Response Analysis and Control of Structures to Multi-dimensional Seismic Motions

Hong-Nan Li1),2) and Lin-Sheng Huo1)

1) Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian,

116024, China 2)School of Civil Engineering, Shenyang Jianzhu University, Shenyang, 110168, China

1) hnli@dlut.edu.cn

ABSTRACT

In recent years, much attention has been paid to research and development of structural aseismic analyses and control techniques with particular emphasis on the alleviation of seismic response of buildings and bridges. In this paper, the affect of distributing of the S wave velocity along depth to surface under ground the wave dispersion are discussed base on the elastodynamic theory to calculate the velocity of the Rayleigh wave and Love wave. A simplified dispersion curve, a bilinear curve with a horizontal line in higher frequency region and bias in the lower frequency region, is suggested. A modal combination method is presented for the earthquake-resistant design of structures to multidimensional seismic excitations. With the assumption that an earthquake is a stationary random vibration, the correlation among the input components is considered in the proposed method. The relationship coefficients between the translational component and rotational component is then derived in the frequency domain. The combination method of response spectrum for structural response to multidimensional earthquakes is proposed based on the random vibration theory. With the help of the derived modal correlation coefficients, the formulation for structural response to the two-dimensional earthquake excitations can be obtained. Numerical examples demonstrate the effectiveness and high precision of the proposed methods. Structural control has been developed from the concept into a workable technology and applied into practical engineering structures. The aim of this paper is to review a state-of-the-art of researches and applications of structural control in civil engineering, which includes the passive control, active control, hybrid control, and semi-active control. The researches on the structural vibration control have made great achievements and some control devices have been applied to practice in recent years. 1. INTRODUCTION It has been validated that rotational ground motion is a factor in structural damage from earthquakes (Luco, 1976; Trifunac, 2006)．However, it is not considered directly

Keynote Paper

due to a lack of recorded rotational ground motion，knowledge of the rotational wave field and the code for structure design (Rutenberg and Heidebrecht, 1985)．Seismologists could only record the translational motions by the 1980s．Some researchers considered that the translational motions are combined with rotational motions，but there was no effective way to extract the rotational motions from the translational motions (Vladimir, 2005)．Some devices have been developed to record the rotational ground motion，such as solid state devices (Nigbor, 1994)，ring lasers measurements (Stedman et al., 1995) and broadband rotation meters (Igel et al．, 2007)．However, these devices have not been widely used．Dense arrays can be used to estimate the rotational components of motion indireetly (Spudich et al., 1995; Suryanto et al．, 2006; Ghayamghamian and Nouri，2007)，but the estimation is only valid for waves with long wavelengths compared with the distances between sensors．The rotational components of ground motion have also been estimated theoretically using kinematic source models and elastodynamic theory of wave propagation in elastic solids(Bouchon and Aki．1982；Takeo．1998)．In civil engineering， simplified theoretical methods for simulating rotational ground motion has been widely studied in the past 40 years．Newmark rl969) established a simple relationship betw een me rotational components and translational components．His idea was pursued further by other investigators(Ham et al., 1975; Nathan and MacKenzie．1975)．An improved approach for obtaining rotational components of seismic motion was presented by Li et al．(2004)． These methods mentioned above need many correlated records of one earthquake，which can be provided mainly by dense arrays．This means that the apparent velocity of a wave recorded by a single seismograph cannot be estimated effectively．In the theory of wave propagation，such as the refraction and reflection theory, the incident angle is an important parameter by which the apparent velocity can be calculated．If the incident wave and the space condition are known，the apparent velocity can be calculated．On the other hand，if the wave field and the apparent velocity are known，the space condition can be concluded．The theory is widely applied in earthquake wave inverse, earthquake exploration and other fields．For earthquake events．it is difficult to use the theory if there are only one or just a few earthquake wave records，and the hypocenter and the type of incident wave are unknown at the same time．However, for practical civil engineering, the S wave of a local earthquake is always considered because it usually causes strong motion on the surface and damage to buildings．The process of using wave propagation theory may be simplified for this case．

This paper estimates the apparent velocity of a local earthquake with a theoretical method．A P-SV ratio method based on elastodynamic theory is presented. The method is confirmed by data from four earthquakes in the SMART1 array．The parameters that may affect the apparent velocity, such as epicenter distance and site condition，are analyzed.

