13th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 2004 Paper No. 236
RESPONSE OF MULTI-STOREY MASONRY BUILDING WITH SLIDING SUBSTRUCTURE SUBJECTED TO BI-DIRECTIONAL EARTHQUAKE GROUND MOTIONS SHAKEEL AHMAD1, M. QAMARUDDIN2, HASSAN IRTAZA3 AND M. Z. ISLAM4
SUMMARY To investigate seismic isolation feasibility, multi-storey masonry building with pure friction sliding support to horizontal bi-directional earthquake ground motions have been considered for the present study. The building with sliding substructure is idealized as a four degree of freedom discrete mass model for computing its earthquake response. The coupled differential equations of motion of the building have been solved using Runga Kutta fourth order method. Effect of bi-directional interaction of the seismic forces on the building response have been compared with those without interaction. The investigations have been made using the time period of the superstructure, mass ratio (base mass to superstructure mass), friction coefficient of the sliding support and damping coefficient as variant parameters. It was found that neglect of bi-directional interaction effects leads to under-estimation of the sliding base displacement and to over prediction of the absolute acceleration of the superstructure. INTRODUCTION Masonry buildings have suffered the heaviest damage during earthquakes of moderate to severe intensities. Their vulnerability to damage results from several factors such as, their short period attracting large spectral acceleration, heavy weight but small tensile and shear strength and poor workmanship. Inspite of the poor performance record of the masonry structures during past earthquakes, they continue to be common option for building construction partly due to low cost. There has been a significant increase in the research activity aimed at improving earthquake resistance by providing reinforcement at critical ___________________ 1 Reader, Department of Civil Engineering, AMU, Aligarh-202002, India, Email:
[email protected] 2 Director, Azad Institute of Engineering and Technology, Lucknow-226002, India, Email:
[email protected] 3 Lecture, Department of Civil Engineering, AMU, Aligarh-202002, India 4 P.G. Student, Department of Civil Engineering, AMU, Aligarh-202002, India
sections of masonry buildings. In-spite of all these measures, masonry buildings cracked during the medium and strong earthquakes. Such damage might be avoided only if it would be possible to find out some other way to dissipate the major portion of earthquake input energy. Most isolation systems are too advanced, as well too expensive for the application to the masonry buildings in developing countries where 80% of the houses are single or double storey and made of brick masonry. An alternative is to provide a base isolation system in which the isolation mechanism is purely sliding friction. Pure friction base isolation is introduced between the superstructure and the substructure to provide lateral flexibility and the energy dissipation capacity. In several major earthquake occurrences in developing countries like India and China, a beneficial behavior of the low–rise buildings which could slide as rigid bodies over their foundations was observed during some past severe earthquakes, viz., the Dhubri Earthquake in Assam in 1930 [1] and in BiharNepal Earthquake in 1934. The damage study by Gee [1] and the observations made in China by Li Li [2] showed that those buildings in which the possibility of movement existed between the superstructure and the substructure suffered less damage than those buildings in which no such freedom of movements existed. Qamaruddin [3], Arya, Chandra and Qamaruddin [4], and Li Li [2] have investigated a base isolation system in which the isolation mechanism is purely sliding friction. Such system utilizes pure friction to allow some parts of the structure to slide relative to the others. It differs from the other categories of sliding system in which there is no restoring force provided by any type of external horizontal springs. A simple mathematical model was introduced by Qamaruddin [3] to compute the seismic response of masonry building with friction base isolation. In this method, a new concept has been proposed for the construction of brick building in which a clear smoothened surface is created just above the damp-proof course at plinth level without any mortar, and the superstructure simply rests at this level and is free to slide except for frictional resistance. The concept of friction seismic isolation system was further strengthened by the damage studies made by Li Li [2] after the Xintai in 1966, Bohai in 1969 and Tangshan in 1976 earthquakes in which it was found that adobe buildings which were free to slide on their foundations (by accident) survived with little or no damage whereas others which were tied on their foundations collapsed. Researchers have made experimental and theoretical studies to incorporate such a system in masonry buildings economically to achieve a