary earthquake excitations were obtained by solving Lyapunov equation.As the stochastic response of a nonlinear system is strongly affected by the nonstationary behaviour of an earthquake [22], the fully nonstationary earthquake model proposed by Conte and Peng [23] is adopted in this paper.

Fully Nonstationary Earthquake Excitation Model.
The earthquake excitation model is considered a sigma-oscillatory process [23], which is a sum of  zero-mean, independent, uniformly modulated Gaussian processes.Each uniformly modulated process consists of the product of a deterministic time modulating function,   (), and a stationary Gaussian process,   ().Thus, the earthquake ground motion ü  () is defined as [23] ü In (22), the modulating function   () is defined as where   and   are positive constants;   is a positive integer;   is the "arrival time" of the th subprocess,   (); and () is a unit step function.
The th stationary Gaussian process,   (), is characterized by its autocorrelation function and its power spectral density (PSD) function in which ]  and   are two free parameters representing the frequency bandwidth and predominant frequency of the process,   (), respectively.Finally, the evolutionary PSD of the earthquake ground excitation, ü  (), is given 3.2.PEM Computational Procedure.The procedure of computing stochastic responses of base-isolated systems is summarized below.
The corresponding nonstationary random vibration response analysis is transformed into an ordinary direct dynamic analysis.Thus k(  ,   ) can be evaluated at a series of equally spaced frequency points   = Δ ( = 1, 2, . . .,   ) at   by Runge-Kutta method, where Δ is the frequency step and   is the total number of frequency steps.
Step  Step 4. Compute the covariance and variance of the responses by Wiener-Khintchine theorem at   .
When the corresponding responses become convergent,   is replaced by  +1 , and Steps 1-5 are repeated for the next time step.Equations ( 13), ( 19), ( 28 Base-fixed building Base-isolated building up the complete formulation of the isolated system.The computational procedure is shown in Figure 1.

Numerical Study
In the base-isolated frame structure shown in Figure 2, each of the 20 storeys is 3.6 m in height, so the total height of the frame structure is 72 m, with 18 m in width and 15 m in depth in the  and  directions, respectively.Thus its height to width ratios are 4 and 4.8 in the  and  directions, respectively.The reinforced concrete beams are all identical, with width of 0.6 m and depth of 0.8 m.The reinforced concrete columns are all of square cross-sections, with side length .The column properties, that is, their side length , extensional rigidities , and flexural rigidities  for the in-plane behavior, are in three different values, depending upon the storey number, as shown in Table 1.The total mass of each storey is distributed uniformly as a lumped mass at each of its nodes, as also shown in Table 1.The damping ratios of the superstructure and isolation slab are 0.03 and 0.10, respectively; the fundamental period of the base-fixed superstructure is   = 1.66 s, and the fundamental period of the base-isolated system is   = 3.50 s.Other values used are the post-to preyielding stiffness ratio  0 = 0.1; and the yielding displacement   = 0.01 m.
A versatile, fully nonstationary earthquake ground-motion model proposed by Conte and Peng [23] is employed here, and this stochastic earthquake model is applied to an actual earthquake, N00W (N-S) component of the San Fernando earthquake of February 9, 1971, recorded at the Orion Boulevard site.The corresponding parameters of the sigma-oscillatory process estimated are given in Table 2 [23].The model parameters are determined by adaptively leastsquares fitting the analytical time varying (or evolutionary)  The PSD function of the earthquake excitation is shown in Figure 3. Obviously, this earthquake model can capture the time variation of both the intensity and the frequency content of the earthquake record at the target.
Figure 4 compares the root mean square (RMS) storey drifts of the isolated structure evaluated by PEM with those given by Monte Carlo simulation (500 samples are used).The relationship between the restoring force and the drift of the isolators is described by the Bouc-Wen model, so each sample of Monte Carlo simulation is a nonlinear time history analysis of the isolated system.Clearly both the drift of the base slab and the storey drifts of the superstructure agree well with the two sets of results, so that the accuracy of the results by the PEM is demonstrated.
The peak RMS of the storey drifts and absolute accelerations of the base-fixed structure and those of the baseisolated one are shown in Figure 5.It demonstrates that the responses decrease significantly after isolation, so they reveal that the isolation technology can still protect the structures from damage during earthquakes even for high-rise buildings with a proper design of the isolators employed.
The improvement of accuracy of the storey drifts by the static correction procedure is given by Figure 6.It shows that the accuracy of the response is improved with employment of this method.Figure 8 shows the influence of the number of the modes participating in the computation of the response of the superstructure.The results are obtained with  = 3, 10, and 40 modes, respectively.They reveal that the storey drifts of the superstructure will be substantially underestimated if only the first few modes are included in the dynamic computation from Figure 8(a), while Figure 8(b) demonstrates that the absolute accelerations vary significantly with the number of modes participating in the dynamic computation.So the influence of the higher modes on the responses of the superstructures should not be neglected for the base-isolated high-rise buildings.
Figure 9 shows the influence of the number of the modes participating in the computation of the drift and absolute acceleration of the base slab.The number of the modes Shock and Vibration participating in the dynamic computation is found to have a small influence on the drift of the base slab but a remarkable influence on the absolute acceleration of the base slab.

Conclusions
The stochastic responses of a base-isolated high-rise building subjected to fully nonstationary ground excitations are analyzed by combining the PEM and the ELM.The conclusions can be drawn as follows.
(1) The results obtained by the PEM agree well with those obtained by the Monte Carlo method and the accuracy of the results of such hysteretic systems evaluated by the PEM is verified.
(2) The static correction procedure is employed for considering the contributions of the higher modes of the structure which causes almost no increase of the computational effort.
(3) An FE model and a shear-type MDOF model are implemented for the superstructure of such highrise buildings.It is found that the storey drifts and absolute accelerations are underestimated if the flexural deformation of the beam components and the axial deformations of the column components are neglected in the shear-type MDOF model used.The peak RMS storey drifts of the superstructure could be underestimated by about 60%, and the peak RMS absolute accelerations of the superstructure could be underestimated by about 7%.
(4) The storey drifts and the absolute accelerations could be underestimated if the higher modes of the superstructure are neglected in the FE model used; and the number of the modes participating in the dynamic computation has a small influence on the response of the base slab but a remarkable influence on its absolute acceleration, and sometimes the absolute acceleration of base slab could be overestimated by about 40%.

Figure 1 :
Figure 1: Flow chart of the computation procedure.

Figure 5 :
Figure 5: Comparison of (a) the peak RMS storey drifts and (b) the absolute storey accelerations of the base-fixed and base-isolated buildings.

Figure 6 :Figure 7 :
Figure 6: Influence of static correction procedure on the storey drifts.

Figure 7 Figure 8 :Figure 9 :
Figure 8: Influence of the number of the modes participating in the computation on the (a) storey drifts and (b) absolute accelerations of the superstructure.
model are each applied to the superstructure, with the first 20 modes participating in the dynamic computation.It shows that storey drifts and absolute accelerations increase when extensions of the columns and flexure of the beams are suitably taken into account in the FE model.

Table 1 :
Properties of columns for the 20-storey building and mass distribution for each storey.

Table 2 :
Estimated parameters of the ground acceleration model for Orion Boulevard 1971 earthquake record.