Even though engineers cannot design a building which is damage proof during

earthquakes and strong winds, the structural control is promising in reducing the vibration of structures. Different from the traditional anti-seismic method, the structural control technique suppresses the structural vibration by installing some devices, mechanisms, sub-structures in the structure to change or adjust the dynamic performance. The structural control system is commonly classified by its device type resulting in four general control types: passive, active, hybrid and semi-active control. An active control system is one in which an external source power control actuators that apply forces to the structure in a prescribed manner including active tendon system (ATS) and active mass damper (AMD). A passive control system does not require an external power source, such as base isolation method, energy dissipation devices, tuned mass damper (TMD), tuned liquid damper (TLD), and so on. The hybrid control implies the combined use of active and passive control systems. Semi-active control systems are a class of active systems in which only small magnitude of external energy is needed to change the parameters of control system, such as active variable stiffness (AVS) system and active variable damper (AVD) system. In recent years, serious efforts have been undertaken in structural control and fruit achievements have been made in China. Structural control has been developed from theoretical analysis and experimental research into engineering application. A set of techniques including isolation, energy dissipation, tuned mass dampers, tuned liquid dampers, active and semi-active control methods have been used in new structure design or existing buildings, bridges, facilities and other structures in China. In this paper, the state-of-art of vibration control techniques, including theoretical and experimental studies, practices in civil engineering are reviewed. The possible future directions of structural control in civil engineering are discussed.

Fig. 1 Plane wave in a multi-layer half space, S wave incident

2. P-SV RATIO METHOD The body waves of an earthquake are incident inclined from deep rock to soft soil above the bedrock．Consider a multi—layer isotropic elastic half space with plane S wave incident from the bedrock (Fig．1)．Haskell (1953) suggested a transfer matrix method to study the elastic wave in a multi—layer isotropic elastic half space．An improvement of the transfer matrix method was introduced by Chen (1974)．Based on the transfer matrix method，the medium displacement spectrum in the space can be expressed as

2( 2 ) ( ) U U U (1)

where U is the displacement spectrum of the medium，λ and μ denote the Lame parameters, ρ represents the density and ω is the frequency. The solution of Eq. (1) is

[ ( ) ( )]

[ ( ) ( )]

( )

B P C

az az bz bz

az az bz bz

bz bz

U U U

k a e a e b b e b e

a a e a e k b e b e

k c e c e

U B P C

B

P

C

(2)

where

xikxie eB - , z

ikxe eP , yikxe eC -

)(22pp kkkkiaia , )(22

ss kkkkibib

pk , sk and k are the wavenumber of P, S waves and apparent wavenumber,

respectively, a , b and c are the amplitudes of the up P, SV and SH waves, if

pkk , a , b and c are the amplitudes of the down P, SV and SH waves, if pkk .

The P-SV ratio method, the travel time analysis and the cross correlation coefficient analysis are used to estimate the apparent velocities of the earthquake waves recorded by the SMART1 array in this paper. The SMART1 array (Fig. 2), a dense digital array of strong-motion seismographs, was built up at the northeastern corner of Taiwan near the city of Lotung on the Lanyang Plain, in 1980. The original SMART1 array consisted of thirty-seven stations with a central station (C00), and configured in three concentric circles with a radius of 200 m, 1000 m and 2000 m, respectively. Each rings with twelve approximately equally spaced stations. Later, four additional stations were added to the south (E0 and E02) and north (E03 and E04) of the array in June 1983 and June 1987, respectively. During the operation period (1980~1991), sixty earthquakes were recorded by the SMART1 array and 4296 accelerograms had been accumulated. The epicenter distances of seismic records are 30-99km and the hypocenter depths are 11-18km. The strong motion segments are S wave mainly. According to the azimuth, the N-E-Z ground motion components are transformed to R-T-Z components and the distances in the R direction between stations are calculated. xA and zA are the amplitudes of R and Z components of ground displacement, respectively. One can obtain it using Fourier transform. The soils beneath the array concsist of 3-18m of grey sandy silt and silty sand with some gravel over recent alluvium down to 30-60m. This layer overlies a 170-540m thick deposit of Pleistocene gravels. The pv and sv between 0-150m are given in Fig.3 (Wen and Yeh,

1984; Abrahamson et al., 1987). The pv 150m under the ground surface are faster

than 1900m/s, so, consider the layers below 150m to be bedrock (Ding and Jin,2000). The equivalent velocity of the upper layers between 0-150m is calculated by Eq. (3).