collapse free if not a damage free performance during the earthquakes. Further studies have also been made by Mostaghel et al. [5] and Qamaruddin et al. [6,7] with encouraging results. The response of a rigid block sliding on a randomly moving foundation has been studied by Ahmadi [8] and Constantinou and Tadjbakhsh [9]. Sliding structures subjected to harmonic support motion have been studied by Mostaghel et al. [5] and Westermo and Udwadia [10]. The response of sliding structure to earthquake excitation was investigated by Mostaghel and Tanbakuchi [11]. Younis and Tadjbakhsh [12] employed a 2-DOF system to study the condition of relative slip between the rigid bodies. Experimental model studies on a shaking table were carried out by Li et al. [13] to verify the analytical result obtained for sliding structures subjected to sinusoidal excitation. The response of a torsionally coupled system with sliding support to both harmonic and real earthquake excitation was investigated by Jangid and Dutta [14]. In spite of several studies on dynamic behaviour of pure friction sliding structures, very few have been reported under bi-directional ground motion namely, seismic response of sliding structures to bidirectional earthquake excitation by Jangid [15] and response of pure friction sliding structures to bidirectional harmonic ground motion by Jangid [16]. Jangid [16] has investigated the response of the structures with sliding support to two horizontal components of Kyona earthquake ground motion. The objectives of the present study may be summarized as: (1) to present a method of dynamic analysis of pure friction sliding structures to bi-directional ground motion which incorporates the interaction between the frictional forces mobilized at the sliding support in two orthogonal directions. (2) to study the effects of bi-directional interaction of frictional forces on the response of sliding structures.
(3) to investigate the influence of important parameters on bi-directional interaction of frictional forces. The parameters include: the time period of super structure, the ratio of super structure mass to the base mass, coefficient of friction of the sliding support and damping coefficient. MATHEMATICAL IDEALIZATION Mathematical Model A multi-storey structure represents the concept of multi-storey structure with base sliding. It is assumed that a layer of a suitable material with known frictional coefficient is laid between the contact surface of bond beam of superstructure and plinth band of substructure. The sliding type building is idealized as a discrete mass model with four degree of freedom for computing the earthquake response (Fig 1). The spring action in the system is provided by the shear walls only, which resists shear force parallel to the direction of earthquake shock. Internal damping is represented by a dashpot that is parallel with spring. The lower mass assumed to rest on a plane with dry friction damping to permit sliding of the system at certain limit. The coefficient of friction (assumed as static) between the sliding surfaces is considered to be constant throughout the motion of the system. Materials used for building construction are linearly elastic within the limit of proportionality, thus the idealized spring is linear elastic. Its stiffness is computed by considering bending as well as shear deformations in the wall element. Sliding displacement at contact surface between the bond beam and the plinth band can occur without overturning or tilting. The building is subjected to two orthogonal horizontal components of ground motion at a time. The effect of vertical ground motion is not considered here. EQUATIONS OF MOTION There are three different phases in the complete motion history of the sliding system of an idealized multistorey masonry building. The equations of motion are written as follows: Phase 1: Initially, so long as the force, which causes sliding of the system, does not overcome the frictional resistance, the bottom mass moves with base since there is no sliding and the system behaves as a two degree of freedom system. In such a case, the equations of motion are: ..
.
.
.
.
M t x t + Cs ( Z 2 − Z1) + K s ( Z2 − Z1) = 0 ..
..
or, Z2 + 2 pξ ( Z2 − Z1) + p 2 ( Z 2 − Z2) = − x g(t) ........................(1) and ..
.
.
M t y t + Cs ( Z 4− Z3) + K s ( Z 4− Z3) = 0 ..
.
.
..
or, Z4 + 2 pξ ( Z4 − Z3) + p 2 ( Z 4 − Z3) = − y g (t) .......................(2) Where, Cs = Coefficient of the viscous damper Ks = Spring constant Mt = Top mass p=
K s / M t = Natural Circular frequency
..
..
x t , Z2 = Absolute and relative accelaerations of the top mass respectively in x - direction .. ..
y t , Z 4 = Absolute and relative accelerations of the top mass respectively in y - direction ..
..
x g (t), yg (t) = ground acceleration in x and y direction respectively Z 2 , Z1 = lateral relative displacements of masses M b , M t in x - direction respectively Z 4 , Z3 = lateral relative displacements of masses M b , M t in y - direction respectively .
.
.
.