N

tii

N

ii

eff

vh

hv

)/( (3)

where ih is the thickness of the layer i , iv implies the velocity( pv or sv ) of the layer i ,

N means the number of layers above bedrock. So, effpv , and effsv , which are used in

Eq.(11) are 1282 m/s and 405 m/s, respectively.

Fig. 2 The layout of SMART1 array Fig. 3 The velocity of the SMART1 site

As an example, Fig. 4 shows k between the frequencies 0-120 rad/s. It will be observed in Fig. 4 that the apparent wavenumbers centralize in a narrow bound. The linear correlation between the apparent wavenumbers and frequencies can be identified. It is coincided with the hypothesis of the P-SV ratio method. Similar behavior is shown in the other earthquakes. Using the fitting formula ak (where a is fitting parameter), fit the data of apparent wavenumbers and frequencies with linear least square method. The fitting parameter a , equal to the slope of the fitting line, can be regarded as the apparent velocity of the wave (Fig. 4). 3. PASSIVE CONTROL 3.1 Base Isolation

Base isolation system was developed as one of remarkable technology to

reduce the seismic load of building and equipment. There are five kinds of materials have been used for isolators in China, including sand layer, graphite lime mortar layer, slide friction layer, roller and rubber bearing. The rubber bearing is the one used most in China. Relatively easy to manufacture, isolation bearings are made by vulcanization bonding of sheets of the rubber to thin steel reinforcing plates. The bearings are very

O01-12

M01-12C00

1000mN

E01

E02

Eq.45

Eq.5

Eq.25

Eq.40

I01-12

Fig. 4 The apparent velocity, equal to the slope of the fitting line

(1) The isolated structures are safer in strong earthquake. The isolators are very effective to reduce the response of structures in earthquake and can prevent the structures from damage or collapse. Compared to the traditional anti-seismic structures, the responses of isolated structures can be reduced to 1/2~1/8 of the one of traditional structures, according to the testing results and the records in real earthquake.

(2) The building cost of isolation structures can be saved 3%~15% of the cost of the general buildings according to the final statistics results of 30 buildings with rubber bearings completed in southern, western and northern China.

(3) The seismic isolation rubber bearing system has wide ranges of application, both new designed structures and existing structures, both important buildings and civil buildings especially for house buildings, both for protecting the building structures an for protecting the facilities inside the building.

(4) The safely working life of rubber bearings is over 70-100 years according to the permanent testing and investigation, which is larger than the working life of structure itself.

The application on seismic isolator in China has proved that the isolation system

is more safe, economic and reasonable than the traditional structural system. More than 500 full-scale implementations of base isolated buildings and bridges have been accomplished to alleviate their earthquake and traffic induced response so far.

3.2 Energy dissipation

Metallic dampers dissipate energy input to a structure from an earthquake

through inelastic deformation of metals. Based on the experimental study and theoretical analysis of the fatigue properties of X-shape steel plate energy dissipaters, the nonlinear method of steel plate of the energy dissipater was established and the strain-fatigue parameters are determined, which provide an important base for the design of mild steel energy dissipaters. In most cases, Metallic dampers dissipate energy through nonlinear property of steel plate after yielding out of plane. Bi-functional mild steel dampers are presented by Gang Li, as shown in Fig. 5 and Fig.6, which is to strengthen the initial stiffness by extending the steel plate in its own plane and to increase the energy-dissipating ability by changing the geometric shape of the steel plate. Results from theoretical analysis and quasi-static experiments showed that these types of mild metallic damper not only provide certain stiffness in normal use, but also are of good ability of the seismic energy dissipation. Bi-functional mild metallic dampers have been installed in the No. 3 Experiment Building of Dalian University of Technology (DUT) in China to increase the stiffness of the first floor and suppress the earthquake induced vibration, as shown in Fig. 7.