Z 2, Z1 = relative velocities of masses M b , M t in x - direction respectively Z 4, Z3 = relative velocities of masses M b , M t in y - direction respectively Cs ξ= = fraction of critical damping 2p M t Phase 2: The sliding of the bottom mass begins when the force which causes sliding, overcomes the frictional resistance at the plinth level. The force to cause sliding, is given by Sx and Sy in x and y direction respectively, .
.
..
.
.
S x = Cs ( Z2 − Z1) + K s ( Z2 − Z1) - M b xb ..
S y = Cs ( Z4− Z3) + Ks ( Z4 − Z3) - M b y b Sliding of bottom mass occurs if 2
(S x ) + (S y) > µ MT g 2
where, g = acceleration due to gravity MT = M t + M b µ = coefficient of friction The system now acts as a four degree of freedom system for which the equations of motion can be written in a simplified form as ..
.
.
..
..
.
.
..
.
.
..
.
.
2 Z1− 2pξθ( Z2 − Z1) − p θ( Z2 − Z1) + Fx = − x g(t) ..
..
Z2 − 2pξθ( Z2 − Z1) + p θ( Z2 − Z1) = −( x g (t) + Z1(t)) 2
..
2 Z3− 2pξθ( Z4 − Z3) − p θ( Z4− Z3) + Fy = − yg (t) ..
..
2 Z4 + 2pξθ( Z4 − Z3) + p θ( Z4 − Z3) = −( yg (t) + Z3(t))
In which .
.
Fx = µg(1 + θ)Sgn(Z1); Fy = µg(1 + θ)Sgn(Z3) .
.
.
.
Z1, Z3 = relative acceleration of mass M b in x and y direction respectively Z2, Z4 = relative acceleration of mass M t in x and y direction respectively M θ = t = mass ratio Mb .
.
Sgn(Z1) = + 1 if Z1 is positive .
= - 1 if Z1 is negative .
.
Sgn(Z3) = + 1 if Z3 is positive .
= - 1 if Z3 is negative Phase 3: At any instant of time during the motion of system if 2
(S x ) + (S y) < µ MT g , 2
then sliding of bottom mass is stopped but top mass continues to vibrate. Therefore, again the system behaves as a two degree of freedom system and its equation of motion is same as governed in Phase1. SOLUTION OF EQUATIONS OF MOTION The equations of motion for the different phases have been solved numerically by Runga-Kutta fourth order method for obtaining the complete seismic response. A computer programme has been developed to compute the time-wise earthquake response of multistory masonry building. PARAMETRIC STUDY A large number of parameters influence the response of ground excitation. In order to reduce the total number of parameters, the time period of the superstructure with a fixed base is kept the same in the two orthogonal directions. The damping ratio of superstructure is taken as 5%, 10% and 15% of the critical damping in both the directions. The time history of Kyona Earthquake is taken same in both the orthogonal directions. The effect of bi-directional interaction of frictional forces is studied against the important parameters, which include: the time period of the super structure, the mass ratio (ms/mb) and the friction coefficient of the sliding support (µ). The bottom mass is kept constant throughout the study. The required time period of the super structure is achieved by changing the stiffness of the superstructure. The response of the system is expressed in the normalized form, which is defined as:
Normalized response = (Peak response of the system considering the interaction of frictional forces)/(The corresponding peak response of the system without interaction of frictional forces)
The normalized response is an index of bi-directional interaction effects and the values are significantly different from unity, which indicate significant bi-directional interaction effects. on the other hand, values close to unity justify the two dimensional idealization of the system and the interaction of frictional forces may be ignored. DISCUSSION OF RESULTS In Figs. 2 to 6, the normalized absolute acceleration and the base displacement are plotted against coefficient of friction. The figures indicate that the normalized absolute acceleration of top mass is less than unity for all parameters indicating that the response corresponding to two-component excitation with interaction is reduced in comparison to those without interaction. The normalized base displacements are greater than unity which indicates that sliding displacement under bi-directional ground motion considering the effect of interaction of frictional forces is higher in comparison to the same, but without interaction. This is due to the fact that when the interaction is taken into consideration the structures start sliding at relatively lower value of the frictional force mobilized at the sliding support as a result, there is more sliding displacement. Thus, if the interaction of the frictional forces at the sliding support is ignored then the superstructure acceleration will be overestimated and the sliding displacement will be underestimated. Effect of Coefficient of Friction It is seen from Figs. 2 to 6 that for a particular time period and coefficient of critical damping, the spectral acceleration and base displacement increases as the friction coefficient increases. This is logical, since the resistance against sliding of the system increase as the coefficient of friction between the sliding surface increases, and