For the some old designed buildings based on the dated building design codes in which the seismic protection was seldom taken into account, both the land developers and engineers have to determine whether to demolish them and then set up new ones or reuse the original buildings through making some innovations to meet the current need. Due to economy, many old existing buildings have been increased with new stories on the top of them in China. A new type of energy-dissipated structural system for the existing building with the story-increased frame is presented, in which

Fig.5 Circle hole type mild steel damper and its hysteresis curve

Fig. 6 Double X type mild steel damper and its hysteresis curve

Fig. 7 Double function mild metallic dampers in the No. 3 Experiment Building of DUT the sliding-friction layer between the lowest increased floor of outer frame structure and roof of original building is applied, and energy-dissipated dampers are used for the connections between the columns of outer frame and each floor of original building, as shown in Fig. 16. The experimental results show that friction and energy-dissipated devices are very effective in reducing the seismic response and dissipating the input energy of model structure. A four-story office building, built up in 1950's without the seismic design along Beiling Street in Shenyang City of China because there was no any seismic codes at that time in China, is designed and analyzed with an increased four-story outer frame structure on its top by using the passive control method presented. The existing building is composed of brick masonry structure; its design is

not in compliance with the current Seismic Code of China. The seismic protection intensity on this building site is VII degree, and the site soil belongs to the type II in China Code. The results show that the good seismic-reduction effectiveness on displacements and accelerations can be achieved by installing the friction and energy-dissipated devices in the story-increased structures.

Original building

Story-increased

frame

Sliding-friction

layer

Energy-dissipated

damper

Fig. 8 Test model of story-increased system Fig. 9 The story-increased building 3.3 Tuned liquid damper

The tuned liquid damper (TLD) and tuned liquid column damper (TLCD) impart

indirect damping to the system and thus improve structural performance. A TLD absorbs structural energy by means of viscous actions of the fluid and wave breaking. In a TLCD, energy is dissipated by the passage of liquid through an orifice with inherent head loss characteristics. TLD are mostly used to reduce the wind induced vibration of structures in engineering. Min-Zheng Zhang carried out shaking table test and verified the vibration control result with TLD subjected to earthquake. The research shows that TLD is applicable for long period structure. However, it is not enough to only consider the first mode of structural response. Hong-Nan Li presented the multi-modal vibration control of structures using multiple TLDs and carried out experimental verification. The experimental results show that reduction effectiveness of the seismic responses for controlling two modes is better than controlling only a single mode. The frequency of TLCD is related to the length of liquid in the TLCD tube. The length of TLCD maybe too large to exceed the available space of the building for structures with low frequencies. Shi Yan presented the adjustable frequency tuned liquid column damper by adding springs to the TLCD system, which can modify the frequency of TLCD and expended its application ranges.

To suppress the torsionally coupled responses of structures, Lin-sheng Huo analyzed the control performance of circular tuned liquid damper (CTLCD). Subsequently, TLCD and CTLCD are used to control the torsionaly coupled vibration of

l

k0

cs

ξ

k0

ks

x

b

z

exey

质量中心

刚度中心(l1x l1y)

(l2x l2y)x

y

x

y

O

O

Mass center

Stiffness center

Fig. 21 Adjustable frequency TLCD Fig. 22 Eccentric structure with liquid dampers structures subjected to multi-dimensional earthquake excitation. The optimal parameters of the liquid dampers are designed for the structures excited by bi-directional seismic based on Genetic Algorithms. 4. CONCLUSIONS The P-SV ratio method, based on the elastodynamic theory in the multi-layer isotropic elastic half space, is presented to estimate the apparent velocity of local earthquake wave. With P-SV ratio method and other methods, the apparent velocities of 4 earthquakes recorded by the SMART1 array are estimated. The results of the P-SV ratio method are consistent with those calculated using travel time analysis, which validates the efficiency of the P-SV ratio method. Furthermore, the factors influencing the incident angle are discussed. The results show that the equivalent incident angle and epicenter distance (40-100km) have no significant relationship. The relationship between the equivalent incident angle and the poisson’s ratio of the site is almost negatively linear. The large poisson’s ratio of the site soil always causes small equivalent incident angle, while the equivalent incident angle is large if the poisson’s ratio is small. Therefore, the equivalent incident angle depends on the site conditions obviously. The researches on structural vibration control have made great achievements and some control devices have been applied to practice in recent years in China. However, most researches focus on linear vibration of structures. Studies on control theories for nonlinear systems are still very limited so far and should be paid more attention in the future. Performance based method for all kinds of control systems should be developed to satisfy the engineering requirements. More smart materials and new technologies should be used to control the structural vibration subjected to dynamic load.

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