the build up of a larger inertia force in the super structure becomes restricted. Thus, the spectral acceleration of such a system decreases. Time Period Effect Figs. 3 and 4 shows the typical frictional acceleration and base displacement spectra for the Koyna earthquake for different parameters. The frictional spectra are generally flat and value does not change much as the period of the system and other parameters vary. In a frictional isolated system, if the ground acceleration coefficient at any instant of the time exceeds the coefficient of friction, the rigid body begins to slide, and the remaining limiting force equals mass times threshold acceleration. Thus, the response of the system is independent of its time period for different values of coefficient of friction. Since only short period structure have been studied, it is logical that the acceleration and displacement response of such structure should not depend much on their period, and only slight variation can be expected. Effect of Mass Ratio From Figs. 5 and 6 it may be observed that as the mass ratio increases, the spectral acceleration and displacement increases in all cases of parametric combinations for Koyna Earthquake. The possible reason for increase in spectral acceleration and displacement is due to increase of mass ratio is that for a system, as the mass ratio increases for a given period and damping, the input dynamic energy is increased and thus the spectral acceleration and displacement increases. Effect of Viscous Damping An increase in viscous damping decreases the spectral acceleration and displacement of the sliding systems for various values of different parameters (Figs. 7 and 8). This is the expected result for conventional spectra, and it indicates the increasing energy dissipation in internal friction of the system as the damping coefficient increases.
CONCLUSIONS The response of friction isolated multi-storey building subjected to two horizontal earthquake component forces have been studied under important parametric variations, like the time period, mass ratio, damping coefficient and coefficient of friction. It was found that bottom mass displacement is more and top mass acceleration is less in the case of bi-axial earthquake horizontal forces taking bi-directional interaction effects than the without taking bi-directional interaction effect. Thus, neglect of bi-directional interaction effect leads to underestimation of bottom mass displacement and over estimation of top mass acceleration with respect to single direction interaction. ACKNOWLEDGEMENT The first three authors wish to acknowledge the financial support given by the All India Council for Technical Education (AICTE) New Delhi, grant No. (F.No. 8020/RID/TAPTEC/38-2001-2002) for this analytical work. REFERENCES 1. Gee, ER “Dhubri Earthquake of 1930.” Memories of Geological Survey of India, Vol. LXV, India, 1934: 1-106. 2. Li Li “Base Isolation Measures for a Seismic in China.” Eight World Conference on Earthquake Engineering, Vol. IV, San Francisco, California, 1984: 791-798. 3. Qamaruddin M. “Development of Brick Building Systems for Improved Earthquake Performance.” Ph..D. Thesis, University of Roorkee, Roorkee, 1978. 4. Arya AS, Chandra B. and Qamaruddin M. “A New Concept for Resistance Masonry Buildings in Severe Earthquake Shocks.” Journal of the Institution of Engineers, India, 1981: 61,302-308. 5. Mostaghel N, Hejazi M and Tanbakuchi J. “Response of Sliding Structures to Harmonic Motion.” Journal of Earthquake Engineering and Structural Dynamics, 1983): Vol. 11, 355-366. 6. Qamaruddin M, Arya AS and Chandra B. “Seismic Response of Brick Buildings with Sliding Substructure.” Journal of Structural Engineering, American Society of Civil Engineers 1986a, 122(3): 558-572. 7. Qamaruddin M, Rasheeduzzafar, Arya AS and Chandra B. “Seismic Response of Masonry Buildings With Sliding Substructure.” Journal of Structural Engineering, American Society of Civil Engineers1986b, 122(9): 2001-2011. 8. Ahmadi G.” Stochastic earthquake response of structure on sliding foundation.” Int. J. Engng Sci,. 1983, 21: 93-102. 9. Constantinou, MC and Tadjbakhsh, IG. “Response of sliding structures to filtered random excitation.”J. Struct. Mech 1984, 12: 401-418. 10. Westermo,B. and Udwadia, F. ”Periodic response of a sliding oscillator system to harmonic excitation.” Earthquake engng Struc. Dyn 1984; 11: 135-146. 11. Mostaghel N and Tanbakuchi J. “ Response of sliding structures to earthquake support motion.” Earthquake engng Struc. Dyn 1983: 11, 729-748 12. Younis, C.J. and Tadjbakhsh, I.G. “Response of sliding structures to base excitation.” J. Engng Mech. ASCE 1984; 12: 401-418 13. Li, Z., Rossow, E.C. and Shah, S.P. ”Sinusoidal forced vibration of sliding masory system.” J. Struct. Engng ASCE . 1989; 115: 1741-1755. 14. Jangid, R.S. and Datta, T.K. ”Seismic response of a torsionally coupled system with a sliding support.” J. Structures Building ICE 1993; 99: 271-280.
15. Jangid, R.S. “Seismic response of sliding structures to bi-directional earthquake excitation.” J. Earthquake Engng Struct. Dyn. 1996; 25: 1301-1306. 16. Jangid, R.S. “Response of pure sliding structures to bi-directional harmonic ground motion.” J. Engng Struct. 1997; 19: 97-104. Z3 Z1 Mt
Cs
Ks Z4
Z2
ÿg(t)
Mb
µgMT
Xg(t)
µgMT
Fig. 1 Mathematical Model For Multi-Storey Masonry Building
Normalised Base Displ.
3.5
Base Displ. in x-dir Base Displ. in y-dir
3 2.5 2 1.5 1 0.1
0.12
0.14 0.16 0.18 Coeff. of friction
0.2
0.22
Fig. 2 Variation of Base Displ. with Coeff. of friction (Damping=5%,MR=4,TP=0.4)
cof=.12 cof=.12 cof=.14 cof=.14 cof=.16 cof=.16 cof=.18 cof=.18 cof=.20 cof=.20
Normalised Base Displ.
3.5 3 2.5 2 1.5 1 0.5
----- in x-dir in y-dir
0 0.2
0.3
0.4
0.5 0.6 Time period
0.7
0.8
0.9
Fig. 3 Variaton of Normalised Base Displ. With Time Period ( Damping 5%,MR=4)
Normalised Top Accl.
0.7
cof=.12 cof=.12 cof=.14 cof=.14 cof=.16 cof=.16 c0f=.18 cof=.18 cof=.20 cof=.20
0.6 0.5 0.4 0.3 0.2 0.1 0.3
0.4
0.5
0.6 Time period
0.7
0.8
Fig. 4 Variaton of Normalised Top Accl. With Time Period ( Damping 5%,MR=4)
0.9
---- in x-dir in y-dir
cof=.12 cof=.14 cof=.16 cof=.18 cof=.20 cof=.12 cof=.14 cof=.16 cof=.18 cof=.20
Normalised Base Displ.
5
3.8
2.6
------- in x-dir in y-dir 1.4 3
3.5
4
4.5 Mass Ratio
5
5.5
6
6.5
Fig 5 Variation of Base Displ. With Mass Ratio ( Damping=5%,TP=0.5)
cof=.12 cof=.12 cof=.14 cof=.14 cof=.16 cof=.16 cof=.18 cof=.18 cof=.20 cof=.20
1
Normalised Top Accl.
0.9 0.8 0.7 0.6 0.5 -----
0.4 0.3 3
3.5
4
4.5 5 5.5 Mass Ratio Fig 6 Variation of Top Accl. With Mass Ratio ( Damping=5%,TP=0.5)
6
6.5
in x-dir in y-dir
Base Displ. in x-dir Base Displ. in y-dir
3.8 Normalized Base Displ.
3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 4
6
8
10
12
14
16
Damping coefficient Fig. 7 Variation of Base Displ. With Damping coefficient (TP=0.5,MR=5,cof=0.20)
Top Accl. In x-dir Top Accl. In y-dir
Normalized Top Accl.
0.8 0.75 0.7 0.65 0.6 0.55 0.5
4
6
8
10
12
14
Damping coeficient Fig. 8 Variation of Top Accl.with Damping coefficient (TP=0.5,MR=5,cof=0.20)
16
